Extended focal depth Fourier domain optical coherence microscopy with a Bessel-beam – LP 02 mode – from a higher order mode fiber

: We present a robust fiber-based setup for Bessel-like beam extended depth-of-focus Fourier-domain optical coherence microscopy, where the Bessel-like beam is generated in a higher order mode fiber module. In this module a stable guided LP 02 core mode is selectively excited by a long period grating written in the higher order mode fiber. Imaging performance of this system in terms of lateral resolution and depth of focus was analyzed using samples of suspended microbeads and compared to the case where illumination is provided by the fundamental LP 01 mode of a single mode fiber. Illumination with the LP 02 mode allowed for a lateral resolution down to 2.5 µm as compared to 4.5 µm achieved with the LP 01 mode of the single mode fiber. A three-fold enhancement of the depth of focus compared to a Gaussian beam with equally tight focus is achieved with the LP 02 mode. Analysis of the theoretical lateral point spread functions for the case of LP 01 and LP 02 illumination agrees well with the experimental data. As the design space of waveguides and long-period gratings allows for further optimization of the beam parameters of the generated Bessel-like beams in an all-fiber module, this approach offers a robust and yet flexible alternative to free-space optics approaches or the use of conical fiber tips. depth of focus and lateral resolution with the performance of a standard (Gaussian-like beam) illumination setup with similar beam waist. Our analysis shows that the Bessel-like beam provides a larger depth of focus and higher resolution than the standard system. Given the simplicity to generate a Bessel-like beam in a fiber-based module and the possibility to explore a large design space for the long period gratings and higher-order mode fibers, this approach could be used to generate Bessel-like beams with parameters allowing to further improve depth of focus and lateral resolution. This can be tailored to developing simpler FD OCT imaging systems in comparison to alternative free-space optics approaches (such as axicon lenses). The use of higher-order mode fibers also offers a flexible and more robust alternative to conical tip fibers.


Introduction
Optical coherence tomography (OCT) is a three-dimensional imaging modality which has proven to be a powerful tool for biological imaging and healthcare diagnostics [1][2][3]. This imaging modality is based on interferometric analysis of backscattered light generated within the medium being imaged [4]. In Fourier domain OCT (FD-OCT) axial resolution is provided via broad-band illumination and a spectral analysis, thus enabling high-speed imaging with high sensitivity [5][6][7][8]. This approach captures a full depth profile (A-scan) within a single detector dwell time without axial scanning, with the imaging speed being essentially limited by the frame rate of the array detector used [9]. In FD-OCT it is important to consider the depth of focus (DOF) of the imaging system, which determines the range at which the lateral extension of the beam remains within predefined limits. For Gaussian beam illumination the DOF is defined as twice the Rayleigh range. The DOF is inversely proportional to the square of the (effective) numerical aperture (NA) of the imaging lens. The beam waist, which determines the lateral resolution of the excitation point spread function (PSF), is also determined by the focusing conditions and is linearly proportional to 1/NA. Increasing the lateral resolution at the focus by use of high NA optics thus leads to a significant shrinking of the DOF. The total PSF of the OCT system and its available signal levels outside the focal plane are further influenced by the detection PSF for the light that is collected after back-scattering.
High lateral resolution implementations of OCT are referred to as optical coherence microscopy (OCM). OCM uses high NA objectives to obtain a higher lateral resolution as compared to standard OCT. The increase in lateral resolution reduces the DOF and therefore limits the multiplexing advantages of FD-OCT. To maintain a high lateral resolution over a larger depth range additional axial scanning can be performed at the cost of compromised imaging speeds [10,11].
The applicability of OCM as a non-invasive technique for 3-dimensional (3D) in vivo structural and functional imaging with micrometer scale resolution was demonstrated in recent years [12][13][14]. Extension of FD-OCT based on illumination and detection of the fundamental spatial laser mode of the light source (so called Gaussian beam illumination) to applications in FD-OCM is hindered by the fact that the DOF in OCM is further limited by confocal gating implied by the spectrometer entrance aperture (or coupling efficiency of the backscattered light into an optical single mode fiber) [15]. For standard OCM implementations using Gaussian beam illumination the DOF is very short (typically a few µm) as compared to OCT implementations where low NAs are used at the expense of lower lateral resolution.
