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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 19 — Sep. 23, 2013
  • pp: 21991–22011
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Numerical characterization of an ultra-high NA coherent fiber bundle part I: modal analysis

Stefaan Heyvaert, Heidi Ottevaere, Ireneusz Kujawa, Ryszard Buczynski, Marc Raes, Herman Terryn, and Hugo Thienpont  »View Author Affiliations


Optics Express, Vol. 21, Issue 19, pp. 21991-22011 (2013)
http://dx.doi.org/10.1364/OE.21.021991


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Abstract

Advances in fiber optics and CCD technology in the last decades have allowed for a large reduction in outer diameter (from centimeters to submillimeter) of endoscopes. Attempts to reduce the outer diameter even further, however, have been hindered by the trade-off, inherent to conventional endoscopes, between outer diameter, resolution and field of view. Several groups have shown the feasibility of further miniaturization towards so called micro-endoscopes, albeit at the cost of a very reduced field of view. In previous work we presented the design of an ultra-high NA (0.928) Coherent FiberBundle (CFB) that, in combination with proximal wave front shaping, could be used to circumvent this trade-off thus paving the way for even smaller endoscopes. In this paper we analyze how the modal properties of such an ultra-high NA CFB determine the required input field to achieve any desired output field. We use the periodicity of the hexagonal lattice which characterizes a CFB, to define a unit cell of which we analyze the eigen-modes. During the modal analysis, we also take into account realistic variations in lattice constant, core size and core shape due to the limitations of the fabrication technology. Realistic values for these types of fabrication-induced irregularities were obtained via SEM images of a CFB fabricated according to the aforementioned design. The presence of these irregularities results, for a desired output, in the required input to be different from the required input for a defect-free CFB. We find that of the different types of fabrication-induced irregularities present in the CFB, variations in core ellipticity have the biggest impact on the required input for a given desired output.

© 2013 OSA

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

1. J. E. A. Wickham, “Endoscopic surgery,” Br. Med. Bull. 42(3), 221–339 (1986), http://bmb.oxfordjournals.org/content/42/3/221.full.pdf. [PubMed]

]. 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

2. S. F. Elahi and T. D. Wang, “Future and advances in endoscopy,” J Biophotonics 4(7-8), 471–481 (2011). [CrossRef] [PubMed]

]. 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

3. C. M. Lee, C. J. Engelbrecht, T. D. Soper, F. Helmchen, and E. J. Seibel, “Scanning fiber endoscopy with highly flexible, 1 mm catheterscopes for wide-field, full-color imaging,” J Biophotonics 3(5-6), 385–407 (2010). [CrossRef] [PubMed]

]. 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

4. C. L. Hoy, N. J. Durr, P. Chen, W. Piyawattanametha, H. Ra, O. Solgaard, and A. Ben-Yakar, “Miniaturized probe for femtosecond laser microsurgery and two-photon imaging,” Opt. Express 16(13), 9996–10005 (2008). [CrossRef] [PubMed]

,5

5. H. C. Park, C. Song, M. Kang, Y. Jeong, and K. H. Jeong, “Forward imaging OCT endoscopic catheter based on MEMS lens scanning,” Opt. Lett. 37(13), 2673–2675 (2012). [CrossRef] [PubMed]

] or a PZT tube [6

6. Y. Wu, Y. Leng, J. Xi, and X. Li, “Scanning all-fiber-optic endomicroscopy system for 3D nonlinear optical imaging of biological tissues,” Opt. Express 17(10), 7907–7915 (2009). [CrossRef] [PubMed]

]) 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

3. C. M. Lee, C. J. Engelbrecht, T. D. Soper, F. Helmchen, and E. J. Seibel, “Scanning fiber endoscopy with highly flexible, 1 mm catheterscopes for wide-field, full-color imaging,” J Biophotonics 3(5-6), 385–407 (2010). [CrossRef] [PubMed]

].

