Fourier optics analysis of phase-mask-based path-length-multiplexed optical coherence tomography

Biwei Yin; Jordan Dwelle; Bingqing Wang; Tianyi Wang; Marc D. Feldman; Henry G. Rylander; Thomas E. Milner

doi:10.1364/JOSAA.32.002169

1. INTRODUCTION

Optical coherence tomography (OCT) is a three-dimensional (3D) interferometric imaging technique [1–4] that has been widely applied in ophthalmology [5,6] and cardiology [7,8]. OCT constructs a depth-resolved intensity image by measuring the optical path-length difference between broadband light backscattered by a sample and a reference surface. By two-dimensional (2D) scanning of incident light, a 3D image of the depth-resolved sample reflectance can be recorded. In most OCT system sample path optics, illumination and backscattered light share a common path and optics; consequently, when a phase mask is inserted into the sample path, both illumination and detection schemes are modified, providing the point spread function (PSF) and coherent transfer function [9] with additional imaging features. An OCT system that uses a path-length-multiplexing element (PME) as a phase mask and depth-encodes subimages for discrete pupil regions is referred to as path-length-multiplexed OCT, and so far its applications have included image depth-of-focus extension, backscattering angular diversity detection, and blood flow velocity measurement [10–12]. A depth-of-focus extension is realized by correcting the defocus-induced wavefront aberration according to subimages acquired from different pupil regions [10]. Similar techniques using a subaperture method to extend depth of focus [13,14] require phase-stable acquisition and are more computationally intensive. Backscattering angular diversity detection is achieved by applying a PME to separate light with different incident and backscattering angles and calculating the signal intensity difference in recorded subimages [11]; therefore, it serves as an alternative to existing scattering angle-resolved OCT realizations [15,16]. This technique has also been applied in Doppler OCT [12]: absolute blood flow velocity is measured from one single B-scan’s path-length-multiplexed image, and has the advantage of rapid measurement and easy implementation.

In addition to conventional intensity OCT, a path-length-multiplexed OCT technique is able to provide useful information from recorded subimages introduced by a phase mask. All subimages are encoded in depth and can be recorded in a single A-scan; therefore, this technique also provides an advantage in acquisition speed. Path-length-multiplexed OCT is a novel implementation and has demonstrated its application in improving image quality and functional OCT development. In this study, we present an analysis of a PME-based OCT system of light propagation in the sample arm of the interferometer using the principles of Fourier optics [17].

2. THEORY

We consider a general optical system with a PME inserted into the OCT sample arm (Fig. 1). OCT probe light emits from a single-mode fiber tip and is collimated by Lens 1 with focal length $f_{1}$ . A PME is placed at the back focal plane of Lens 1. A telescope and scanning system is used to relay the beam to the sample. Objective Lens 2 with focal length $f_{2}$ is used to focus the beam onto the sample. Lens 2 is positioned so that the conjugate plane of the PME through the telescope and scanning system is at the front focal plane of Lens 2. In the following analysis, field distribution in each plane is denoted as $U$ , and Fourier transform of the field distribution is denoted as $F$ . Symbols $o$ , $p 1$ , $p 2$ , and $s$ refer to planes at the fiber tip ( $o$ ), PME ( $p 1$ ), conjugate plane of the PME ( $p 2$ ), and the sample plane ( $s$ ). The coordinate axis systems for planes $o$ , $p 1$ , $p 2$ , and $s$ are taken, respectively, as $(ξ, η)$ , $(x_{1}, y_{1})$ , $(x_{2}, y_{2})$ , and $(u, v)$ (Fig. 1). The following analysis is based on Fresnel diffraction, the paraxial approximation is applied when noted, and the finite extent of a lens aperture is neglected.

Fig. 1. Sample arm optical system for path-length-multiplexed OCT. PME: path-length-multiplexing element.

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Light emitted from the fiber tip with field distribution of $U_{o} (ξ, η)$ propagates through Lens 1 and forms the field distribution $U_{f 1} (x_{1}, y_{1})$ on the back focal plane of Lens 1, which is written as

U_{f 1} (x_{1}, y_{1}) = \frac{1}{j λ f_{1}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} U_{o} (ξ, η) e^{- j \frac{2 π}{λ f_{1}} (ξ x_{1} + η y_{1})} d ξ d η = \frac{1}{j λ f_{1}} F_{o} (\frac{x_{1}}{λ f_{1}}, \frac{y_{1}}{λ f_{1}}) .

