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Abstract
Fiber probes for optical coherence tomography (OCT) recently employ a short section of step-index multimode fiber (SIMMF) to generate output beams with extended depth of focus (DOF). As the focusing region of the output beam is generally close to the probe end, it is not feasible to adopt the methods for bulk-optics with spatial pupil filters to the fiber probes with fiber-based filters. On the other hand, the applicable method of the beam propagation method (BPM) to the fiber probes is computationally inefficient to perform parameter scan and exhaustive search optimization. In this paper, we propose the method which analyzes the non-Gaussian beams from the fiber probes with fiber-based filters using the eigenmode expansion (EME) method. Furthermore, we confirm the power of this method in designing fiber-based filters with increased DOF gain and uniformly focusing by introducing more and higher-order fiber modes. These results using the EME method are in good agreement with that by the BPM, while the latter takes 1-2 orders more computation time. With higher-order fiber modes involved, a novel probe design with increased DOF gain and suppressed sidelobe is proposed. Our findings reveal that the fiber probes based on SIMMFs are able to achieve about four times DOF gain at maximum with uniformly focusing under acceptable modal dispersion. The EME method enables fast and accurate simulation of fiber probes based on SIMMFs, which is important in the design of high-performance fiber-based micro-imaging systems for biomedical applications.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical coherence tomography (OCT) is an attractive imaging modality due to its ability to acquire in vivo high resolution structural and/or functional information of internal tissues and organs using miniature probes [1]. In contrast to conventional optical imaging systems, the axial resolution of OCT is mainly determined by the coherence length of the adopted light sources. Using the state-of-the-art broadband light sources, axial resolutions of a few micrometers have been demonstrated [2]. However, if increasing their transverse resolutions to similar values, the useful depth range will be severely compromised because the light beam diverges rapidly under strong focusing.
To solve this dilemma, various approaches have been proposed to maintain high transverse resolution over an increased depth range, including numerical refocusing [3], depth-encoded synthetic aperture [4], dynamic focusing [5], and quasi-Bessel beam [6]. However, these methods rely on phase stability [3,4], mechanical scanning [5], or separated illumination/detection configurations [6]; therefore, their implementations in miniature probes are difficult. Micro-axicons [7] and binary-phase spatial filters [8] have been proposed to increase the depth of focus (DOF) of these miniature probes, but their DOF gains are reduced compared to their bulk-optic counterparts. Rather than trying to mimic bulk-optic spatial filters, we explored the use of two-mode interference in the step-index multimode fiber (SIMMF) to modulate the complex amplitude field, and the complex amplitude field with expected radial distribution was realized to extend the DOF [9,10]. Compared to the conventional fiber probe with the same lateral resolution, the proposed fiber probe achieves two times of DOF with experimental verification. By introducing more fiber modes, the DOF gain is expected to be further increased.
The relative ease of fabrication makes the fiber-optic beam shaper based on multimode interference an ideal substitute for complex optical elements designed for the same purpose. However, because the foci of miniature fiber probes exist in a region that is quite close to the probe end, Fresnel approximation commonly adopted in the design of bulk-optic spatial filters may introduce significant computation error [11,12]. Additionally, the phase distributions of fiber-based pupil fibers vary continuously in the radial direction, so that the design methods to obtain a maximally-flat response for binary filters are not suitable. To simulate the non-Gaussian beam from fiber probes accurately, beam propagation method (BPM) is a commonly used method and can yield results in good agreement with experimental data [10,13–15]. It takes about 8 to 16 seconds to simulate a waveguide structure with 256*256*260 grid points [14], which is generally thought to be a very efficient method. But as the simulation region and accuracy requirement increase, the simulation time could easily increase to several minutes, and the parameter scan and exhaustive search optimization for three or more variables of the probe structure are difficult if not impossible. On the other hand, since the beam propagation in z-invariant fibers consists of the propagations of finite orthogonal eigenmodes, it’s a natural way to treat the field evolution inside and outside the fiber probes in terms of fiber modes. The eigenmode expansion (EME) method re-calculates the modified parts of the structure only when performing parameter scans [16], potentially allowing very efficient optimization of fiber-optics probes. Zhu etc. have analyzed the nondiffracting beams from multimode optical fibers in terms of linearly polarized (LP0n) modes [17]. However, the beam propagation in free space after leaving the fiber is approximated by the superposition of multiple Bessel fields, while the diffraction of the apertured Bessel fields is not considered. Additionally, some parameters that are important for OCT imaging, such as the modal dispersion, are not analyzed. Yin etc. have analyzed a coaxially focused multimode beam by geometric projection of the propagation modes to the lens aperture and diffracting the projected pattern to the free space outside the probe using Fresnel diffraction formula [15], but the interference between higher-order modes are not investigated, especially the destructed interference that potentially degrades the axial uniformity of the focusing.
