Depth-of-focus extension in optical coherence tomography via multiple aperture synthesis

In this paper, we report what we believe is a novel technique to overcome the depth-of-focus (DOF) limitation in optical coherence tomography (OCT). Using confocal optics on a sample arm, we scanned the illumination beam across the under-filled objective lens pupil plane by steering the beam at the pinhole using a microcylindrical lens. The detected interferometric signals from multiple distinctive apertures were digitally refocused, which is analogous to synthetic aperture radar (SAR). Using numerical simulations and imaging experiments, we verified that this technique can maintain a diffraction-limited transverse resolution along a DOF that is ∼10 times larger than the confocal parameter. The ability to extend the DOF without signal loss and sidelobe artifacts may ultimately overcome the DOF limitation in high-resolution OCT.


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©2017 Optical Society of America (OSA). This paper was published in Biomedical Optics Express and is made available as an electronic reprint (preprint) with permission of Optical Society of America (OSA). The published version is available at: [http://dx.doi.org/10.1364/OPTICA. 4.000701]. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law.

INTRODUCTION
Optical coherence tomography (OCT) is a cross-sectional, threedimensional (3D) imaging technique that provides non-invasive, high-speed, high-resolution images of scattering media, such as inhomogeneous biological tissues [1]. Until now, this technique has been widely applied in ophthalmology [2] and cardiology [3]. In OCT, the transverse resolution Δx is determined by the diffraction-limited spot size of the sample focused beam and defined as the beam waist diameter under the assumption of a Gaussian input beam. This parameter can be expressed as Δx 2λ∕πNA, where NA d ∕2f is the numerical aperture (NA) of the objective lens. The corresponding depth of focus (DOF) can be defined as the confocal parameter and is expressed as DOF 4λ∕πNA 2 , which represents the axial distance between the values of z (axial depth) where the size at the 1∕e 2 beam is two times larger than that at the beam waist. According to the definitions stated above, the transverse resolution is inversely proportional to the NA of the objective lens, and the DOF is proportional to the square of the transverse resolution. Consequently, a tradeoff occurs between the transverse resolution and the DOF in which a lower transverse resolution results in a larger DOF and vice versa.
To achieve high transverse resolution over an extended DOF, various approaches have been proposed. Axicon lenses and phase and amplitude pupil filters can generate a focus with Bessel beam characteristics, including a more uniform main lobe diameter in the transverse direction over an extended DOF compared with the confocal parameter [4,5]. However, in the spatial frequency domain, Bessel beams suffer a major loss of spatial frequency components in the coherence transfer function (CTF) compared to that of Gaussian beams, which results in a significant decline of sensitivity and sidelobe artifacts in the OCT images [6]. Although these problems can be mitigated by dark-field Gaussian detection, the fundamental problems of a suboptimal CTF must still be resolved [6][7][8]. Digital refocusing has drawn considerable research interest because an optimal CTF can be maintained, which in principle prevents reductions in sensitivity and sidelobe issues [9][10][11]. A previously reported digital refocusing approach, called interferometric synthetic microscopy (ISAM), computationally reconstructs diffraction-limited transverse resolutions by solving the inverse scattering problem. However, the ISAM approach requires absolute phase stability during 3D image acquisition, which is difficult to realize in vivo, especially in endoscopic applications [9]. Recently, depthencoded synthetic aperture, which we believe is a novel approach, has been reported. This approach extends the DOF by synthesizing different apertures and correcting the wavefront curvature caused by the defocus effect [10,11]. It does not require a phase-stabilizing setup, and it has low computational costs and good compatibility with conventional Fourier domain OCT (FD-OCT) hardware. However, an inherent issue of signal coupling loss caused by the mismatch between the NA of the fiber pinhole and apertures generated by the wavefront divider/splitter is observed, and an overall 50% signal loss can be attributed to the implementation of aperture division and the multiplexing apparatus [10]. Here, we demonstrate a multiple aperture synthesis (MAS) technique to achieve a digital refocusing function without signal loss or sidelobe artifacts.

