## Abstract

Recently, we have experimentally demonstrated a new form of cross-sectional, coherence-gated fluorescence imaging referred to as SD-FCT (’*spectral-domain fluorescence coherence tomography*‘). Imaging in SD-FCT is accomplished by spectrally detecting self-interference of the spontaneous emission of fluorophores, thereby providing depth-resolved information on the axial positions of fluorescent probes. Here, we present a theoretical investigation of the factors affecting the detected SD-FCT signal through scattering media. An imaging equation for SD-FCT is derived that includes the effects of defocusing, numerical-aperture, and the optical properties of the medium. A comparison between the optical sectioning capabilities of SD-FCT and confocal microscopy is also presented. Our results suggest that coherence gating in fluorescence imaging may provide an improved approach for depth-resolved imaging of fluorescently labeled samples; high axial resolution (a few microns) can be achieved with low numerical apertures (*NA*<0.09) while maintaining a large depth of field (a few hundreds of microns) in a relatively low scattering medium (6 mean free paths), whereas moderate NA’s can be used to enhance depth selectivity in more highly scattering biological samples.

©2007 Optical Society of America

## 1. Introduction

Coherence gating in optical imaging has proven to be useful for biological and biomedical imaging of tissue reflectance through scattering media [1]-[7]. Of particular importance has been the ability of coherence gating techniques, such as optical coherence tomography (OCT) [1, 7], to simultaneously provide micron-scale optical sectioning while maintaining a large depth of field.

Recently, we have developed a new technique termed *fluorescence coherence tomography* (FCT) [8] that utilizes coherence gating to obtain wide-field optically sectioned images of fluorescent samples. In FCT, low numerical-aperture (NA) objectives are used in a 4Pi configuration [9] to detect fluorescence self-interference from a labeled sample. Self-interference fringes are only detected when the two paths of the interfereometer are matched to within the coherence length of the fluorophore emission [10]. This configuration allows micron-resolution fluorescence optical sectioning with low NA objectives, resulting in cross-sectional images that can be obtained over a large depth of field. Experimental realizations of FCT can be accomplished either in time-domain (TD) or spectral-domain (SD). The former requires scanning the coherence gate through the sample whereas the latter utilizes a spectrometer.

The aim of this paper is to present a detailed theoretical analysis of the image formation in SD-FCT through scattering media. In Section 2 of this manuscript, we derive the imaging equation of SD-FCT which takes into account the attenuation of the sample, the defocus point-spread response of the SD-FCT imaging system as well as the fluorescence autocorrelation. The impact of defocusing, NA values and sample attenuation on spectral fringe formation, theoretical sensitivity limits, and optical sectioning performance are discussed in detail in Section 3. Additionally, a comparison between depth selectivity in SD-FCT and that of confocal imaging is described. Based on this theoretical analysis, conclusions regarding the potential role of SD-FCT for biological imaging are drawn in Section 4.

## 2. SD-FCT: The imaging equation

Consider the geometry of the SD-FCT system shown in the left panel of Fig. 1. An object comprising a fluorescently labeled specimen is located between two opposing objectives, near the zero differential path length point (*z*
_{0_diff}) of an interferometer and is illuminated with the excitation beam (green). Fluorescence emission (orange) is collected by the two objectives and combined at the beam splitter. The resulting interference as a function of emission wavelength is detected by a spectrometer. The location of excited fluorophores along the illumination beam is encoded by an interferometric frequency modulation of the emission spectrum, where the frequency modulation is proportional to the fluorophore’s distance from *z*
_{0_diff}. The axial locations of these fluorophores may then be retrieved by computing the modulus of the inverse Fourier transform of the spectral interferogram intensity. Finally, a cross sectional ‘*y*-*z*’ fluorescence image can be obtained by scanning the sample or excitation beam along the ‘*y*’ coordinate in the object plane (confocal configuration). Alternatively, cross sectional images can be obtained using an excitation line focus together with an imaging spectrometer (non-confocal configuration) [9]. Hereon, we present an analysis of the confocal arrangement since it leads to closed form expressions and provides important insights into the process of fluorescence coherence-gated imaging. Our analysis is also relevant to the non-confocal case under the assumption that transversal and spectral overlaps detected by the imaging spectrometer are small.

