1. Introduction

Minimally Invasive Procedures (MIP) are a set of medical procedures which are based on the tenet that minimizing the pain and discomfort of the patient is as important as achieving the desired therapeutic or diagnostic result [1]. To achieve this, MIP access the internal organ or tissue of interest using small incisions which result in smaller scars, less post-operative pain and a shorter revalidation time. However, due to the limited size of the incisions in MIP, direct visualization of the organ or tissue of interest during the procedure is no longer possible making the use of endoscopes (e.g. flexible imaging devices with small outer diameter) which are to be inserted alongside the surgical tools, mandatory. Transmitting images from the distal end of the endoscope (inside the patient) to the proximal end (outside the patient) with a sufficiently high resolution and Field-Of-View (FOV) while limiting the outer diameter to a centimeter or less has proven to be a technological challenge that could not adequately be met until the advent of two enabling technologies namely fiber optics and the Charged Coupled Device (CCD) [2]. In the CCD based endoscope, image capture is achieved via a CCD video chip (in combination with micro-optics) placed at the distal end of the endoscope with white light illumination provided by optical fibers. The fiber optics implementation of the endoscope uses a Coherent Fiber Bundle (CFB) that consists of tightly packed, regularly ordered high-refractive index cores in a common low-refractive index cladding. Though fundamentally different, both classes of endoscope share a commonality in the way they pixelate the area within the FOV to be imaged: in both cases each pixel or image element corresponds uniquely with a sensing element, whether it is optical (a high-refractive index core in the CFB) or electrical (a photosensitive element of the CCD video chip). As a result of this one-to-one relationship, image quality in conventional endoscopes is strongly dependent on the packing density of the sensing elements. Thanks to rapid advances in fiber optics and CCD manufacturing technology endoscopes with an outer diameter of less than 1mm are now commercially available. But as the sensing elements become smaller and more tightly packed, cross-talk becomes more prevalent thereby degrading the image quality. This imposes a lower limit on the outer diameter of the standard endoscope: with the current manufacturable packing density, a standard endoscope with an outer diameter of 0.5mm would have a visual acuity lower than that of a person is who is considered legally blind [3]. In the past few years, several groups have shown the feasibility of high-resolution micro-endoscopic imaging by using alternative endoscopic imaging principles that are not based on the bijective relationship between image element and sensing element. However, it should be noted that in all cases, this reduction in diameter came at the cost of a reduced FOV with respect to the large FOV achievable with the wide-field illumination and light collection of standard endoscopy.

Within the micro-endoscopic imaging principles, two diametrically opposite approaches can be distinguished: the distal approach and the proximal approach. In both approaches the illumination light exiting the endoscope is raster scanned along the image plane. The difference between both approaches lies in the way this beam steering is achieved. In the distal approach, the beam steering is done via a micro-mechanical scanner (either a MEMS type device [4,5] or a PZT tube [6]) integrated in the distal tip of the endoscope. Though this approach is very promising in terms of achievable resolution and FOV, its scalability remains to be proven since, to the best of our knowledge, the smallest diameter achieved is currently 1mm [3].

In the proximal approach on the other hand, scanning and focusing the exiting light is achieved via appropriate spatial modulation of the light before it is coupled into the endoscope. Since this method, scanning endomicroscopy, does not require any additional micro-optics or micro-mechanics at the distal end it holds, in the opinion of the authors, the greatest potential for miniaturization. In the past two years several groups have shown the feasibility of the concept using either a commercial CFB [7] or a commercial Multi Mode Fiber (MMF) [8,9]. The achievable FOV reported in [7] is small however (less than 12° full cone angle) and while [8] and [9] do not explicitly mention a FOV we can estimate it to be less than 26° full cone angle using the NA (0.22 in both studies). In previous work [10] we proposed the design of a custom designed CFB with ultra-high NA (0.928) which in combination with Proximal Spatial Light Modulation (PSLM) could achieve a FOV of 46° full cone angle. Simulations in [10] did assume a CFB with perfectly circular, identical cores, on a perfect hexagonal grid. However, due to the current limitations in the fabrication technology, fabricated CFBs will always exhibit some form of deviation either in the size of the cores, the shape of the cores or in its lattice. All of these have an impact on the CFBs guided eigen-modes and should consequently be taken into account if a more realistic model of the CFB’s propagation properties is to be achieved.

