## Abstract

Laser speckle imaging (LSI) techniques provide important functional information about tissue perfusion and mechanical properties. To perform LSI *in vivo*, laser speckle patterns are transmitted via optical fiber bundles incorporated within small-diameter endoscopes. Inter-fiber crosstalk due to mode coupling in fiber bundles can result in erroneous speckle statistics and therefore reduces the accuracy of LSI analysis. In this paper, we investigate the influence of multiple parameters that influence crosstalk between neighboring cores within optical fiber bundles and govern the modulation of transmitted laser speckle patterns. Our results show that in addition to large core-to-core separation, large refractive index contrast between core and cladding material, reduced number of propagating modes and variability in core size are essential parameters for accurate speckle pattern transmission to conduct endoscopic LSI.

©2014 Optical Society of America

## 1. Introduction

Laser speckle [1,2] is a granular intensity pattern that arises from the interference of coherent light scattered from randomly distributed light scattering particles. The dynamic variation of optical phase shifts between different partial waves caused by scatterer motion leads to time-varying speckle intensity fluctuations that can be analyzed to recover physiologic information about the scattering medium [3–7]. For instance, laser speckle imaging (LSI) techniques have been applied for measuring tissue perfusion [3,8] by assessing speckle contrast variations caused by the rapid movements of red blood cells in vasculatures. Our group has applied the laser speckle imaging techniques to evaluate the mechanical properties of tissues by analyzing Brownian motion displacements of light scattering particles [4–7]. To perform LSI in vivo, coherent light is delivered to tissue and the reflected laser speckle patterns are collected and transmitted often using optical fiber bundles incorporated within small diameter endoscopes [4–7,9–11].

Due to their small transverse dimensions and flexibility, optical fiber bundles have been widely used in medical endoscopy [12–16] and other minimally-invasive approaches to permit visualization of coronary arteries or other conduits in human body. In optical fiber bundles, the relatively larger numerical apertures (NA>0.3) compared to optical fibers and high fill density of optical cores are generally desired to permit sufficient light collection and spatial sampling to image internal structures through a small diameter endoscope. However in LSI, light is highly coherent unlike the white light endoscopy [16] and the effects of crosstalk between neighboring cores cannot be neglected [5]. In this case, the high density of cores could introduce strong coupling between adjacent fibers. It has been shown that the inter-core coupling can significantly impact image quality by introducing blurring artifacts and therefore degrade image resolution achieved via fiber bundles used for standard intensity imaging [16–19]. In our earlier work, we have shown the inter-core coupling can severely affect the capability to accurately retrieve laser speckle fluctuations transmitted through fiber bundles [5]. The coupling between fibers is because each fiber of a fiber bundle supports multiple guided modes and the field of these modes can extend into the cladding and overlap with mode fields of surrounding fibers [20]. Such overlapping leads to the coupling between modes of individual fibers and inter-fiber power exchange between adjacent fibers, broadly termed as optical crosstalk. Consequently, the transmitted laser speckles are modulated by the inter-fiber crosstalk in fiber bundles due to mode coupling. During *in vivo* LSI, the movement of the fiber bundle caused by bulk motion of surrounding tissue (for instance, cardiac motion) could cause the dynamic coupling of light between neighboring fiber cores and the time-varying modulation of the transmitted speckle intensities independent of tissue physiology. Thus, optical crosstalk between cores can cause erroneous speckle temporal statistics and therefore reduce the accuracy of LSI analysis [5].

The coupling between the fibers modes of different fibers in fiber bundles has been previously studied based on fiber core size, core spacing, NA and non-uniformity of core sizes [19,21,22] to evaluate the effect of these parameters on the strength of core-to-core coupling. In our previous experimental work, we demonstrated that leached fiber bundles could effectively reduce the crosstalk between cores and could permit measurement of relatively accurate temporal speckle statistics during bundles’ motion because of large core-to-core separation due to manufacturing processes of leached fiber bundles [5]. However the effect of optical mode coupling influenced by a combination of multiple relevant fiber bundle parameters on speckle pattern transmission has not been previously quantified and is highly relevant to the LSI field.