To establish the mathematical formalism for the SD-FCT signal, we first write the spectrally-resolved electric field focused on the spectrometer’s detection plane (or, image plane) as

with K_{L} and K_{R} being losses in the left and right arms of the interferometer, respectively, and u_{L} and u_{R} representing the spectrally-resolved, scalar field distributions of the emitted light from the object, emerging from the left and right arms of the interferometer through the focusing lens, respectively. *ω* is the angular frequency, *ω _{0}* corresponds to the angular frequency of the fluorescence emission center, and

*r*describes the radial dependence of the electric field across the detector pixels of the spectrometer. Next, the photoelectron spectral density (PSD) of (1) recorded on the detector array of the spectrometer reads as

_{i}$$+{2K}_{L}{K}_{R}\mathrm{Re}\left\{\u3008{u}_{L}\left({r}_{i},\Omega \right){u}_{R}^{*}\left({r}_{i},\Omega \right)\u3009\right\}],$$

where *$\tilde{\rho}$* represents losses due to the diffraction grating and spectrometer geometry, *η* is the quantum efficiency of the detector array, ħ is the reduced Planck’s constant, Ω = *ω* - *ω _{0}*, and

*T*is the exposure time of the detector array. Also, 〈 〉 stands for ensemble average and integration over the pixel detector area. Note that the units of

*G*(Ω) are

*1*/angulaer frequency. Expressions for the electromagnetic fields, u

_{L}and u

_{R}, follow. For a collimated Gaussian light beam focused on the fluorescent sample, the excitation profile (

*h*) in the sample is

_{exc}where

and

Similarly,

In Eq. (3), *P* is the total excitation laser power and *k _{exc}* is the wavenumber corresponding to the excitation center wavelength. As shown in the object plane sketched in Fig. 1 (top-right panel),

*W*is the radius of the Gaussian beam at the objective’s back focal plane,

_{0}*z*

_{fexc}and

*z*

_{fem}denote the effective geometric focal point (in air) of the excitation and emission beams, respectively, resulting from the air-biological medium interface,

*f*is the objective lens focal length (in air) and

*D*is the objective’s diameter. Note that

*z*

_{f, exc}and

*z*

_{f, em}are different for

*n*≠ 1 and

*W*≠

_{0}*D*but equal (both) to

*f*for

*n*= 1.

*d*represents the distance between the objectives and the sample’s surface in a symmetric configuration where this distance is equal for both lenses. For the asymmetric case,

*d*should be replaced with

*d*+

*z*

_{shift}in Eq. (3a) and Eq. (3c) and with

*d*+

*z*

_{shift}and

*d*-

*z*

_{shift}for the left and right fluorescence fields, respectively, in Eq. (3d). Here,

*z*

_{shift}represents the axial offset of the sample’s surface from

*d*. Furthermore, μ

_{t}is the attenuation coefficient of the sample where it is assumed that light is attenuated ballistically in the sample. Also,

*n*denotes the refractive index of the biological specimen. Finally,

*r*and

_{o}*z*are the radial and longitudinal coordinates in the object plane, respectively.

_{o}Now, assume that the induced fluorescence of a point object positioned at
$(\overline{{r}_{o}},{z}_{o})=(\overline{{r}_{s}},{z}_{s})$
is proportional to the incidence intensity of the excitation light at the same
point (i.e., we neglect local saturation effects); then the fluorescence distribution in the object space (*P _{f}*) is given by

where K_{exc}
^{2} is the power loss of the illumination beam due to optical elements in the excitation path, and *A _{f}* is the effective cross-section of the fluorescent marker. We used Fresnel integrals [12] to describe the 4Pi fluorescence collection in FCT and derived the spectrally-resolved, fluorescent image field distribution emerging from each arm of the interferometer through the focusing lens. Assuming balanced dispersion in the two arms of the interferometer, the resultant expression for u

_{L}(

*r*) is

_{i}, Ω$$\bullet \underset{0}{\overset{1}{\int}}\underset{0}{\overset{2\pi}{\int}}\mathrm{d\rho d\theta \rho}{J}_{0}\left(\rho \left(1+\frac{\Omega}{{\omega}_{0}}\right){v}_{i}\right){e}^{{\mathrm{iu}}_{i}\frac{{\rho}^{2}}{2}\left(1+\frac{\Omega}{{\omega}_{0}}\right)-\frac{{\mu}_{t}}{2}\sqrt{{\left(n\left({z}_{f,\mathrm{em}}-d\right)+{z}_{s}\right)}^{2}+{\mid {\stackrel{\u0305}{r}}_{f,\mathrm{em}}-{\stackrel{\u0305}{r}}_{s}\mid}^{2}}},$$

with

where *NA* denotes the effective numerical aperture of the objective lens (in air), λ_{0} and k_{0} are the fluorescence emission center wavelength and its corresponding wavenumber, respectively, *M* is the magnification of the imaging system, *S _{f}(Ω)* is the normalized fluorescence emission spectrum (with units of