A fundamentally different, but also very attractive, approach to extend the DOF beyond the limit given by Gaussian optics that was proposed for FD-OCT is based on wavefront engineering [9,[27][28][29] to generate so-called diffraction-less (self-healing) beams (most prominently Bessel beams), to be used instead of the traditional Gaussian beam illumination. Because of this feature Bessel beams are also considered to be advantageous for imaging within scattering media. A combination of computational methods and Bessel-like illumination has also been applied for simultaneous optimization of lateral resolution, signal-to-noise-ratio and DOF [30].
Traditionally, free-space axicon lenses have been the most popular method for the generation of Bessel-beams for FD-OCT [9,[31][32][33][34]. Optical fiber-based methods for Bessel-like beam generation are desirable for imaging applications because they potentially allow for building more robust and compact setups, enable remote delivery, and because they are inherently more compatible with endoscopic applications.
Bessel-like beams have been generated in a fiber by focusing a ring mode with a lensed fiber tip [35], or by fabricating a micro-axicon directly onto a fiber core [36]. These methods improve the robustness and reliability of Bessel-beam generation but offer limited design flexibility [37]. Alternatively, illuminating a large-core multimode fiber on-axis can also provide a Bessellike output albeit with a strong degree of axial variation in the near field and inherent strong wavelength-dependent performance [37,38].
In fiber optics, due to rotation symmetry, the degeneracy of the HE and TE eigenmodes allows to transform the mode solutions to Bessel-like LP nm modes. Selectively excited LP 0m cladding modes by means of a long period grating (LPG) have been shown to behave like diffraction-resistant self-healing Bessel beams in free-space [37]. LPGs allow to achieve high mode conversion efficiency and accurate control of the number of rings in the mode. To achieve coupling to cladding modes LPGs were written in a H 2 -loaded high-NA single mode fiber [37] and due to the inherent sensitivity of LPGs to fiber bends or temperature fluctuations, a practical implementation would be written in a double clad fiber that can reliably propagate LP 0m modes [39]. In this type of fiber instead of light propagating in the fundamental, approximately Gaussian-shaped (LP 01 ) mode, the light can be forced to travel in a single sought-after higher order mode (HOM). Since the Bessel-like LP 0m mode is directly delivered from the fiber facet, accidental damage to the fiber tip can be repaired by simply cleaving off a short piece of fiber or polishing the fiber tip. Fabrication of a new module, as would be the case with a micro-axicon fiber tip, is not required.
In this work we exploit the favorable properties of higher order mode fibers. We demonstrate a proof of principle realization on the feasibility and imaging capabilities of an FD-OCT imaging system where the Bessel-like beam is generated in a stable higher order mode fiber LPG module. The HOM fiber used here is designed to support four LP modes including LP 01 and LP 02 at 1030 nm (fiber 2 of [40]). The HOM fiber is spliced to a fiber that is single moded at 1030 nm (OFS ClearLite 980-14) to launch the LP 01 mode of the HOM fiber. To get a pure LP 02 out of the fiber, an LPG is UV written into the HOM fiber with a period (Λ) matching the difference in propagation constant β of the two modes.
The LPG will then convert the LP 01 mode to the LP 02 mode. The HOM fiber is furthermore designed so the difference in propagation constant versus wavelength is flat around 1030 nm, which results in very broadband conversion [40][41][42]. It is important to reiterate that the LP 02 mode propagating in the HOM fiber is not a cladding mode but a guided core mode, thus providing a very robust and stable approach to generate a Bessel-like beam at the output. The resolution power of this system is characterized by imaging an optical phantom target containing randomly distributed point scatterers (Ø 3 µm polystyrene beads) embedded in agarose gel and a standard resolution test target. We compare the performance of the system with the fiber-generated Bessel-like LP 02 beam to the performance of the same system where the HOM fiber is replaced by a single mode fiber providing a Gaussian-like beam.