In this paper, we apply coupled mode theory (CMT) [23], which provides an approximate analytical approach to analyze coupling between guided modes of parallel waveguides to investigate the influence of optical crosstalk on laser speckle transmission. The role of multiple fiber bundle parameters such as fiber core size, core spacing, NA and non-uniformity of core sizes on the inter-fiber crosstalk and the modulation of transmitted laser speckles are quantified. Furthermore, we define and recommend a set of fiber bundle parameters to conduct endoscopic LSI that would considerably reduce the modulation of transmitted speckle patterns caused by mode coupling between fiber cores.

## 2. Methods

Optical fiber bundles typically incorporate thousands of hexagonally arranged individual optical fiber cores as shown in Fig. 1(a). The analysis of mode coupling between all of the fiber cores is far too complicated and numerically intensive to be calculated. However a simplified system of 7 parallel fibers (Fig. 1(b)) can be used to model the coupling between the modes of these fibers and the results can be easily extended to an entire fiber bundle [19,21,22]. We demonstrate the influence of multiple fiber bundle parameters by varying core size, core spacing and NA, on the transmission of speckle patterns via a 1m fiber bundle typically used in commercial endoscopes.

CMT [20,23–25] is a common theoretical model used to obtain approximate solutions to the coupling between waveguides of multiple waveguides systems. Compared to the normal mode expansion method [21], in which the field is expanded in terms of normal modes solved from Maxwell’s equations with the boundary conditions of the entire complicated structure, CMT provides a simplified solution in which the electric and magnetic fields, **E** and **H**, are decomposed into the modes of each individual waveguides [20]:

*a*is the complex amplitude of the

_{ν}*ν*

^{th}mode;

**e**

*and*

_{ν}**h**

*are the electric and magnetic components of the normalized mode field of each individual fiber, respectively;*

_{ν}*z*is the propagation distance along the fiber bundle and the summation over

*ν*runs through all modes of all individual fibers. The mode propagation constant,

*β*, is the measure of the phase of the mode

_{ν}*ν*varying with

*z*. The effective refractive index of the mode

*ν*is defined as

*n*

_{eff}=

*β*/

_{ν}*k*, where

*k*= 2π/

*λ*is the wave number and

*λ*is the wavelength. The complex amplitude of modes can be obtained by solving the coupled mode equation [20] which describes how the mode amplitude varies with propagation distance

*z*along with the length of the coupled waveguides:

*ν*and

*μ*. The coupling coefficient is directly related to the degree of overlapping of the mode fields. The coupling coefficient

*κ*along with the difference in mode propagation constant, ∆

_{νμ}*β*=

_{νμ}*β*, governs the coupling strength of the

_{μ}−β_{ν}*μ*

^{th}mode to the

*ν*

^{th}mode. The mode coupling coefficient

*κ*is determined by the overlap coefficient

_{νμ}*c*of the mode fields (

_{νμ}**e**

_{ν},

**h**

_{ν}) and (

**e**

_{μ},

**h**

_{μ}) and the perturbation ${\tilde{\kappa}}_{\nu \mu}$ of the mode

*μ*to the mode

*ν*. Here the element of matrix

**c**is ${c}_{\nu \mu}={\displaystyle \iint ({e}_{\nu}^{\ast}\times {h}_{\mu}+{e}_{\mu}\times {h}_{\nu}^{\ast})\cdot zdxdy}$ and

*c*= 1 for the normalized mode field by definition. The element of matrix $\tilde{\kappa}$is given by${\tilde{\kappa}}_{\nu \mu}=\omega {\displaystyle \iint \Delta {\epsilon}_{\mu}{e}_{\nu}^{\ast}\cdot {e}_{\mu}dxdy}$, where

_{νν}*ω*is the angular frequency of the laser light and ∆

*ε*(

_{μ}*x*,

*y*) =

*ε*(

*x*,

*y*)

*−ε*(

_{μ}*x*,

*y*) is the difference between the dielectric constant of the whole multi-core structure and the dielectric constant of the structure with only the individual fiber supporting the mode

*μ*.