The PME is placed at the back focal plane of Lens 1; as the fiber tip is positioned at the front focal plane, the phase quadratic terms are eliminated on this plane. We define the transmittance function of the PME as $t (x_{1}, y_{1})$ , so the field after light passes through the PME $U_{p 1} (x_{1}, y_{1})$ is written as

U_{p 1} (x_{1}, y_{1}) = U_{f 1} (x_{1}, y_{1}) t (x_{1}, y_{1}) = \frac{t (x_{1}, y_{1})}{j λ f_{1}} F_{o} (\frac{x_{1}}{λ f_{1}}, \frac{y_{1}}{λ f_{1}}) .

For a typical OCT system, a telescope and oscillating mirrors form the 2D scanning system; for a catheter system, scanning mirrors are usually replaced by a motorized stage and a rotary junction to translate/rotate miniaturized probe optics. We assume the lateral magnification of the telescope system is M, and since planes $p 1$ and $p 2$ are conjugates, the field distribution $U_{p 2} (x_{2}, y_{2})$ at plane $p 2$ is

U_{p 2} (x_{2}, y_{2}) = \frac{1}{| M |} U_{p 1} (\frac{x_{2}}{M}, \frac{y_{2}}{M}) .

Lens 2 Fourier transforms $U_{p 2}$ and projects the field onto the sample plane $s$ , forming field distribution $U_{f 2} (u, v)$ :

U_{f 2} (u, v) = \frac{1}{j λ f_{2}} F_{p 2} (\frac{u}{λ f_{2}}, \frac{v}{λ f_{2}}) .

U_{f 2} (u, v)

is backscattered by the sample, and we assume the sample backscattering angular diversity function is

s (u, v)

, so that the field distribution

U_{s} (u, v)

after light backscattered by the sample is

U_{s} (u, v) = s (u, v) * U_{f 2} (u, v),

where the sample backscattering angular diversity function is written in Eq. (6). A primed superscript is used to distinguish between backward and forward propagating light fields and the point coordinates on the

p 2

plane, and

*

is used to represent ordinary multiplication. Coefficient

w (Δ α, Δ β)

is the backscattering angle distribution describing the possibility of change in backscattering angles for

α

and

β

on the meridional and sagittal planes, as depicted in Fig. 1; the phase term is the scalar form for the vector multiplication of wave number and propagation path, representing a directional propagation that shifts the incident field’s spatial frequency when backward propagating onto its Fourier plane

p 2

. A paraxial approximation is applied for simplifying

Δ α

and

Δ β

. The integral of

Δ α

,

Δ β

within a paraxial approximation regime represents superimposed fields at various backscattering angles:

s (u, v) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} w (Δ α, Δ β) e^{j \frac{2 π [u Δ α + v Δ β]}{λ}} d Δ α d Δ β, Δ α = \frac{x_{2} - x_{2}^{'}}{f_{2}}, Δ β = \frac{y_{2} - y_{2}^{'}}{f_{2}} .

The backscattered field from the sample is Fourier transformed by Lens 2 so the field distribution $U_{f 2}^{'} (x_{2}, y_{2})$ on the Fourier image plane is

U_{f 2}^{'} (x_{2}, y_{2}) = \frac{1}{j λ f_{2}} F_{s} (\frac{x_{2}}{λ f_{2}}, \frac{y_{2}}{λ f_{2}}) .

The telescope and scanning system has lateral magnification 1/M for backward propagating light so that, on plane $p_{1}$ , the field distribution $U_{f 1}^{'} (x_{1}, y_{1})$ is

U_{f 1}^{'} (x_{1}, y_{1}) = | M | U_{f 2}^{'} (M x_{1}, M y_{1}) .

Backward propagating light passes through the PME, and has the field distribution $U_{p 1}^{'} (x_{1}, y_{1})$ :

U_{p 1}^{'} (x_{1}, y_{1}) = U_{f 1}^{'} (x_{1}, y_{1}) t (x_{1}, y_{1}) .

With Lens 1, the backscattered field is finally projected onto the fiber tip plane and coupled back into the single-mode fiber. The field distribution on the fiber tip $U_{o}^{'} (ξ, η)$ is written as

U_{o}^{'} (ξ, η) = \frac{1}{j λ f_{1}} F_{p 1}^{'} (\frac{ξ}{λ f_{1}}, \frac{η}{λ f_{1}}) .