Optical setup
The FD-OCM imaging system used in this study is shown in Fig. 1. The illumination source is an all-polarization maintaining femtosecond Ytterbium-doped fiber oscillator [43]. The free-space output of the oscillator is coupled into a single mode fiber via fiber collimator L1 (Thorlabs F240APC-1064). The single mode fiber is spliced to the single-mode pigtail of the HOM fiber (half)-module for imaging using the LP 02 output mode of the (∼2 m long) HOM fiber. The HOM fiber output is fixed on a stainless-steel v-groove mounted on a 3-axis translation stage, and the output beam is collimated using an aspheric lens L2 (Thorlabs C240TME-B, f = 8mm). The insertion loss of the HOM module is 0.8dB. It was designed for use as broadband intracavity dispersion compensation (of single mode fiber) in ultrafast Yb:fiber oscillators [44], and hence supports about 60 nm bandwidth around 1040 nm.
For our comparative measurements using a Gaussian-like beam, the oscillator output is also coupled to a single mode fiber using L1, and the output end of this fiber is placed at the same v-groove mounted and collimated by L2. This approach allows us to quickly change from Bessel-like illumination to Gaussian-like illumination and vice-versa without the need to realign any system component other than the 3-axis translation stage.
The collimated output is relay-imaged with a 4f-telescope (L3 and L4, both Thorlabs AC254-100-B-ML, f = 100 mm) to the Michelson interferometer, which consists of a 50/50 beamsplitter (Thorlabs BS011). The sample arm consists of two galvanometric scan mirrors and scan lens (L5, Thorlabs LSM02-BB, f ∼ 18 mm). The reference arm contains a glass block for dispersion compensation (Thorlabs LSM02DC) and a plane silver mirror for reflection. L4 is placed one focal length away from the center point between the two galvanometric mirrors, which coincides with the back-pupil of L5. The output of the Michelson interferometer is focused by lens L6 The effective spectral resolution of our spectrometer is better than 0.1 nm, resulting in a depth sensing range of more than 2 mm. The axial resolution, resulting from the spectral bandwidth covered by the spectrometer, is about 20 µm. The use of different lenses at the position of lens L6 allows changing the point spread function (PSF) (i.e. resolution and signal attenuation away from the geometric focus of lens L5) of the detection pathway, without affecting the spectrometer resolution. Generally, shorter focal lengths of lens L6 enlarge the detection PSF. The detection PSF can also be changed by using different sized pinholes; however, changing the pinhole also influences the resolution of the spectrometer.
The diffraction efficiency of our gold-coated grating favors light polarized perpendicular to the grating grooves. To ensure optimal detection efficiency, a fiber polarization controller before the HOM fiber and a half-wave plate between L2 and L3 are used to fix the polarization state of the light in the setup.
Note that the effective mode-field area of the LP 02 mode in the HOM fiber module is comparable to the LP 01 mode in the standard single mode fiber used (PM980). As a result, the diameter of the outer ring lobe of the LP 02 mode is comparable in size to the LP 01 mode at the fiber output as well as in the far-field -collimated -beam profile, and the same collimation and focusing optics can be used for optimal performance of both the Bessel-like and Gaussian-like illumination. The collimation lens is chosen to utilize most of the available NA of the scan lens L5 (and clear aperture of the galvanometric mirrors), without clipping the beam.
The insets in Fig. 1 show the collimated far-field beam profiles (intensities, not electric fields) of both the SMF and HOM fiber output. Note that the far-field beam profile of the HOM fiber (Bessel-like) output is very different from the annular far-field profile (i.e. without a central lobe) of a Bessel-like beam from an axicon. The ring-shaped far-field profile from an axicon can be used in a dark-field illumination setup when combined with a low-pass filter in the detection pathway [9,31]. Dark field illumination can be advantageous when bright direct reflections (e.g. from a cover glass at the top of the sample) cause artifacts that limit the ability to detect weaker reflections from diffuse scatterers. In contrast, the LP 02 mode has a bright central spot, and as a result, (bright) direct reflections are not suppressed in the detection path, thus representing a bright field detection scheme. This facilitates the direct performance comparison to Gaussian-like illumination.

