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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 4, Iss. 2 — Feb. 10, 2009
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Experimental and theoretical analysis of core-to-core coupling on fiber bundle imaging

Xianpei Chen, Kristen Lantz Reichenbach, and Chris Xu  »View Author Affiliations


Optics Express, Vol. 16, Issue 26, pp. 21598-21607 (2008)
http://dx.doi.org/10.1364/OE.16.021598


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Abstract

Flexible endoscopes commonly use coherent fiber bundles with high core density to facilitate in vivo imaging. Small, closely spaced cores are desired for achieving a high number of resolvable pixels in a small diameter fiber bundle. On the other hand, closely spaced cores potentially lead to strong core-to-core coupling. Based on numerical simulations, it was previously explained that image fiber bundles can successfully transmit images because of nonuniformities in the core size that reduce coupling. In this paper, we show numerically and experimentally that, due to the randomness of the structural nonuniformity, significant core-to-core coupling still exists in fiber bundles that are routinely used for imaging. The coupling is highly dependent on the illumination wavelength and polarization state. We further show that the resolution achievable by a fiber bundle depends not only on the core density, but also on the inter-core coupling strength. Finally, we propose that increasing the core-cladding index contrast is a promising approach to achieve a fiber bundle with low core coupling, high core density, and effectively single moded propagation in individual cores.

© 2008 Optical Society of America

1. Introduction

Coherent fiber bundles or image fibers are a common component of both conventional medical endoscopes and endoscopes with research applications involving one-photon confocal and multiphoton microscopy [1

1. A. F. Gmitro and D. Aziz, “Confocal microscopy through a fiber-optic imaging bundle,” Opt. Lett. 18, 565 (1993). [CrossRef] [PubMed]

8

8. B. Flusberg, E. Cocker, W. Piyawattanametha, J. C. Jung, E. Cheung, and M. J. Schnitzer, “Fiber-optic Fluorenscence Imaging,” Nature Methods 2, 941 (2005). [CrossRef] [PubMed]

]. They have also been employed in areas such as optical sensing [9

9. C. Amatore, A. Chovin, P. Garrigue, L. Servant, N. Sojic, S. Szunerits, and L. Thouin, “Remote Fluorescence Imaging of Dynamic Concentration Profiles with Micrometer Resolution Using a Coherent Optical Fiber Bundle,” Anal. Chem. 76, 7202–7210 (2004). [CrossRef] [PubMed]

] and optical coherence tomography [10

10. T. Xie, D. Mukai, S. Guo, M. Brenner, and Z. Chen, “Fiber-optic-bundle-based optical coherence tomography,” Opt. Lett. 30, 1803–1805 (2005). [CrossRef] [PubMed]

]. Through the use of an image fiber, a distal image is transported to the proximal end of a flexible endoscope allowing in vivo observation and measurement of internal tissue or organs. The images have an inherent pixilation due to the individual cores that transmit the information, and the quality of the image is limited by the core sizes and separations. It was generally assumed that image fiber bundles with small closely-spaced cores (i.e., high core density) are desired because they reduce endoscope sizes while retaining a large number of resolvable pixels.

2. Experimental setup and results

The experimental and numerical results presented here are based on two image fibers from Fujikura (distributed by Myriad Fiber Imaging Technology, Inc.), FIGH-10-350S and FIGH-10-500N, hereon referred to as 350S and 500N in Table 1. The fiber parameters are summarized in table 1. These two fiber bundles are chosen in our investigation because they represent a majority of commercially available fiber bundles, and their dimensions are typical of practical use. Both fibers contain ten thousand pixels with core and cladding indices of 1.5 and 1.446 respectively, thus a single core has a numerical aperture (NA) of approximately 0.39. The fiber bundles are approximately one foot in length.

Table 1:. parameters of the Fujikura fiber bundles 500N and 350S

table-icon
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Because variations in the cores of the image fiber are random in nature, strong coupling involving nearly one hundred percent power transfer between adjacent cores is still possible and can be readily observed in the image fibers we tested. The experimental setup is illustrated in Fig. 1. The output of a single-mode fiber coupled tunable diode laser (EOSI 2010, linewidth ~ MHz) was focused onto the fiber bundle. The NA of the focusing objective matched the NA of the image fiber, such that the spot size was approximately the size of an individual core or pixel. The light transmitted through the fiber bundle was imaged onto a CCD camera.