The overlapped field coupled into the fiber ( $U_{fiber}$ ) and coupling coefficient $μ$ are expressed as

μ = {| \iint_{\infty} U_{fiber} (ξ, η) d ξ d η |}^{2}, U_{fiber} (ξ, η) = U_{o}^{'} (ξ, η) U_{o}^{*} (ξ, η), = U_{o}^{'} (ξ, η) U_{o} (ξ, η) .

The superscript $*$ in Eq. (11) refers to the complex conjugate. Equation (11) describes the mode coupling between the single-mode fiber and the field returned from the tissue sample. As $U_{o} (ξ, η)$ is a real function (Gaussian beam waist field amplitude distribution), Eq. (11) is simplified, and it states that the signal transmitted by the single-mode optical fiber is the integral of the detected field weighted by a field distribution determined by the fiber mode, indicating a coupling efficiency degradation for the returned field with a phase variation across the mode (e.g., light scattered off-axis).

Equations (1)–(11) describe the transformation of the field in each image plane beginning at the light emitted from the fiber tip to the light coupled back into the single-mode fiber in the sample path of the interferometer. The sample arm illumination and detection layout (Fig. 1) is that utilized in most conventional OCT imaging systems.

We combine Eqs. (1)–(4) so that the field illuminating the sample $U_{f 2} (u, v)$ is written as

U_{f 2} (u, v) = - \frac{| M |}{λ^{2} f_{1} f_{2}} {T (\frac{M u}{λ f_{2}}, \frac{M v}{λ f_{2}}) \otimes λ^{2} f_{1}^{2} U_{o} (- \frac{f_{1}}{f_{2}} M u, - \frac{f_{1}}{f_{2}} M v)},

where function

T

is the Fourier transform of the PME transmittance function

t (x_{1}, y_{1})

. The field incident on the sample is the convolution of the field distribution of light emitted from the fiber tip and the transfer function of the PME scaled by the telescope lens system. The incident angular spectrum is encoded in the PME transfer function

T

.

By combining Eqs. (7)–(10), the detected backscattered field from the sample $U_{o}^{'} (ξ, η)$ is written as

U_{o}^{'} (ξ, η) = - \frac{| M |}{λ^{2} f_{1} f_{2} M^{2}} {T (\frac{ξ}{λ f_{1}}, \frac{η}{λ f_{1}}) \otimes λ^{2} f_{2}^{2} U_{s} (- \frac{f_{2}}{f_{1}} \frac{ξ}{M}, - \frac{f_{2}}{f_{1}} \frac{η}{M})} .

By substituting Eqs. (5) and (6) into Eq. (13) and combining with Eq. (12), the field can be expressed as a convolution of the PME transfer function $T$ and the field emitted from the single-mode fiber tip $U_{o}$ :

U_{o}^{'} (ξ, η) = U_{o} (ξ, η) \otimes T (\frac{ξ}{λ f_{1}}, \frac{η}{λ f_{1}}) \otimes [s (- \frac{f_{2}}{f_{1}} \frac{ξ}{M}, - \frac{f_{2}}{f_{1}} \frac{η}{M}) * T (- \frac{ξ}{λ f_{1}}, - \frac{η}{λ f_{1}})] .

According to Eq. (14), when we assume that the optical system has inverse symmetry and the input field is a point source, we set $U_{o} (ξ, η)$ as a delta function, and obtain the impulse response of the system $h (ξ, η)$ as

h (ξ, η) = T (\frac{ξ}{λ f_{1}}, \frac{η}{λ f_{1}}) \otimes [T (\frac{ξ}{λ f_{1}}, \frac{η}{λ f_{1}}) * s (\frac{f_{2}}{f_{1}} \frac{ξ}{M}, \frac{f_{2}}{f_{1}} \frac{η}{M})] .

The transfer function of the system $H (f_{ξ}, f_{η})$ for spatially coherent light is

H (f_{ξ}, f_{η}) = \frac{λ^{2} f_{1}^{4} M^{2}}{f_{2}^{2}} t (λ f_{1} f_{ξ}, λ f_{1} f_{η}) * [t (λ f_{1} f_{ξ}, λ f_{1} f_{η}) \otimes S (\frac{f_{1} M}{f_{2}} f_{ξ}, \frac{f_{1} M}{f_{2}} f_{η})],

where the function

S

is the Fourier transform of sample backscattering angular diversity function

s (u, v)

, which represents a spatial frequency shift introduced by the sample:

S (f_{u}, f_{v}) = \iint_{Δ α, Δ β} w (Δ α, Δ β) δ (f_{u} - \frac{Δ α}{λ}, f_{v} - \frac{Δ β}{λ}) d Δ α d Δ β .