MULTIPLE APERTURE SYNTHESIS
The MAS technique extends the DOF in a spectral domain OCT (SD-OCT), which is analogous to synthetic aperture radar (SAR). In Figs. 1(a) and 1(a′), a microcylindrical lens (MCL) is positioned ∼10 μm away from the tip of the sample fiber with an axisymmetric configuration along the optical axis. All light paths of the spatial (angular) frequencies originating from the fiber pinhole are refracted sequentially by MCL, L1, and L2, and they intersect at the focal point of L2 with a uniform phase or optical path length. Under this ideal condition, a maximal constructive interference occurs at this focal point, which results in a diffraction-limited transverse point spread function (PSF). When the MCL is linearly shifted along a transverse direction perpendicular to the cylindrical axis by a piezoelectric transducer (PZT) [Figs. 1(b) and 1(b′)], the spatial or angular frequency of the sample beam with respect to the full aperture of the focusing lens L2 is swept as the incidence angle at the curved surface of the MCL changes. The aberration caused by the cylindrical lens is not noticeable and the resolution in the x and y transverse dimensions is the same. Figure 1(c) depicts the effective optical apertures in the objective lens (L2) pupil plane. Apertures 1-5 denote five apertures generated by transversely shifting the MCL. Figure 1(d) illustrates the spatial or angular frequency of focused beams 1-5, which correspond to apertures 1-5 in Fig. 1(c) in the focal plane of the objective lens L2. During image acquisition, we chose to acquire an axial-line (A-line) at each of the five equally spaced spatial frequencies and synthesize five apertures together as shown in Fig. 1(e). Compared with the standard SD-OCT, more high spatial frequency light signals of the sample reflectivity can be coherently synthesized, which will theoretically and experimentally generate a larger aperture and an extended DOF.
The total optical path difference (OPD) between A-lines acquired in Figs. 1(a) and 1(b) is defined as Δz Δz s Δz c and composed of two components: Δz s , which is induced by the transverse shift of the MCL; and Δz c , which is the defocusing term with Δz c 0 at the focus of L2. For a scatter located away from the focal point, when these OPDs are corrected to 0, the Alines with distinctive apertures can be coherently summed together to form a digitally synthesized aperture.
In the classical theory of FD-OCT, the detected interference signal Ik in k-space includes a direct-current (DC) term, a cross-correlation (CC) term, and an auto-correlation (AC) term, and they are expressed in the following simplified formulation: where Sk is the spectral distribution of the light source and I r k and I s k are the electric field reflectivity from the reference arm and the sample arm, respectively, at depth z. Because only the CC encodes depth-resolved information for OCT imaging, Sk, DC and AC can be omitted in the following derivation. The original CC term for the nth cross-sectional scan (B-scan) is where C.C. is the abbreviation for the complex conjugate, which will also be omitted in the following derivation. The subscript n represents the sequence number of distinctive apertures. To correct the OPDs and the defocus term, we define k k 0 Δk and z z n Δz n , where z is the corrected optical path length for rendering a perfect focus. Then, Eq. (2) is rewritten as where α n −k 0 · Δz n and β n −Δk · Δz n . Equation (3) includes a constant phase α n and an oscillation phase β n , which are induced by multiple transverse shifts of the MCL and the defocusing term. The MAS process consists of two main steps to correct α n and β n in sequence. First, according to the Fourier shift theorem, we correct the axial-delayed OPDs and axially shift them to the aligned depth by multiplying the B-scan by exp−iα n . This step is called the axial shift. The nth axial-shifted B-scan is calculated as where the superscript ax represents the axial-shift operation. When the criterion j P m n1 I n k · exp−iα n j reaches its maximum value, α n achieves its optimal value α op n ; here, the superscript op represents the optimal value and m represents the total number of apertures.
Second, the constant phase β n can be eliminated by multiplying the Fourier spectrum of Eq. (4) by a constant phase coefficient exp−iβ n . This step is called the defocusing correction. The nth defocusing-corrected B-scan is calculated as where the superscript de represents the defocusing-correction operation. To precisely correct the defocusing term, we must obtain the optimal β n at depth z for each B-scan. The optimal β n z is strictly contained in the phase set fπ∕2; π; 3π∕2; 2πg. When the criterion maxfj P m n1 I ax n k · exp−iβ n zjg reaches its maximum value, β n z achieves its optimal value β op n z. Combining Eqs. (4) and (5), the digitally refocused B-scan is calculated as This MAS achieves optimal constructive interference at the center of the scatter and destructive interference away from the center, which not only increases the total backscattered intensity but also preserves the diffraction-limited transverse resolution over a DOF extended multiple times.