To evaluate the modulation of laser speckle patterns during transmission through optical bundles
due to mode coupling, the laser speckle fields are first numerically generated [2] by calculating the Fourier transform of a complex field with
random phase (Fig. 2(a)). The polarization state of speckles is chosen along with the linear polarization of the
fundamental mode of fibers. The generated speckle fields are then decomposed into the HE, EH, TE
andTM modes of individual fibers. The complex amplitude of each guided fiber mode at z = 0,
*a _{ν}*(0), is given by:

**E**

_{0}is the numerically generated speckle electric field. By solving the Eq. (2) for each propagating mode with the initial value of

*a*(0), the complex amplitude at propagation distance

_{v}*z*can be obtained. The transmitted speckle patterns are then reconstructed by linearly combining the fields of all fiber modes:While the phase term, exp(

*iβ*), is taken into account in Eq. (1), for the purpose of speckle reconstruction in Eq. (4) above, the phase term is ignored. In the absence of mode coupling,

_{ν}z*a*(

_{ν}*z*) remains constant with

*z.*Yet, due to the different

*β*, the phase term of different modes, exp(

_{ν}*iβ*) will vary with propagation distance, causing the field

_{ν}z**E**in Eq. (1) to vary with z. Thus, the phase term, exp(

*iβ*), can modulate the calculated electric field,

_{ν}z**E**, even in the absence of inter-fiber mode coupling. By ignoring the phase term for speckle reconstruction purposes in Eq. (4), we can accurately isolate and determine the influence of inter-fiber mode coupling without the confounding influence of the z-dependent phase variations of different modes on the speckle pattern modulation.

A numerically generated speckle pattern (Fig. 2(a)) is transmitted into a fiber bundle (3 µm core size, 6 µm core spacing and 0.40 NA) and speckle patterns are reconstructed at propagation distances of *z* = 0 and *z* = 1 m (Fig. 2(b) and 2(c)). The randomly generated speckle patterns excite all guided modes of each core to varying degrees. The summation of these mode fields with varying mode amplitudes, *a _{v}*(

*z*), will produce different speckle structures in each core varying with

*z*(Fig. 2(b) and 2(c)). For example, the large speckle in the center of Fig. 2(a) excites all modes of the central core, with the fundamental mode excited most efficiently. Because the intensity distribution of the leading modes (HE, EH, TE, TM), except the HE

_{1µ}(µ = 1, 2,…) modes, have a ring structure appearance, the reconstructed speckle pattern at

*z*= 0 caused by the summation of all excited modes within the central core also demonstrates a ring structure [20]. However, with varying propagation distance, the coupling strength of the higher order modes is much stronger than that of the fundamental mode. Therefore, at z = 1m (Fig. 2(c)) the higher order modes are coupled more strongly from the central core to the surrounding cores, while the fundamental mode (HE

_{11}) is only weakly coupled into the surrounding fibers. As a result, the intensity distribution in the central core is dominated by the fundamental mode (HE

_{11}) and displays a large central peak instead of the ring structure in this case (Fig. 2(c)). We observe strong modulation of the speckle pattern reconstructed at

*z*= 1 m compared with that at

*z*= 0 m as discussed below. The extent of modulation of the transmitted speckles caused by optical crosstalk is quantified by calculating the correlation coefficient,

*C*(

*z*), of the intensity patterns between reconstructed speckle patterns at different positions along the length of bundles and the speckle pattern at z = 0 [26],

*I*(

*x*,

*y*,

*z*) is the intensity of the speckle electric field,$\overline{I}(z)$ and ${\sigma}_{I}(z)$ are the spatial average and the standard deviation of the intensity patterns at different

*z*, respectively, and

*x*,

*y*are the transverse coordinates of the points within the 7 core areas.