*sec/rad*),

*J*is the zeroth-order Bessel function of the first kind,

_{0}*ρ*represents the normalized radial coordinate, $({\overline{r}}_{f,\mathit{\text{em}}}$

*r*̅

_{f,em}denotes the position of emission fluorescence rays residing within the collection cone of the objective lens at the air-sample interface, and Δz stands for a possible path-length difference between the left and right arms of the interferometer with respect to

*z*=0. Finally, notice that the phase term of the electric field propagating in the optical system includes nonlinear radial and linear longitudinal contributions due to the deviation of the object position from the geometric focal plane as well as an additional linear longitudinal contribution resulting from the mismatch of the two arms of the interferometer.

_{s}The corresponding expression for u_{R}(*r _{i},Ω*) can be shown to be equivalent to Eq. 5 with ‘

*z*’ and ‘

_{s}*Δz*’ replaced by ‘-

*z*’ and ‘-

_{s}*Δz*’, respectively. It is worth pointing out that the radial- and angular-frequency dependencies in Eq. (5a) can be decoupled for sufficiently low NA objectives with $NA\text{\hspace{0.17em}}\ll \sqrt{0.5(n+0.5\Delta z/{z}_{s})}$ (

*NA*≪ 0.84 for

*n*=1.4 and

*Δz*=0). In this case, u

_{L}(

*r*) reduces to

_{i},ωNote that the integral in Eq. (6) is simply the defocused amplitude point-spread function (APSF) of a circular lens [12]. Finally, Eq. (5) and Eq. (6) can be extended for the asymmetric case in which *d* is not equal for the two objectives by replacing *d* with *d* + *z*
_{shift} and *d* - *z*
_{shift} for the left and right fluorescence fields, respectively. Substitution of Eq. (5a) in (2) under the assumption that light attenuation is dominant along the longitudinal propagation direction yields the main result of this work -- the imaging equation for SD-FCT through scattering media, which reads as

$$+2{\mathrm{K\u0303}}_{L}{\mathrm{K\u0303}}_{R}\phantom{\rule{.2em}{0ex}}\mathrm{cos}(2\angle {h}_{i}({\stackrel{\u0305}{r}}_{i},\Omega ;{\stackrel{\u0305}{r}}_{s},{z}_{s})-{2k}_{0}\left({\mathrm{nz}}_{s}+\Delta z\right)\left(1+\frac{\Omega}{{\omega}_{0}}\right)\left)\right]2\pi \left(d\mid {\stackrel{\u0305}{r}}_{i}-M{\stackrel{\u0305}{r}}_{s}\mid \right),$$

with

and

with

Also, *r _{d}* denotes the pixel radius. It is important to point out that the weak interference signals produced by the specular reflectance at the surface of the sample are not included in Eq. (7). Finally, for the asymmetric case where d is not equal for the two objectives,

*k*+

_{0}(nz_{s}*Δz)*appearing inside the argument of the cosine function in Eq. (7a) is replaced by

*k*+

_{0}(nz_{s}*Δz*+

*z*.

_{shift}(n-l))The first term inside the integral (7a) is the DC fluorescence light where the second term describes the spectral modulation. Both terms are affected by the imaging PSF and the attenuation inside the sample. The term outside the integral is determined by the emission spectral shape, the effective excitation power and spectrometer characteristics (i.e., losses, exposure time and quantum efficiency).

To gain useful insight regarding the SD-FCT signal, we further simplified Eq. (7) and invoked the approximation of a sufficiently low NA to obtain the following expression

$$+{2\mathrm{K\u0303}}_{L}{\tilde{K}}_{R}\mathrm{cos}(2\angle {h}_{c}({\stackrel{\u0305}{r}}_{i};{\stackrel{\u0305}{r}}_{s},{z}_{s})-2{k}_{0}\left({\mathrm{nz}}_{s}+\Delta z\right)\left(1+\frac{\Omega}{{\omega}_{0}}\right)\left)\right]2\pi \left(d\mid {\stackrel{\u0305}{r}}_{i}-M{\stackrel{\u0305}{r}}_{s}\mid \right),$$

where

With Eq. (8) at hand, the SD-FCT signal was derived by simply calculating the modulus of the inverse Fourier Transform (FT^{-1}) of the detected interferogram term. The final expression for the SD-FCT signal is then