NUMERICAL ANALYSIS
The image formation in an OCT system can be described by a CTF, which is the Fourier transform of the effective PSF [12][13][14][15]. Here, we define pm; n as the pupil function (PF) of the objective lens. m and n are the corresponding radial spatial frequencies in two transverse directions. Because the light passes the objective lens twice, the 2D CTF can be obtained by convolving PFs of the illumination and detection aperture [15][16][17][18] as Cm; n pf λm; f λn ⊗⊗ pf λm; f λn; where ⊗⊗ represents the 2D convolution operation. f and λ represent the focal length of the objective lens and the wavelength of the illumination light. The PSFs are the Fourier transform of the CTFs in two transverse directions and are calculated as Cm; n exp−2πimx nydmdn; (8) where x and y denote the variables on the plane perpendicular to the optical axis and z is the axial variable. We performed numerical simulations using MATLAB software (MathWorks, Natick, Massachusetts, USA) to characterize the PFs, PSFs, and CTFs of the MAS technique with respect to available methods. Here, we consider three aperture types in the illumination and detection paths: full aperture, annular apodized aperture, and MAS. We define the radius of the full aperture as 1 [ Fig. 2(a)] and the radii of the annularly apodized aperture as 0.7 [ Fig. 2(b)]. In the case of MAS, five apertures were equally spaced with a transverse spacing of 0.25 as indicated by the bold numbers 1-5 in Fig. 2(c). Under the condition of Gaussian illumination, CTFs show that although the annular apodized aperture produces the largest frequency range, the cutoff spatial frequency of the MAS and full aperture is much greater than that of the apodization. The MAS technique has a better transverse CTF than the full aperture in the focal plane [ Fig. 2(m) [8]. However, the DOF of the MAS [ Fig. 2(l)] is much greater than that of the full aperture [ Fig. 2(j)]. The FWHM transverse spot size of the full aperture doubles at depth b, whereas that of the apodized aperture and MAS are expanded by 13.9% and 73.0% [ Fig. 2(p)] at depth 10b, respectively. These results

Research Article
indicate that the MAS technique can extend the DOF by a value that is more than 10 times that of the full aperture.

A. Optical Coherence Tomography
The schematic of a MAS SD-OCT system is illustrated in Fig. 3. A superluminescent diode (SLD) array (T850-HP, Superlum, Carrigtwohill, Ireland) provides a broadband spectrum with an FWHM bandwidth of 165 nm centered at 850 nm. The light source, whose total output power is 18.0 mW, is split into a sample light and a reference light by a 2 × 2 wideband fiber-optic coupler (TW850R2A2, Thorlabs, Newton, New Jersey, USA) with a splitting ratio of 90∶10. The light beam in the reference arm is The spectrometer consists of a 1765 line/mm diffraction grating (PING-Sample-020, Ibsen Photonics, Farum, Denmark), a camera lens (85 mm, f/1.4, Zeiss, Oberkochen, Germany) and a 4096pixel charge-coupled device (CCD) camera (AViiVA EM4, e2V, Chelmsford, UK). The detected spectrum is digitized at a 12bit resolution and transferred to a computer via an image acquisition board (KBN-PCE-CL4-F, Bitflow, Woburn, Massachusetts, USA). Two-directional transverse scanning is implemented by two galvanometer-mounted mirrors driven with sawtooth pulses, which are generated by a 16-bit analog output of a data acquisition (DAQ) board (PCI-6221, National Instruments, Austin, Texas, USA). Image acquisition and galvo mirror scanning are synchronized by the external trigger from a DAQ digital output. A discrete Fourier transform is performed on each frame of the 1024 A-lines obtained by the CCD to resolve the axial depth profile of the sample.

B. Microcylindrical Lens Fabrication
We selected a closed-loop PZT with four embedded strain gauges and a travel range of 15.7 μm. The output diverging beam of the sample fiber has a full angular range (defined by the 1∕e 2 power) of 7.45 4λ∕πΔx deg , which is determined by the spot diameter Δx of the focused beam. The desired radius of the MCL is 60-70 μm, which is based on the travel range of the PZT and fiber NA. The MCL is precisely grinded from the coreless silica termination fiber (FG125LA, Thorlabs, Newton, New Jersey, USA). As shown in Fig. 4(a), the PZT is mounted on a two-axis goniometer (GNL20, Thorlabs, Newton, New Jersey, USA) and a three-axis compact stage (MBT616D, Thorlabs, Newton, New Jersey, USA). The inset provides a magnified view, which shows that two MCLs are cured on the silica. The crosssectional width of the MCL is measured at 110 μm as shown in Fig. 4(b). Figure 4(c) is a side view showing the position of the MCL relative to the tip of the sample arm fiber.

