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The coupling properties of multi-core fibers are analyzed using the multipole method and coupled mode theory in order to gain insights into the performance of image fibers that are commonly used in flexible endoscopes. It is explained that coherent fiber bundles with high core density are able to transport images because nonuniformity in the pixel size reduces the inter-core coupling that causes crosstalk. The wavelength, average core size and separation, and degree of core size variation determine the strength of coupling between adjacent cores, such that fibers with a smaller core size and separation at longer wavelengths require more nonuniformity in order for reliable image transmission. Guidelines are given for assessing the performance of image fibers in a particular system.
©2007 Optical Society of America
1. Introduction
A fiber bundle contains thousands of high index cores in a common lower index cladding. The core sizes and separations are on the order of a few microns, thus differing from a simple bundle of step-index fibers. The index contrast is increased from that of standard single mode fiber in order to more tightly confine the light and reduce crosstalk between cores. In coherent fiber bundles, or image fibers, the input and the output are spatially correlated such that an image transmitted through the fiber can be reconstructed from either end face in an identical manner. Image fibers are used primarily for confocal and multiphoton endoscopic imaging [1–10] but have also been employed in areas such as optical interconnects [11–13], sensing [14], and optical coherence tomography [15]. In imaging systems, the resolution of the transmitted image is limited by the pixilated nature of the fiber bundle as well as by the degree of coupling between cores.
Applications for image fibers motivate the production of higher resolution or higher information density fibers with smaller cores that are more closely packed. Decreasing the fiber dimensions, however, will increase the strength of coupling or crosstalk between cores, resulting in blurred images and a lower signal to background ratio. Available image fibers manufactured by Fujikura and Sumitomo, for example, already have dimensions well below the acceptable theoretical limit for crosstalk. Numerical simulations of these fibers reveal short coupling lengths relative to the typical length of fiber utilized. That is, the strength of the coupling between adjacent cores is predicted to be such that power will begin to disperse amongst the cores after a propagation distance of much less than a meter. Yet these image degrading effects are not dramatic in practice. A closer look at these fibers reveals a significant degree of nonuniformity in the size and shape of individual pixels or cores. In this paper, we will describe how the introduction of nonuniformities disrupts coupling by creating a mode mismatch between adjacent cores. In certain fiber structures, despite high core density, even a small mismatch is sufficient to cause the cores to de-couple. We further verified this behavior using coupled mode theory, suggesting methods for assessing the performance of a particular image fiber and giving guidelines for choosing an image fiber for a particular wavelength. The authors anticipate that a greater understanding of coupling in multi-core fibers and the effects of nonuniformities on crosstalk will not only motivate the production of more reliable image fibers but also facilitate the more effective use of existing fibers.
2. Methods
The numerical simulations in this paper are based on two image fibers manufactured by Fujikura, distributed by Myriad Fiber Imaging Technology, Inc. The fiber specifications are given in Table 1 and, unless otherwise noted, the values are those given by the distributor. The samples are approximately one foot in length. These two fiber types were chosen because their dimensions are typical of imaging applications and of current research [3, 4, 10]. SEM imaging was performed on each fiber in order to verify their dimensions and the pixel size for the 350S fiber was corrected. The final column in Table 1 provides an estimate for the variation in pixel diameter taken from the standard deviation of elliptical fits to the cores in the SEM images using an image processing program. The nonuniformity in the pixel cores is evidenced by the SEM image of Fig. 1. The degree of nonuniformity in the core shape and size observed in these fiber appears to be typical of other commercially available fiber bundles, such as Sumitomo, and has been noted previously [9, 16–21].
Because the coupling behavior between the thousands of cores of an image fiber would be extremely complicated to model, smaller systems of two and seven cores are examined in detail in order to make informed assumptions about the larger system. The nature of the coupling between cores can be understood by calculating the power or energy distribution in each core at different points along the fiber length. In a multi-core system, the power oscillates between cores as a function of the propagation distance, z. The percentage of power transferred between cores is referred to as the coupling efficiency, related to crosstalk, and the rate of power transfer is described by the coupling or beat length. The coupling length and the coupling efficiency are used to quantify the strength of coupling and depend on the interactions between cores or on the beating of the modes of the fiber. The power can be determined from a normal mode expansion of the field solution in the fiber given a Gaussian single core input field. The method used here employs the multipole method to solve for the fiber modes and follows Ref. [22].