In this paper, we use the EME method and an exhaustive search method to simulate and design the all-fiber probes with a fiber-based filter for OCT. A fabricated all-fiber probe with fiber-based filter is adopted to demonstrate the advantages of the EME method over BPM in terms of simulation speed and accuracy. Then, an exhaustive search method is applied to optimize fiber-based filters that are formed by up to ten LP0n modes. Finally, the potential of the fiber-based filters in the DOF extension for OCT is discussed.
2. Numerical modeling technique
A typical all-fiber probe with fiber-based filter for OCT is shown in Fig. 1. It consists of a length of single mode fiber (SMF) for light delivering, a section of graded-index fiber (GIF) labeled as GIF1 to alleviate the light divergence and increase the coupling efficiency between the SMF and the following large core fiber (LCF). The LCF is a SIMMF that supports multiple modes. Due to the rotational symmetry of the probe structure and the limited V number of the LCF, only LP01 mode and LP02 mode are excited. As a result of dual-mode interference, the complex amplitude field on the end facet of the LCF is controllable by tunning the lengths of the GIF1 and the LCF. The complex field is expanded by the GIF2 and imaged to the entrance pupil of the GIF3. This magnified complex field acts as the final pupil filter. The GIF3 is used to focus the output beam and control the beam diameter. In our previous work, the BPM is used for light simulation and the simulation results are verified by the experiment data [10]. In this section, we will demonstrate how to simulate this kind of probes efficiently by the EME method and compare the simulation results of the EME method to that by the BPM.
2.1 Eigenmode expansion method
In view of the EME, the light field outside the probe is expressed by a superposition of the diffracted mode fields of the last fiber in the probe:
Since the mode fields of z-invariant fibers do not vary with the lengths of the fibers, the diffraction field of each mode can be evaluated before the start of the simulations to save computation. For this pre-computed mode, the complexity of the EME method is not dependent on the choices of the diffraction formulas. Thus, the Rayleigh–Sommerfeld diffraction formula that gives correct results for both far-field and near-field diffraction is used to solve the diffraction problem [18]:
To obtain the mode coefficients at the end facet of the GIF3, we start from mode coupling on the SMF-GIF1 interface. Due to the modal orthogonality, the coupling coefficients between the fundamental mode of the SMF and the LP0n mode of the GIF1 are given by:
where r is the radial coordinate, ${E_{\textrm{SMF}}}$ is the normalized fundamental mode of the SMF, ${\hat{e}_{1n}}$ is the normalized LP0n mode filed of the GIF1. Let us consider that ${N_1}$ is the number of the modes,${\beta _{1n}}$ is the mode propagation constant, ${L_{\textrm{GIF1}}}$ is the fiber length. Then, the electric field after propagating in the GIF1 is given by:Here, the direction of matrix multiplication is from the right to the left. ${{\bf C}_{01}}$ is the N1×1 join matrix between the SMF and GIF1. The elements of ${{\bf C}_{01}}$ is simply:
where ${\hat{e}_{0j}}$ is the normalized LP0j mode filed of the SMF. ${{\textbf P}_1}({L_{\textrm{GIF1}}})$ in Eq. (5) is the propagation matrix of the GIF1. It’s a N1×N1 diagonal matrix of the form:If we apply Eq. (3) and Eq. (4) to the rest fibers sequentially, the electric field on the end facet of the GIF3 is obtained:
Since the mode fields in GIF2 are diffracted in the following NCF before coupled into the GIF3, the calculation of their join matrix is slightly different from the Eq. (6):
2.2 Numerical accuracy of the EME method
In the aid of the EME method, the computation time of the output beam from the probe is decreased to be 0.64 second per simulation, which makes it practical to optimize the probe parameters by an exhaustive search method. However, since only finite guide modes are considered, the computation error in the EME method remains to be analyzed.