$$+{\tilde{K}}_{L}{\tilde{K}}_{R}\mid {I}_{2}\left({\stackrel{\u0305}{r}}_{s},{z}_{s}\right){R}_{f}\left(z-2\left({\mathrm{nz}}_{s}+\Delta z\right)\right)\mid ],\phantom{\rule{.2em}{0ex}}z\ge 0,$$

where $\phantom{\rule{.2em}{0ex}}{I}_{1}={\int}_{0}^{{r}_{d}}{\mid {h}_{c}({\stackrel{\u0305}{r}}_{i};{\stackrel{\u0305}{r}}_{s},{z}_{s})\mid}^{2}2\pi \left(d\mid {\stackrel{\u0305}{r}}_{i}-{M\stackrel{\u0305}{r}}_{s}\mid \right),{I}_{2}={\int}_{0}^{{r}_{d}}{h}_{c}^{2}({\stackrel{\u0305}{r}}_{i};{\stackrel{\u0305}{r}}_{s},{z}_{s})2\pi \left(d\mid {\stackrel{\u0305}{r}}_{i}-M{\stackrel{\u0305}{r}}_{s}\mid \right),{R}_{f}\left(z\right)={\mathrm{FT}}^{-1}\left\{{S}_{f}\left(\Omega \right)\right\}\left(z\right).$

As seen from Eq. (9), the SD-FCT signal carries useful information about the position of an excited fluorophore (z_{s}), which is related to the spectral fringe periodicity imprinted on the emission spectrum. To extract *z _{s}*, note that the measured spectral fringe periodicity should be compensated against the constant mismatch between the two arms of the interferometer (

*Δz*). Most importantly, Eq. (9) also reveals that for a pixel array consisting of point detectors the SD-FCT signal is the product of the confocal signal (∣

*h*∣

_{c}^{2}) and the fluorescence autocorrelation function (

*R*(z)). The fluorescence autocorrelation function improves the confocal depth resolution to an extent which depends on the coherence length of the fluorophore emission (L

_{f}_{c}).

It should be pointed out that the generalized expression for the SD-FCT signal which corresponds to an arbitrary axial distribution of fluorophores should include the complex argument of *R _{f}*(z), unless the separation of the fluorescent probes is greater than 0.5L

_{c}(typically a few microns). Then, the SD-FCT signal is simply the linear combination of the signals for individual fluorophores which are governed by Eq. (9).

The undesired overlap between the autocorrelation and interferogram signals (left and right terms inside the brackets in Eq. (9), respectively) can be avoided by simply placing the fluorescent object at a distance of several coherence lengths from the zero path-length point of the interferometer (*z _{0_diff}*). Furthermore, the interferogram signal is conjugate symmetric, resulting in wrapping the data corresponding to negative and positive distances of fluorescent probes with respect to

*z*. This can be resolved by designing the SD-FCT system such that the distance of all fluorophores from

_{0_diff}*z*is either positive or negative. Also, techniques that measure the complex spectral density [13] can be implemented, thereby eliminating wrapping as well as doubling the maximal axial range provided by SD-FCT.

_{0_diff}## 3. SD-FCT: Spectral fringe visibility, Spectral fringe periodicity, Signal-to-noise ratio and Optical sectioning response

This Section examines in detail the dependencies of numerous SD-FCT parameters on defocusing and NA of the two matched objective lenses used inside the interferometer (see Fig. 1) as well as the attenuation coefficient of the sample.

- I. Fringe visibility For the sake of brevity and without loss of generality, let us define
*N*_{max}as the maximum number of photoelectrons across the sampled photoelectron spectrum*N(*Ω^{n}*)*for a point fluorescent marker located along the optical axis (i.e.,*r*=0). That is,_{s}where

*N(*Ω_{n}*)*=*G*⊗*h*Ω_{pixei}(_{n}*)*with ⊗ denoting the convolution operator and*h*Ω_{pixel}(*)*being a rectangular function with a width that matches the spectral spectrometer’s resolution (Δ*Ω*). Ω_{n}=*ω*-_{n}*ω*where_{0}*ω*is the center frequency of the_{n}*n*-th detector pixel. We note that the*n*-th detector pixel covers a spectral range of*ω*± Δ_{n}*ω*. Also, notice that when the spectrometer’s resolution is much higher than the interferometric frequency modulation of the emission spectrum, the deterioration of the fringe visibility through fringe averaging is negligible. Next, let*N*_{min}represent the detected number of photoelectrons corresponding to the local minimum point adjacent to*N*_{max}. The spectral fringe visibility is then defined as$$\mathrm{Fringe}\phantom{\rule{.2em}{0ex}}\mathrm{visibility}=\frac{\left({N}_{max}-{N}_{min}\right)}{\left({N}_{max}+{N}_{min}\right)}.$$It should be mentioned that the fringe visibility depends also on the width of the excited fluorophore axial distribution. It can be shown that for uniformly distributed fluorophores, the visibility scales approximately as ∣sinc(2π

