## Abstract

In this paper, we introduce a new form of cross-sectional, coherence-gated fluorescence imaging, which we term ‘*spectral-domain fluorescence coherence tomography*’ (SD-FCT). SD-FCT is accomplished by spectrally detecting self-interference of the spontaneous emission of fluorophores located along the axial (depth) dimension of the sample. We have built a first generation SD-FCT system that utilizes two opposing low numerical-aperture objective lenses in an interferometer and an imaging spectrometer for detecting self-interference of fluorescence emitted from a sample. Here, in proof-of-principle experiments we demonstrate cross-sectional profiling of layered fluorescence phantoms. Narrow (a few micrometers FWHM) axial point-spread functions, large ranging depths (a few hundreds of micrometers) and wide fields of view (>1 mm) were measured. Initial results suggest that SD-FCT may be a viable tool for the investigation of semi-transparent and selectively labeled fluorescent samples.

©2006 Optical Society of America

## 1. Introduction

The investigation of optical imaging techniques for obtaining information within a biological specimen is a rapidly evolving field, with applications in diverse areas of science and technology. Many existing and proposed techniques aim to detect signal at one given plane or point within the sample, while rejecting all other light. Among these methods are spatial filtering, nonlinear excitation, and coherence gating. Coherence gating is advantageous over the other methods in that it does not require an imaging lens with a high numerical aperture (NA). Resultant coherence-gated images may be therefore obtained over a wider field of view and deeper into biological specimens as has been demonstrated by optical coherence tomography (OCT) [1], [2] and optical coherence microscopy (OCM) [2], [3] which are probably the most widely employed coherence gating techniques nowadays.

In this paper, we introduce a new paradigm of fluorescence coherence-gated tomography termed ‘*spectral-domain fluorescence coherence tomography*’ (SD-FCT) which adopts concepts of optical coherence gating and fluorescence self-interference in order to provide axial (depth) profiling of fluorescent molecules and labels. We demonstrate large-scale (hundreds of microns) ranging depth capabilities of SD-FCT in a proof-of-principle experiment employing layered fluorescence phantoms. Cross-sectional images are obtained without beam, sample, or objective lens scanning. These initial results suggest that SD-FCT may be a viable tool for studies that would benefit from wider fields of view (millimeter-scale) and cross-sectional information (over hundreds of microns with high axial resolution of a few micrometers) including imaging of specifically fluorescently labeled semi-transparent specimens in a larger, more organismal context [4] as well as tracking of fluorescent labels in semi-transparent media across large areas. Finally, it should be noted that while demonstrated here for fluorescence, the same principles of SD-FCT can be applied to phosphorescence and quantum dots.

It is worth mentioning that the phenomenon of fluorescence self-interference has been investigated in fundamental studies of molecules in front of reflecting surfaces [5], [6] as well as used to determine nanometer displacements of a fluorescent molecule above reflectors, by measuring emission intensity [fluorescence interference contrast microscopy, 7] and spectral phase shift [spectral self-interference fluorescence microscopy, 8]. It is important to recognize that while spectral self-interference fluorescence microscopy measures nanometer displacements based on phase information, FCT measures axial fluorophores profiles using intensity information. Note that phase information of the FCT signal can also be exploited to measure nanometer displacements. Finally, fluorescence self-interference has been employed in 4-Pi(B/C) microscopes [9] and I^{2/5}microscopes [10] for decreasing the spatial extent of the microscope point-spread function (PSF).

The manuscript is organized as follows. In Section 2, we describe the operational principles of SD-FCT and in Section 3 we present the experimental set-up. Section 4 is devoted to the experimental results and discussion. Finally, conclusions are drawn in Section 5.