Three probes with different structure parameters are simulated by the EME method and the BPM for comparison. The fiber specifications and the lengths of the fibers used in the simulations are listed in Table 1 and Table 2 respectively. To perform the BPM simulation as accurately as possible, the lateral and axial grid sizes are set to be 0.1 μm and 1 μm respectively. The padé orders in the simulation regions of GIF1, GIF2 and GIF3 are set to be (1,0) for better simulation accuracy for wide-angle beam propagation. The beam propagation in the homogeneous medium (such as the NCF and the air) is calculated by the Rayleigh–Sommerfeld diffraction formula. With the transverse simulation area of 120 μm × 120 μm, it takes about 9 minutes to conduct a single simulation by BPM in the same computer (i7-2600K CPU @ 3.4GHz, 16GB RAM).
The simulated field intensity of the output beams of the three probe designs in Table 2 are presented in Fig. 2. The panels in the first row are the simulation results by the BPM, and the panels in the second row are the simulation results by the EME method. The output beam characteristics, such as the minimal beam diameter (MBD), DOF, the working distance (WD) and the DOF gain, are also evaluated according to the simulated intensity distributions. They are listed in Table 2, where the values in the parentheses are obtained by the EME method. The simulated intensity distributions by the EME method and by the BPM are in good consistency. However, some channeled artifacts can be observed in Figs. 2(e)–2(g), especially in the range of 0 < z < 100 μm in Fig. 2(g). These artifacts introduce errors to the calculation of ≥ the MBD and the DOF. Since the DOF is determined by the depth range over which the beam diameter is smaller than twice of the MBD, a slight error in the MBD causes quite different results of the DOF for design II in Table 2. But the artifacts have minor effect on the calculated WDs and the DOF gains. Since the DOF gains are utilized in the optimization, the potential impact of the computation errors on the design optimization process is minor. Thus, the EME method is suitable to design the fiber-based filters with a maximum DOF gain.
Fig. 2. Simulated field intensity of the output beams in air normalized to the peak intensity in SMF for three typical cases of the designed probe. (a-c) The simulation results for probe I, II and III by the BPM (Ref. [10], Fig. 2); (e-g) The simulation results for the same probe designs by the EME method.
3. Design of fiber-based filters formed by more fiber modes
In our previous work, a section of SIMMF with low NA and small core diameter was chosen for the fiber-based filter. The V number of such MMF is relatively small, so that only two LP0n modes are capable to propagate in the fiber. By selecting a SIMMF supporting more guide modes (≥ 3), the DOF gain of the fiber-based filter is expected to be significantly increased, which is analogous to increase the number of zones in annular multi-phase plates [19].
However, since it provides different paths with distinct optical path lengths (OPLs) to the light delivery, the SIMMF potentially degrades the axial resolution and causes artifacts on the OCT images. To prevent the noticeable deterioration on the imaging quality, the modal dispersions of the probes should at least be controlled within the axial resolution of the imaging system. Since the modal dispersion of the SIMMF generally increases with its V number, there must be a limit on the maximum DOF gain without severely compromising the axial point spread function of OCT.
In this section, we discuss the criteria for choosing the satisfactory candidates from various non-Gaussian optical field distributions for OCT. Then, the EME method is used to simulate the probes with filters containing increased fiber modes. The exhaustive search method is utilized to optimize the probe parameters according to the proposed criteria.
3.1 Optimization criteria for the output beams from the probes
In general, the probes for OCT require an output beam with small beam diameter, long DOF and working distance, low sidelobe level, minimal modal dispersion, high Strehl ratio, and uniform axial focusing. Reasonable definitions of these metrics are important to simplify the optimization code and obtain satisfactory results.
