Ординатура / Офтальмология / Английские материалы / Handbook of Optical Coherence Tomography_Bouma, Tearney_2002

is the difficulty of implementing a lateral scanning mechanism within the restricted volume available at the working end of the probe. A novel solution to this problem that has recently been proposed uses a miniaturized diffraction grating–lens combination with no moving parts [23]. This system encodes the reflectivity of the sample as a function of lateral extent into the spectral content of the detected signal light, using a broadband optical source.

10.4COHERENCE-GATED CONFOCAL MICROSCOPY

Although confocal microscopy has been used successfully for imaging through relatively thin layers of living biological tissues [24], the mechanism of image formation in the axial dimension immediately gives rise to two important limitations to confocal imaging in a turbid media. First, because turbid media are exponentially scattering (i.e., their reflectivity varies exponentially as a function of depth), an imaging system that aims to image deeply into turbid media should preferably feature an axial PSF, which is a stronger function of distance from the object plane than an exponential. The envelope of the sinc functions in Eqs. (4a) and (5) do not satisfy this criterion. Second, confocal microscopy features no intrinsic mechanism for rejection of multiply scattered light, which inevitably leads to image clouding at sufficient depth [20].

Both of these concerns are alleviated by the addition of coherence grating to confocal microscopy. As will be shown below, the effective axial PSF of an optical coherence microscope approaches a Gaussian function of distance from the object plane (in the ideal case of a source with a Gaussian spectrum), which is a stronger function of distance than an exponential and thus achieves stronger theoretical spatial discrimination than a confocal gate alone. Second, the addition of the coherence gate to the confocal gate provides an additional barrier for increased rejection of non-image-bearing multiply scattered light.

To extend the quantitative analysis of the previous section to the case of coherence-gated confocal microscopy (i.e., OCM), we must first expand the theoretical description of OCT derived in previous chapters to include the axial PSF of the confocal gate, which is always present (but rarely considered because it is usually broad) in OCT. The lateral PSF of an OCM system is not affected by the coherence gate and is given by Eq. (4b) for a hard-edged (i.e., bulk-optic system) confocal aperture. The axial response of OCM depends upon both confocal and coherence gates. The detector current in an OCT/OCM system is in general given by [25]

where lr is the reference arm length, ls is the sample arm length, RsðlsÞ represents the depth-dependent reflectivity of the sample, hsðlsÞ is the axial PSF of the sample arm confocal optics [given by Eqs. (4)–(7) depending upon the normalized aperture of the

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pinhole or fiber and the nature of the object], and Riiðlr lsÞ is the source autocorrelation function, which can be thought of as the axial PSF of the coherence gate. The tildes above the detector current and autocorrelation functions denote that these quantities are undemodulated; typically the detector current is demodulated at the carrier frequency of the autocorrelation function (usually given by the Doppler shift of the reference arm light) before being digitized and displayed [26]. The square root

sign denotes that in OCM/OCT the detector current is proportional to the electric field amplitude returning from the sample arm rather than its power. The salient feature of Eq. (8) is that the confocal PSF hsðlsÞ multiplies with the depth-dependent

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reflection coefficient profile RsðlsÞ, whereas the low coherence interferometric PSF Rii ðlr lsÞ convolves with the reflection coefficient profile [27,28].

The Weiner–Khinchin theorem states that the autocorrelation function of a signal is given by the inverse Fourier transform of that signal’s power spectral density [29]. Although many sources used in OCT are not ideal, many sources approach the ideal case of having a Gaussian power spectral density (PSD) represented by

sample reflectivity and confocal gate PSF is a Gaussian function.

The interaction of the confocal and coherence PSFs is illustrated by considering the case of a single planar reflector at depth ls0 in the sample arm. In this limit,

location and reference arm position thus provide independent control over the positions of the peaks of the confocal and coherence gates, respectively. Maximum signal acquisition occurs for lr ¼ ls0, when both functions are overlapped at the desired detection depth. Maintaining perfect overlap of these two functions can be a challenging experimental concern in random media such as biological tissues (see below).

One mode of imaging in OCM is to overlap the confocal and coherence gates and then raster scan in the x–y direction, i.e., at fixed sample arm depth ls0 [9,25,30– 32]. Another mode is to perform dynamic focusing wherein the focus position is scanned within the tissue in synchrony with reference mirror scanning [33–35]. In this case, Eq. (8) takes the form

In this case, both the coherence and confocal PSFs convolve with the depth-depen- dent sample reflectivity.