2. Optical beam steering with a CFB: concept

The idea of coherent beam steering originated in radar theory [12

12. E. Brookner, “Phased-array radars: Past, astounding breakthroughs and future trends,” Microwave J. 51(1), 30 (2008).

] 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
sin(θ)=λa
(1)
with λ the wavelength of the light. So in order to have the beam at the distal end leaving under an angle θ, the phase of the individual cores should be adjusted in such way to have a linear phase shift with a 2π phase difference achieved over a distance a. This is illustrated in Fig. 1
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.
, which also illustrates how, at the distal end of the CFB, the discrete phase of the cores (with lattice constant Λ) form a sampled version of the wave front necessary for steering and/or focusing.

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.

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

13. D. Lorenc, M. Aranyosiova, R. Buczynski, R. Stepien, I. Bugar, A. Vincze, and D. Velic, “Nonlinear refractive index of multicomponent glasses designed for fabrication of photonic crystal fibers,” Appl. Phys. B 93(2–3), 531–538 (2008). [CrossRef]

]) 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
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.
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.

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

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

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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

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

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, together with the corresponding nominal design values.

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
Fig. 4 Schematic of the method used to determine the necessary input field Eproximal in order to have a desired Eiexiting the CFB.
and is based on the orthogonal mode decomposition in which the propagating field is represented as a superposition of orthogonal modes [17

17. T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4(6), 388–394 (2010). [CrossRef]

]. In our case, the eigen-modes of the CFB will serve as the orthogonal modes.

Suppose we have a straight piece of CFB of length L of which we know the eigen-modes E0 and we would like to know which field Eproximal should be coupled into the CFB in order to have a field Ei exiting the distal end. In the first step we determine Edistal as the linear combination of the CFB’s guided eigen-modes E0which best approximates Ei
Edistal=q=1N0χqEq0,
(2)
with complex coefficientsχqcalculated via the linear least squares method. If the CFB is unperturbed, we can assume that the CFB’s refractive index distribution n(x,y,z) is invariant along the z-axis (the direction of propagation) so that propagation of Edistal towards the proximal end can be calculated using the propagation constant associated with each eigen-mode so that the necessary input Eproximal can be written as
Eproximal=q=1N0χqEq0ejβqL.
(3)
with βq the propagation constant of the eigen-mode with ordinal number q.

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
Fig. 5 Unit cell for a CFB without defects.
while Fig. 6
Fig. 6 |Ex|, |Ey| and |Ez| 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.
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 Ez component.

With the eigen-modes of the unit cell now known, we can determine Eproximalfor any desiredEi. Since the goal is to have the distal field coherently come to focus in the observation plane, Eishould be linearly polarized. So we define the desired Ei 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

7. A. J. Thompson, C. Paterson, M. A. A. Neil, C. Dunsby, and P. M. W. French, “Adaptive phase compensation for ultracompact laser scanning endomicroscopy,” Opt. Lett. 36(9), 1707–1709 (2011). [CrossRef] [PubMed]

]) 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 Ei are shown in Fig. 7
Fig. 7 |Ex| and Ex of the desired Ei.
while its representation as a linear combination of eigen-modes Edistal is shown in Fig. 8
Fig. 8 The top row shows |Ex| and Ex of Edistal while the bottom row shows |Ey| (left) and |Ez| (right).
.

Comparing Fig. 7 with Fig. 8, it is clear that there exists a linear combination Edistal of the CFB’s orthogonal eigen-modes which is a good approximation of our desired fieldEi. Note that since all the unit cell’s eigen-modes possess a small but non-negligible Ez , Edistaldoes too, contrary toEi. Using the knowledge of the eigen-modes and their corresponding propagation constants, we can now propagate Edistal towards the proximal end to determine Eproximal. The resulting proximal field for L = 0.5m, is shown in Fig. 9
Fig. 9 Shown here is the input Eproximal required to have the desired Edistal (shown in Fig. 7) exit an ideal CFB of length L = 0.5m.
. 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.