To find the angular spectrum of the backscattered field, the spatial frequencies in Eq. (18) are written in terms of the focal length of Lens 1 ( $f_{1}$ ) and the lateral position in the $p 1$ plane $(x_{1}, y_{1})$ , corresponding to the angular spectrum of Lens 1:

f_{ξ} = \frac{x_{1}}{λ f_{1}}, f_{η} = \frac{y_{1}}{λ f_{1}} .

By substituting Eqs. (17) and (18) into Eq. (16), normalizing the system transfer function gives

H (\frac{x_{1}}{λ f_{1}}, \frac{y_{1}}{λ f_{1}}) = \iint_{Δ α, Δ β} [\begin{array}{l} w (Δ α, Δ β) t (x_{1}, y_{1}) \\ t (x_{1} - \frac{x_{2} - x_{2}^{'}}{M}, y_{1} - \frac{y_{2} - y_{2}^{'}}{M}) \end{array}] d Δ α d Δ β .

Because planes $p 1$ and $p 2$ are conjugates, the lateral position located in $(x_{1}, y_{1})$ on plane $p 1$ corresponds to position $(x_{2} = M x_{1}, y_{2} = M y_{1}, x_{2}^{'} = M x_{1}^{'}, y_{2}^{'} = M y_{1}^{'})$ on the $p 2$ plane, and Eq. (19) is further simplified as

H (\frac{x_{1}}{λ f_{1}}, \frac{y_{1}}{λ f_{1}}) = \iint_{Δ α, Δ β} w (Δ α, Δ β) t (x_{1}, y_{1}) t (x_{1}^{'}, y_{1}^{'}) d Δ α d Δ β, x_{1}^{'} = x_{1} - Δ α f_{2} / M, y_{1}^{'} = y_{1} - Δ β f_{2} / M .

Equation (20) describes the principle of using a PME to modify the angular spectrum of the backscattered field and suggests the feasibility of recording the angular spectrum distribution of the backscattered field with an appropriate PME design. From Eq. (20), one can also observe that the PME functions as a spatial frequency filter.

3. RADIAL-ANGLE-DIVERSE PME

The analysis presented in Section 2 is valid for a general phase-mask-based OCT system. In this section, we consider a specific PME design, which is used to estimate the backscattering angular diversity.

The radial-angle-diverse PME is a ring structured phase mask, and in simplest form is constructed of a glass window with a central circular clear aperture, so that the inner circle has a refractive index of air and the outer ring has refractive index $n$ of the glass substrate. The different refractive indices in the two regions introduce a path-length difference for light propagating through the inner and outer apertures. We assume the inner circle has radius $r_{1}$ and the outer ring has radius $r_{2}$ , and the thickness of the PME is $d$ (Fig. 2).

Fig. 2. Radial-angle-diverse PME.

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The PME transmittance function $t (x_{1}, y_{1})$ is written as

t (x_{1}, y_{1}) = circ (\frac{\sqrt{x_{1}^{2} + y_{1}^{2}}}{r_{1}}) + (circ (\frac{\sqrt{x_{1}^{2} + y_{1}^{2}}}{r_{2}}) - circ (\frac{\sqrt{x_{1}^{2} + y_{1}^{2}}}{r_{1}})) e^{j k (n - 1) d},

where

circ (r) = \begin{matrix} 1 & r < 1 \\ \frac{1}{2} & r = 1 \\ 0 & r > 1 \end{matrix} .

The first term in Eq. (21) represents transmittance of the inner circle, and the second term represents the outer ring, which introduces a path-length difference due to propagation through glass. By substituting Eq. (21) into Eq. (20), the spatial frequency selection attribute of this PME type and path-length-multiplexing effect is shown:

H_{low-low} (\frac{x_{1}}{λ f_{1}}, \frac{y_{1}}{λ f_{1}}) = \iint_{\infty, \infty} [\begin{array}{l} w (Δ α, Δ β) * circ (\frac{\sqrt{x_{1}^{2} + y_{1}^{2}}}{r_{1}}) \\ * circ (\frac{\sqrt{x_{1}^{' 2} + y_{1}^{' 2}}}{r_{1}}) \end{array}] d Δ α d Δ β,

where

H_{low-low}

represents the low-spatial frequency illumination and detection, while high-spatial frequency illumination and detection is denoted as

H_{high-high}

:

H_{high-high} (\frac{x_{1}}{λ f_{1}}, \frac{y_{1}}{λ f_{1}}) = e^{j k 2 (n - 1) d} * \iint_{\infty, \infty} \begin{array}{l} w (Δ α, Δ β) * [circ (\frac{\sqrt{x_{1}^{2} + y_{1}^{2}}}{r_{2}}) - circ (\frac{\sqrt{x_{1}^{2} + y_{1}^{2}}}{r_{1}})] * \\ [circ (\frac{\sqrt{x_{1}^{' 2} + y_{1}^{' 2}}}{r_{2}}) - circ (\frac{\sqrt{x_{1}^{' 2} + y_{1}^{' 2}}}{r_{1}})] d Δ α d Δ β \end{array} .

The phase term in Eq. (24) is introduced by the double passage of light through the PME outer ring. As OCT records depth-resolved images by Fourier transforming the k-space power spectral density function to obtain the correlation function between sample and reference light, high-spatial-frequency components in the sample spectrum $S$ are path-length shifted by $2 (n - 1) d$ in the OCT image relative to the low-spatial-frequency component.

Moreover, two additional terms in the sample transfer functions exist in the acquired OCT image, which correspond to high-spatial-frequency-illumination–low-spatial-frequency-detection and low-spatial-frequency-illumination–high-spatial-frequency-detection. The transfer functions corresponding to these two conditions result from the case that incident light is passing through the PME outer ring but backscattered light is passing through inner circle or the case that incident light is passing through the inner circle but backscattered light is passing through the outer ring. These two terms contribute to a degeneracy in the recorded OCT image as they have equal path-length difference $(n - 1) d$ . For these two cases, sample scattering introduces spatial frequency of the beam transferring from one band to the other:

H_{high-low} (\frac{x_{1}}{λ f_{1}}, \frac{y_{1}}{λ f_{1}}) = e^{j k (n - 1) d} * \iint_{\infty, \infty} \begin{array}{l} w (Δ α, Δ β) * (circ (\frac{\sqrt{x_{1}^{2} + y_{1}^{2}}}{r_{2}}) - circ (\frac{\sqrt{x_{1}^{2} + y_{1}^{2}}}{r_{1}})) \\ * circ (\frac{\sqrt{x_{1}^{' 2} + y_{1}^{' 2}}}{r_{1}}) d Δ α d Δ β \end{array}, H_{low-high} (\frac{x_{1}}{λ f_{1}}, \frac{y_{1}}{λ f_{1}}) = e^{j k (n - 1) d} * \iint_{\infty, \infty} \begin{array}{l} w (Δ α, Δ β) * circ (\frac{\sqrt{x_{1}^{2} + y_{1}^{2}}}{r_{1}}) \\ * (circ (\frac{\sqrt{x_{1}^{' 2} + y_{1}^{' 2}}}{r_{2}}) - circ (\frac{\sqrt{x_{1}^{' 2} + y_{1}^{' 2}}}{r_{1}})) d Δ α d Δ β \end{array} .

Equations (23)–(25) describe the spatial frequency and corresponding angular spectrum encoded in depth for the path-length-multiplexed OCT image.

To investigate the PSF on the sample plane, we recall Eq. (12), which states that the field illuminating the sample is the convolution of the transfer function of the PME with the field amplitude on the fiber tip. Since the beam emitted from a single-mode fiber tip has a nearly Gaussian amplitude distribution, according to the convolution theorem, Eq. (12) suggests a Gaussian apodization on the collimated beam passing through the PME. For simplicity, we assume a power normalized amplitude Gaussian distribution of $U_{o} (ξ, η)$ :

U_{o} (ξ, η) = \sqrt{\frac{2}{π}} \frac{1}{r_{m}} e^{- \frac{(ξ^{2} + η^{2})}{r_{m}^{2}}},

where

r_{m}

is the half-width of the single-mode fiber’s mode field diameter.