Table 1. Image fiber specifications.
When solving for the modes of multi-core fibers, the mode group with the largest propagation constants contains 2N modes, where N is the number of cores in the system and the factor of two is consistent with two polarization states. The modes in this group will be referred to as the fundamental modes because the distribution of energy within the individual cores for each of the modes is approximately Gaussian with azimuthal symmetry, as characteristic of a fundamental mode. The normal mode expansion is truncated after these 2N lowest order modes since they represent the most significant contribution to the sum due to their large overlap with the Gaussian input field. Our modeling showed that ignoring the higher order modes results in negligible error because the modal amplitudes, aj, become extremely small.
A two-core fiber has nondegenerate mode solutions of even and odd nature for each polarization, analogous behavior to a coupled oscillator system. The energy coupled into each of these modes at the fiber endface, represented by the coefficients aj, will propagate at slightly different speeds along the fiber due to the splitting in the mode effective indices. The resulting effect on the power in each core is the “beating” shown in Fig. 2, where the power in each core, as determined from a normal mode expansion, is plotted over z for two-core fibers with dimensions taken from the two image fiber samples of Table 1. The energy distribution in the cores at different points along the z-axis is included below Fig. 2(a). The input field is an x-polarized Gaussian, Ey = 0, that is centered on the left core with a spot diameter (1/e2 intensity) equal to the radius of the core. As shown in Fig. 2, the coupling efficiency is 100% when the two cores have identical diameters, indicating that all the power coupled into the fiber oscillates back and forth between the two cores. The coupling length is defined in Fig. 2(b) as the distance after which all power, when incident initially on one of the cores, will be coupled into the second core. For the case when the cores are identical, the coupling length for each polarization can be calculated directly from the beat frequency of the modes in the following manner:
Fig. 2. The normalized power in each core of a two-core fiber is plotted as a function of the propagation distance. The two cores have identical diameters. a) FIGH-10-500N, Lc = 0.25 m; b) FIGH-10-350S, Lc = 1.39 cm. λ = 600 nm.
where e and o indicate the even and odd modes. This distance is 0.25 m for fiber 500N and 1.4 cm for fiber 350S. Because practical applications with fiber bundles use lengths much longer than these distances, coupling between two cores should be readily observed in endoscopic systems.
In an image fiber, a single core has not just one but six nearest neighbor cores and the impact of these additional cores is to increase the rate at which power is transferred between the cores. In the plot of Fig. 3, the power in each core of a seven-core fiber is plotted versus the propagation distance when light is incident on the central core. The core dimensions correspond to the FIGH-10-350S fiber and the wavelength is 600 nm. Along the length of the fiber, the system oscillates between a state where all the power is in the central core (the initial state) and a state where one-seventh of the power is in each of the outer cores and the central core. The slightly differing behavior between cores in the outer ring, visible in Fig. 3 at the point where the power in the outer cores is labeled, is a result of the linear polarization of the input field and the six-fold symmetry of the seven-core system. Note that the coupling length, or the distance to the first minimum of the power in the input core, is reduced from that of a two-core system [compare with Fig. 2(b)], providing further evidence that severe crosstalk is predicted to greatly reduce the practical use of these fibers.
Numerical simulations obviously demonstrate a discrepancy between theoretical predictions of strong crosstalk and the experimentally demonstrated successful image transmission of the image fibers in Table 1. Considering the case of only two cores, in order for at least 90% of the propagating power to remain in the initial core, simulations predict that the fiber FIGH-10-500N could be no longer than 10 cm while a length less than 0.5 cm would be necessary for fiber FIGH-10-350S. In addition, the trend demonstrated by Fig. 3 indicates that this length will decrease when more neighboring cores are considered. Typical flexible endoscopes utilize fibers on the order of a meter long; thus, core coupling or crosstalk should produce detrimental effects on the transmitted images. Yet, experimental results with similar sized fibers [8–10] do not show the significant blurring predicted here. This obvious contradiction between demonstrated fact and numerical results indicates that the simulated system does not accurately represent the fiber used in the lab. As is evident in the images of Fig. 1, available image fibers are not composed of identical cores. Numerical simulations lead to drastically different conclusions when nonuniformity is introduced into the cross-sections of the simulated multi-core fibers [22, 23].