*n*W/*λ*)∣ where sinc(_{0}*x*)=sin(*x*)/*x*and ‘W’ is a measure of the width of the axial distribution of the excited fluorophores (e.g., FWHM). This implies that SD-FCT would benefit from a narrow axial fluorophore distribution with an extent smaller than*λ*/2_{0}*n*. Clearly, working with higher NA’s reduces this fringe averaging effect due to the smaller excitation volume. Figure 2 shows the fringe visibility as a function of the attenuation coefficient of the sample medium and the position of the point fluorescent marker for three different objective NA’s. The deep blue color denotes areas for which (a) the excitation power exceeded 10 mW (horizontal boundary), (b) the sensitivity limit per pixel was reached (curved boundary), or (c) the fluorophore was placed outside a range which equals to the average of the confocal range and the depth of field (vertical boundary). The parameters used for Fig. 2 were*n*=1.4,*λ*=610 nm,_{0}*λ*=532 nm, K_{ex}_{L}=K_{R}=0.57, K_{exc}=0.9,*T*=1 sec,*ρ*=0.7,*η*=0.9,*A*=1.8×10_{f}^{-15}m^{2},N_{pixels}=512,*r*=7.5 μm,_{d}*D*=7 mm and*W*=1.75 mm (unless otherwise stated). The maximum fluorescence emission power was 4.9 nW. The magnification (_{0}*M*) was selected such that a diffraction-limited spot filling 90% of the pixel size was focused on the spectrometer’s pixels array. The distance between the objectives and the sample surface –*d*– was determined such that the sample thickness was 1 mm. Finally,*S*was modeled as a Gaussian function with full-width at half maximum of 70 nm and the spectrometer’s resolution was selected to be 0.03 nm so that fringe washout was negligible along the examined axial range. For_{f}(λ)*NA*=0.54 the Gaussian beam was expanded in order to fill the back aperture of the objective. This avoids the undesired mismatch between the focal planes of the excitation and emission beams, which becomes large for relatively high objective NA’s.Several important characteristics are revealed from Fig. 2: (1) for all NA’s, the visibility reduces as the fluorescent probe moves away from the geometric focal plane due to the defocusing effect [12]. A slight increase in the visibility can be observed for ∣

*z*∣0.25 mm and_{s}*NA*=0.06 due to the larger response of the objective APSF; (2) the visibility is asymmetric with respect to*z*=0 (at which the geometric focal plane of the emission light is located) due to larger attenuation experienced by the excitation light. This effect is more pronounced at low NA’s; (3) the visibility scales with the NA such that it becomes narrower and taller as the NA increases due to the smaller depth of field and the higher collection efficiency, respectively._{s}Finally, it is instructive to point out that for an SD-FCT system employing sufficiently low NA objectives and where

*Δω*_{FWHM}≫∣*πc/(nz*+_{s}*Δz)*∣ (that is, a spectral fringe period that is much smaller than the FWHM bandwidth of the emission spectrum), the spectral fringe visibility for a point fluorescent marker is approximately$$\frac{{2\tilde{K}}_{L}{\tilde{K}}_{R}}{{\tilde{K}}_{L}^{2}+{\tilde{K}}_{R}^{2}}.\frac{\mid {\int}_{0}^{{r}_{d}}{h}_{c}^{2}({r}_{i};0,{z}_{s}){\mathrm{dr}}_{i}\mid}{{\int}_{0}^{{r}_{d}}{\mid {h}_{c}({r}_{i};0,{z}_{s})\mid}^{2}{\mathrm{dr}}_{i}}.$$where the first ratio term is determined by the losses in the interferometer, the medium thickness, attenuation and refractive index, while the second ratio term results from the defocusing effect.