The sections of this paper are as follows: in section 2 we shortly reiterate the concept of optical beamsteering with PSLM. In section 3, we present our model of the optical propagation properties of an ultra-high NA CFB with and without fabrication irregularities. Several prototypes of the CFB were made at the Institute of Electronic Materials Technology (ITME) [11] according to the design presented in [10]. In Subsection 3.1 we use SEM images to look at and quantify the most common defects found in these different prototypes. We find that the prevalent defects are variations in core size, core shape and lattice constant. Next in 3.2, we outline the method used to determine the necessary input field for any desired output field. Following this, in 3.3, we use the spatial periodicity of the hexagonal lattice to define a unit cell of which will form the basis for the modal analysis. We then determine the required input field for a ‘flawless’ unit cell meaning perfectly circular cores on a perfect hexagonal lattice. We then analyze in Subsections 3.4-3.6 how each type of defect individually, modifies the electrical field as it propagates through a CFB with defects and determine the electrical field which should be used at the input if a certain output field is to be achieved. Finally, in Subsection 3.7 we used this method to look at the required input field for a CFB when all three types of fabrication error are present.

2. Optical beam steering with a CFB: concept

The idea of coherent beam steering originated in radar theory [12] where it was found that by adjusting the phase of each antenna from an array of equidistant antennas, the output beam could be steered in a well-defined direction depending on the wavelength and the distance between neighboring antennas. The fiber optics translation of this concept uses a CFB consisting of tightly packed, regularly ordered single mode fibers (or alternatively single mode cores in a common cladding) as the phase adjustable radiating elements. Via appropriate spatial modulation of the light at one end of the CFB, the electrical field of adjacent fibers/cores at the opposite end of the CFB can be given the necessary phase relationship required for beam steering and/or focusing. For example, if the 0-order peak of the output beam needs to directed in a direction characterized by the angle θ, then from scalar diffraction theory we know that the distance a over which the 2π linear phase shift should be accomplished, should obey the relation

 

Fig. 1 Calculation of the necessary phase of each single mode fiber/core is achieved via sampling of the continuous wave front with sampling distance equal to the lattice constant Λ.

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In a similar manner the beam can be focused by adjusting the phases of the cores in such way that a spherical wave front is achieved at the distal end of the CFB. Focusing and steering the beam at the same time can be done by simply adding the required wave fronts.

In previous work [10] we investigated what the requirements and constraints would be for a CFB if it were to be used as a micro-endoscope using PSLM. By investigating the relationship between core size, lattice constant and numerical aperture on the one hand and imaging characteristics (such as the FOV) on the other hand, we were able to determine a design methodology based on the trade-off between imaging characteristics and sensitivity to fabrication defects. Using this design methodology, we presented the design of two CFBs (one with focusing and scanning capabilities and one with focusing only). From those two designs, we take the CFB with both scanning and focusing capabilities as the starting point of this paper. This CFB consists of step index, single mode cores (circular) in a common cladding on a hexagonal lattice. The design parameters of this CFB are summarized in Table 1.

Table 1. Design Parameters of the CFB from [10]

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3. Modelling the propagation properties of the CFB

3.1 Identification and quantification of defects in a fabricated CFB

Ideally, the fabricated CFB would be conform with the nominal design of Table 1 i.e. identical, perfectly circular cores on a perfect hexagonal lattice. In reality, the stack-and-draw technique used to fabricate CFBs has its limitations. For example, during the different drawing stages the temperature distribution over the cross-section will never be completely homogeneous resulting in for example cores with different sizes and shapes. In addition, the material combination SF6 (from Schott) and NC21 (a multi component glass developed by the Institute of Materials Technology in Warsaw [13]) we chose for core and cladding of the CFB respectively has, to the best of our knowledge, never been used, making the fabrication challenging since the stack-and-draw method is equal parts science and craftsmanship. As a result, multiple iterations of the drawing process (each with different process parameters such as drawing speed and temperature) were required before the fabricated CFB came close to the desired design parameters. Of all these fabricated prototypes, we will only take into consideration one CFB, the one which best approximates the nominal design of Table 1.