The beam diameter is defined as the full-width at half-maximum (FWHM) focal spot diameter of the central lobe of the output beam. We choose to use the beam diameter of the main lobe rather than the whole output beam because the former yields a smoother curve of the beam diameters versus the z coordinate, which makes the following DOF calculation more reliable. The MBD is searched in the image space where z ≥ 75 μm, to exclude the pseudo foci near the probe end caused by the simulation error of the EME method. The average beam diameter is given by the mean of the beam diameters within the DOF range.
The DOF is defined as the length of the longest continuous region over which the beam diameter is smaller than twice its minimum value. This definition allows more tolerance for beam width fluctuations in the focal region of non-Gaussian beam profiles than the Rayleigh-range criterion, and is thought to be more consistent with the perceived useful imaging depth range in OCT [14]. Since the longest continuous region is considered only, the output beams with uniformly focusing instead of the separated foci are preferable. The DOF gain is defined as the DOF ratio between the probe with the filter and the Gaussian beam with the same MBD. The modal dispersion of the probe is simply evaluated by the multiplication of the length of the LCF and the group index difference between the lowest order mode and the highest order mode with considerable power (at least 50% of the peak mode power). The maximum lengths of the LCF are determined on the fly during the simulation according to modal dispersion limit and the mode excitation condition.
To evaluate the Strehl ratio, a conventional probe with the structure of SMF-NCF-GIF is used as the reference. The length of the NCF is fixed to 400 μm, so that the light beam from the SMF is expanded sufficiently to cover the aperture of the GIF. While the length of the GIF is chosen to focus the light beam to the same MBD as the probe with the filter. The Strehl ratio is then given by the ratio between the light intensity at the focus of the probe with the filter and that without the filter.
The working distance is defined by the distance of the center of the DOF region from the probe. The relative working distance is given by the ratio of the working distance of the probe with the filter over that without the filter.
Sidelobes are the lobes (local maxima) of the lateral intensity profile of the output beam, except the main lobe. The sidelobe level is the ratio between the maximum peak of all the sidelobes and the peak of the main lobe.
The goal of the optimization is to achieve maximum DOF gain with the following constraints:
- 1. The sidelobe levels within the DOF range are less than 50%;
- 2. The average beam diameter is 5 ± 0.5 μm;
- 3. The modal dispersion of the probe is less than 2.5 μm;
- 4. The peak location of the on-axis intensity is greater than 75 μm.
3.2 Probe with a filter formed by lower-order fiber modes
To increase the DOF gain of the fiber-based filter, LCF1 (FG050LGA, Thorlabs, Inc.) with larger core diameter is adopted, as demonstrated in Fig. 3. The GIF2 and the NCF for beam expansion are removed because the core diameter of the LCF1 is large enough to match that of the GIF3.
In the fiber-based filter, GIF1 is introduced to manipulate the excited modes in the following LCF1. Without the GIF1, the incident power tends to be distributed in the LP02 mode in the LCF1. As the length of the GIF1 increases to a quarter of its pitch length, the divergence angle of the beam decreases to the minimum value, and the incident power is mainly distributed in the fundamental mode of the LCF1. The length of the LCF1 is used to tune the phase differences among the fiber modes on the end facet of the LCF1. The length of the GIF3 is used to adjust the average beam diameter of the output beam. In the parameter scan, the scan range of the GIF1 length is between 0 to 285 μm (quarter-pitch length). Once the length of the GIF1 is given, the maximum length of the LCF1 is determined by the 3rd optimization constraint in Subsection 3.1. Once the length of the LCF1 is determined, the length of the GIF3 is found by a binary search algorithm to satisfy the 2nd optimization constraint in Subsection 3.1.