## 2. SD-FCT: Principles of operation

SD-FCT performs coherence-gated ranging by using an imaging spectrometer to detect self-interferences of fluorescent emitters located along the axial (depth) dimension of a specimen. In SD-FCT, the sample is located near the zero differential path length point (*z*
_{0}) of a two opposing low-NA objectives interferometer and is illuminated with a line focus at the excitation wavelength as shown in Fig. 1(a). Fluorescence emission is collected by the two matched, opposing objectives. Following transmission through the interferometer’s beam splitter, the fluorescence from the sample is imaged onto a CCD along one dimension, and spectrally resolved in the other direction as depicted in Fig. 1(b). The location of each fluorophore is encoded by an interferometric frequency modulation of the emission spectrum, where the frequency is proportional to the fluorophore’s distance from *z*
_{0}. For any given point on the line, the axial locations of the fluorophores may then be retrieved by calculating the modulus of the inverse Fourier transform of the spectral interferogram intensity. A cross-sectional fluorescence image is obtained when all points along the line are processed in an identical manner.

To establish the mathematical formalism of SD-FCT let *f _{zf}* (

*ζ*

_{F}) = ${\sum}_{m=1}^{M}$

*P*

_{m}

*δ*(

*ζ*

_{F}-

*ζ*

_{m}) represent a one-dimensional distribution of emitting fluorophores along a pencil excitation beam propagating parallel to the optical axis (

*z*) with

*P*

_{m}(

*m*=

*1,2 ...,M*) being the percentage of emitting fluorescent markers (out of the total

*N*radiating fluorophores along the illumination axis) located at position

*ζ*

_{m}(

*m = 1, 2, ...,M*).

It is assumed throughout the following analysis that the zero path-length point (*z*_{0}
) of the interferometer is positioned at *z* = 0 and that *ζ*_{m}
> 0, (*m = 1, 2, ...,M*). Treating the fluorescence emission in the framework of classical electro-magnetic fields, employing the paraxial approximation and assuming that the fluorescent labels in the sample were stationary as well as ignoring multiply scattered fluorescent light, defocusing, dispersion, attenuation and scattering of the fluorescence and excitation radiation propagating in the sample as well as polarization effects, the detected power spectrum at the interferometer exit, *I*_{D}*(ω)*, can be expressed as (see for example the derivation of Eq. (4.16) in [2])

$$\phantom{\rule{.2em}{0ex}}=N\bullet {I}_{F}\left(\omega \right)\bullet {E}_{{Z}_{F}}\left[\left({a}^{2}\left({\zeta}_{F}\right)+{b}^{2}\left({\zeta}_{F}\right)\right)\bullet \left(1+V\left({\zeta}_{F}\right)\mathrm{cos}\left(\frac{2{\zeta}_{F}}{c}\omega \right)\right)\right]$$

where *N* is the total number of the total number of emitting fluorophores along the illumination axis, *A*_{m}*(t)* is the fluorescence radiating electric field of the *m*-th fluorophore, *I*_{F}*(ω)*=<|ℑ*{A*_{m}*(t)}(ω)*|^{2}> is the detected fluorescence emission spectrum and *c* is the speed of light. In Eq. (1) *a(ζ)* and (1) *b(ζ)* are field amplitudes accounting for the possible difference in losses between the two interferometer arms, and *V=2ab/(a*^{2}*+b*^{2}*)* is the visibility of the spectral fringes. Finally, *E _{ZF}* (∙) is the ensemble average operator with respect to

*f*(

_{ZF}*ζ*

_{F}).

The axial positions of the emitting fluorophores (i.e., {*ζ*_{m}
: 0<*P*_{m}
≤ 1${\}}_{m=1}^{M}$) are encoded in the oscillation period *πc/ζ*_{m}
of the cosine function and are extracted by computing the squared modulus of the inverse Fourier transform (FT) of Eq. (1), yielding the following expression

with *Γ*_{F}*(z)*=ℑ^{-1}{*I*_{F}*(ω)*}*(2z/c)* representing the complex self-coherence function (or, autocorrelation function) of the fluorescence emission light.