Fig. 3. The power in each core of a seven-core FIGH-10-350S fiber when the input is in the central core, λ = 600 nm. The three unique energy distributions of the fourteen fundamental modes of the sevencore fiber are shown on the right, along with an illustration of the fiber endface.
3. Numerical analysis
Although both the shape and size of the image fiber cores are quite irregular, only variations in the diameters of adjacent circular cores will be examined here. The effects of other types of nonuniformity, such as core ellipticity and irregular core separation will be briefly discussed later in the paper. In order to numerically study random variations in the core diameter, each core in a multi-core fiber is randomly assigned a diameter from a Gaussian distribution with an average value of do (taken from Table 1) and a standard deviation of δd. The degree of nonuniformity in the fiber is quantified by the ratio of the standard deviation to the mean of this distribution (δd/do) times 100, referred to as a percentage of variation. If the average value used for the diameter is 2.0 μm, a variation of 1% indicates a standard deviation of 20 nm. The core separation is the average value from Table 1.
Returning first to the simple system of only two cores, asymmetry resulting from random variations in the core diameters will alter the coupling properties of a fiber differently depending on the average size of the cores, the wavelength, and the degree of nonuniformity. In general, both the coupling efficiency and the coupling length decrease in magnitude from their value when the cores are identical. Figure 4 shows an example of how the power in each core is altered from that of Fig. 2(b) when the core diameters of a FIGH-10-350S two-core fiber are no longer equal. Both types of fibers were analyzed at three different wavelengths, 600 nm, 980 nm and 1300 nm (material dispersion was neglected), for several degrees of variation. Because 100% power transfer may not occur, the coupling length is defined as the distance to the first power minimum of the incident core and the efficiency is the maximum power transferred out of the incident core. In Fig. 5, the average coupling efficiency is shown to decrease as the degree of variation increases and as the wavelength decreases for both types of fibers at three different wavelengths. Each marker represents the average efficiency for a set of 99 two-core fibers whose core diameters were randomly assigned from a Gaussian distribution of values. Notice the FIGH-10-500N image fiber is more sensitive to nonuniformity; smaller variations produce a more drastic reduction in the coupling efficiency for the FIGH-10-500N fiber as compared to the FIGH-10-350S type. When the wavelength is 600 nm, a variation of 1% in a FIGH-10-500N fiber results in an average efficiency of approximately 2e-3, indicating essentially independent core propagation or greatly reduced crosstalk. A variation of at least 10% would be needed in the FIGH-10-350S type fiber in order to obtain this same condition. If the coupling efficiency is small, very little power oscillates between the cores of a fiber and the coupling length ceases to be a relevant estimate for crosstalk or the strength of coupling. The introduction of nonuniformities in core size can therefore reduce the crosstalk in cores that would otherwise be strongly coupled. The conclusions made from studying two-core fibers are applicable to multi-core fibers because they correctly predict the response of larger seven-core systems when the core diameters are no longer equal. For a situation where the diameters of seven cores have been randomly selected with a percentage variation of 2% and one outer core happens to have a diameter nearly identical to that of the central core, the power evolution in all seven cores is shown in Fig. 6(a). The potentially complicated behavior of a seven-core system is essentially reduced to that of a two-core system. The two cores with almost identical diameters—the difference is less than a tenth of a nanometer—transfer power in a manner similar to the two-core system of Fig. 2, while the remaining cores exhibit behavior similar to Fig 4, participating very little in the power exchange. Additional simulations reveal that when the diameter of a second core in the outer ring is forced to match that of the already participating outer core, the system behaves like the respective three-core system, depending on the orientation of the three similar cores, either in a line, Fig. 6(b), or a triangle, Fig. 6(c).
Fig. 4. The normalized power is plotted versus propagation distance for a FIGH-10-350S two-core fiber with a percentage variation of 1%. λ = 600 nm. The difference in the diameters of the two cores is 16 nm.
Fig. 5. The average efficiency for data sets of 99 two-core fibers is plotted versus the percentage variation for a) FIGH-10-350S and b) FIGH-10-500N at three different wavelengths. The lines have been added to more clearly show the trends in the data. Note that the horizontal scales differ.
Fig. 6. Power in a seven-core system when a) one of the outer cores has a similar radius to the central core and when b) and c) two outer cores have a similar diameter to the central core. The insets illustrate which cores are active in the coupling.