- II. Fringe periodicity The spectral modulation frequency of the detected photoelectron distribution is determined by the complex argument of the interference signal, which is$$\angle \left[{\int}_{0}^{{r}_{d}}{h}_{i}^{2}({\stackrel{\u0305}{r}}_{i},\Omega ;{\stackrel{\u0305}{r}}_{s},{z}_{s})2\pi \left(d\mid {\stackrel{\u0305}{r}}_{i}-M{\stackrel{\u0305}{r}}_{s}\mid \right)\right]-{2k}_{0}\left({\mathrm{nz}}_{s}+\Delta z\right)\left(1+\frac{{\Omega}_{n}}{{\omega}_{0}}\right).$$
As observed from Eq. (13), the spectral interferogram is generally chirped due to the defocusing effect (left term). The deviation of the fringe periodicity from the constant period

*πc*/*(nz*+_{s}*Δz)*(with c being the speed of light in vacuum) is within 5% for the parameters used in Fig. 2. In general, operation of SD-FCT in the regime where*NA*≪1 and^{2}z_{s}*πc/(nz*+_{s}*Δz)*≠0 results in only slight deviation of the fringe periodicity from*πc/(nz*+_{s}*Δz)*. - III. Signal-to-noise ratio The signal-to-noise ratio (SNR) of an SD-FCT system can be calculated following the lines reported by Leitgeb
*et al*. [11]. Let*N*_{DC}(Ω_{n}) and*N*_{AC}(Ω_{n}) represent the DC and modulating photoelectron distributions, respectively, such that*N*(Ω_{n})=*N*_{DC}(Ω_{n})+*N*_{AC}(Ω_{n}). Then, without loss of generality, the SNR of a fluorescent point object located along the optical axis is expressed as$$\mathrm{SNR}=\frac{\underset{n>0}{max}{{\mid \mathrm{DFT}}^{-1}\left[{N}_{\mathrm{AC}}\left({\Omega}_{n}\right)\right]\left(n\right)\mid}^{2}}{{\sigma}_{\mathrm{Rec}}^{2}+{\sigma}_{\mathrm{Shot}}^{2}\phantom{\rule{.2em}{0ex}}+{\sigma}_{\mathrm{RIN}}^{2}},$$where

$${\sigma}_{\mathrm{Rec}}^{2}=\frac{{\sigma}_{\mathrm{read}}^{2}\phantom{\rule{.2em}{0ex}}+\phantom{\rule{.2em}{0ex}}{\sigma}_{\mathrm{dark}}^{2}}{{N}_{\mathrm{pixels}}},$$$${\sigma}_{\mathrm{Shot}}^{2}=\frac{1}{{N}_{\mathrm{pixes}}}{\mathrm{DEF}}^{-1}\left[{N}_{\mathrm{DC}}\left({\Omega}_{n}\right)\right]\left(0\right)$$and

$${\sigma}_{\mathrm{RIN}}^{2}=\frac{1}{\mathrm{T\Delta}{\omega}_{f}}{\left[{\mathrm{DFT}}^{-1}\left[{N}_{\mathrm{DC}}\left({\Omega}_{n}\right)\right]\left(0\right)\right]}^{2}\mathrm{,}$$with σ

^{2}_{read}and σ^{2}_{dark}denoting the read and dark noise terms of the detector array, respectively. N_{pixels}is the number of pixels in the spectrometer’s detector array onto which the fluorescence emission is dispersed and DFT^{-1}is the inverse discrete Fourier transform. The SNR analysis assumes negligible fringe washout effects due to the spectrometer’s finite resolution. This assumption holds in cases where the spectrometer’s resolution is much finer than the spectral fringe period (*∣π/(nz*+_{s}*Δz)*∣). Note that in general, an appropriately designed spectrometer would suffer theoretically from up to ∼ 4 dB decrease in SNR at the maximal axial range due to fringe washout [11].Figure 3 shows the SNR of an SD-FCT system for a point object positioned at

*z*=0 (that is, at the geometric focal plane of the emission light) as a function of the emission power and the mean free path (MFP) for three different objective NA’s. The deep blue color represents areas for which either the excitation power exceeded 10 mW (concave boundary) or the sensitivity limit per pixel was reached (convex boundary). The parameters used for this figure were the same as those in Fig. 2. Finally, as was previously noted, for_{s}*NA*=0.54 the Gaussian beam was expanded in order to prevent the undesired mismatch between the focal planes of the excitation and emission beams. As can be seen, the SNR for a given MFP value increases with the objective’s NA due to the improved collection efficiency. Note that this is also the reason for the larger tolerable emission power in SD-FCT systems employing higher NA objectives. Finally, the larger attenuation of the excitation light for fluorophores positioned at*z*>0 results in a lower SNR relative to that obtained for fluorophores located at_{s}*z*<0. This effect becomes more pronounced for systems using low NA objectives._{s}To gain more insight on the SNR of the SD-FCT signal, we used Eq. (8) and Eq. (14) to derive the SNR of a fluorescent point object located along the optical axis for an SD-FCT system employing sufficiently low NA objectives and a pixel array consisting of point detectors. The resulting SNR expression is