In either scanning mode, however, the Gaussian shaped autocorrelation function falls off more rapidly than the axial response of the confocal microscope; thus the overlap of both functions in OCM provides enhanced scatter rejection of out-of- focus light compared to confocal microscopy alone. As an example, experimental plots of the confocal axial response (for a planar mirror sample) and source autocorrelation function of a laboratory OCM system are illustrated in Fig. 2. The

Figure 2 (a) Linear and (b) logarithmic plots of the confocal axial response of an OCM using a 40 , 0.65 NA objective and the source autocorrelation function under conditions of exact spatial overlap. The plots were obtained by translating a mirror through the focal region with the reference arm blocked (confocal only) and unblocked (autocorrelation function). (From Ref. 25.)

confocal axial response was obtained by scanning a mirror through the focus of the objective with the reference arm blocked. The source autocorrelation function was obtained by scanning a mirror through the focal region of the OCM with weak sample arm focusing and is close to a Gaussian function as expected for a semiconductor superluminescent diode source [36]. It is apparent from the plots that near the focus the combined axial response of the OCM is dominated by the axial response of the confocal microscope. However, far from the focus (better seen on the logarithmic plot) the sharper autocorrelation function shape acts to reject scattered light, which is incompletely eliminated by the aberrated wings of the confocal-only axial response. In this case where the confocal PSF dominates on a linear scale, one can think of the coherence gate as essentially isolating a region of depth in the sample within which the confocal microscope operates as usual.

The advantage of OCM over confocal microscopy alone for scattered light rejection in turbid media is illustrated in Fig. 3. This figure illustrates en face imaging of absorbing structures placed in highly scattering media, which were not visualized using confocal microscopy alone but which were clearly visualized with the addition of coherence gating. The confocal image in this case was obscured by out-of-focus light rather than electronic noise, which the confocal PSF was incapable of penetrating.

10.5IMAGE FORMATION IN SCATTERING MEDIA

A complete analytical description of image formation using either confocal or coherence gating that is valid in highly scattering media has not yet been developed. A number of Monte Carlo simulations of confocal microscopy, OCT, and OCM have appeared in the literature [20,37,38], but it is difficult to obtain physical insight from these results. A preliminary theoretical description of single and multiple scattering

Figure 3 Two-dimensional images of absorbing bars embedded 700 m deep in a scattering medium (0:3 m diameter polymer microspheres) with scattering coefficient s 6 mm 1. Top left, confocal image without coherence gating. Bottom left, confocal plus coherence gating reveals the embedded structures. Right, plots of backscatter versus depth at positions both between (A) and over (B) the absorbing bars. (From Ref. 9.)

contributions to OCT has been adapted from atmospheric scattering theory primarily by Schmitt and coworkers [19,38]. Although this model is in no sense complete, it does give considerable physical insight into image formation in scattering media and compares favorably with both experimental and Monte Carlo results published to date.

10.5.1 Single-Backscatter Theory

A single-backscatter model, adapted from atmospheric scattering theory, describes the heterodyne signal from a homogeneous scattering medium under the assumption that only single-backscattering events need to be considered [9,18,19]. Although this model is clearly not valid deep in biological or other turbid media, it does provide experimentally verified accurate results near tissue surfaces and also serves as a starting point for more sophisticated models, which include multiple scattering.

The single-backscatter model depends on the following assumptions [19]:

1.The distance between scatterers is much greater than the wavelength of the light source.

2.The polarization of scattered light is the same as that of the incident beam by the single-backscatter events.

3.The time-averaged return signal from a single scatterer is incoherent with respect to the time-averaged signal from any other scatterer.

4.The scattering phase shifts vary randomly, because the scattering particles are not all identical.

Figure 4 depicts the geometry of three coordinate systems including a transmitter at plane x0y0, a scattering plane xy, and a receiver at plane x00y00. All have a common z axis, which is the direction of beam propagation. An objective is located at plane x0y0, which is overlapped with the x00y00 plane. A collimated beam with a Gaussian amplitude distribution (having diameter D at intensity equal to 1=e2 of its center intensity) passes through the objective with focal length f and diameter D and is focused at a distance F from the transmitter. The medium between the objective and the sample has refractive index n1. The surface of the sample, having refractive index n2 and extinction coefficient t, has a coordinate (r; z ¼ 0) in the xy plane and is located at a distance a from the transmitter. The time-averaged heterodyne signal power contributed by the scatterers at a distance LðzÞ ¼ n1a þ n2z from the transmitter can be calculated by summing the scattered power over the scattered plane r and the axial distance z from the sample surface. Distances a and z are measured in air. F and LðzÞ are the distances of the focal plane and the scattering plane from the objective expressed in optical pathlengths. After integrat-

energy, P0 is the incident power, and b is the backscattering coefficient. j ð Þj is the self-coherence function of a heterodyne detection system, where is the optical time delay due to the pathlength difference of two arms. j ð Þj is equal to 1 for ¼ 0 and diminishes rapidly when the optical pathlengthÐdifference exceeds the coherence length lc of the light source. lc is defined by lc 11 j ð Þj2 d .