But even with an optimized drawing process, the fabricated CFB will exhibit several types of defects. Identification and quantification of each type of defect is important since they influence the way the electrical field is modified during propagation in the CFB. To identify these defects, we cleaved the fabricated CFB manually with a ceramic knife, coated the cleaved surface with a layer of Pt/Pd of 3 nanometers thick using a Cressington 208HR with high resolution thickness controller, and used a Jeol JSM-7000F Scanning Electron Microscope (SEM) to image the cross-section. Figure 2 shows the SEM images of one of the prototypes at different magnifications. By analyzing the SEM images of the CFB, we found that the main types of defect are variations in core size, core shape (cores tend to have a certain ellipticity) and lattice constant.

 

Fig. 2 SEM images of one of the prototypes with magnification 500 (a), 5000 (b) and 10000 (c). The dark, slanted segment on the left image was caused by the ceramic knife used to introduce a fracture necessary for cleaving the CFB.

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To obtain quantitative data on core sizes, core shapes and lattice constant from these SEM images we used Fiji, an open source platform for image analysis [15]. Using Fiji, we first filtered the images using a Gaussian convolution and converted them from 8-bit grey scale to black-and-white using the ‘Threshold’ function (which is based on the isodata algorithm [16]). In the next step we used the ‘Analyze Particles’ function, which automatically identified the cores on the SEM image and extracted data such as core area, core shape (via fitting of an ellipse to each core) and position (coordinates of the core center with respect to the top left corner). Cores that are only partially visible on the SEM images, are excluded from the data set as can be seen by comparing Fig. 3(c) and 3(d). Another advantage of the ‘Analyze Particles’ function is that dust particles, which are usually smaller and more irregular in shape than the cores, can also be excluded from the data set. The results of the different steps of the procedure are shown in Fig. 3 which starts with a raw SEM image and ends up with the outlines of the cores the ‘Analyze Particles’ function extracted.

 

Fig. 3 To extract data from the SEM images, the raw SEM image (a) is filtered (b) and converted to a black-and-white image (c). The numbered outlines in (d) represent the cores of which the data will be extracted. Dust particles or cores which are only partially visible in the SEM image are excluded as can be seen by comparing (c) with (d).

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The output of the ‘Analyze Particles’ function is a table with info on core size, core shape and core position. Part of such table is shown in Table 2 which contains the extracted data from cores 1 through 5 of Fig. 3(d). ‘X’ and ‘Y’ denote the coordinates of the center of the core with relation to the top left corner of the full image, ‘major’ and ‘minor’ denote the length of the major and minor axes of the fitted ellipse and ‘angle’ is the angle between the horizontal x-axis and the major axis of the fitted ellipse. Applying this procedure on the SEM images of the different prototypes we can determine the average μ and standard deviation σ of the core area, the (equivalent) core diameter d, the lattice constant Λ, the major/minor axes of the fitted ellipse and the angle. The equivalent core diameter d was calculated using the core area assuming the core was perfectly circular. Note that all angles were reduced to the [-90°, 90°] interval. An overview of the resulting statistics are shown in Table 3, together with the corresponding nominal design values.

Table 2. Data Output for Cores 1 through 5 from Fig. 3(d) Using the ‘Analyze Particle’ Function

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Table 3. Statistics on Core Size, Core Shape and Lattice Constant for the Prototype CFB

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Each of the listed defects influences the way the E-field is modified as it propagates through the CFB, resulting in an output E-field which can be very different from the input E-field. In the next Subsections we will analyze for the fabricated CFB what field is required at the input in order to achieve an output field with the desired amplitude and wave front taking into account the different defects.