Simulations
We have simulated the beam propagation of the Gaussian-like LP 01 , the Bessel-like LP 02 mode and an ideal high-NA Gaussian beam throughout the optical imaging system of Fig. 1. These create the excitation PSFs shown in Figs. 2(a), 2(b) and 2(c), respectively. We simulated the high NA Gaussian by over-filling the back aperture of lens L5, thus that it reaches a similar lateral width as the LP 02 mode in focus. However, it quickly diverges outside of the focus. We further simulated the detection PSF for light that is back-scattered by the sample. The detection PSF results from the combination of the finite numerical aperture of the scan lens L5 and the rejection of out-of-focus light by the pinhole after lens L6. The effective detection mode in Fourier space can be modeled as a Gaussian of finite width and as if the sample would be illuminated by light that is passing through the pinhole from the other direction [15]. It can be shown that in object space (i.e. real space) the total PSF is simply the product of the excitation and the detection PSF [15]. We assumed a low-NA Gaussian for the detection mode, determined by the size of the pinhole. The corresponding total PSFs are shown in Figs. 2(d)-(f). The detection PSF is not explicitly shown in Fig. 2, but its 1/e width is shown is shown as the magenta curves in Figs. 2(g) and 2(h).
The evolution of the waist of the total PSFs for the high-NA Gaussian, the LP 01 mode, and the corresponding width of the central lobe in the LP 02 mode, as a function of distance from the focus are representative of the transverse resolution of our FD-OCT setup. Figures 2(g) and 2(h) show the waists of the excitation beams and total PSFs, respectively, for the high-NA Gaussian (green), the LP 01 mode (blue), and the LP 02 mode (red) as a function of distance from the focus. Furthermore, in Fig. 2(i) we show the signal dampening outside the focus in units of dB, which can be relevant if the scattered light does not provide a high SNR.
For the calculations we started with the near-field profiles at the fiber facet that can be calculated as a solution of the Maxwell equation for each fiber respectively. The high NA Gauss is modeled as a Gauss distribution with small waist at the fiber facet. Then we applied a 2D Fourier transform to obtain the far-field (collimated) beam profiles. Taking into account the rotational symmetry of the fiber modes, the 2D Fourier transform can be expressed as a zeroth order Hankel transform, yielding the far-field beam profile as a function of angle φ from the center of the beam [45]: With r being the distance from the optical axis in the near-field profile, k=2π/λ the wavenumber and the wavelength λ being 1030 nm. Appropriate (linear) scaling of the angular distribution of the far-field profile E FF allows to obtain the correct beam size on the scan lens, by matching the simulated far field profile to the experimentally observed profile.
The 3D focus distributions for the given far-field beam profiles can be calculated by evaluating the following formula [46]: where r and z are the radial and axial coordinate, respectively, and α reflects the finite numerical aperture of the focusing element. The resulting excitation PSFs are depicted in Figs. 2(a)-(c). Under some conditions (under-or over-filling of the pupil of the scan lens L5) it can be observed that the propagation of the LP 01 and LP 02 modes both do not display a monotonously decreasing beam-size (or central spot size) towards the focus, especially around the focus, whereas an under-filled Gaussian-shaped beam displays a monotonously decreasing beam size when propagating towards the focus. Similar effects have been observed in multiphoton-microscopy experiments with the output from a HOM fiber [47]. A simple explanation for this behavior can be found in the fact that both the LP 01 and LP 02 modes have positive and negative electric field values in their far field profile. In addition, overfilling of the back pupil creates a hard cut-off for the beam profiles which typically leads to small ripples in the focal distributions both in r and z (cf., for instance, the well-known Airy disk in 2D and its corresponding 3D focal distribution [46]).