## 3. Results and discussion

The range of the fiber bundle parameters of the core size, the core-to-core spacing, the NA and
the variability in core size selected for investigation in this paper are based on the
specifications of typical leached fiber bundles used to conduct LSI [4–7]. We first investigate the
mode coupling between all guided modes in the different individual cores of a 7-core structure
with 3 μm core size, 6 μm core spacing and 0.40 NA. Since each core in this
structure can support 16 guided modes at wavelength 690 nm, the total number of guided fiber
modes of 7 fibers is 112 and therefore the dimensions of the matrix of coupling coefficient
**κ** between all modes are 112 by 112. The matrix, **κ**,
describes the coupling coefficients *κ _{νμ}* between
any two guided modes,

*ν*and

*μ*, of all cores (Eq. (2)). The coupling coefficient,

*κ*, is directly related to the degree of mode field overlap and in part governs the coupling strength between the

_{νμ}*μ*

^{th}and

*ν*

^{th}mode. For an ideal straight fiber, since the propagating modes are orthogonal to each other, the coupling coefficients and resultant coupling between modes within each core are negligible. The coupling coefficients between the non-orthogonal modes of two different cores are determined by the degree of mode field overlap which is in turn directly related to the distance between cores. Thus given their close proximity, the coupling coefficients between modes of the nearest neighboring cores are much higher than those of the non-neighboring cores. As a result, the modes of the central core equally couple with the same modes of all the 6 surrounding cores, whereas in each surrounding cores, mode coupling is strongest between the 3 nearest neighbors. Figure 3(a)-3(f) show that the intensity in each core, given by the summation of squared mode amplitudes over all guided modes in the core, oscillates between the central fiber and the surrounding fibers with propagation distance

*z*. In Fig. 3(a)-3(f), each of the leading modes of the central fiber, such as the HE

_{11}, TE

_{01}, HE

_{21}TM

_{01}, HE

_{31}and HE

_{12}mode, is initially excited at

*z*= 0. The amplitude of the coupled mode over a distance of 1 m (typical length of a medical endoscope) is then calculated from Eq. (2). As the order of excited mode of central fiber increases with Fig. 3(a)-3(f), the coupling distance defined as the oscillation period of intensity along the propagation distance z becomes shorter. When excitation is limited to the fundamental mode of the central fiber, the coupling to the surrounding cores is minimal as shown in Fig. 3(a). While Fig. 3(b)-3(f) shows that when higher order modes (TE

_{01}, HE

_{21}TM

_{01}, HE

_{31}and HE

_{12}) of the central fiber are excited, then strong coupling is observed within the surrounding fibers at short distance <<1m. This is because, for the same order modes in adjacent cores, the difference between the propagation constant ∆

*β*of these modes is 0 and the coupling strength depends purely on the modecoupling coefficient. Therefore, in this case, coupling between fundamental modes which is tightly confined within each core is minimal. However, the fields of higher order modes extend more into the cladding, the overlapping of higher order mode fields is stronger and therefore the coupling between higher order modes of adjacent cores is stronger. For different orders of modes in adjacent identical cores, since a small mismatch in the mode propagation constants could greatly suppress coupling between modes [21], the coupling between different order modes with different propagation constant is extremely weak. Therefore, the crosstalk between adjacent identical cores is dominated by the coupling between higher order modes and reducing the number of guided modes may greatly inhibit the crosstalk between modes.