We assume mode field diameter of the single-mode fiber is 9.2 μm, numerical aperture (NA) is 0.14, the focal lengths of the collimator and objective are 15 and 18 mm, respectively, magnification is 1, a PME with $r_{1}$ of 0.5 mm and $r_{2}$ of approximately 2.1 mm ( $r_{2}$ is determined by the incident beam size, here estimated as $NA * f_{1}$ ), and the center wavelength of the incident beam is 1.3 μm. Substituting Eq. (26) into Eq. (12), the PSF of low-spatial-frequency and high-spatial-frequency illumination for a diffraction-limited condition is numerically simulated [Fig. 3(a)]. The beam passing through outer ring of the PME introduces a PSF with full width at half-maximum (FWHM) of 7.3 μm, which is approximately 1/3 of the FWHM of the PSF formed by the inner circular aperture (24.3 μm). Compared with a 2.1 mm radius circular aperture with FWHM of the PSF of approximately 7.5 μm, the PSF of the PME outer ring illumination presents a finer PSF central peak. A finer PSF central peak for the PME outer ring is expected by increasing the inner circle radius as the lower bound of the frequency band is increased, but, similar to the effect introduced by a central obstruction in a circular aperture, more energy will leak into sidelobes as a result of a narrowed spatial-frequency bandwidth.

Fig. 3. (a) Normalized simulated PSF of illumination on the sample plane. (b) Normalized detected PSF ( $U_{fiber}$ ) projected on the sample plane (lateral dimension is scaled by $f_{2} / M f_{1}$ ) after the single-mode fiber confocal gating effect [Eq. (11)]. Intensity in the plot is in arbitrary units and the lateral dimension is in micrometers.

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The difference introduced by the PME exists in both the detection and illumination schemes. The impulse response of the system in Eq. (15) combined with Eq. (14) indicates that, for a backscattering angle-invariant reflector $(s = 1, Δ α = 0, Δ β = 0)$ in the sample plane, the detected field is the convolution of the forward-propagating PME transfer function (with Gaussian apodization) and backward propagating PME transfer function filtered by confocal gating imposed by the single-mode fiber [Eq. (11)]. With the simulation for the same PME design considered above, Fig. 3(b) illustrates the normalized detected PSF in the sample plane for each OCT subimage. The single-mode fiber’s confocal gating effect significantly sharpens the PSF for each subimage, high-spatial-frequency-illumination–high-spatial-frequency-detection (high–high subimage) exhibits the finest PSF ( $FWHM = 4.7 μm$ ), and low-spatial-frequency-illumination–high-spatial-frequency-detection/high-spatial-frequency-illumination–low-spatial-frequency-detection (high–low subimage) presents slightly better PSF than low-spatial-frequency-illumination–low-spatial-frequency-detection (low–low subimage) ( $FWHM = 6.1 μm$ ); as indicated in Fig. 3(b), two curves are close enough to overlap.

4. BACKSCATTERING ANGULAR DIVERSITY DETECTION

Next, we examine the case of utilizing a PME for backscattering angular diversity detection. We assume the backscattering angle distribution is a standard Gaussian distribution and the OCT system is a rotationally symmetric optical system, so that the backscattering angular diversity function can be written as

s (r) = \int_{Δ θ} w (Δ θ) e^{j \frac{2 π r Δ θ}{λ}} d Δ θ, w (Δ θ) = \frac{1}{σ \sqrt{2 π}} e^{- \frac{{(Δ θ - μ)}^{2}}{2 σ^{2}}},

where

μ

and

σ

are the mean and deviation of the backscattering angle distribution, respectively. For comparison, we assume three distributions with equal deviation but increased average backscattering angle. As in Fig. 4(a), mean value for each distribution is 0, 0.0175 rad (1 deg), and 0.0349 rad (2 deg), respectively, and the deviation is 0.0175 rad (1 deg). The range of the Gaussian distribution is confined by the acceptance angle

r_{2} / f_{2}

. According to Eqs. (11) and (14), the coupling coefficient (

μ

) can be calculated and the signal intensity of each subimage in

10 * \log_{10}

scale when assuming original signal level is 100 dB is plotted in Fig. 4(b). The signal intensities of low–low, high–low, and high–high subimages for each Gaussian distribution are (66.54, 87.33, 97.64), (64.90, 87.59, 97.72), and (59.87, 88.11, 97.94) dB, respectively, for PME with

r_{1} = 0.5 mm

. Similarly, a simulation is done for PME with

r_{1} = 1 mm

in Fig. 4(c), where low–low, high–low, and high–high subimage signal intensities are (84.53, 94.06, 90.98), (84.03, 94.42, 91.29), and (82.52, 95.27, 92.01) dB, respectively. As expected, when increasing the average backscattering angle, energy transfers from the low–low subimage to the high–low and high–high subimages. The signal intensity differences among the high–high and low–low subimages are 31.10, 32.82, and 38.07 dB, respectively, for a PME with

r_{1} = 0.5 mm

with increased mean backscattering angle. Therefore, a 0.0349 rad mean scattering angle difference introduces an approximate 7 dB change in the signal intensity difference. The signal intensity differences are 6.45, 7.26, and 9.49 dB for a PME with a bigger inner circle aperture, which correspond to a 3 dB change for 0.0349 rad mean scattering angle difference. An increase in the aperture size of the inner circle leads to a higher signal in the low–low subimage, but the signal intensity change in response to different backscattering angles becomes less obvious.