Figure 4 Geometry of the illumination and collection optics used in the single-backscatter model.

Therefore, the time-averaged heterodyne signal current within a coherence length of the center position at LðzAVÞ ¼ L from the transmitter (or at zAV from the surface of the sample) can be approximated as

RsðzÞ is the on-axis reflectance of the sample in the z plane. This expression contains an exponential attenuation term describing light propagation to the scattering site and back and another term describing a local reflectivity peak at the position of the focus, L ¼ F.

The focal point of a collimated light beam passing through an objective (focal length f ) is extended to nf when the objective is immersed in a medium having refractive index n. In this condition, the optical pathlength of the focal point from the objective, F, and the distance of the scattering plane from the objective, L, are

When there is an air gap between the objective and the scattering medium, Eqs. (16) are modified to be

Experimental verification of these relationships is provided in Fig. 5. The plots depict experimental profiles of the heterodyne backscatter signal obtained in a polymer microsphere suspension along with the theoretical behavior calculated from Eq. (15). Although the single-backscatter model overestimates the signal magnitude at the focus and near the sample surface, the general behavior of Rs as a decreasing exponential punctuated by a confocal focus is demonstrated.

10.5.2Comparison of OCM and Confocal Microscopy Under the Single-Scattering Assumption

The single-backscatter model was adapted by Izatt et al. [9] to compare the behavior and calculate the imaging depth limits of OCM and confocal microscopy. Two interesting limits can be readily calculated. First, the imaging depth limit of a confocal microscope was estimated under the condition of coherent detection by assuming that the received signal in a confocal microscope equals the integral of the time-averaged heterodyne current [Eq. (14)] over all depths. When light scattered from other planes exceeds the collected signal from the focal plane, image contrast drops dramatically. This contrast limit to confocal microscopy was estimated by solving Eq. (14) for the depth where the signal from the focal plane equals the signal at the surface. Given in terms of tz1 or the number of scattering mean free paths, this limit is

Figure 5 Plots of the heterodyne backscatter signal normalized to the measured reflection coefficient from a mirror versus the imaging depth in a calibrated polymer microsphere suspension ( s ¼ 6 mm 1). Plots are shown for different focal plane depths of a 20 microscope objective. Dashed curves are predictions of the single-backscatter model. (From Ref. 9.)

"# " #

where NA is the numerical aperture and M (the lens aperture diameter divided by the focal spot diameter) is the geometrical magnification due to the lens.

A second limit is the depth to which single-backscattered light can be detected given quantum detection and tissue tolerance limits. After Hee et al. [39] and assum-

where E is the total energy incident upon the sample for each resolution element. Tissue damage thresholds and reasonable image acquisition times limit the energy that can be delivered to biological samples to a few hundred millijoules per pixel [9,39]. Linear increases in optical power, data acquisition time, and signal averaging affect this limit only logarithmically.

The range in imaging depths between these two limits predicts when OCM outperforms confocal microscopy according to single-scattering theory. Figure 6

Figure 6 Parametric plot of theoretical limits to OCM in the single and multiple backscattering models. Solid lines: Predictions of the single-backscatter model. Model parameters: D ¼ 3:7 mm, n0 ¼ 1:33, lc ¼ 20 mm, ¼ 800 nm, b ¼ 0:3191 s, E ¼ 130 mJ. Dashed lines: Predictions of Huygens–Fresnel multiple scattering theory [38] for maximum single-scattered imaging depth. Model parameters: n ¼ 1:3, f ¼ 5 mm, ¼ 800 nm.

depicts these limits, assuming typical microscope and tissue scattering parameters. Also plotted on the figure are more realistic predictions from consideration of multiple scattering, which is considered in the following sections.