In order to better understand the effect of different fiber bundle parameters on the coupling efficiency of all modes, we investigate the total intensity coupled from central core to surrounding cores along with propagation distance for varying core sizes, core spacings and NAs as shown in Fig. 4. The coupling between cores for 3 different core sizes (2 µm, 3 µm and 4 µm), core spacings (6 µm, 7 µm and 8 µm), and NAs, (0.20, 0.30 and 0.40) are presented. In Fig. 4(a)-4(c), we find the coupling is stronger with increasing core size because larger cores support larger numbers of higher order modes that are strongly coupled and the overlap of lower order mode is also stronger due to a reduction in the effective spacing between cores. Figure 4(d)-4(f) show that with the core size kept constant, an increase in core-to-core spacing leads to reduced coupling due to the large separation between mode fields. Figure 4(g)-4(i) show that an increase in larger NA indicating larger refractive index contrast between core and cladding material leads to stronger confinement of mode fields and therefore reduces overlapping of modal fields of neighboring fibers.

An additional parameter that influences mode coupling is the non-uniformity or variability of fiber specifications such as fluctuations of core size and irregular core shape. This non-uniformity could introduce the mismatch in propagation constant *β* between the modes of adjacent cores and a small amount of the mismatch could extensively reduce the mode coupling between fibers [21,24]. To demonstrate this effect we select a 7-core structure with the core size of 3 μm, the core spacing of 6 μm and NA = 0.40, which elicited significant speckle modulation as discussed above in Fig. 2. However, by introducing a slight variation of ± 0.03 μm and ± 0.1 μm in core size, the mode coupling is significantly reduced. This reduction can be observed in Fig. 4(k) and 4(l) in which the total intensity transferred from central fiber to surrounding fibers for fiber bundles with same core size and ± 0.03 μm and ± 0.1 μm randomness in core size respectively are shown. Figure 4(k) shows one instance that one mode of central fiber and one mode of a neighboring core have almost the same *β*, so that there is strong coupling between these two modes. Although introducing a small variation in core size between neighboring fibers can significantly reduce inter-core mode coupling (Fig. 4(j)-4(l)), in some cases a non-negligible probability of strong inter-core coupling may exist. In these cases, strong mode coupling is introduced due to because the nearly identical propagation constants of different modes of neighboring cores with different diameters could lead to zero phase mismatch between these modes. If there are larger numbers of modes propagating in each core of a fiber bundle, the possibility of strong inter-core coupling due to occasional match of *β* of two modes in neighboring cores is higher. Therefore, a fiber bundle with few modes propagating in each core is thus preferred because there may be lower probability that a mode of a core has the almost same propagation constant as a mode of a nearby core with a slightly different diameter. The number of modes in each core can be changed by altering NA or core size. However lower NA also reduces the ability of sufficiently confining the mode field within the core area, leading to stronger mode field overlapping and stronger inter-core coupling. A reduction in core size can reduce the number of modes and also leads to increased effective spacing between the mode fields of neighboring cores resulting in weaker mode coupling. Therefore reducing core size could reduce cross talk since it not only reduces the number of guided modes and but also simultaneously impacts the strength of inter-core coupling of the guided modes. The above points when taken together suggest that a fiber bundle with high NA, small core size, large core to core spacing and small variability in core size can sufficiently limit crosstalk to achieve robust speckle transmission.