3.2 Modelling the influence of fabrication defects on the propagation properties of the CFB

When an E-field is coupled into the CFB, several eigen-modes (or in the case of multi core fibers, supermodes) are excited. These eigen-modes have different propagation constants resulting in a periodic fluctuation of the power within the cores (assuming there is no mode coupling during propagation). CFBs with the same design but with different types of fabrication defects or varying degrees of the same defect will have different eigen-modes and propagation constants and thus a different output E-field for the same input E-field. Or conversely, CFBs with different fabrication defects will require different input fields in order to achieve the same output field. In this subsection, we will outline the method we used to determine, for a given length of CFB, which input field is required if a desired output field is to be achieved. Next, we determine the distal output when the necessary proximal input field cannot exactly be generated by the Spatial Light Modulator (SLM) and compare it with the case where the proximal input is exact. We first apply the method to an ‘ideal CFB’ (i.e. a CFB without defects which adheres perfectly to Table 1). Following, we compare the required input field of the ideal CFB with the required input field of CFBs with defects as characterized by Table 3.

The method we used in order to determine which E-field is required at the input of the CFBs is illustrated in Fig. 4 and is based on the orthogonal mode decomposition in which the propagating field is represented as a superposition of orthogonal modes [17]. In our case, the eigen-modes of the CFB will serve as the orthogonal modes.

 

Fig. 4 Schematic of the method used to determine the necessary input field ${\overrightarrow{E}}_{proximal}$ in order to have a desired ${\overrightarrow{E}}_i$exiting the CFB.

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Suppose we have a straight piece of CFB of length L of which we know the eigen-modes ${\overrightarrow{E}}^0$ and we would like to know which field ${\overrightarrow{E}}_{proximal}$ should be coupled into the CFB in order to have a field ${\overrightarrow{E}}_i$ exiting the distal end. In the first step we determine ${\overrightarrow{E}}_{distal}$ as the linear combination of the CFB’s guided eigen-modes ${\overrightarrow{E}}^0$which best approximates ${\overrightarrow{E}}_i$

3.3 Definition of a unit cell

This straightforward method, however, suffers from a practical problem: for a CFB with a large number of (single-mode) cores (as is the case for our CFBs which have thousands of cores), determining all the eigen-modes for the complete structure as required by Eqs. (2) and (3), leads to prohibitively high computation times and memory demands. Instead, we used the spatial periodicity of the hexagonal lattice and defined a ‘unit cell’ that can be seen as a spatial period of the complete CFB structure. To determine a unit cell’s eigen-modes we used Lumerical MODE solutions, a fully vectorial mode solver based on a finite difference engine. Using periodic boundary conditions in both x and y direction, a total of 18 eigen-modes can be found. The unit cell for an ideal CFB is shown in Fig. 5 while Fig. 6 shows the unit cell’s eigen-modes with ordinal number 1 and 18, which are the eigen-modes with the highest and lowest propagation constant respectively (the eigen-modes are numbered in decreasing order of propagation constant). Note that because of the ultra-high NA, the eigen-modes have a small but non-negligible E_z component.

 

Fig. 5 Unit cell for a CFB without defects.

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Fig. 6 |E_x|, |E_y| and |E_z| of eigen-mode 1 ((a)-(c) respectively) and eigen-mode 18 ((d)-(f) respectively). The eigen-modes were numbered in decreasing order of propagation constant.

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With the eigen-modes of the unit cell now known, we can determine ${\overrightarrow{E}}_{proximal}$for any desired${\overrightarrow{E}}_i$. Since the goal is to have the distal field coherently come to focus in the observation plane, ${\overrightarrow{E}}_i$should be linearly polarized. So we define the desired ${\overrightarrow{E}}_i$ to be linearly polarized along the x-axis, with the same amplitude for all the cores, and a linear phase shift so as to direct the exiting light onto an observation plane at a distance of 500μm (as in [7]) under an angle of 10° (chosen arbitrarily within the FOV) with regard to the optical axis. We assume the length L of the CFB to be 0.5m. Since medical endoscopes display a large variety in lengths (rigid ones are usually shorter, while the flexible ones can be much longer), we would like to emphasize that the proposed method can be used for any other length as long as the eigen-modes are known. The amplitude and phase of the desired ${\overrightarrow{E}}_i$ are shown in Fig. 7 while its representation as a linear combination of eigen-modes ${\overrightarrow{E}}_{distal}$ is shown in Fig. 8.