Fig. 1. Experimental set-up: Obj1 is an objective with NA=0.30, Obj2 is a 100x objective. The image inset is an SEM micrograph of a fiber bundle endface.

When a single core of the image fiber is illuminated, the fiber output shows a distribution of power among multiple cores, indicating that inter-core coupling is occurring. We have observed strong intercore coupling in both the 350S and 500N Fujikura image fibers. Similar characteristics were also observed in Sumitomo image fibers (IGN-02-03 and IGN-035-06), although the results with Sumitomo fibers are not shown here.

To examine the wavelength dependence of inter-core coupling, we capture one image for each step change in the illumination wavelength. Three example images taken at different wavelengths are shown in Figs. 2(E)–2(G). These images demonstrate that the power distribution in different cores changes sensitively with the illumination wavelength. Even a fraction of a nanometer change in illumination wavelength can result in totally different output pattern. The fourth image, Fig. 2(H), is taken with a femtosecond Ti:Sapphire laser (Spectra-Physics), with a spectral bandwidth covering the entire tuning range of Figs. 2(E)–2(G). The input power of the pulsed illumination is kept low so that there is no nonlinear optical effects in this experiment. We found that, in general, the images taken with the pulsed source are comparable to the weighted sum of individual images taken using the CW diode laser at wavelengths within the pulse spectral bandwidth.

Experimentally obtained plots of the power versus the wavelength are shown in Figs. 2(C) and 2(D) for the two types of fibers. The power in a particular core was determined from the images by summing over the pixel values in a particular core area and dividing by the sum for the entire image. The period of power oscillations is indicative of the coupling length, which is shorter for the higher density 350S fiber; thus, a smaller change in wavelength results in a more drastic change in the power distribution. Simulated plots that approximately match the periodicity of the experimental data are shown in Figs. 2(A) and 2(B). The core size and separation for these plots are, respectively, 4.2 µm and 2.84 µm for Fig. 2(A), and 2.8 µm and 2.1 µm, respectively, for Fig. 2(B). The fact that these parameters do not match the average values for each type of fiber is not surprising given the amount of variation present in the fiber cross-section.

Our result clearly indicates that one must exercise great caution when illuminating the fiber bundle with a broadbandwith, pulsed source. Because core-to-core coupling depends sensitively on the illumination wavelength, the coupling behavior can change drastically within the spectral bandwidth of a 100-fs pulse, as shown in Fig. 2(H). If a pulsed laser is used as an imaging source, such a core-to-core coupling will not only split the illumination power into a distribution of cores, but also split the pulse within a single core into temporally separated multiple pulses. These findings are particularly important for multiphoton imaging using image fiber bundles.

In order to experimentally assess the polarization dependence, the polarization of the input light is changed by inserting and rotating a half-wave plate in the beam path. Figures 3(A) and 3(B) show, respectively, the simulated and experimental data for fiber type 350S. The experimental data has been fit to a sine-squared function with a 90 degree period. The oscillating power in each of the two cores indicates that the coupling depends on the input polarization state. For the data used in Fig. 3(B), the total power was distributed in several additional cores not shown in the plot; therefore, the total power does not equal to one.

Fig. 2. Numerical (A) and (B), and experimental (C) and (D), results for the power in two cores as a function of wavelength. The numerical results are calculated at a propagation distance of z=0.3 m. Fiber type 500N is shown in (C), while type 350S is shown in (D). Images (E)-(H) are of the transmitted light through the image fiber 350S at different wavelengths; the images are approximately 7 µm square. The illumination source for images (E)-(G) is a CW tunable laser, while in image (H) the source is broadband.

Our observation of the polarization dependence of the core-to-core coupling is of practical importance. In endoscopic applications, fiber bundles are usually twisted and bent when passing through the body cavity in order to reach the remote site. Thus, the polarization state of the input light typically cannot be maintained in a flexible endoscope system, resulting in a dynamic coupling behavior along the fiber, and a changing output pattern. Such a dynamic core-to-core coupling will be difficult to compensate in endoscopic imaging applications unless feedback control mechanisms are employed.