For Gaussian beams with focused 1/e 2 beam waist w 0 , and wavelength λ, a much simpler expression can be also used to calculate the 1/e 2 width w(z) as a function of distance z from the focus in a medium with refractive index n. For the case of our LP 02 beam, addition of a scaling parameter C = 4 -effectively 4-fold expanding the Rayleigh length of an equally sized Gaussian beam -into that well-known expression allows matching both the size of the focus and divergence of the central spot of the beam: When comparing the results from this equation to the results of our simulations, the LP 01 beam -while using C = 1 -appears to have a slightly larger spot size in the focus than a Gaussian beam with equal far-field divergence. Estimating the confocal range using the beam-evolution (of the excitation beams -dashed curves in Fig. 2(g), and a virtual detection beam -magenta curve in Fig. 2(g)) calculated with the above approximation yields a confocal range of 324 µm for the LP 02 beam, and a confocal range of 96 µm for the extended Gaussian beam, both with a minimum 1/e radius of 2.8 µm. The confocal range for the PSF obtained with the LP 01 beam approximated with Eq. (4) is estimated to be 440 µm, with a minimum 1/e width of 5.3 µm. Note that the 1/e width of the PSF obtained with the LP 02 beam is smaller than this size over a range of more than 500 µm.

Results and discussion
To test and compare the resolution performance of the HOM fiber-based FD-OCT system (LP 02 ) to the performance of the FD-OCT system seeded with a standard single mode fiber output (LP 01 ), we prepared a phantom sample by dispersing 3 µm Ø polystyrene beads in Agarose gel (1% weight/volume low-melt in dH 2 0). We acquired A-scans on a 180 × 180 pixel grid, with a pixel-pitch of 0.78 µm, to obtain an OCT image of the phantom sample. The geometric focus of the scan lens was located about 0.5 mm below the surface of the phantom sample. Figures 3(a) and 3(b) show 3D (isosurface) renderings of the reconstructed volume, for the Gaussian-like and Bessel-like illumination respectively. Before drawing the isosurfaces, each plane in both volumes is normalized to its respective maximum signal. The red (iso)surfaces are chosen to appear approximately at the 1/e value of the beads (as the detected signals vary between beads and measurements). Figure 3(c) shows the measured width of 81 and 61 beads detected in the volumes reconstructed from the LP 02 and LP 01 FD-OCT measurements, respectively. The measured width stems from a Gaussian fit to the total distribution (LP 01 ) or the central peak (LP 02 ), respectively. More beads than the respective 81 and 61 beads of which we determined the size could be identified in the reconstructed volumes, however the size of these beads could not be determined with sufficient accuracy because of interference of their signature with the signal from nearby other beads. As the spot size in the LP 01 measurements is larger, this effect allowed to determine the size of less beads in these measurements.
The apparent width of the beads as a function of distance z from the focus is very different in the two measurements. In the measurement with the LP 02 mode, beads appear with a profile reflective of the LP 02 mode profile, i.e., as a bright central spot enclosed by a ring. The width of the central spot slowly increases with increasing distance from the focal plane. The central spots of the beads observed in the focal plane have about the same size as the central spot of the beam profile of the LP 02 mode in the focus. In the measurements with the LP 01 mode, beads appear as approximately Gaussian shaped spots, which increases in size significantly with increasing distance from the focal plane. For the LP 02 mode the size of the beads (∼2.5 µm) measured in focus agrees well with the simulated size of the central peak of the LP 02 mode. For the LP 01 mode the in-focus size of the beads (∼ 4.5 µm) is significantly smaller than the size of the LP 01 mode. Away from the focus the extracted bead size as a function of distance from the focal plane does not coincide with the calculated size of the excitation beam. The reason for this is that the resolution of our OCT system is not only determined by the spot size of the imaging beam, but also by the confocal gating of the detection system. In addition to the above measurements of our 3 µm bead phantom sample, we have also taken OCT measurements off a USAF 1951 resolution test target (Thorlabs R3L1S4P). Images taken with the Gaussian-like beam illumination and the Bessel-like beam illumination are shown in Figs. 4(a) and 4(b), respectively. Figure 4(c) shows the measured steepness of the edge of the square in-between size groups 6 and 7 as a function of shift of the surface away from the geometric focal plane of the scan lens. In this measurement, different lenses were used at the position of lenses L2 and L6 (Thorlabs C280TME-B, f = 18.4 mm and Thorlabs AC254-100-B-ML, f = 100 mm, respectively), resulting in a smaller focal spot size of the excitation beams and increased detection NA. Nonetheless, the analysis of the apparent sizes of the beads in our phantom sample and the mechanical displacement of the USAF 1951 resolution test target yield similar results for the resolution as a function of distance from the geometric focus. While the latter measurement allowed to reconstruct the resolution over a slightly larger depth range, the measurement with the 3 µm bead phantom sample shows that the LP 02 (Bessel-like) illumination not only provides an extended depth of focus, but also provides sufficient signal-to-noise ratio to detect and quantify structures across this extended depth range. The measurements with the USAF 1951 resolution test target confirm the expansion of the PSF beyond this extended range.