The optimized parameter pairs are plotted in Fig. 4(a), with the output beam metrics plotted in Fig. 4(b). The modal dispersion values are relatively small (≤ 2.5 μm) according to the optimization criteria in Subsection 3.1, so they are not displayed here. It shows that the working distance of the probe decrease with the increase of the length of the GIF1, while the DOF gain does not present a straightforward relationship with the length of the GIF1. When the length of the GIF1 is between 60 μm to 170 μm, the DOF ratio reaches to its maximum of 3.4 with working distance enhancement of 1.5. Three probes with structure parameters in this region are selected for further investigation. The selected probe design I, II and III are marked by small boxes in Fig. 4(b). Their structure parameters are listed in Table 3. The simulated intensity distributions of their output beams in the air are presented in Figs. 5(b)–5(d), and the Gaussian beam in Fig. 5(a) is for reference. The detailed variations of light intensity and the beam diameters are shown in Figs. 5(e)–5(h). It is noted that the Strehl ratio equals to the relative on-axis intensity at the focus location. The simulations suggest that the output beams with better average lateral resolutions and uniform axial focusing also maintain an in-focus beam profile over a depth range that is 3-4 times that of the Gaussian beam. The side lobe levels are typically 40%-50% at the working distances of the probe designs. Modifying the 1st constraint in Subsection 3.1 would lead lower side lobes, but the achieved DOF gain might decrease to some extent. The characteristics of the output beams are also listed in Table 3.
Fig. 4. The results of the parameter study for the probes with a filter formed by lower-order fiber modes. (a) The lengths of the LCF1 and the GIF3 versus the length of the GIF1 for the optimized condition. (b) The DOF gain, Strehl ratio and the working distance ratio of the output beams versus the length of the GIF1 for the optimized condition. Three small boxes (red, orange and purple) refer to the three optimized probe designs (design I, II and III) respectively.
Fig. 5. The two-dimensional intensity distributions of (a) the Gaussian beam and the output beams of (b) the probe design I, (c) the probe design II and (d) the probe design III in the air. The intensity is normalized by the peak intensity, and displayed in a log scale with a dynamic range of 15 dB. (e) The on-axis intensity distributions. (f) The beam diameters versus the axial distance. (g) zoom in view of the boxed area in (f). (h) The lateral intensity distributions at the working distances.
Table 3. Fiber lengths and output beam characteristics corresponding to the optimized probe design I, II and III with lower-order fiber modes and the probe design IV with higher-order modes.
3.3 Probe with a filter formed by higher-order fiber modes
Because high-order mode fields resemble more closely exact Bessel fields, the nondiffracting propagation or the DOF gain is most pronounced when higher order LP0n modes are excited. To excite higher order modes, the divergence angle of the input beam needs to be increased. Thus, the light beam from the SMF is firstly expanded in the NCF1, and then focused by the GIF4 to a tighter spot. The length of the GIF4 is determined so that the beam is focused just on the interface between the GIF4 and the LCF1. Then, the divergence angle of the focused beam increases monotonically with the length of the NCF1. The layout of the probe is depicted in Fig. 6.
The procedure of the parameter study is similar to that of Subsection 3.2. However, the constraint on the modal dispersion is relaxed to 5-10 μm instead, because the maximum group index difference among the fiber modes increases rapidly as the orders of the modes increase. The optimized parameter pairs are plotted in Fig. 7(a), with the output beam metrics including the relative modal dispersion plotted in Fig. 7(b). The relative modal dispersion is the ratio between the modal dispersion and the average beam diameter. The latter one is about 5 μm according to Subsection 3.1. It shows that the working distance of the probe has an overall increase with the increase of the length of the NCF1, while the DOF gain does not present a straightforward relationship with the length of the NCF1. The output beam with DOF gain of 3.8, working distance ratio of 2, relative small modal dispersion of 3.8 μm is observed at L_NCF1 = 215 μm in Fig. 7(b), which is adopted as the optimized probe design IV for further investigation. The structure parameters of the probe design IV are listed in Table 3.
Fig. 7. The results of the parameter study of the probes with a filter formed by higher-order fiber modes. (a) The lengths of the GIF4, the LCF1 and the GIF3 versus the length of the NCF1 for the optimized condition. (b) The DOF ratio, Strehl ratio, the working distance ratio and the relative modal dispersion of the output beams versus the length of the NCF1 for the optimized condition. The small red box refers to the optimized probe design IV.