The first term inside the modulus operator of Eq. (2) describes the autocorrelation signal, which is located around *z*=0 and contains no direct information about the axial location of the fluorophores. It has a magnitude that equals the average fluorescence emission stemming from all fluorophores successfully excited and radiating along the illuminating pencil beam, and is collected from both arms of the interferometer. The second term inside the modulus operator in Eq. (2) is the interferogram signal, which provides the axial position of the fluorophores (with respect to *z*_{0}
) convolved by Γ
_{F}*(z)*. The width of Γ
_{F}*(z)*, termed the coherence length, therefore defines the axial resolution of the FCT system. For most molecular probes, the coherence length is typically on the order of a few micrometers. Eq. (2) enables depth-resolved retrieval of fluorophore location for a single point on the excitation line focus; cross-sectional fluorescence information is obtained when all points along the line are processed in an identical manner as schematically depicted in Fig. 1(b). It is important to note that the undesired overlap between the interferogram and autocorrelation signals can be avoided by simply placing the fluorescent object at a distance of several coherence lengths from the zero path-length point (*z*_{0}
) of the interferometer. Furthermore, the interferogram signal is conjugate symmetric, therefore under the assumption that *ζ*_{m}
>0 (*m=1,2 ...,M*) only the positive *z*-domain of the interferogram should be retained for determining the axial positions of the fluorophores.

Interestingly, a comparison between Eq. (2) and corresponding expressions derived for spectral-domain optical coherence tomography (SD-OCT) [2] reveals that SD-FCT can be viewed as a hypothetical SD-OCT system in which the examined sample also acts as a spatially incoherent source with low temporal coherence. Since the source for SD-FCT is spatially incoherent, coherent crosstalk that degrades image quality in SD-OCT, manifested as speckle noise, is suppressed in SD-FCT [11]. As in OCT, FCT employs low-NA objective lenses to obtain a long Rayleigh range. However, unlike OCT, FCT suffers from low light collection efficiency since its reference signal results from self-interference and not from a separate strong reference signal as in OCT. Hence, the heterodyne gain commonly employed by OCT to place the detection system in the shot-noise limited regime can not be utilized by SD-FCT and as a result SD-FCT requires low-noise, high sensitivity CCD cameras and bright fluorescent emitters. Moreover, the absence of heterodyne gain in FCT results in a signal-to-noise-ratio (SNR) curve that monotonically increases with fluorescence power, in contrast to SNR curves of OCT that have a global maximum at a particular reference power level. Finally, the sensitivity of SD-FCT is determined by the specific radiating fluorophore distribution along the excitation beam. Consequently, for SD-FCT systems operating in the shot-noise or intensity-noise detection-limited regime, a high intensity fluorophore at any point inside the sample increases the noise floor, potentially making it difficult to detect weaker fluorophores located along the same axial line. Furthermore, like SD-OCT, the phase ambiguity that occurs for fluorophores located at positive and negative distances from the zero differential path length point (*z*_{0}
) of the interferometer can be readily resolved by recording the complex spectral density [12], thereby doubling the maximal ranging depth provide by SD-FCT.

FCT can also be implemented in the time domain by moving the sample along the axial dimension. However, similar to the advantages of SD-OCT in comparison to time-domain OCT [2], SD-FCT does not require moving parts and provides an increased SNR for a shot-noise or intensity-noise limited-detection system due to the decorrelation of noise detected by the different spectral channels.

## 3. Experimental arrangement

Figure 2 shows a schematic of the SD-FCT experimental setup. It included four main parts:

*Excitation optics*consisting of a cylindrical lens (L_{C}: f_{C}=45 mm), a vertical slit (S), a spherical lens (L_{1}: f_{1}=50 mm), and an objective lens (O_{1}: NA_{O1}=0.06), which produced an excitation line focus of 11.7 μm × 1.2 mm along the*‘x-y’*plane of the fluorescent sample (FS). A dichroic mirror (DM_{1}) directed the excitation beam toward the sample. Consequently, a set of axial distributions of fluorophores was simultaneously excited where each distribution was associated with a similar*‘x’*location but a different*‘y’*position on the specimen. Note that this excitation configuration enables*‘x-z’*imaging without scanning the beam or the sample, resulting in high-speed and robust cross-sectional fluorescence imaging. A similar line illumination concept has been recently proposed for realizing parallel SD-OCT [13].*Two opposing low-NA objective lenses interferometer*. Fluorescence emission was collected by two low-NA objectives (O_{1}and O_{2}: NA_{O1}=NA_{O2}=0.06) and recombined by a 50/50 non-polarizing beam-splitter (NPBS_{1}). DM_{1}filtered back-reflected excitation light and transmitted fluorescence emission from the right-hand arm of the interferometer to NPBS_{1}and a second dichroic mirror (DM_{2}), positioned in the left-hand arm of the interferometer, balanced dispersion and filtered excitation light.*Imaging spectrometer*comprising a diffraction grating (DG) having 600 lines/mm, an achromatic lens (L_{S}: f_{S}=80 mm) and a cooled electron multiplying CCD camera (EMCCD_{S}: Photometrics® Cascade® II, imaging array=512 × 512, pixel size=16 μm × 16 μm, readout rate=1, 5, 10 MHz, 16-bit digitization). The line focus excited fluorescence that was imaged along one dimension of 1.9 mm of the EMCCD array, while the spectral interference was detected along the other dimension of the array. The spectral range of the spectrometer was approximately 150 nm distributed over 512 pixels, thus resulting in a wavelength spacing between pixels of*Δλ*=0.29 nm. This sampling interval yielded a maximum depth range of*z*_{max}=*λem*^{2}*/(4Δλ)*∼ 320 μm where*λ*_{em}=610 nm was the optimal emission wavelength of the fluorophores. To simultaneously evaluate the axial depth profiles of fluorophores distributed in a specific*‘y-z’*plane of the specimen, one dimensional DFT’s and corrections for the nonuniform frequency sampling introduced by the spectrometer [14] were performed on the set of 512 power spectral densities acquired by the EMCCD camera. Specifically, the numerical corrections included linear interpolation by computing the inverse DFT of the sampled spectrum, dual zero-padding, and DFT. The interpolated spectrum was then resampled to generate an array of samples regularly spaced in the frequency domain.The exposure time of the EMCCD camera was set to 0.1 sec in all the experiments. The camera was cooled to -50°C, the read-out rate was 5 MHz and no multiplication gain or pixel binning was used.

*Imaging system*comprising an imaging lens (L_{2}: f_{2}=50 mm) and a CCD camera (CCD_{I}: V-1070: Marshall Electronics), which was used to initially align the two counter-propagating fluorescence images.To better understand the propagation of excitation light and fluorescence emission in the anamorphotic optical arrangement shown in Fig. 2, ray tracing diagrams of the vertical and horizontal planes in the direction of the imaging spectrometer path are drawn in Fig. 3.

## 4. Results and Discussion

To study the axial PSF of SD-FCT, we used a unimodal fluorophore distribution realized by drying a drop of a fluorescent bead suspension comprising nanospheres with a diameter of 100 nm (Red Fluorescing Polymer Microspheres, Duke Scientific; optimal excitation and emission wavelengths 540 nm and 610 nm, respectively) onto a 170 μm glass coverslip. Then, a second identical glass coverslip was glued on top of the fluorescent beads. The beads were excited at 532 nm, close to the optimal excitation wavelength, using a frequency doubled Nd:Yag Coherent Verdi V5 laser. The excitation power focused on the sample was 80 mW.

Figures 4(a) and 4(b) show, respectively, axial PSF’s as well as typical recorded spectral interferogram signal and fluorescence emission spectrum measured with a single fluorescence layer. The layer was positioned between the objective lenses to vary the path-length differences between the two arms of the interferometer from 50 μm to 250 μm. The curves were obtained by averaging 10 measurements and were numerically corrected for the non-uniform frequency sampling of the acquired power spectral densities as described above. Sharp depth-dependent axial PSF’s were clearly observed. For comparison with the noise floor of the recorded PSF’s, we also measured the electrical noise level of the cooled EMCCD camera as depicted in Fig. 4(a) in black line. From the observed difference of approximately 10 dB between the electrical noise of the CCD, the noise floor of the PSF’s, and the low signal focused on the EMCCD (a few thousands of electrons per 0.1 sec), we deduce that the shot-noise process generated by the total number of emitting fluorophores distributed along the measured axial line constitutes the dominant noise contribution.