Figure 5 shows the decorrelation of the transmitted speckle patterns due to core coupling. The ensemble average over 20 speckle realizations of correlation functions for fiber bundles with different core sizes, core spacings and NAs are shown in Fig. 5(a), 5(b) and 5(c) respectively. From the results presented above, we observe that larger core-to-core separation, small core size and large refractive index contrast between core and cladding material are essential to enhance the correlation between transmitted speckle patterns so that it can reliably transmitted speckle patterns. However, small core sizes with large core separation reduce the light collection efficiency and limits spatial resolution for conducting LSI. Since the non-uniformity of core sizes could greatly suppress the core-to-core mode coupling as shown in Fig. 4(j)-4(l), introducing core size variability could help to preserve the large size ratio of core-to-cladding and hence maintain the light collection efficiency and resolution while suppressing mode coupling to conduct LSI. We generated 48 sets of 7-core structure of 3 ± 0.1 μm-core-size, 6 μm-core-spacing, and 0.40-NA (Fig. 6). The value of core sizes are retrieved from numerically generated random numbers with a normal distribution having an average of 3 μm and standard deviation of 0.1 μm. The core spacing and NA is fixed at 6 μm and 0.40 respectively. In over 80% of cases, speckle transmission was robust with minimum speckle modulation over 1m (Fig. 6(a)-6(c)). In the remaining 20% of cases strong coupling was observed between at least one mode of the central fiber and the modes of surrounding fibers (Fig. 6(d)-6(f)). Based on the results shown in Fig. 4, 5 and 6, we find the fiber bundles with 3 ± 0.1 µm core size, 6 µm core spacing and 0.40 NA has moderate crosstalk between fibers and will likely permits robust transmission of laser speckle patterns to conduct LSI on 690 nm.

The studies described here are limited to stationary straight fiber bundles. For bent or twisted fiber bundles, the mode coupling within each core may cause energy transfer between modes [27] and modulate the transmitted speckle patterns. If there is severe bending or twisting, light is no longer confined within the core which can cause speckle modulation. In this case the total intensity along the length of the bend is not constant. However, prior work has shown that bending loss is negligible when the normalized bending radius (given by the radius of curvature of the bend ÷ core radius) ≥10^{3} [24]. Most in vivo endoscopic applications satisfy this condition; for instance for intracoronary applications the normalized bending radius is ~10^{4}. In conjunction with a large NA (> 0.35) which confines the light ray more tightly within the core over relatively short propagation distance (~1 m) typically required for endoscopy, bending losses can be approximated to be negligible [24]. As a result, the total transmitted intensity within each core of fiber bundles with minimum crosstalk remains constant and the bending introduced mode coupling within a core will only redistribute the energy within the modes of each core. Thus, the transmitted speckle intensity within each core area can be spatially averaged to minimize the influence of intra-core mode coupling. In our earlier work, we have experimentally demonstrated that the spatially averaged intensity within each core remains constant during fiber bundle bending over a normalized bend radius ~10^{4}) [5]. During fiber bundles motion, the coupling between cores could dynamically change with time because the motion could change the mode overlapping and mode propagation constants, and hence introduce further modulation due to bending and twisting of fiber bundles [28,29]. In our earlier experimental work [5], we observed that laser speckle patterns are robustly transmitted via leached fiber bundles with small core size, larger core-core spacing and higher NA [5]. Our calculations in this paper corroborate the findings of our prior experimental work by quantifying the effects of various bundle parameters on speckle intensity decorrelation over the length of the bundle. Based on our calculations we find that fiber bundles with small core (~3 µm), large core separation (>6 µm) with an NA of > 0.3 may reliably transmit speckle patterns provided that the variability in core size is at least 1%. This may be reliably accomplished using commercially available leached fiber bundles that allow sufficiently robust speckle pattern transmission to conduct LSI in vivo [5,6].

## 4. Conclusion

Optical fiber bundles have been demonstrated to be a key component to conduct endoscopic laser speckle imaging (LSI) [5,6]. However, the transmitted laser speckles are modulated by inter-fiber coupling reducing the accuracy of speckle temporal statistics. In this study, we apply coupled mode theory and analyze the influence of fiber core size, core spacing, numerical aperture and variations in core size on mode coupling and speckle modulation. The analysis of the speckle intensity autocorrelation of time-resolved speckle frames showed that fiber bundles with small core sizes, larger core spacing and NA and non-uniformity in core sizes may permit reliable speckle transmission to conduct endoscopic LSI. Our results provide solutions and recommendations for the design, selection and optimization of fiber bundles to conduct endoscopic LSI.

## Acknowledgements

This work is supported in part by funding support from Canon USA Inc.

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