The simulated intensity distribution and the characteristics of the output beam of the optimized probe IV are shown in Fig. 8. The Gaussian beam with the same MBD is also depicted as reference in Figs. 8(b)–8(e). By increasing the order of the fiber modes excited in the SIMMF, the output beam in Fig. 8 achieves slight increase in the DOF gain and significantly reduced sidelobe level compared to that in Fig. 5. According to Fig. 8(e), the first sidelobe level at the working distance of the novel design is reduced to 26%. The sidelobe level is mitigated because more energy spreads to the higher order sidelobes. Further increase in the DOF gain is limited by the maximum length of the LCF1, which is determined by the upper limit of the modal dispersion. The characteristics of the optimized output beam are also summarized in Table 3.
Fig. 8. The characteristics of the Gaussian beam and the output beam of the probe design IV. (a) The two-dimensional intensity distribution of the output beam in the air. The intensity is normalized by the peak intensity, and displayed in a log scale with a dynamic range of 15 dB. (b) The on-axis intensity distributions. (c) The beam diameters versus the axial distance. (d) zoom in view of the boxed area in (c). (e) The lateral intensity distributions at the working distance.
The mode excitation in the LCF1 for the four optimized probe designs are depicted in Fig. 9. For the first three probe designs with a filter formed by lower-order fiber modes, the power is mainly distributed in the first 3 lower-order modes. For the probe design IV with a filter formed by higher-order fiber modes, significant power is distributed in the modes with order > 3.
Fig. 9. Calculated excitation power of azimuthally symmetric modes in the LCF1 for the probe design I, II, III with lower-order fiber modes and IV with higher-order fiber modes.
4. Conclusion
In summary, we have demonstrated the usefulness of the EME method for fast simulation and design of fiber probes with fiber-based filter for future micro-endoscopic OCT applications where optimum resolution, increased depth range and uniformly focusing will be crucial for enabling minimally invasive imaging with quality comparable to state-of-the-art benchtop OCTs.
Our simulations have revealed that the EME method yields optical fields in good consistence with the BPM, which used to be the main technique for accurate modeling of miniature fiber probes with non-Gaussian beam output for OCT. The important metrics such as the MBD, the DOF and the working distance obtained from the EME method are almost the same as that by the BPM. Moreover, the EME method provides significantly enhanced simulation speed by 1-2 orders of magnitude, enabling efficient optimization of the probe structure parameters based on an exhaustive search method.
Furthermore, we have demonstrated the power of the EME method for studying advanced probe designs incorporating novel optical elements such as fiber-based filters formed by more fiber modes. By adopting a SIMMF with core diameter of 50μm, more fiber modes are involved in the optimization of the fiber-based filter. The optimized probe with the filter achieves 3.4 times of DOF gain, 1.5 times of working distance enhancement and uniformly focusing. We also propose to use a section of NCF-GIF to reduce the beam diameter of the output beam from the SMF, so that the light power tends to distribute among higher-order modes of the following SIMMF. It is observed about four times DOF gain, two times of working distance enhancement, reduced sidelobe level (26% at the working distance of the probe design) and uniformly focusing in the novel design. We would like to point out that one major obstacle of the development and application of such technique is the modal dispersion. To prevent perceptible imaging artifacts caused by multimode propagation, the length of the SIMMF should be strictly controlled within about 1.5mm, which reduce the amounts of freedom in optimization and limit the achievable DOF gain in return.
In this paper, we have focused on fiber probes with fiber-based filters for OCT imaging, but the methods described here can also be applied to other imaging systems incorporating step-index multimode fibers, such as micro-endoscopes for photoacoustic imaging [20]. Some elements of the work presented here could also be useful for non-imaging applications, such as optical trapping and transport of particles [21].
Funding
National Natural Science Foundation of China (62035011, 11974310, 31927801, 61905214); National Key Research and Development Program of China (2017YFA0700501, 2019YFE0113700); Natural Science Foundation of Zhejiang Province (LR20F050001); Fundamental Research Funds for the Central Universities.
Disclosures
The authors declare no conflicts of interest.
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