The detection performance of SD-FCT for the single fluorescence layer was studied by analyzing the SNR of the detected PSF’s as a function of the axial location as shown in Fig. 5(a) (closed circles). The SNR was defined as the difference between the peak and the noise floor of the detected axial PSF. The SNR values varied from 33.75 dB for a fluorescent layer positioned at a distance of 50 μm from the zero path-length of the interferometer to 15.5 dB for a distance of 250 μm. The decrease of SNR with axial position stems from averaging of the spectral fringes due to the finite spectrometer’s resolution. The averaging effect becomes stronger as the oscillation period of the spectral fringe increases and it is modeled as a convolution of a rectangular function, describing the square shape of the camera pixels, with a Gaussian beam profile focused on the spectrometer. The normalized squared modulus of the inverse FT of the convolution signal yields the following expression for the magnitude of SNR reduction with axial position

with sinc(*x*)=sin(*πx*)/*πx* and *z*
_{max} being the maximum ranging depth. Also, *W=dλ/Δλ* where *dλ* is the spectral resolution of the spectrometer (FWHM). Eq. (3) was fitted to the SNR decay data with *W* being the fitting parameter. The resultant curve is shown in Fig. 5(a) in a dashed line. The value for *W* was 1.81 resulting in a practical spectral resolution of *dλ*=0.52 nm. This indicates that the spectral resolution of the spectrometer is a limiting factor for the usable axial range in our SD-FCT setup and can be theoretically improved up to ∼4dB at z=*z*
_{max} by reducing *dλ* such that *W*≪1. Typical sensitivity reduction values in SD-OCT due to the finite spectrometer’s resolution are 5-18 dB across *z*
_{max}∼2 mm which corresponds to *W*=0.5-1.85 [15], [16].

As mentioned in Section 2, SNR of SD-FCT is determined by the specific radiating fluorophore distribution along the excitation beam. Assuming shot-noise limited detection and *S*(*S*=1,2,3,...) identical and thin (i.e., similar emission power and axial width smaller than one half of the fluorophore’s center emission wavelength) fluorescent layers separated axially by greater than one half of the fluorophore’s coherence length, the SNR of a layer at any given axial location is given by SNR = *κa*^{2}*b*^{2}*P*_{f}*τ / (a*^{2}*+b*^{2}*)S* where *P*_{f}
is the fluorescence power emitted by a single layer, *τ* is the camera exposure time and *κ* is a proportionality constant with units of W^{-1}. For *S=2*^{l}
(*l*=1,2,...), the SNR of a single layer at any given axial location is 3*l* dB lower than that corresponding to a single fluorescent layer (i.e., *S*=1) at the same axial position, due to the higher noise floor along the excitation beam.

A continuous distribution of fluorophores that extends along the axial dimension over a continuous range greater than one half of the fluorophore’s center emission wavelength may also degrade SD-FCT sensitivity since spectral fringes produced by fluorophores distributed along the axial dimension are linearly combined [Eq. (1)]. FCT is therefore most sensitive to discrete fluorophore distributions (where the extent of each discrete mode is less than one half of the fluorophore’s center emission wavelength and the separation between adjacent discrete modes is greater than one half of the fluorophore’s coherence length), similar to the manner in which OCT is sensitive to well-defined specularly reflecting interfaces (produced by discontinuities in the scattering potential [2]) that are separated by more than one half of the coherence length of the OCT source. As a result, FCT is most suitable for profiling structures using selective fluorescence labeling, rather than non-specific labeling. Once more, note that selective fluorescence labeling in SD-FCT comprises thin (i.e., axial width smaller than one half of the fluorophore’s center emission wavelength) fluorescent probes separated axially by greater than one half of the fluorophore’s coherence length (typically, a few micrometers) and distributed across a few hundreds of micrometers.