Fig. 3. The numerical (A) and experimental (B) results for the power distribution in two cores of fiber 350S when a half-wave plate is rotated 360 degrees.

We have also examined the inter-core coupling effects on imaging quality. As shown in Fig. 4, transmitted images of a United States Air Force (USAF) target are taken by scanning the CW illumination at the input face of an imaging fiber bundle. The output face of the fiber bundle is held in close proximity to the USAF target. The transmitted light is then collected and detected by a silicon photodiode. To demonstrate the wavelength dependence of inter-core coupling strength, images are taken at three different wavelengths (630, 768 and 978 nm) using both fiber bundles. For the purpose of resolution comparison, images taken without a fiber bundle are also shown in Fig. 5.

Fig. 4. Experimental set-up: Obj is an objective with NA=0.50. Both fibers (Fujikura 500N and 350S) are tested at three different wavelengths 630, 768, and 978 nm

As shown in Fig. 5, the inter-core coupling has a strong impact on the imaging performance of the fiber bundles tested. Inter-core coupling significantly degrades the fiber bundle resolution and image qualities, especially at the longer wavelength tested. It was generally assumed that the resolution of a fiber bundle is defined by its core diameter and core spacing. Because fiber bundle 350S has a smaller core diameter and spacing than 500N, fiber bundle 350S was expected to have a better resolution than fiber 500N. However, our results in Fig. 5 show that fiber 500N can resolve the air force target better than 350S, particularly at the longer illumination wavelengths (978 nm). Our results show that the imaging performance of a fiber bundle is determined not only by the core density, but also by the core-to-core coupling strength. The denser the fiber bundle is, the higher the number of potentially resolvable pixels; on the other hand, high core density can result in strong inter-core coupling, significantly reducing the “effective pixel numbers”.

To quantitatively understand the wavelength dependence of inter-core coupling, we performed simulations, and our numerical results shows that inter-core coupling increases approximately exponentially as the illumination wavelength increases. The relation between inter-core coupling and wavelength is presented in Fig. 6. The inter-core coupling increases at least 8 times from 630 nm to 978 nm at 4 % core-size variation. At short wavelengths, such as 630 nm, the inter-core coupling is relatively weak. The performance of the fiber bundle is thus mostly determined by the core size and spacing, making 350S a better fiber bundle for imaging. At long wavelengths, such as 978nm, the inter-core coupling plays a major role in image quality. Images obtained using fiber bundle 500N actually show better quality than those obtained using 350S. Our findings are particularly important for imaging at long wavelengths, such as multiphoton imaging. Our results show that, depending on the illumination wavelength, a fiber bundle with less core density may in fact outperform a fiber bundle with closely packed cores. The optimal choice of fiber bundles depends on the interplay of the core density and the inter-core coupling.

Fig. 5. Images taken at three different wavelengths. A: Images taken with Fujikura 350S. B: Images taken with Fujikura 500N. C: Images taken without a fiber bundle. Images in the left, middle, and right columns in each panel are taken at 630nm, 768nm, and 978nm, respectively.

3. A potential solution to improve fiber bundle performance

Our experimental results clearly show that inter-core coupling in a fiber bundle has a dramatic impact on its performance, especially at long wavelengths. In order to improve the fiber bundle performance, a fiber bundle with low inter-core coupling and high core density is clearly desired. Increasing the core-to-core distance to reduce the inter-core coupling will inevitably lower the core density and enlarge the fiber size. While the coupling efficiency decreases rapidly with increasing structural variation (Fig. 6), the structure nonuniformity is difficult to control precisely in the fabrication process.

Fig. 6. The average coupling efficiency for data sets of 30 two-core fibers is plotted versus the percentage core-size 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 in the two plots differ.