Fig. 4. (a) Backscattering angle distributions with means of $μ = 0$ , 0.0175, and 0.0349 rad, respectively, and deviation of $σ = 0.0175 rad$ . (b) Histogram of signal intensity distribution for PME $r_{2} = 0.5 mm$ . (c) Histogram of signal intensity distribution for PME $r_{1} = 1 mm$ .

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Simulations have also been completed for backscattering angle distributions with equal mean backscattering angle 0, but increased deviation: 0.0175, 0.0349, and 0.0524 rad, as plotted in Fig. 5(a). Figure 5(b) is the signal intensity histogram for a PME with $r_{1} = 0.5 mm$ , at which the low–low, high–low, and high–high subimage signal intensities are (66.54, 87.33, 97.64), (63.37, 87.84, 97.84), and (61.23, 88.11, 97.98) dB, respectively. Figure 5(c) is the signal intensity histogram for a PME with $r_{1} = 1 mm$ , and the low–low, high–low, and high–high subimage signal intensities are (84.53, 94.06, 90.98), (83.17, 94.88, 91.68), and (81.95, 95.47, 92.19) dB, respectively. A 5.65 dB and a 3.79 dB change in signal intensity difference between high–high and low–low subimages can be detected for two PMEs due to a 0.0349 rad difference in backscattering angle deviation. The result suggests that a sample with a broader range of potential backscattering angles tends to have lower signal intensity in a low–low subimage, while it has a higher signal intensity in a high–high subimage. The backscattering angle distribution discussed in Fig. 5(a) is noted to be similar to the backscattering property of a typical biological tissue system.

Fig. 5. (a) Backscattering angle distributions with mean of $μ = 0$ and deviations of $σ = 0.0175$ , 0.0349, and 0.0524 rad. (b) Histogram of signal intensity distribution for PME $r_{2} = 0.5 mm$ . (c) Histogram of signal intensity distribution for PME $r_{1} = 1 mm$ .

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In summary, in the sample field that introduces a stronger disturbance to the incident field (e.g., large backscattering angle and deviation), field intensity increases in the high-spatial-frequency band and decreases in the low-spatial-frequency band, which can be represented as an increase in signal intensity difference between high–high and low–low subimages by calculating the contrast between the two. Therefore, we conclude that without compromising image resolution, signal collection efficiency, or acquisition speed, sample backscattering angular diversity detection is achieved.

5. SIGNAL SPATIAL FREQUENCY DEPENDENCE

Examination of the transfer function [Eqs. (23)–(25)] shows that subimages cover different backscattered signal spatial frequency bands; therefore, we further investigate the signal formation in each subimage in terms of sample spatial frequency and its effect on backscattering angular diversity detection sensitivity.

We assume the sample contains a sinusoidal feature with spatial frequency $f_{s}$ ; the sample function $I_{s}$ can be written as

I_{s} (u, v) = 1 + \cos (2 π f_{s} \sqrt{u^{2} + v^{2}}) .

By substituting Eq. (28) into Eq. (14) and using a convolution for image construction based on the PSF, the modified field distribution on fiber tip can be written as

U_{o}^{'} (ξ, η) = U_{o} (ξ, η) \otimes T (\frac{ξ}{λ f_{1}}, \frac{η}{λ f_{1}}) \otimes [T (- \frac{ξ}{λ f_{1}}, - \frac{η}{λ f_{1}}) * (s (- \frac{f_{2}}{f_{1}} \frac{ξ}{M}, - \frac{f_{2}}{f_{1}} \frac{η}{M}) \otimes I_{s} (- \frac{f_{2}}{f_{1}} \frac{ξ}{M}, - \frac{f_{2}}{f_{1}} \frac{η}{M}))],

where we assume the periodic structure of the sample has a backscattering angular diversity function as in Eq. (27), and the Gaussian distribution has mean 0 and deviation 0.0175 rad. Figure 6 illustrates the signal intensity difference between the high–high and low–low subimages with respect to sample spatial frequency for PMEs with

r_{1} = 0.5

and 1 mm, respectively. At low sample spatial frequency regime, the signal intensity difference between the high–high and low–low subimages increases with increasing sample spatial frequency. This corresponds to the fact that the backscattered signal continuously transfers from the low–low subimage to the high–low and high–high subimages. The increasing trend is nonmonotonic. This is because, when the loss of the backscattered signal in the high–high subimage due to limited bandwidth is greater than the signal transferred into it, the signal intensity difference is reduced, which corresponds to the occurrence of ripples in the figure. A peak emerges at approximately