Fig. 7. The average coupling efficiency of the central core is plotted versus the percentage variation for λ = 600nm. Each marker represents an average over thirty structures and the fiber dimensions are those of FIGH-10-350S.
The average coupling behavior of a seven-core system with randomly varying core diameters is similar to that of the two-core fibers in Fig. 5. When the diameters of the cores are no longer identical, the maximum amount of power remaining in the central core is, in general, increased. In Fig. 7, the average coupling efficiency of the central core is plotted versus the percentage variation for type FIGH-10-350S seven-core fibers at λ = 600 nm. Notice that the average efficiency in Fig. 7, for each degree of variation shown, is approximately six times the corresponding value in Fig. 5(a) for the same wavelength. Figure 5(a), therefore, provides, in the weak coupling regime, an average efficiency for each two-core interaction in a many core system. When a single core is surrounded by six cores, and on average the interactions with each of these cores have a coupling efficiency of 10%, then approximately 60% of the power will oscillate back and forth between the center core and the outer cores while the remaining 40% will be retained in the central core. Obviously, this argument is no longer relevant when the coupling is strong; for fibers with strongly coupled cores, the coupling behavior is similar to that of the symmetric structure in Fig. 3. Thus, as in the two-core system, a large percentage of variation will increase the power that remains in the input core; however, this amount will be decreased by the interactions with each neighboring core.
A more practical way to view the data in Figs. 5 and 7 is as a function of the diameter difference between the input core and the other cores in the fiber. In this way, the coupling properties of two cores can be related directly to their size mismatch. These relationships are plotted in Fig. 8 for the same fibers used for the data in Fig. 5. Each marker represents a different randomly generated fiber, including all the degrees of variation. As the difference between the diameter of the input core and that of the other core, or Δd, increases, both the coupling length and the efficiency decrease. This decrease occurs more or less quickly depending on the wavelength and the average core size. For example, the efficiency decreases more dramatically at shorter wavelengths and for fiber type FIGH-10-500N, indicating that sensitivity to nonuniformity increases as the wavelength decreases and as the average core size increases. In addition, the coupling length is nearly wavelength independent when the mismatch is large. The spread in the data is due to the fact that the diameters of both cores are varied randomly about an average value; therefore, each Δd does not refer to a unique structure and can be associated with a different coupling length and coupling efficiency.
Fig. 8. The coupling efficiency a),c) and the coupling length b),d) are plotted versus the difference in diameters of the two cores for two fiber types FIGH-10-350S a),b) and FIGH-10-500N c),d). For a percentage variation of 2%, the Δd for a two-core fiber at the standard deviation of the distribution would be 0.04 μm for FIGH-10-350S and 0.058 μm for FIGH-10-500N.
Fig. 9. The coupling efficiency, a), and the coupling length, b), for the outer cores of a seven-core system plotted versus the difference between the diameter of an outer core and that of the central core. λ = 600 nm and fiber dimensions match those of FIGH-10-350S. Only fibers with a percentage variation up to 4% are included.
Similar trends appear when the efficiency and the coupling length of each of the outer cores of a seven-core fiber are plotted versus their difference in diameter from the central core, as shown in Fig. 9 for the 350S fiber at a wavelength of 600 nm. For these fibers, the central core is kept at d0, the average value, and the diameters of the outer cores are randomly assigned according to a percentage variation of up to 4%, thus reducing the spread in the data points from the two-core data. The close resemblance of Figs. 8(a) and 8(b) to Figs. 9(a) and 9(b) indicate that the behavior of a seven-core fiber can be predicted by studying two-core fibers. Provided an estimate can be made for Δd in a particular fiber, understanding these relationships could lead to a method for predicting the coupling behavior of image fibers with variations.
4. Coupled mode theory
Coupled mode theory (CMT) is commonly used for obtaining approximate analytical solutions to systems of coupled waveguides, such as multi-core fibers. In CMT, each core is solved for independently and interactions with neighboring cores are treated as a perturbation. The total field solution of a system of two waveguides, label them a and b, is approximated as a linear combination of the mode fields of the individual waveguides [24, 25], as shown in Eq. (2).
The power as a function of z is calculated from Eq. (2) in the standard way as shown in Eq. (3).
The amplitude coefficients of the modes of the two cores obey the coupled amplitude equations of Eq. (4) as derived from the general reciprocity relation [24, 26].