We propose here a possible new fiber bundle to achieve simultaneously low inter-core coupling and high core density. Through extensive numerical modeling, we found that an increase in core-cladding index contrast will effectively reduce core coupling and extend the application of the fiber bundle to long wavelength range. Numerical simulations based on the multipole method [12

12. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. d. Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002). [CrossRef]

] and the coupled mode theory [13

13. A. Snyder and J. Love, Optical Waveguide Theory (Kluwer, London, 1983).

18

18. E. Marcatili, “Improved coupled-mode equations for dielectric guides,” IEEE J. Quantum Electron. 22, 988–993 (1986). [CrossRef]

] have been performed on a two-core fiber system. Although only two-core fibers were investigated, we have shown previously that a two-core fiber system is an effective representation of the multicore fiber bundle [19

19. K. L. Reichenbach and C. Xu, “Independent core propagation in two-core photonic crystal fibers resulting from structural nonuniformities,” Opt. Express 13, 10336 (2005). [CrossRef] [PubMed]

]. Our simulations show that the index contrast between core and cladding plays an important role in the inter-core coupling within a fiber bundle. By increasing the index contrast (for example, from 3% in existing fiber bundle to 15% in a microstructured fiber), coupling between neighboring cores decreases by several order of magnitude (Fig. 7) at the same level of core nonuniformity. The core spacing in Fig. 7 is significantly smaller than that in existing fiber bundles. In addition, only a small structural variation (~1%) is necessary to ensure essentially non-coupled propagation in such a fiber bundle, making the proposed design feasible in practice. A larger index contrast also leads to tighter mode confinement, thus reducing the core spacing and increasing the core density.

Fig. 7. The average inter-core coupling efficiency for data sets of 30 two-core fibers (pitch 2.5 µm, core diameter 2 µm at wavelength 0.8 µm) is plotted versus the core radius percentage variation at four different index contrasts.

Fig 8. Simulation results of the relation between effective index difference and core-cladding index contrast of a fiber core with a 2.5 µm pitch, 2 µm core diameter at wavelengths from 630nm to 980nm.

The proposed fiber bundle has a more compact dimension, but the light collection efficiency in each core is not reduced when compared to existing commercial fiber bundles. The light collection efficiency is dependent on both fiber core size and its numerical aperture (NA) [21

21. E. Beaurepaire, M. Oheim, and J. Mertz, “Ultra-deep twophoton fluorescence excitation in turbid media,” Opt. Commun. 188, 25–29 (2001). [CrossRef]

]. An increase in core-cladding index contrast increases the NA of the fiber core. For example, Fujikura 350S has an estimated NA of 0.39; while the proposed fiber bundle has an estimated NA of 0.65 for each core at 10% index contrast level. Thus, the light collection efficiency of the proposed fiber bundle can actually be higher than that of existing fiber bundles.

4. Conclusion

In this paper, we experimentally show that inter-core coupling in current image fiber bundles plays a significant role in the fiber bundle performance. Core-to-core coupling depends sensitively on the input wavelengths and polarization states. The performance of an image fiber bundle is determined by its core density and inter-core coupling. For long wavelength imaging applications, our experiments show that a fiber bundle with less core density may in fact outperform a fiber bundle with closely packed cores. Based on numerical simulations, we further propose a new fiber bundle with a large core-cladding index contrast, achieving simultaneously low inter-core coupling and a high core density.

References and links

1.

A. F. Gmitro and D. Aziz, “Confocal microscopy through a fiber-optic imaging bundle,” Opt. Lett. 18, 565 (1993). [CrossRef] [PubMed]

2.

A. F. Gmitro, A. R. Rouse, and A. Kano, “In vivo fluorescence confocal microendoscopy,” in Biomedical Imaging, 2002. Proceedings. 2002 IEEE International Symposium on (2002), pp. 277–280. [CrossRef]

3.

Y. S. Sabharwal, A. R. Rouse, L. Donaldson, M. F. Hopkins, and A. F. Gmitro, “Slit-Scanning Confocal Microendoscope for High-Resolution In Vivo Imaging,” Appl. Opt. 38, 7133–7144 (1999). [CrossRef]

4.

J. Knittel, L. Schnieder, G. Buess, B. Messerschmidt, and T. Possner, “Endoscope-compatible confocal microscope using a gradient index-lens system,” Opt. Commun. 188, 267–273 (2001). [CrossRef]

5.

V. Dubaj, A. Mazzolini, A. Wood, and M. Harris, “Optic fibre bundle contact imaging probe employing a laser scanning confocal microscope,” J. Microsc. 207, 108–117 (2002). [CrossRef] [PubMed]

6.