 

Fig. 7 $\left|E_x\right|$ and $\angle E_x$ of the desired ${\overrightarrow{E}}_i$.

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Fig. 8 The top row shows $\left|E_x\right|$ and $\angle E_x$ of ${\overrightarrow{E}}_{distal}$ while the bottom row shows $\left|E_y\right|$ (left) and $\left|E_z\right|$ (right).

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Comparing Fig. 7 with Fig. 8, it is clear that there exists a linear combination ${\overrightarrow{E}}_{distal}$ of the CFB’s orthogonal eigen-modes which is a good approximation of our desired field${\overrightarrow{E}}_i$. Note that since all the unit cell’s eigen-modes possess a small but non-negligible E_z , ${\overrightarrow{E}}_{distal}$does too, contrary to${\overrightarrow{E}}_i$. Using the knowledge of the eigen-modes and their corresponding propagation constants, we can now propagate ${\overrightarrow{E}}_{distal}$ towards the proximal end to determine ${\overrightarrow{E}}_{proximal}$. The resulting proximal field for L = 0.5m, is shown in Fig. 9. Using PSLM, an input field needs to be generated which matches this required proximal field as closely as possible in order to have the desired distal field exiting the CFB.

 

Fig. 9 Shown here is the input ${\overrightarrow{E}}_{proximal}$ required to have the desired ${\overrightarrow{E}}_{distal}$ (shown in Fig. 7) exit an ideal CFB of length L = 0.5m.

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Looking at Fig. 9 however, we notice that the required proximal field has a small but non-negligible E_z component, which in terms of spatial light modulation is not trivial to generate. To determine how crucial E_z is in the proximal input in order to obtain the desired distal output, we defined ${\overrightarrow{E}}_{SLM}$ which has the same E_x as shown in Fig. 9 but in which E_z and E_y are 0. To simulate the coupling of ${\overrightarrow{E}}_{SLM}$into the CFB, it is then decomposed into eigen-modes and this new proximal input ${\overrightarrow{\tilde{E}}}_{proximal}$ is propagated towards the distal end of the CFB. The resulting ${\overrightarrow{\tilde{E}}}_{distal}$ is shown in Fig. 10, and as is clear from comparing Fig. 10 with Fig. 8, ${\overrightarrow{E}}_{distal}$and ${\overrightarrow{\tilde{E}}}_{distal}$ closely match. This is numerically confirmed when the E_x, E_y, E_z components of ${\overrightarrow{E}}_{distal}$and ${\overrightarrow{\tilde{E}}}_{distal}$ are subtracted as shown in Fig. 11. This means that, at the input, a linearly polarized field without E_z can be used as long as it is a good approximation of E_x (or E_y, depending on the polarization of the desired output) of ${\overrightarrow{E}}_{proximal}$ as shown in Fig. 8. Note that the amplitudes in Fig. 11 were scaled relative to the maximum of the corresponding amplitudes in Fig. 10 meaning ‘1’ in Fig. 11 is equal to the maximum of its corresponding counterpart in Fig. 10.

 

Fig. 10 Shown here is ${\overrightarrow{\tilde{E}}}_{distal}$, obtained by taking ${\overrightarrow{E}}_{proximal}$ from Fig. 8, setting its E_y and E_z equal to 0 and propagating the resulting ${\overrightarrow{\tilde{E}}}_{proximal}$ back to the distal end.

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Fig. 11 The differences between the components of ${\overrightarrow{E}}_{distal}$and ${\overrightarrow{\tilde{E}}}_{distal}$ shown here, are small and so when coupling light into the proximal end, linearly polarized light can be used. Note that the figures were scaled relative to those in Fig. 8.

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We should remark that Fig. 10 may give the false impression that at the input control of amplitude and phase of only one polarization direction is required. This is only the case when all the cores have a perfectly circular cross-section, identical along the CFB. Any deviation from perfect circularity (e.g. elliptical cores) will in general result in the excitation of two birefringent eigen-modes as will be illustrated later. Thus, using only one polarization would mean that in the observation plane only 50% (in the best case) of the power would be able to coherently add up to a focus and so for the general case, control of phase and amplitude of two orthogonal polarizations is necessary in order to obtain optimal results [18].