Looking at Fig. 9 however, we notice that the required proximal field has a small but non-negligible Ez component, which in terms of spatial light modulation is not trivial to generate. To determine how crucial Ez is in the proximal input in order to obtain the desired distal output, we defined ESLM which has the same Ex as shown in Fig. 9 but in which Ez and Ey are 0. To simulate the coupling of ESLMinto the CFB, it is then decomposed into eigen-modes and this new proximal input E˜proximal is propagated towards the distal end of the CFB. The resulting E˜distal is shown in Fig. 10
Fig. 10 Shown here is E˜distal, obtained by taking Eproximal from Fig. 8, setting its Ey and Ez equal to 0 and propagating the resulting E˜proximal back to the distal end.
, and as is clear from comparing Fig. 10 with Fig. 8, Edistaland E˜distal closely match. This is numerically confirmed when the Ex, Ey, Ez components of Edistaland E˜distal are subtracted as shown in Fig. 11
Fig. 11 The differences between the components of Edistaland 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.
. This means that, at the input, a linearly polarized field without Ez can be used as long as it is a good approximation of Ex (or Ey, depending on the polarization of the desired output) of Eproximal 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.

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

18. T. Čižmár and K. Dholakia, “Shaping the light transmission through a multimode optical fibre: complex transformation analysis and applications in biophotonics,” Opt. Express 19(20), 18871–18884 (2011). [CrossRef] [PubMed]

].

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 Eproximal, 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 Eproximal and compared this with the Eproximal 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 Eproximal is shown in Fig. 12
Fig. 12 The Eproximalfor 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 Eproximal of an ideal CFB shown in Fig. 8 due to high Δneff/Δdiameter.
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 Edistal now gone.

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 Ex, Ey and Ez and still have very different propagation constants resulting in a wholly different dependency of Eproximal 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 Eproximal, Ey and Ez are very small compared to Ex as was the case for the ideal CFB and as a result, if we use a proximal input which approximates this Ex well but has an Ey and Ez equal to zero, the resulting Edistal(shown in Fig. 13
Fig. 13 The E˜distalfor a CFB of length L = 0.5m with different core diameters obtained by taking Eproximal from Fig. 12, assuming Ey and Ez are 0, and propagating it back to the distal end, is still a good approximation for the desired output of Fig. 7.
) 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.

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
Fig. 14 The Eproximalfor 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.
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 Ex and Ey 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
Fig. 15 The Eproximal for a CFB of length L = 0.5m with elliptical cores from Fig. 12 aligned along the x-axis has a negligible Ey (since only one eigen-mode per core is excited) as opposed to the Eproximal for randomly oriented elliptical cores.
which shows the required Eproximal 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 Ex (of Fig. 15) as the proximal input to generate a distal field that closely matches the desired distal output. The resulting E˜distal for the CFB with aligned elliptical cores is shown in Fig. 16
Fig. 16 The E˜distalfor a CFB of length L = 0.5m with aligned elliptical cores, is still a good approximation for the desired output of Fig. 7.
and is a good match for the desired output field. Note that in Fig. 14 there are some irregularities in the wave fronts of Ex and Ey (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.

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

19. V. Ramaswamy, W. G. French, and R. D. Standley, “Polarization characteristics of noncircular core single-mode fibers,” Appl. Opt. 17(18), 3014–3017 (1978). [CrossRef] [PubMed]

]. 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
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.
). Looking at Fig. 18
Fig. 18 The Eproximalfor 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.
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 Lc would already be reduced to 0.85mm and increasing the ellipticity to 1.2 would reduce Lc even further by a factor 4 (Fig. 17, inset).

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

8. R. Di Leonardo and S. Bianchi, “Hologram transmission through multi-mode optical fibers,” Opt. Express 19(1), 247–254 (2011). [CrossRef] [PubMed]

] and [9

9. T. Cizmar and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Comm. 3, Article number: 1027 (2012). http://www.nature.com/ncomms/journal/v3/n8/full/ncomms2024.html

] 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

20. J. P. Moore and M. D. Rogge, “Shape sensing using multi-core fiber optic cable and parametric curve solutions,” Opt. Express 20(3), 2967–2973 (2012). [CrossRef] [PubMed]

] 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 Eproximal for such a unit cell with the combined types of fabrication irregularities is shown in Fig. 19
Fig. 19 The Eproximalfor a CFB of length L = 0.5m with combined fabrication irregularities (core area, core ellipticity and lattice irregularities).
. 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
Fig. 20 The Eproximalfor 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.
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.