To quantify the free-space axial resolution achieved by SD-FCT, we fitted a Gaussian curve to the measured axial amplitude PSF of the single fluorescent layer and extracted the FWHM of the Gaussian fit. As shown in Fig. 5(b), the range of the axial resolution values was 3.29-3.45 μm over an axial (depth) range of 250 μm and is comparable to the 3.2 μm estimated from the detected fluorescence emission spectrum. These values confirm that the 100 nm fluorescent beads indeed produced a single fluorescent layer which was thinner than one half of the emission coherence length, thus enabling us to use the axial PSF FWHM as an adequate measure for the axial resolution of SD-FCT. Other common fluorescent markers can potentially provide even higher axial resolution, for instance, green fluorescent protein (GFP) will yield a theoretical axial free-space resolution of 2.8 μm, cyan fluorescent protein (CFP), 1.8 μm, and 4′,6-Diamidino-2-phenylindole (DAPI), dimethylsulfoxide, 0.9 μm. These calculations assumed Gaussian shape of the fluorescence emission spectra. Also, the spectra used in the computations can be found in http://www.mcb.arizona.edu/IPC/spectra_page.htm To demonstrate the non-scanning large depth and wide cross-sectional profiling capabilities of SD-FCT, we prepared a fluorescent sample comprising two layers of 100 nm fluorescent beads separated by 120 μm as shown schematically in Fig. 6(a). A gasket (Adhesive 120 μm gaskets; Bioscience Tools) was used to form a closed chamber of 120 μm between the two fluorescent coverslips. To minimize back-reflections inside the chamber, the gap between the fluorescent coverslips was filled with optically transparent UV-curing epoxy with a refractive index of 1.56. Figures 6(b) and 6(c) show the cross-sectional fluorescence distribution and the corresponding tomogram of the two-layered fluorescent sample measured by the SD-FCT system. These measurements were performed by averaging five consecutive images, each acquired in 0.1 sec, in order to reduce noise floor fluctuations. Two fluorescence layers can be clearly observed over a wide transverse *‘y’*-field (> 1 mm). The mean distance between the layers was measured to be 120.1 μm. Note that the low-NA objectives employed in SD-FCT limit the lateral resolution to the range of 5-30 μm, therefore making SD-FCT suitable for profiling of large (mm^{2}) transversal area samples. Alternatively, higher NA lenses and spatial filtering or nonlinear excitation may be utilized to confine the axial extent of excitation/detection volume, at the expense of ranging depth.

## 5. Conclusions

In this work, we have introduced a paradigm for cross-sectional fluorescence imaging termed ‘*spectral-domain fluorescence coherence tomography*’ (SD-FCT). SD-FCT uses principles of coherence gating and spectral self-interferometery to obtain micrometer-resolution, depth-resolved images of fluorescent probes distributed across a wide field of view and over large axial range. Based on these results, we believe that SD-FCT may be useful for wide-field axial profiling of semi-transparent specimens that are selectively labeled with fluorescent markers. Once more, selective fluorescence labeling consists of thin (i.e., axial width smaller than one half of the fluorophore’s center emission wavelength) fluorescent probes separated axially by greater than one half of the fluorophore’s coherence length (typically, a few micrometers) and distributed across the usable ranging depth of SD-FCT.

Future experimental and theoretical studies of SD-FCT performance in turbid media (e.g., using Monte-Carlo simulations [17]) are planned. Specifically, the effect of scattered fluorescence waves on the FCT interference signal will be explored. Furthermore, we anticipate that extensions of SD-FCT may result in other additional biological applications. First, while SD-FCT described here uses low-NA objectives in order to have large depth of focus and wide field of view, employing higher-NA objectives may provide improved depth sectioning, higher lateral resolution, and sensitivity. Second, phase information of the SD-FCT signal can be also exploited, to allow nanometer-scale localization of fluorescent probes.

## 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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