3.4 Influence of variations in core diameter

So for a given length of ideal CFB, we now know which proximal field is necessary in order to obtain a desired field at the distal end. But as is clear from Table 3, fabricated CFBs will contain different types of defects in varying degrees with each type and amount of defect influencing the required proximal input. In order to determine how each of the major types of defects (core size, core shape and lattice irregularities) influences the required input ${\overrightarrow{E}}_{proximal}$, we introduced each type of defect into the uniform unit cell of Fig. 5 separately (with the magnitude of the defect dictated by Table 3), determined the new ${\overrightarrow{E}}_{proximal}$ and compared this with the ${\overrightarrow{E}}_{proximal}$ of an ideal unit cell. Note that, for every type of defect, the four ‘half’ cores of Fig. 5 were adapted in such way to ensure the validity of the periodic boundary conditions used to calculate the eigen-modes. First, we will look at how variations in core area or diameter will influence the required proximal input as compared to an ideal CFB, for the same desired distal output (shown in Fig. 8) and length (L = 0.5m). To do so, we started from the ideal unit cell Fig. 5 and gave the cores different diameters according to a Gaussian probability density function (pdf) with average and standard deviation taken from Table 3 (μ = 0.542μm and σ = 0.013μm), determined the new eigen-modes and recalculated the required proximal input. The resulting ${\overrightarrow{E}}_{proximal}$ is shown in Fig. 12 and as expected for the case with variations in core diameter, the required proximal input is markedly different from the input for an ideal CFB with the linear phase relationship from ${\overrightarrow{E}}_{distal}$ now gone.

 

Fig. 12 The ${\overrightarrow{E}}_{proximal}$for a CFB of length L = 0.5m with different core diameters (with the average and standard deviation taken from Table 3) shown here, is markedly different from the ${\overrightarrow{E}}_{proximal}$ of an ideal CFB shown in Fig. 8 due to high Δn_eff/Δdiameter.

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This large difference is due to the high contrast in refractive index of core and cladding, resulting in a high sensitivity of the propagation constant β as function of the refractive index distribution n(x,y) and thus also of the core diameters. Eigen-modes from an ideal unit cell and eigen-modes from a unit cell with variable core sizes can have similar looking E_x, E_y and E_z and still have very different propagation constants resulting in a wholly different dependency of ${\overrightarrow{E}}_{proximal}$ on the CFB’s length. The fluctuation of the field within a certain core is therefore very dependent on the variation in core diameter of that core’s neighbors. Also notable is that the required proximal input can still be seen as being mainly polarized along the x-axis something which is not the case when dealing with asymmetrical cores. In this ${\overrightarrow{E}}_{proximal}$, E_y and E_z are very small compared to E_x as was the case for the ideal CFB and as a result, if we use a proximal input which approximates this E_x well but has an E_y and E_z equal to zero, the resulting ${\overrightarrow{E}}_{distal}$(shown in Fig. 13) is a good approximation of the desired distal output. This means that even for a unit cell that is asymmetrical with respect to the x-axis, an input linearly polarized along that x-axis will result in an output which is (nearly) linearly polarized along the same x-axis.

 

Fig. 13 The ${\overrightarrow{\tilde{E}}}_{distal}$for a CFB of length L = 0.5m with different core diameters obtained by taking ${\overrightarrow{E}}_{proximal}$ from Fig. 12, assuming E_y and E_z are 0, and propagating it back to the distal end, is still a good approximation for the desired output of Fig. 7.