4. Conclusion

Acknowledgments

This research was funded by Stefaan Heyvaert’s Ph.D. grant of the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT- Vlaanderen). R. Buczynski and I. Kujawa were supported by the project operating within the Foundation for Polish Science Team Programme, co-financed by the European Regional Development Fund, Operational Program Innovative Economy 2007-2013. This work was also supported in part by the FWO, the 7th FP European Network of Excellence on Biophotonics Photonics 4 Life, the MP1205 COST Action, the Methusalem and Hercules foundations and the OZR of the Vrije Universiteit Brussel (VUB). The authors would also like to thank Pierre Wahl and Yannick Lefevre for the discussions concerning optical waveguide eigen-modes.

References and links

1.

J. E. A. Wickham, “Endoscopic surgery,” Br. Med. Bull. 42(3), 221–339 (1986), http://bmb.oxfordjournals.org/content/42/3/221.full.pdf. [PubMed]

2.

S. F. Elahi and T. D. Wang, “Future and advances in endoscopy,” J Biophotonics 4(7-8), 471–481 (2011). [CrossRef] [PubMed]

3.

C. M. Lee, C. J. Engelbrecht, T. D. Soper, F. Helmchen, and E. J. Seibel, “Scanning fiber endoscopy with highly flexible, 1 mm catheterscopes for wide-field, full-color imaging,” J Biophotonics 3(5-6), 385–407 (2010). [CrossRef] [PubMed]

4.

C. L. Hoy, N. J. Durr, P. Chen, W. Piyawattanametha, H. Ra, O. Solgaard, and A. Ben-Yakar, “Miniaturized probe for femtosecond laser microsurgery and two-photon imaging,” Opt. Express 16(13), 9996–10005 (2008). [CrossRef] [PubMed]

5.

H. C. Park, C. Song, M. Kang, Y. Jeong, and K. H. Jeong, “Forward imaging OCT endoscopic catheter based on MEMS lens scanning,” Opt. Lett. 37(13), 2673–2675 (2012). [CrossRef] [PubMed]

6.

Y. Wu, Y. Leng, J. Xi, and X. Li, “Scanning all-fiber-optic endomicroscopy system for 3D nonlinear optical imaging of biological tissues,” Opt. Express 17(10), 7907–7915 (2009). [CrossRef] [PubMed]

7.

A. J. Thompson, C. Paterson, M. A. A. Neil, C. Dunsby, and P. M. W. French, “Adaptive phase compensation for ultracompact laser scanning endomicroscopy,” Opt. Lett. 36(9), 1707–1709 (2011). [CrossRef] [PubMed]

8.

R. Di Leonardo and S. Bianchi, “Hologram transmission through multi-mode optical fibers,” Opt. Express 19(1), 247–254 (2011). [CrossRef] [PubMed]

9.

T. Cizmar and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Comm. 3, Article number: 1027 (2012). http://www.nature.com/ncomms/journal/v3/n8/full/ncomms2024.html

10.

S. Heyvaert, C. Debaes, H. Ottevaere, and H. Thienpont, “Design of a novel multicore optical fibre for imaging and beam delivery in endoscopy,” Proc. SPIE 8429, Optical Modelling and Design II, 84290Q, 84290Q-13 (2012). [CrossRef]

11.

I. Kujawa, R. Buczynski, T. Martynkien, M. Sadowski, D. Pysz, R. Stepien, A. Waddie, and M. R. Taghizadeh, “Multiple defect core photonic crystal fiber with high birefringence induced by squeezed lattice with elliptical holes in soft glass,” Opt. Fiber Technol. 18(4), 220–225 (2012). [CrossRef]

12.

E. Brookner, “Phased-array radars: Past, astounding breakthroughs and future trends,” Microwave J. 51(1), 30 (2008).

13.

