In our analytical model, we disregarded the effect of the variations of core radius, cross-talks between adjacent fibers, and polarizations so that optical fibers could be simply viewed as slab waveguides with cylindrical coordinates where the waves were trapped in the core of the fiber by total internal reflection at the core-cladding layer interface. In fact, there are specific ways that the waves travel in optical fiber, referred as modes of the waveguide, as unique transverse field patterns. Therefore, we were able to estimate the effect of multimodal coupling in a single core fiber on OCT imaging (A-mode) based on the fiber bundle imager even though there could have been a fraction of minor errors not including structural deviations, polarization changes by bending, and cross-talk in the real fiber bundle imager. The normalized frequency V was defined as,
was the core radius, λ
the free space wavelength, and n1
the refractive indices of the core and cladding, respectively.
For the modeling and experiment of the imaging fiber, we introduced a couple of different types of imaging bundle fibers. As shown in , types I and II (Edmund Optics: NT53-846 and NT53-840, respectively) which were both rigid conduit fibers and had the same total diameter (3.2 mm) and length (76.2 mm) but with different core sizes (50 μm and 12μm) and slightly different numerical apertures (NA) approximately 0.53~0.55. Type III fiber (Fujikura: FIGH-10-500N, distributed by Myriad Fiber Imaging Tech) was flexible (minimum bending radius: 25 mm) with a length of 30.5 cm and a total diameter of 0.5 mm (image circle diameter: 460 ± 25 μm). It had a smaller NA (0.39) and core spacing (4.5 μm) than types I and II. The image fiber was coated by silicone resin for protection (thickness: 100 ± 10 μm).
Types of fiber bundle optical imager and their parameters.
Representative images showing type I (50μm core spacing) and III (4.5 μm core spacing) fiber bundles have been provided in , respectively. We observed that the fiber pixels were arranged in an irregular grid rather than a strict regular relay and also that the shape of individual fibers varied from a perfectly circular fiber (uncircularity < 5%) - especially significant for flexible bundled fiber (estimated variation based on elliptical fitting to each core: 10 ~ 15 % [21
]). Damaged and contaminated fibers that could have influenced the imaging results were also clearly observable which depended on the fabrication process and the handling (lattice defect < 0.1 %). The structural dimensions of the fiber pixels were measured where the radius is 1.47 ± 0.08 μm and the pixel separation was 4.56 ± 0.28 μm, which closely matched the original fiber design (1.45 μm and 4.50μm).
Sample fiber bundle images: (a) Scanning Electron Microscope (SEM) images for type I fiber bundle (core spacing: 50μm); (b) Digital microscope image for type III fiber bundle (core spacing: 4.5μm).
Using the setup detailed in with a 0.8 μm broadband source (SLD) to mimic the OCT setup instead of employing a single longitudinal mode source, it was possible to selectively couple the beam into a single pixel as shown in (colored spots marked with arrows) where a high-resolution digital microscope (450× ~ 5000×) was utilized to examine the type III fiber bundle, which has been shown in . The transmitted beam can deviate depending on the characteristics of each coupled fiber pixel. In the setup in , the beam waist was carefully formed using various objectives (magnification and numerical aperture) and the beam position was precisely controlled by introducing a 3-D adjustable fiber mount. Here, all the components were stabilized using a single piece (stage) of fiber mount. Thus, it was possible to efficiently select the position of coupled fiber by moving the entrance side of the fiber bundle. A fiber feed-through was placed to be directed to the digital microscope and later to the beam profiler for the measurement. The inter-core coupling due to the narrow pitch between cores and the polarizations in the flexible fiber influenced the overall beam coupling and the image performance.
Setup for fiber bundle characterization test: (a) setup for beam coupling to the single core of fiber bundle imager; (b) sampled results of digital microscope image (Selective beam coupling into single fiber core).