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3.5 Influence of core ellipticity

To determine the influence of elliptical cores, we assumed all the elliptical cores of the unit cell to have the same area (in this case the average core area) but different ratios of major to minor axis as dictated by Table 3. As for the orientation of the elliptical cores, Table 3 shows there is little or no preferred direction for the orientation of the ellipses, so the cores of the unit cell were randomly orientated. As a result of this random orientation, two orthogonal eigen-modes (one each along major and minor axis) will be excited in the majority of the elliptical cores which means that, depending on the length of the CFB, the polarization of the required proximal field will differ from the desired x-oriented polarization of the distal field. This is shown in Fig. 14 which shows that the state of polarization varies for all the cores, a consequence of the birefringence (between the two eigen-modes of an elliptical core) being strongly dependent on the core’s ellipticity (defined here as the ratio between major and minor axis). Where in the case of perfectly circular cores (even with varying core diameters) polarization during propagation was maintained, the non-degenerate nature of the elliptical core’s eigen-modes now makes for a difference in amplitude and phase between E_x and E_y for each core. And since the SEM images from the fabricated CFBs show that most if not all cores are elliptical to some degree, in general control over amplitude and phase of two orthogonal polarizations will be required. The intrinsic birefringence of elliptical cores however, offers a way of ensuring that a linear input results in a linear output. If the fabrication process could be optimized so as to ensure that the major axes of the elliptical cores are aligned along a common axis (which was not the case for the CFB presented here), and the input field is linearly polarized along that common axis (or an axis perpendicular to this common axis), then polarization would be maintained during propagation as only one eigen-mode per core would be excited. This can be seen in Fig. 15 which shows the required ${\overrightarrow{E}}_{proximal}$ for a unit cell with aligned elliptical cores (obtained by aligning the cores of Fig. 14 along the x-axis). With such a CFB, we can again use only E_x (of Fig. 15) as the proximal input to generate a distal field that closely matches the desired distal output. The resulting ${\overrightarrow{\tilde{E}}}_{distal}$ for the CFB with aligned elliptical cores is shown in Fig. 16 and is a good match for the desired output field. Note that in Fig. 14 there are some irregularities in the wave fronts of E_x and E_y (near the top two and bottom two half cores respectively) which can be neglected as the amplitude of the respective fields at these discontinuities is very small.

 

Fig. 14 The ${\overrightarrow{E}}_{proximal}$for a CFB of length L = 0.5m with randomly oriented elliptical cores diameters (core area and ellipticity taken from the prototype as shown in Table 3) shown here, has a state of polarization which varies from core to core.

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Fig. 15 The ${\overrightarrow{E}}_{proximal}$ for a CFB of length L = 0.5m with elliptical cores from Fig. 12 aligned along the x-axis has a negligible E_y (since only one eigen-mode per core is excited) as opposed to the ${\overrightarrow{E}}_{proximal}$ for randomly oriented elliptical cores.

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Fig. 16 The ${\overrightarrow{\tilde{E}}}_{distal}$for a CFB of length L = 0.5m with aligned elliptical cores, is still a good approximation for the desired output of Fig. 7.

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An added advantage of using elliptical cores would be the insensitivity to external mechanical perturbations that induce energy transfer between orthogonally polarized eigen-modes. This unwanted energy transfer is largest when the perturbation is on the same length scale as the coupling length determined by the propagation constants of the ellipse’s orthogonal eigen-modes [19]. Since the coupling length is inversely proportional to the difference in propagation constants, having a large birefringence would ensure that the coupling length would be much smaller than the most common mechanical perturbations. Using Lumerical MODE, we determined the birefringence for an elliptical core with area = 0.0238μm² (the same area as an ideal circular core with diameter 0.55μm) as function of the ellipticity (see Fig. 17). Looking at Fig. 18 we see that a large birefringence of Δn = 0.001 can already be achieved with a small ellipticity of 1.05. With such a birefringence of Δn = 0.001, the coupling length L_c would already be reduced to 0.85mm and increasing the ellipticity to 1.2 would reduce L_c even further by a factor 4 (Fig. 17, inset).

 

Fig. 17 Birefringence as function of the ellipticity (or ratio between major and minor axis) for elliptical cores with the same area as a circular core with diameter 0.55μm.

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Fig. 18 The ${\overrightarrow{E}}_{proximal}$for a CFB of length L = 0.5m with circular cores from Fig. 5 on an irregular lattice. Size of the displacement was determined by a Gaussian pdf with σ = 0.036μm and direction of the displacement according to uniform pdf in the [0,2π] interval.