Comparing the case of Bessel-like (LP 02 ) illumination to Gaussian-like (LP 01 ) illumination, the Bessel-like illumination allows an improved lateral resolution (in the case of our phantom sample measurement ∼2.5 µm for the LP 02 beam and 4.5 µm for the LP 01 beam) and a 3-fold extension of the depth of focus compared to a high-NA Gaussian-like beam with similar focal spot size. Note that such a hypothetical high-NA Gaussian-like beam requires overfilling the back pupil of the imaging lens, such that a large portion of the excitation beam power is discarded.
Compared to Bessel beams generated using different methods, such as using an axicon lens, the LP 02 mode has only one ring outside the main peak. Because the Bessel-like LP 02 mode is generated in the higher order mode fiber module, the Bessel-like beam is not dependent on the alignment of any components in the optical setup. Whereas there is virtually no bandwidth limitation to the generation of Bessel-like beams using axicon lenses, LPG based mode converters make use of phase-matching, which can limit the bandwidth of the generated Bessel-like beam. With the appropriate design of the LPG, efficient conversion from the LP 01 mode to the LP 02 mode can be achieved over large bandwidths. The LPG used in our setup provided about 60 nm bandwidth, and larger bandwidths have been demonstrated [42]. State-of-the-art OCT systems use fiber-based interferometers with detection through the same fiber used to deliver the excitation light, rather than the free-space interferometer used in our setup. Because of the inefficient coupling into the fiber delivering the excitation light when traveling back through the axicon lens, in axicon lens-based OCM systems this detection geometry is favored over one where light is coupled back into the delivery fiber. In our system, because the Bessel-like beam is delivered directly from the fiber, this problem can be avoided. Because light coupled to different modes in our HOM fiber will experience different dispersion, the relatively long length of HOM fiber after the LPG would cause additional artifacts in the detection, as the backscattered light can couple to each of the modes supported by the fiber. This would be avoided in a setup with a shorter HOM fiber. A mode filter (possibly preceded by a second mode converter, as back-scattered light may couple more efficiently to the fundamental mode to be converted into a back-propagating LP 02 mode by the LPG) before combining the back-propagating light with the reference arm, may be needed to ensure proper overlap of the sample arm and reference arm in a fiber-based interferometer.
In conclusion, we have demonstrated for the first time to our knowledge the feasibility of an extended focus OCM imaging system where the Bessel beam illumination is achieved through a HOM fiber module, making use of an LPG as mode converter to selectively excite the LP 02 mode in a fiber especially designed to support the LP 02 mode.
We compared the OCT imaging capabilities of this system in terms of depth of focus and lateral resolution with the performance of a standard (Gaussian-like beam) illumination setup with similar beam waist. Our analysis shows that the Bessel-like beam provides a larger depth of focus and higher resolution than the standard system. Given the simplicity to generate a Bessel-like beam in a fiber-based module and the possibility to explore a large design space for the long period gratings and higher-order mode fibers, this approach could be used to generate Bessel-like beams with parameters allowing to further improve depth of focus and lateral resolution. This can be tailored to developing simpler FD OCT imaging systems in comparison to alternative free-space optics approaches (such as axicon lenses). The use of higher-order mode fibers also offers a flexible and more robust alternative to conical tip fibers. Disclosures. The authors declare no conflicts of interest. Data availability. Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.