C. Phantom Preparation
A phantom of polystyrene calibration microparticles was constructed by mixing agarose solution (No. PC0701-100 g, Vivantis, Oceanside, California, USA) with polystyrene microparticles (No. 64090-15, nominal size 6 μm, Sigma-Aldrich, St. Louis, Missouri, USA). This mixture was stored in a vial and  Research Article placed in an ultrasonic bath for 10 min to remove residual clusters. The sample (10 g) was poured into a cell culture dish, cured for 30 min at 100°C and then cured for 24 h at room temperature.

EXPERIMENTAL RESULTS
To demonstrate the performance of the DOF extension using the MAS technique, we conducted imaging experiments using the microparticle polystyrene calibration sample. Five B-scans m 5 were acquired using the MAS SD-OCT system, and they corresponded to five transverse shifts of the MCL with five distinctive apertures. One of B-scans was dispersion compensated using the algorithm in [19]. It is shown in Fig. 5(a), which indicates that the B-scans suffered from defocusing beyond the focus. According to Eqs. (4) and (5), the MAS process consisted of two steps: an axial shift and a defocusing correction. The results of the axial shift and defocusing correction are shown in Figs. 5(b) and 5(c), respectively. In Figs. 5(a′)-5(c′), magnified views of two calibration microparticles are indicated by the dashed boxes. We measured the transverse FWHMs of 50 microparticles at different depths in Figs. 5(a) and 5(c) and performed the 10-order polynomial-fitting as shown in Fig. 5(d). Because the MFD (defined by the 1∕e field) was 5.0 0.5 μm, the theoretical transverse FWHM of the PSF at focus was estimated at 2.92 μm and the corresponding DOF was 16.01 μm at the center wavelength of 850 nm. The finest transverse FWHM of the refocused microparticle was 9.06 μm; thus, the finest nominal transverse FWHM of the PSF was 3.06, which corresponded to a DOF of 17.30 μm. The experimental results are consistent with the theoretical results. The DOF was estimated within a depth range from 910 to 927 μm, where two fitted lines crossed in Fig. 5(d). Over the full axial depth range of 1030 μm n 1.33, the maximum, averaged, and minimum transverse FWHMs of the microparticles in Fig. 5(a) were 39.34 μm, 20.16 μm, and 9.26 μm, respectively; therefore, the maximum, averaged, and minimum transverse FWHMs of the PSFs were 33.34 μm, 14.16 μm, and 3.26 μm, respectively. In contrast, the maximum, averaged, and minimum transverse FWHMs of the microparticles in Fig. 5(c) were 13.64 μm, 11.21 μm, and 9.06 μm, respectively; therefore, the maximum, averaged, and minimum transverse PSFs were 7.64 μm, 5.21 μm, and 3.06 μm, respectively. Thus, the MAS SD-OCT system appears to preserve the diffraction-limited transverse resolution over the full axial depth, and it obtained a DOF extension of ∼10-fold, which is consistent with the results of the numerical simulation in Section 3.

DISCUSSION AND CONCLUSIONS
We developed the MAS technique to address the fundamental problem of limited DOF in high transverse resolution FD-OCT. MAS is free from signal loss and sidelobe artifacts, which are caused by a suboptimal CTF function and inherent in the currently available methods. These merits of the MAS method have been demonstrated theoretically and experimentally in this work. Therefore, we believe the MAS technique has the potential to provide a sufficient DOF to stably acquire subcellularresolution images in vivo. Moreover, the ability to manipulate the complex PF makes MAS a powerful tool that can eliminate various types of optical aberrations beyond the defocus induced by the sample or focusing optics, such as aberrations induced by human eye optics. However, the current study has two limitations. First, we used transverse priority scanning, which is susceptible to motion artifacts. This limitation can be easily resolved by changing to angular frequency priority scanning. Second, the current MAS form is implemented at the cost of scanning speed. In this work, we coherently sum five A-scans with five distinctive apertures together to obtain one DOF extended A-line, which decreases the imaging speed by five times. This limitation may be mitigated by the use of faster spectrometers or an ultrahigh-speed swept source. Moreover, a phase ramp could be digitally created along the fast scanning direction in the PF to achieve digital transverse scanning. The proposed optical apparatus is applicable to desktop OCT imaging systems for DOF extension, but a significant research effort is required to develop a miniaturized optical design for endoscopic and intravascular applications. In conclusion, the proposed MAS technique overcomes the inherent tradeoff between DOF extension and signal loss/sidelobe artifacts and may ultimately overcome the DOF limitation in high-resolution OCT. Future works will focus on the development of a faster scanning MAS method for in vivo applications.