A schematic view of the CPOCT system with an optical fiber bundle probe has been illustrated in [9
]. The OCT system was comprised of a low coherence super luminescence diode (SLD), a 2×1 fiber directional coupler, and a spectrometer. A broadband light source with 5 mW output and a high-resolution spectrometer were used to obtain signal spectrum which was subsequently processed for the depth imaging. A customer configured CCD (charge coupled device)-based spectrometer had a minimum integration time of 3.8 ms, corresponding to an acquisition speed (maximum line scan rate) of 260 Hz. Each CCD pixel was digitized at 1 MHz and the total number of pixels was 3648. The fiber coupler was used to route the SLD output to an X/Y scanner which was used to scan the beam across the proximal entrance of the fiber bundle. Thus, there were no moving parts nor driving/motorized means at the specimen site because the true scanning mechanism was placed at the proximal fiber bundle entrance resulting in a compact imaging probe that could be integrated with medical catheters. One of the outputs from the fiber coupler was placed at the proximal input of the fiber bundle and was attached to the X/Y scanner for the transverse 2-D scanning. A microscope objective was used to couple the light from the coupler into each fiber pixel.
Experimental setup for optical coherence tomography with a fiber bundle imager.
Considering the single fiber dimension to be a perfect circle, refractive indices of core and cladding, and the operating wavelengths, the employed fiber bundles accommodated numerous linearly polarized higher-order multimodes (LP11
, …) as well as a fundamental mode (LP01
) based on their normalized frequencies (V
). The results have been summarized in , categorized by the common operating wavelengths used in OCT (either 1.3μm or 0.8μm). In order to calculate the effective refractive indices of all possible coupled fiber modes and their corresponding fields based on the above basic assumptions and parameters in the tables, we used the following characteristic equation for the optical fibers with a perfect circular core and a step index profile waveguide which was written as [22
Normalized frequencies (V) of fiber bundles.
(·) and Kv
(·) were the Bessel function of the first kind and modified Bessel function of the second kind of order v
, respectively, and the apostrophe (′) indicated the differentiation with respect to the argument of the Bessel or modified Bessel functions, i.e.,
. The two unknown variables
not only satisfied the above Eq. (1)
for the boundary condition at the core-cladding boundary but also had to satisfy,
was the normalized frequency previously defined in Eq. (1)
By numerically solving the Eqs. (1)
, we obtained the solution for LPvm
mode, whose effective refractive index neff
could be expressed as,
was the propagation constant in free space and βvm
was the longitudinal propagation factor for mode LPvm
For instance, the dependence of wavelength on the coupled fiber modes and their corresponding effective refractive indices was plotted in for type III fiber bundle (core diameter 2a = 2.9 μm; n1 = 1.500; n2 = 1.446) in 0.7 μm ~ 0.9 μm region (There were fewer modes for 1.3 μm region due to the reduced V value in the same fiber dimension). In this case, there were only four possible modes (LP01, LP11, LP21, LP02) for most of the wavelength range based on the fiber parameters listed on and except the fifth mode (LP31) which could be generated at shorter wavelength below 0.7 μm due to the increased V parameter as shown in . As expected, the lower order mode showed a refractive index closer to that of the core, and the higher order mode had a refractive index closer to that of the cladding of the fiber. In addition, the coupled axial (z-axis) propagating field of mode LPvm could be written as,
Guided (coupled) effective refractive indices of fiber modes in an imaging bundle (in case of type III fiber).
Thus, the total field and the intensity of beam was obtained as,
The regular step index single mode fiber (SMF) at 0.8 μm showed a perfect circular or Gaussian shape beam profile as in for v
= 0, m
= 1, which contained only the fundamental mode (LP01
). However for a-few-mode fiber including other modes such as LP11
, the perfect circular shape was modified due to these higher order modes included in Eqs. (6)
. The analysis result for the used fiber in at z = 0 (λ = 0.8 μm) has been shown in and the measured mode intensity profile in , where the results were comparable to each other. The measurement were simply taken by 2-D profiling the output beam with the infrared CCD camera (Ophir, model: BeamStar FX) at the end of the used fiber based on the setup in instead of using the digital microscope. Because of the higher order modes, the beam profile changed to an ellipsoidal shape from a circular one with only fundamental mode and during the measurement in a background noise was observable due to the cladding mode in a short step index single core fiber. In the figures, intensity changed from dark red (circle center with high intensity) to dark blue (low circle boundary with low intensity). Here, the exact traditional degenerated modes solved from circular waveguides without employing weakly guided approximation such as HE21
could be grouped as LP11
because the propagation constants were almost identical or too small to be separated during the propagation in the short fiber bundle for OCT imaging.
Fig. 5 Beam profiles of fiber modes: (a) measured result for a single mode fiber (mode includes: LP01); (b) simulations result for a few-mode fiber (dimension: 2a × 2a where a is core radius, modes include LP01, LP11, LP21, and LP02); (c) measured result (more ...)