$$\frac{\frac{1}{{N}_{\mathrm{pixels}}}{\left(\frac{\mathrm{\rho \eta}}{\mathit{\u0127}{\omega}_{0}}{\mathrm{TR}}_{f}\left(0\right)\right)}^{2}{\mid {h}_{\mathrm{exc}}{h}_{c}\mid}^{4}{\mathrm{K\u0303}}_{L}^{2}{\mathrm{K\u0303}}_{R}^{2}}{{\sigma}_{\mathrm{read}}^{2}+{\sigma}_{\mathrm{dark}}^{2}+\frac{1}{{N}_{\mathrm{pixel}}}\frac{\mathrm{\rho \eta}}{\mathit{\u0127}{\omega}_{0}}{\mathrm{TR}}_{f}\left(0\right){\mid {h}_{\mathrm{exc}}{h}_{c}\mid}^{2}\left({\mathrm{K\u0303}}_{L}^{2}+{\mathrm{K\u0303}}_{R}^{2}\right)\left(1+\frac{\mathrm{\rho \eta}}{\mathit{\u0127}{\omega}_{0}}\frac{{R}_{f}\left(0\right)}{\Delta {\omega}_{f}}{\mid {h}_{\mathrm{exc}}{h}_{c}\mid}^{2}\left({\mathrm{K\u0303}}_{L}^{2}+{\mathrm{K\u0303}}_{R}^{2}\right)\right)},$$where ${R}_{f}(0)={\displaystyle {\int}_{-\infty}^{\infty}{S}_{f}(\Omega )d\Omega =1}$ and

*h*are evaluated, without loss of generality, for a fluorescent point object located along the optical axis (i.e., at (_{c}*r*)=(0,_{s},z_{s}*z*)) and_{s}*r*=0. It can be shown that ∣_{i}*h*(0;0,_{c}*z*)∣=(2_{s}*πNA*∣M∣)∙sinc(^{2}/λ_{0}*u*/4)). SNR values computed using Eq. (15) deviated by maximum 7% from the accurate values over a range of fluorophore positions covering the entire depth of focus. Ignoring defocusing effects, it is also possible to account for the approximated effect of an axial distribution of uniformly excited fluorophores on the distributed SNR by simply multiplying Eq. (15) by sinc_{i}^{2}(2π*n*W/*λ*) and replacing_{0}*R*(0) with the total number of axially excited fluorophores._{f}Similar to spectral-domain optical coherence tomography (SD-OCT), the SNR in SD-FCT for shot-noise/RIN limited detection is independent of the number of pixels whereas it is not for receiver noise limited detection [cf. Eq. (15)]. However, it is important to note that unlike SD-OCT, where the large reference arm power (which is practically much larger than the sample arm power) provides heterodyne gain and results in a negative parabolic-like SNR dependence on sample arm power with a maximum achieved in the shot-noise limited detection regime (cf. Fig. 4 in [11]), the SNR in SD-FCT for comparable losses in the two interferometer arms monotonically increases with emission power and asymptotically approaches the RIN detection limit. Therefore, the use of a low noise detector array is critical for placing SD-FCT in the shot-noise limit operation regime. Finally, note that in contrast to practical SD-OCT where the detection of photons in each spectral channel is on average shot-noise limited, the absence of a large reference arm power in SD-FCT may result in information that cannot be extracted from the spectral channels. Therefore, the SNR per pixel rather than the integrated SNR [Eq. (15)] is the practical limit to the performance of SD-FCT.