20 {mm}^{- 1}

spatial frequency for both PME designs, which is interpreted as a 50 μm sample structure period. This value is of the same order as the objective PSF, and indicates that this is the best regime for backscattering angular diversity detection as it provides the highest contrast between the high–high and low–low sub-images for this case. Also, the PME with a smaller inner circle size has a higher sensitivity in backscattering angular diversity detection. After passing the peak, the intensity difference decreases monotonically, which is understood to be because most of the backscattered signal is beyond the bandwidth of the high–high subimage, so the loss of signal reduces the high–high subimage’s signal intensity, resulting in a decrease in signal intensity difference between the high–high and low–low subimages. Finally, the curve reaches a steady state when the spatial frequency of the backscattered light is too high to provide any useful information for backscattering angular diversity detection. According to Fig. 6, this frequency is located at approximately

50 {mm}^{- 1}

, which corresponds to a 20 μm structure period, slightly larger than twice the PSF of the imaging system (Fig. 3). Biological tissue can be considered to consist of a multiplicity of various spatial frequency components. Only when the objective and PME designs meet the requirement that the spatial frequency of the targeted tissue structure is within the detectable regime can this technique provide sufficient sensitivity for backscattering angular diversity detection.

Fig. 6. Signal intensity difference (decibels) between high–high and low–low subimages with respect to sample spatial frequency ( ${mm}^{- 1}$ ) for (a) PME $r_{1} = 0.5$ and (b) PME $r_{2} = 1$ .

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6. DISCUSSION AND CONCLUSION

In this study, without loss of generality, we have introduced a Fourier optics analysis for a phase-mask-based path-length-multiplexed OCT technique. Algebraic expressions for the impulse response, PSF, and transfer function for each multiplexed subimage are derived. Signal intensities corresponding to different backscattering angle distributions are simulated and compared to demonstrate the feasibility of using a radial-angle-diverse PME to observe backscattering angular diversity. Furthermore, an OCT signal, in terms of sample spatial frequency, is simulated and analyzed, indicating the operational regime of this technique. Two PME designs with different inner circle diameters are discussed and compared in terms of sensitivity in backscattering angular diversity detection. This technique has been implemented for human retinal nerve fiber layer (RNFL) backscattering angular diversity measurement [11], where the observed backscattering angle variation around the optic nerve head matches the variation of the degree of polarization (DOP) degradation [18]. This is consistent with the hypothesis that the difference in degradation of the DOP in the human RNFL is associated with scattering angles, and the measurement result is also consistent with the known retinal ganglion cell (RGC) neural anatomy of the RNFL [19].

In the analysis of the PSF of the path-length-multiplexed image, we observe that the transfer functions for subimages are modified according to different combinations of illumination and detection pupil regions. Techniques that use a phase mask for pupil shaping to improve OCT image lateral resolution and depth of focus have been reported [20–22]. To achieve the same goal, methods such as using an axicon lens have also been demonstrated [23–25]. The high-spatial-frequency illumination provided by a PME is similar to a Bessel beam illumination scheme; therefore, an extended depth of focus is also expected [21].

Similar to the reported space division multiplexing scheme [26], a PME-based path-length-multiplexed system acquires multiple images simultaneously and presents them as subimages within one conventional OCT image. Therefore, no extra acquisition time is required, which enables the system’s application for high-speed imaging.

Funding

American Society for Laser Medicine and Surgery (ASLMS); National Institutes of Health (NIH) (RO1EY016462); The Clayton Foundation; CPRIT (DP150102).

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Fourier optics analysis of phase-mask-based path-length-multiplexed optical coherence tomography

Abstract

1. INTRODUCTION

2. THEORY

3. RADIAL-ANGLE-DIVERSE PME

4. BACKSCATTERING ANGULAR DIVERSITY DETECTION

5. SIGNAL SPATIAL FREQUENCY DEPENDENCE

6. DISCUSSION AND CONCLUSION

Funding

REFERENCES

Cited By

Figures (6)

Equations (29)

Journal of the Optical Society of America A