In the weak coupling regime, or conventional CMT, the cross field power (the terms in Eq. (3) containing the cross field integrals, Cab and Cba) is ignored. Although this is strictly only true when Kab = Kba * and the cores are identical, this approximation is valid when Cab and Cba are very small and Kab ≈ Kba *. In this case, βa and βb are the propagation constants for the individual, independent waveguides and the coupling coefficients, Kba and Kab, are proportional to the overlap integral of the mode fields of the individual waveguides in each core [24, 27]. Improved CMT considers the situation where Kab and Kba are very different, therefore second order terms are added to the constants in Eq. (4) and the cross field power is included in order to conserve power [24, 27]. There has been much discussion over the last two decades concerning the limitations of conventional CMT and the regimes for which ICMT is necessary [27–29]. At this time, the authors continue to ignore the cross field power as the simplicity of conventional CMT has its advantages. The validity of these assumptions will be assessed later in the paper.
Solving the system of equations in Eq. (4) and applying the initial condition that light is incident on waveguide a at z = 0, or a(0) = 1 and b(0) = 0, produces the expressions for the power given in Eq. (5) [27]. The notation used here is that of Snyder [27] and differs from that used by the authors in previous work [22] because the definitions of the constants in Eq. (5) are better suited to the arguments presented in this paper.
The maximum power transferred out of waveguide a, or the coupling efficiency, is F 2 and βd is related to the beat frequency or the rate of power transfer. The coupling length is the distance, Lc, at which the power in waveguide b has oscillated to its first maximum, ie. βd/Lc =π/2 . As the mismatch between the modes of the individual cores, βa–βb or Δβ, increases, these relationships also dictate that the coupling efficiency and the coupling length will decrease, as observed previously.
The expressions in Eq. (5) can now be applied to the data presented in Fig. 8. Specifically, the trend in the coupling efficiency in Figs. 8(a) and 8(c) should be described by F 2, or
and the coupling length is π/(2βd) or
The mismatch, or Δβ, is then considered to be a function of Δd and the product KabKba is treated as a constant which is typically a valid assumption [30]. Because Δβ is determined from the propagation constants of the modes of the individual waveguides, this value can be approximated by calculating β over a range of core diameters for a standard step index fiber [31, 32]. For the degree of variation considered here, β is an approximately linear function of diameter, and Δβ can be written as the slope times Δd. Equations (6) and (7) are then rewritten as the following:
where the constant a is equal to slope2/(4Kab Kba) and b is the coupling length for the system when the two cores are identical. The fitting of Eq. (8) to the data in Fig. 8 is shown for fiber FIGH-10-500N in Fig. 10. Table 2 displays the product Kab Kba as solved for from the fit parameter a for the four plots in Fig. 8. The values for Kab Kba derived from the two fit parameters in the coupling length plots are approximately equal; therefore, only Kab Kba as derived from the a parameter are shown in Table 2 for Figs. 8(b) and 8(d). The final column of Table 2 gives the value for Kab Kba that is calculated from a two-core system with identical cores by setting Eq. (1) equal to Eq. (7) for Δβ = 0 and solving for Kab Kba.
The values for Kab Kba in Table 2 are very similar demonstrating that simple conventional CMT is adequate for predicting the coupling behavior for image fibers with core diameter mismatch and that the product Kab Kba is essentially constant for the degree of nonuniformity studied here. In addition, these fits are consistent for the long wavelength data as well, indicating that the expressions in Eq. (6) and Eq. (7), derived using conventional CMT, retain accuracy even in a stronger coupling regime. Therefore, these equations can be used for predicting the coupling behavior of nonuniform two-core fibers, which we have shown are a good model for multi-core fibers such as fiber bundles.
Fig. 10. Data from Fig. 8(c) and 8(d) replotted and fitted from Eq. (8) for fiber FIGH-10-500N. The coupling efficiency is plotted in a) while the coupling length is plotted in b).
Table 2. Values for Kab Kba based on the fits of data in Fig. 8, solved for from the parameter a.