D. Lorenc, M. Aranyosiova, R. Buczynski, R. Stepien, I. Bugar, A. Vincze, and D. Velic, “Nonlinear refractive index of multicomponent glasses designed for fabrication of photonic crystal fibers,” Appl. Phys. B 93(2–3), 531–538 (2008). [CrossRef]

14.

Schott website: http://www.schott.com/advanced_optics/english/abbe_datasheets/schott_datasheet_sf6.pdf?highlighted_text=SF6

15.

J. Schindelin, I. Arganda-Carreras, E. Frise, V. Kaynig, M. Longair, T. Pietzsch, S. Preibisch, C. Rueden, S. Saalfeld, B. Schmid, J. Y. Tinevez, D. J. White, V. Hartenstein, K. Eliceiri, P. Tomancak, and A. Cardona, “Fiji: an open-source platform for biological-image analysis,” Nat. Methods 9(7), 676–682 (2012). [CrossRef] [PubMed]

16.

T. W. Ridler and S. Calvard, “Picture thresholding using an iterative selection method,” IEEE Trans. Syst., Man, Cybern. Syst. 8(8), 630–632 (1978).

17.

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4(6), 388–394 (2010). [CrossRef]

18.

T. Čižmár and K. Dholakia, “Shaping the light transmission through a multimode optical fibre: complex transformation analysis and applications in biophotonics,” Opt. Express 19(20), 18871–18884 (2011). [CrossRef] [PubMed]

19.

V. Ramaswamy, W. G. French, and R. D. Standley, “Polarization characteristics of noncircular core single-mode fibers,” Appl. Opt. 17(18), 3014–3017 (1978). [CrossRef] [PubMed]

20.

J. P. Moore and M. D. Rogge, “Shape sensing using multi-core fiber optic cable and parametric curve solutions,” Opt. Express 20(3), 2967–2973 (2012). [CrossRef] [PubMed]

21.

S. Heyvaert, H. Ottevaere, I. Kujawa, R. Buczynski, and H. Thienpont, “Numerical characterization of an ultra-high NA coherent fiberbundle part II: point spread function analysis,” Opt. Express in press (2013).

OCIS Codes
(060.2270) Fiber optics and optical communications : Fiber characterization
(060.2350) Fiber optics and optical communications : Fiber optics imaging
(060.2400) Fiber optics and optical communications : Fiber properties
(170.2150) Medical optics and biotechnology : Endoscopic imaging

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: July 10, 2013
Revised Manuscript: August 23, 2013
Manuscript Accepted: August 27, 2013
Published: September 11, 2013

Virtual Issues
Vol. 8, Iss. 10 Virtual Journal for Biomedical Optics

Citation
Stefaan Heyvaert, Heidi Ottevaere, Ireneusz Kujawa, Ryszard Buczynski, Marc Raes, Herman Terryn, and Hugo Thienpont, "Numerical characterization of an ultra-high NA coherent fiber bundle part I: modal analysis," Opt. Express 21, 21991-22011 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-19-21991