- IV. Optical sectioning response As was already noted in Section 2, the optical sectioning capability of SD-FCT is realized by employing coherence gating. To characterize the optical sectioning performance, we computed the axial response of an SD-FCT system for a fluorescent point object located at (
*r*)=(0,_{s},z_{s}*z*) by calculating the modulus of the FT_{s}^{-1}of the spectral interferogram signal. The left panel in Fig. 4 describes the normalized depth response of an SD-FCT system employing 0.18-NA objectives. The parameters used for this figure were the same as those in Fig. 2. Also shown is the normalized depth response of a confocal microscope computed for a fluorescent point object that is axially scanned through the focus and detected by a point detector (*r*=0) and a detector with a radius of_{d}*r*=7.5 μm. The right panel in Fig. 4 shows the axial FWHM extent of the depth response for SD-FCT and confocal microscope for NA values ranging from 0.05 to 0.65. The parameters used for this figure were the same as those in Fig. 2. As observed in Fig. 4, SD-FCT shows superior sectioning performance compared to confocal microscopy. As a result, for NA’s lower than ∼0.2 SD-FCT benefits from a large depth of field and a narrower axial PSF. When imaging specifically fluorescently labeled samples, this translates into a significantly improved axial resolution (one to two orders of magnitude) along large depths (∼20-400 μm). Note that for NA’s higher than 0.52 and 0.58 and emission bandwidths of 50 nm and 70 nm, respectively, the extent of the depth response is primarily dictated by the objective PSF [cf. ∣_{d}*I*_{2}∣ in Eq. (9)].Lastly, we examined the degree to which coherence gating enhances confocal microscopy for fluorescence imaging through scattering media using selective labeling. Figure 5 shows the theoretical limits to fluorescence coherence tomography and microscopy when imaging a point fluorescent probe (that is, W <

*λ*/2_{0}*n*= 217 nm for*λ*=610 nm,_{0}*n*=1.4) that resides deep (500 μm) inside a scattering sample consisting of a set of identical point fluorescent markers that are separated by one and one-fifth of the fluorophore coherence length (∼2 μm in*n*=1.4) along 1 mm. This type of axial fluorophore distribution is ideal for SD-FCT since it avoids averaging of the spectral fringes and enables SD-FCT to localize each probe given that the receiver noise and the shot/RIN noise generated by the entire fluorophore distribution are lower than the power of the single probe; using more continuous fluorophore distributions would make SD-FCT less practical for imaging due to the reduced visibility and SNR. Additional parameters used for this figure were the same as those in Fig. 2. As can be seen, one limit to SD-FCT is imposed by the detection sensitivity and the second limit is due to scenarios in which confocality is sufficient for rejecting out-of-focus light. Figure 5 reveals that coherence gating improves confocal microscopy for low and moderate NA values (∼0.05-0.45) and for the range of 0-12 scattering MFP, therefore predicting superior depth imaging of specifically labeled biological structures. Low NA’s (<0.09) can be attractive for achieving high axial resolution (a few microns) along a large depth of field (a few hundreds of microns) in a relatively low scattering medium, whereas moderate NA’s can be used to enhance depth selectivity in a moderately scattering medium. Finally, note that the reduction in sensitivity for NA’s>0.21 is due to the increased mismatch between the focal planes of excitation and emission light. This could be circumvented by expanding the excitation beam for NA’s>0.21 which would, on the one hand, extend the viability of SD-FCT to regions of higher scattering; on the other hand, it would make confocal microscopy sufficiently efficient at lower NA’s.

## 4. Conclusions

In this work, we have described in detail the image formation for a new form of coherence-gated fluorescence imaging -- fluorescence coherence tomography (FCT) and microscopy (FCM). Our analysis shows that the combination of coherence-gated fluorescence techniques with specific fluorescence labeling methods may be suited for imaging into scattering samples. For low NA objectives (<0.09), the coherence gate determines the axial extent of the SD-FCT point-spread function, making FCT attractive for accomplishing high axial resolution (a few microns) along a large depth of field (a few hundreds of microns) in a relatively low scattering medium (6 MFPs). For moderate NA’s, the coherence gate can assist in improving the optical sectioning performance in moderately scattering media.

The visibility of the SD-FCT signal becomes narrower and taller as the NA increases, decreases for fluorescent markers positioned far from the focal plane and shows an asymmetric dependence on the location of the fluorophores. The SNR of an SD-FCT system is larger for higher NA objectives and for fluorophores located closer to the side from which excitation light emerges and increases monotonically with emission power. Both SNR and visibility in SD-FCT are affected by the axial fluorophore distribution: An ideal selective fluorescence labeling for SD-FCT consists of thin (<*λ _{0}*/2

*n*) fluorescent probes separated axially by greater than one half of the fluorophore’s coherence length (typically, a few micros) and distributed across the usable ranging depth of SD-FCT.

We point out that since fluorescence coherence-gated imaging is a path-length resolved method there can be additional coherent and incoherent contributions to the FCT/FCM signal due to fluorescent multiply scattered light. These effects would probably broaden the SD-FCT point-spread function in highly scattering media and will be studied in the framework of Monte Carlo simulations [14]. Finally, an experimental verification of the theoretical predictions presented in this work is planned.

## Acknowledgments

A. Bilenca gratefully acknowledges the support of the Commission of the European Communities under the Marie Curie Outgoing International Fellowship.

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