The quality of images transmitted by an image fiber depends on the amount of crosstalk between cores, or the coupling efficiency; however, because crosstalk can be reduced through the introduction of mismatch between adjacent cores, the performance of a particular fiber depends on the amount of nonuniformity, the wavelength, and the average core size. In order to better understand the interplay of these different parameters and the sensitivity of a particular fiber to core diameter mismatch we compare the two parameters that determine the efficiency, Kab Kba and μβ 2/4, see Eq. (6). The relative magnitude of these values determines how sensitive a system is to asymmetry, and also the average amount of crosstalk in a fiber. Because Kab Kba is essentially constant over variations of approximately 10%, it is therefore a property of the symmetric system and can be calculated reliably from Eq. (1) for any two-core fiber. The parameter Δβ can be estimated from the degree of nonuniformity in a particular image fiber. If a particular two-core fiber is weakly coupled with a small value for Kab Kba, a small Δβ caused from nonuniformities can significantly decrease the coupling efficiency; this fiber would be considered as very sensitive. In >Fig. 11, these two parameters are plotted versus wavelength for the fiber type FIGH-10-350S in Fig. 11(a), and type FIGH-10-500N in Fig. 11(b). The solid lines indicate Kab Kba, as calculated from Eq. (1). The non-solid lines are Δβ 2/4 for different degrees of asymmetry, where Δβ = βb (λ)-λa (λ) . For simplicity, the core labeled, a, maintained a diameter of d0, the average core diameter given in Table 1, while the diameter of core b was increased by 1%, 4% and 10%. The propagation constant, β, was calculated over wavelength for each core separately and subtracted in order to obtain Δβ.
Fig. 11. Kab Kba is plotted with a solid line and Δβ 2/4, as defined in the text, is represented by non-solid lines for different Δd. The dotted lines are for a diameter difference of 1%, the dashed 4%, and dot-dashed 10%. Fiber FIGH-10-350S is shown in (a) while FIGH-10-500N is in (b). When the wavelength and value for Δβ fall in the gray shaded region, the two cores will have less than 1.67% coupling efficiency.
The utility of these plots can be understood by examining, for example, the point where a solid line crosses a non-sold line. At the wavelength of this crossing point, the predicted efficiency, from Eq. (6), will be 50%. Above this wavelength, the efficiency will increase and at lower wavelengths it will decrease. As the wavelength and Kab Kba decrease, a fiber becomes more sensitive to nonuniformities, approaching a region of independent core propagation. For example, when the wavelength and value for Δβ fall in the gray shaded region, the two cores will have less than 1.67% coupling efficiency. For the opposite trend in wavelength and Kab Kba, a fiber enters a strongly coupled regime where reducing crosstalk becomes more and more difficult. Notice also that fiber type FIGH-10-500N has a larger wavelength range with low efficiency coupling demonstrating again that this fiber is more sensitive to nonuniformity. Due to the scale invariance of Maxwell’s equations, the results in Fig. 11 can be applied to situations not within the range shown and to fibers of different core sizes, if the wavelength and fiber dimensions are scaled appropriately. These plots can also be generated for any two-core fiber given a step-index fiber mode solver and the average coupling length when the cores are identical.
5. Discussion
The dramatic impact of nonuniformity on the coupling properties of multi-core fibers can be understood more intuitively by examining the changes that occur in the modal fields when variation in core size is present. When the core diameters of a seven-core fiber differ significantly, the modes of the multi-core system decouple into those of the individual cores. The existence of localized modes in a disordered waveguide system such as a fiber bundle has also been observed by others [33]. Figure 12 shows the seven unique energy distributions of the fourteen nondegenerate modes of a decoupled seven-core system, which can be compared to the modes of the identical core fiber shown in Fig. 3. Light incident on a single core of the fiber in Fig. 12 will couple almost completely into just one mode of the system and therefore not mix with other cores or modes along the length of the fiber. The field expansion will contain essentially one mode and the energy distribution is no longer a strong function of z. The cores of this image fiber behave independently of their neighbors because differing adjacent core diameters have created a mode mismatch that will inhibit coupling.
Fig.12. The decoupled modes of a seven-core system with a percentage variation in core diameters of 4% and dimensions of FIGH-10-350S, λ = 600 nm.