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References

  1. J. E. A. Wickham, “Endoscopic surgery,” Br. Med. Bull.42(3), 221–339 (1986), http://bmb.oxfordjournals.org/content/42/3/221.full.pdf . [PubMed]
  2. S. F. Elahi and T. D. Wang, “Future and advances in endoscopy,” J Biophotonics4(7-8), 471–481 (2011). [CrossRef] [PubMed]
  3. C. M. Lee, C. J. Engelbrecht, T. D. Soper, F. Helmchen, and E. J. Seibel, “Scanning fiber endoscopy with highly flexible, 1 mm catheterscopes for wide-field, full-color imaging,” J Biophotonics3(5-6), 385–407 (2010). [CrossRef] [PubMed]
  4. C. L. Hoy, N. J. Durr, P. Chen, W. Piyawattanametha, H. Ra, O. Solgaard, and A. Ben-Yakar, “Miniaturized probe for femtosecond laser microsurgery and two-photon imaging,” Opt. Express16(13), 9996–10005 (2008). [CrossRef] [PubMed]
  5. H. C. Park, C. Song, M. Kang, Y. Jeong, and K. H. Jeong, “Forward imaging OCT endoscopic catheter based on MEMS lens scanning,” Opt. Lett.37(13), 2673–2675 (2012). [CrossRef] [PubMed]
  6. Y. Wu, Y. Leng, J. Xi, and X. Li, “Scanning all-fiber-optic endomicroscopy system for 3D nonlinear optical imaging of biological tissues,” Opt. Express17(10), 7907–7915 (2009). [CrossRef] [PubMed]
  7. A. J. Thompson, C. Paterson, M. A. A. Neil, C. Dunsby, and P. M. W. French, “Adaptive phase compensation for ultracompact laser scanning endomicroscopy,” Opt. Lett.36(9), 1707–1709 (2011). [CrossRef] [PubMed]
  8. R. Di Leonardo and S. Bianchi, “Hologram transmission through multi-mode optical fibers,” Opt. Express19(1), 247–254 (2011). [CrossRef] [PubMed]
  9. T. Cizmar and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Comm. 3, Article number: 1027 (2012). http://www.nature.com/ncomms/journal/v3/n8/full/ncomms2024.html
  10. S. Heyvaert, C. Debaes, H. Ottevaere, and H. Thienpont, “Design of a novel multicore optical fibre for imaging and beam delivery in endoscopy,” Proc. SPIE 8429, Optical Modelling and DesignII, 84290Q, 84290Q-13 (2012). [CrossRef]
  11. I. Kujawa, R. Buczynski, T. Martynkien, M. Sadowski, D. Pysz, R. Stepien, A. Waddie, and M. R. Taghizadeh, “Multiple defect core photonic crystal fiber with high birefringence induced by squeezed lattice with elliptical holes in soft glass,” Opt. Fiber Technol.18(4), 220–225 (2012). [CrossRef]
  12. E. Brookner, “Phased-array radars: Past, astounding breakthroughs and future trends,” Microwave J.51(1), 30 (2008).
  13. D. Lorenc, M. Aranyosiova, R. Buczynski, R. Stepien, I. Bugar, A. Vincze, and D. Velic, “Nonlinear refractive index of multicomponent glasses designed for fabrication of photonic crystal fibers,” Appl. Phys. B93(2–3), 531–538 (2008). [CrossRef]
  14. Schott website: http://www.schott.com/advanced_optics/english/abbe_datasheets/schott_datasheet_sf6.pdf?highlighted_text=SF6
  15. J. Schindelin, I. Arganda-Carreras, E. Frise, V. Kaynig, M. Longair, T. Pietzsch, S. Preibisch, C. Rueden, S. Saalfeld, B. Schmid, J. Y. Tinevez, D. J. White, V. Hartenstein, K. Eliceiri, P. Tomancak, and A. Cardona, “Fiji: an open-source platform for biological-image analysis,” Nat. Methods9(7), 676–682 (2012). [CrossRef] [PubMed]
  16. T. W. Ridler and S. Calvard, “Picture thresholding using an iterative selection method,” IEEE Trans. Syst., Man, Cybern. Syst.8(8), 630–632 (1978).
  17. T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics4(6), 388–394 (2010). [CrossRef]
  18. T. Čižmár and K. Dholakia, “Shaping the light transmission through a multimode optical fibre: complex transformation analysis and applications in biophotonics,” Opt. Express19(20), 18871–18884 (2011). [CrossRef] [PubMed]
  19. V. Ramaswamy, W. G. French, and R. D. Standley, “Polarization characteristics of noncircular core single-mode fibers,” Appl. Opt.17(18), 3014–3017 (1978). [CrossRef] [PubMed]
  20. J. P. Moore and M. D. Rogge, “Shape sensing using multi-core fiber optic cable and parametric curve solutions,” Opt. Express20(3), 2967–2973 (2012). [CrossRef] [PubMed]
  21. S. Heyvaert, H. Ottevaere, I. Kujawa, R. Buczynski, and H. Thienpont, “Numerical characterization of an ultra-high NA coherent fiberbundle part II: point spread function analysis,” Opt. Express in press (2013).

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