10.5.3 Role of Multiple Scattering

Multiple scattering becomes important when imaging deeply into dense biological tissues [38,40]. According to the analysis of the previous section, multiple scattering reduces image contrast in confocal microscopy when imaging at a sufficient depth that the probability of multiply scattered light entering the detection pinhole exceeds the extinction of singly scattered light. Analogously, multiple scattering can reduce OCT image contrast (and possibly resolution) when imaging at a sufficient depth that the dual probabilities of multiply scattered light both entering into the detection pinhole and having traveled the correct distance exceed the extinction of singly scattered light. Since OCM is the limiting case of OCT at high numerical aperture, it would be important to understand how multiple scattering affects image formation in this limit. Unfortunately, this is still an open question.

Experimental evidence of multiple scattering in OCT has been presented by several authors. These evidences have included observations of reductions of image contrast and resolution deep in highly scattering media [40–43], the observation of phantom scattered light signals from nonscattering layers in otherwise highly scattering phantoms [40], deviations of A-scans from exponential decay in highly scattering media [40,41,43] (see Fig. 7), and observations of detected electronic power spectrum broadening due to light multiply scattered from particles undergoing Brownian motion [44,45]. In several of these experiments, multiply scattered light was detected in the range of 5–10 scattering mean free paths deep into turbid samples. However, other experimental variables (scattering anisotropy of the sample, numerical aperture of the confocal optics, wavelength and coherence length of the source) varied widely, and to date no comprehensive study of multiple scattering in OCT as a function of these variables has been published.

A few theoretical treatments of multiple scattering in OCT have also appeared [19,38,42,43]. Although largely untested experimentally, the most complete of these

Figure 7 Experimental data illustrating multiple scattering in OCT. Deviations of the detected signal from exponential decay with depth indicate the detection of multiply scattered light for depths greater than 0:5 mm. The samples were various dilutions of Intralipid solution (g 0:8, scattering coefficients as indicated), imaged with confocal optics having a numerical aperture of 0:1. (From Ref. 41.)

treatments considers image formation in OCT by adapting the extended Huygens– Fresnel formulation from atmospheric scattering theory [38]. In this formulation, which is more general than the one-dimensional treatment typically applied in OCT, the detector current is described as an integral over the product of the mutual coherence functions (MCFs) of the reference and sample arm beams:

reference and sample arm beams, respectively, which describe the spatial correlation of the electric fields of the beams at a given depth z as a function of the lateral

In a turbid medium, the detector current can be expressed as a sum of contributions from single and small-angle multiple scattering events. The single-back- scattered component results from a coherent summation over the light field singly backscattered from each of the sample volume scatterers in the vicinity of the beam focus in the sample, as in Eq. (14). Because the phases of signals having experienced pathlength dispersion resulting from multiple scattering are random, the multiply scattered component results from an incoherent sum over the multiply scattered signal powers received.

An interesting limit to OCT and OCM in scattering media suggested by Schmitt occurs at an imaging depth when the magnitudes of the multiply scattered

signal overtakes that of the singly scattered signal. This occurs when

!

Here s is the scattering coefficient of the sample, R is the 1=e radius of the incident probe beam, and 0 is the transverse spatial coherence length of the beam. An approximate relation for 0 valid for small-angle scattering is [46]

Here g is the scattering anisotropy of the specimen, rms is the mean scattering angle in the tissue, and the approximate relation rms2 ¼ 2ð1 gÞ was used [47].

In expression (23) it is notable that the quantity R2=20 approximately corresponds to the number of speckle spots resulting from lateral beam incoherence (due to multiple scattering) that are projected onto the detector. This result leads to the surprising conclusion that increasing R (equivalent to increasing the numerical aperture of the sample arm optics for a fixed focal length) decreases the sensitivity of the OCM to multiply scattered light. The increase in the coherent (singly scattered) signal amplitude outstrips the increase in uncorrelated (multiply scattered) signal amplitude as a function of numerical aperture. Also, increases in the mean scattering angle (decreases in g) also lead to reduced sensitivity to multiple scattering, because increasing rms reduces the lateral coherence length and thus also increases the number of speckle spots projected onto the detector. The dependence of zmax on numerical aperture (NA) and scattering anisotropy g is plotted in Fig. 6 for typical OCM imaging parameters. In rough correlation with experimental results referred to above, the predicted range of zmax is in the range of 5–10 scattering mean free paths for typical imaging parameters. It should be emphasized, however, that the functional dependences of the theoretical predictions have not yet been systematically investigated experimentally.

10.6EXPERIMENTAL APPROACHES

The challenges encountered in constructing OCM systems include essentially all of the challenges of building confocal microscopes plus the necessity to perform coherence gating with high signal-to-noise ratio. As in conventional confocal microscopy, obtaining high quality objectives is expensive and always involves a trade-off