In addition to variations in size, the cores of image fibers also vary in shape and separation. The impact of these nonuniformities can be estimated to be less than that of variation in core size. The increase in Δβ caused by core ellipticity can be estimated from studies on the geometric birefringence of elliptical core fibers [34–36]. For example, if the variation in core diameter is 10%, a maximum birefringence can be estimated by defining the major and minor axes of an elliptical core as 10% more and 10% less than the average core diameter, resulting in an ellipticity of approximately 0.182. This shape distortion would cause a Δβ on the order of 10-4 [34–36]. Comparing the square of this value to the non-solid lines in Fig. 11, it is evident that ignoring the effect of nonuniformity in core shape is appropriate due to the relatively small size of this parameter compared to the Δβ caused by variation in the core diameters. Ellipticity results in a minimal amount of mode mismatch because the overall core area is not altered as dramatically as when the core radius is changed; the propagation constant is effected more by a change in the modal area than by a change in the modal shape. Irregularity in the core spacing will affect the coupling length rather than the coupling efficiency when the core diameters are the same. For non-identical cores, changing the separation will make the two-core system more or less sensitive to core diameter mismatch, depending on whether the cores become closer or farther apart. Because an increase in the core separation has the opposite consequence from decreasing the core separation, the net effect is expected to be small. Core diameter mismatch is, therefore, the dominant effect in these multi-core fibers.
Our numerical and analytical assessments of multi-core fibers provide a method for analyzing the crosstalk in a particular fiber bundle. The plots of Fig. 11, based on the parameters in Eq. (6), can be used to estimate the coupling efficiency of two core interactions given a certain degree of nonuniformity. Based on arguments present in this paper, this twocore analysis can be applied to cores with six neighboring cores. In this manner, the reliability of a pixel in an image fiber can be estimated give a certain acceptable amount of crosstalk and an average degree of nonuniformity. For example, if the pixel size of the FIGH-10-350S fiber type is preferred for an experiment, yet a coupling efficiency of no more than 0.1 can be tolerated at λ = 780 nm—in other words, 90% of the signal power is expected to be retained in each core—the average coupling efficiency for each core pair should then be approximately 0.0167, when six near neighbor interactions are considered. The value for Δβ that would produce this efficiency would need to be at least sixty times larger than Kab Kba (using Eq. (6) ); the shaded region in Fig. 11 indicates the values for Δβ that satisfy this condition. Referring to Fig. 11, a diameter difference of at least 4% would result in a difference of this order of magnitude. Since this type of image fiber has approximately a 10% variation in core diameter, as measured by SEM (Table 1), and given a Gaussian distribution, there is a 70% chance that two cores will have a diameter difference of exactly 4% or higher. Therefore, approximately 30% of the pixels will be limited by crosstalk given this situation and tolerance. Due to the higher sensitively of the FIGH-10-500N fiber, better performance would be anticipated given the same efficiency restrictions. This example illustrates the fact that the coupling will typically not be reduced below a desired limit in one hundred percent of the cores in an image fiber; some degree of error is inevitable. Table 3 provides an estimate of the pixel accuracy of the two fibers studied given two different values for the total percentage of power coupled out of the input core that can be tolerated in an experiment. The estimates also assume the fiber exhibits 10% variation in the core size and that the wavelength is 780 nm. A percentage of 95% indicates that around 500 of the total ten thousand cores will exhibit stronger coupling than desired. In fact, strong core coupling in these image fibers has been demonstrated experimentally and is described elsewhere [37]. These effects will be manifested by an overall reduction in the image resolution and the reliability of each pixel value.
Table 3. Pixel accuracy based on 10% variation in pixel diameter.
6. Conclusion
Theory and simulation indicate that due to the small core size and separation of current image fibers, crosstalk between pixels should make the transmission of images through flexible endoscopes severely blurred. However, nonuniformity in the core size is shown to reduce the efficiency of the coupling between adjacent cores, substantially suppressing this crosstalk. The manufacture of functional image fibers with closer and smaller pixels requires the pixels to be nonuniform in order to enhance mismatch between neighboring cores. The degree to which two cores will interact or couple depends on the wavelength, the degree of variation, and the average core size. Coupled mode theory was used to generate expressions that can predict the amount of nonuniformity that will reduce crosstalk to acceptable levels in addition to assessing the pixel reliability of available image fibers.
Acknowledgments
The authors would like to acknowledge the Cornell Nanofabrication Facility (CNF) staff for assisting with SEM images, Jim McDonald at Myriad Fiber Imaging Tech., Inc., for the fiber samples and information about the fibers, Larry Chen and other members of the Xu group for support and assistance, and finally the University of Sydney through its ARC Centre of Excellence - Ultrahigh-bandwidth Devices for Optical Systems, School of Physics as the exclusive Licensor of the CUDOS MOF UTILITIES Software used by the authors as a mode solver for multi-core fibers.
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