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Thus, using aligned elliptical cores would not only ensure the distal output to be linearly polarized (if the input polarization is correctly aligned), but would also make the CFB’s output polarization (but not its amplitude and wavefront) insensitive to all but the smallest (in terms of spatial frequency) of mechanical perturbations as opposed to [8] and [9] where the use of an MMF results in the necessary input polarization (as well as the input amplitude and wavefront) to change when the MMF is being bent. This brings us to one of the main challenges in fiber optic PSLM: the geometry of the fiber (both CFB and MMF) needs to be known at all times since every change in fiber geometry requires a new input in order to achieve the same output. One possibility would be to integrate the optical fiber for PSLM with a 3-core fiber optic shape sensor [20] into a common catheter allowing for a real-time knowledge of the catheter’s shape.

3.6 Influence of lattice irregularities

Next, we investigated how irregularities in the lattice influence the required proximal input. To determine a representative unit cell we started from a unit cell with perfect hexagonal lattice (as in Fig. 5) with lattice constant equal to the average one measured on SEM images (Λ = 1.478μm, see Table 3). We then added a different displacement to each (circular) core according to a Gaussian probability function with standard deviation (σ = 0.036μm). Also, each core was displaced in a different direction using a uniform distribution within the [0,2π] interval as no preferential direction for the lattice errors was observed in the SEM images. Due to the boundary conditions used to determine the unit cell’s eigen-modes, care was taken to ensure the four ‘half’ cores were displaced in such way their centers after displacement still formed a rectangle. The resulting proximal input is shown in Fig. 18 which shows that in the presence of the lattice irregularities, the required proximal input is different from the distal output but in a much less dramatic way than was the case with variation in core diameters and elliptical cores.

3.7 Combined fabrication errors

Lastly, we combined all the fabrication irregularities into one unit cell meaning each core was given a different area and ellipticity and a lattice displacement all according to the aforementioned methods. The necessary ${\overrightarrow{E}}_{proximal}$ for such a unit cell with the combined types of fabrication irregularities is shown in Fig. 19. The combination of fabrication irregularities makes for a required proximal input with large variations in amplitude and phase for each core with control of amplitude and phase in two orthogonal polarization directions necessary. Here also, aligning the elliptical cores would allow the use of only one polarization direction. This is shown in Fig. 20 which shows the required proximal input for a unit cell with all the types of fabrication irregularities but in which the elliptical cores are aligned. Just as in Fig. 15, we see that polarization during propagation can be maintained if the input polarization lies along the alignment axis or the axis perpendicular to the alignment axis.

 

Fig. 19 The ${\overrightarrow{E}}_{proximal}$for a CFB of length L = 0.5m with combined fabrication irregularities (core area, core ellipticity and lattice irregularities).

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Fig. 20 The ${\overrightarrow{E}}_{proximal}$for a CFB of length L = 0.5m with combined fabrication irregularities (core area, core ellipticity and lattice irregularities) but with ellipses aligned along a common axis.

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4. Conclusion

In this paper we presented our modal analysis of an ultra-high NA CFB to be used in combination with proximal wave front shaping in order to achieve micro-endoscopic imaging. Using SEM images of the fabricated prototypes of the CFB, we determined the most common fabrication defects or irregularities and quantified these. Based on the eigen-modes of a unit cell, both with and without defects, we determined the field necessary at the input of the CFB in order to achieve a desired input. We found that the required input field is the most sensitive to variations in core ellipticity due to the large difference in refractive index between core and cladding. However, we also found that the high birefringence found in the CFB’s elliptical cores could potentially be used to improve both the CFB’s propagation and bending properties: if the fabrication process could be optimized so that all cores would be elliptical and oriented along a common axis, then the CFB’ output polarization would be insensitive to all but the smallest (in terms of spatial frequency) mechanical perturbations. In a future publication we will present the results of our numerical analysis of the imaging properties of such a ultra-high NA CFB [21]. We find that even if the actual distal output field deviates from the ideal output field (identical amplitude for all cores and the wave front the combination of the appropriate spherical and tilted plane wave front), the central peak of the point spread function remains mostly unaffected.