W. Göbel, J. N. D. Kerr, A. Nimmerjahn, and F. Helmchen, “Miniaturized two-photon microscope based on a flexible coherent fiber bundle and a gradient-index lens objective,” Opt. Lett. 29, 2521–2523 (2004). [CrossRef] [PubMed]

7.

K.-B. Sung, R. Richards-Kortum, M. Follen, A. Malpica, C. Liang, and M. R. Descour, “Fiber optic confocal reflectance microscopy: a new real-time technique to view nuclear morphology in cervical squamous epithelium in vivo,” Opt. Express 11, 3171(2003). [CrossRef] [PubMed]

8.

B. Flusberg, E. Cocker, W. Piyawattanametha, J. C. Jung, E. Cheung, and M. J. Schnitzer, “Fiber-optic Fluorenscence Imaging,” Nature Methods 2, 941 (2005). [CrossRef] [PubMed]

9.

C. Amatore, A. Chovin, P. Garrigue, L. Servant, N. Sojic, S. Szunerits, and L. Thouin, “Remote Fluorescence Imaging of Dynamic Concentration Profiles with Micrometer Resolution Using a Coherent Optical Fiber Bundle,” Anal. Chem. 76, 7202–7210 (2004). [CrossRef] [PubMed]

10.

T. Xie, D. Mukai, S. Guo, M. Brenner, and Z. Chen, “Fiber-optic-bundle-based optical coherence tomography,” Opt. Lett. 30, 1803–1805 (2005). [CrossRef] [PubMed]

11.

K. L. Reichenbach and C. Xu, “Numerical analysis of light propagation in image fibers or coherent fiber bundles,” Opt. Express 15, 2151 (2007). [CrossRef] [PubMed]

12.

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. d. Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002). [CrossRef]

13.

A. Snyder and J. Love, Optical Waveguide Theory (Kluwer, London, 1983).

14.

A. W. Snyder, “Coupled-Mode Theory for Optical Fibers,” J. Opt. Soc. Am. 62, 1267 (1972). [CrossRef]

15.

S.-L. Chuang, “A coupled-mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation,” J. Lightwave Technol. 5, 174–183 (1987). [CrossRef]

16.

J. Fini, “Perturbative numerical modeling of microstructure fibers,” Opt. Express 12, 4535–4545 (2004). [CrossRef] [PubMed]

17.

K. Saitoh, Y. Sato, and M. Koshiba, “Coupling characteristics of dual-core photonic crystal fiber couplers,” Opt. Express 11, 3188–3195 (2003). [CrossRef] [PubMed]

18.

E. Marcatili, “Improved coupled-mode equations for dielectric guides,” IEEE J. Quantum Electron. 22, 988–993 (1986). [CrossRef]

19.

K. L. Reichenbach and C. Xu, “Independent core propagation in two-core photonic crystal fibers resulting from structural nonuniformities,” Opt. Express 13, 10336 (2005). [CrossRef] [PubMed]

20.

X. Feng, T. Monro, P. Petropoulos, V. Finazzi, and D. Hewak, “Solid microstructured optical fiber,” Opt. Express 11, 2225–2230 (2003). [CrossRef] [PubMed]

21.

E. Beaurepaire, M. Oheim, and J. Mertz, “Ultra-deep twophoton fluorescence excitation in turbid media,” Opt. Commun. 188, 25–29 (2001). [CrossRef]

OCIS Codes
(110.2350) Imaging systems : Fiber optics imaging
(170.2150) Medical optics and biotechnology : Endoscopic imaging

ToC Category:
Imaging Systems

History
Original Manuscript: October 21, 2008
Revised Manuscript: December 3, 2008
Manuscript Accepted: December 4, 2008
Published: December 15, 2008

Virtual Issues
Vol. 4, Iss. 2 Virtual Journal for Biomedical Optics

Citation
Xianpei Chen, Kristen L. Reichenbach, and Chris Xu, "Experimental and theoretical analysis of core-to-core coupling on fiber bundle imaging," Opt. Express 16, 21598-21607 (2008)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-26-21598


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References

  1. A. F. Gmitro and D. Aziz, "Confocal microscopy through a fiber-optic imaging bundle," Opt. Lett. 18, 565 (1993). [CrossRef] [PubMed]
  2. A. F. Gmitro, A. R. Rouse, and A. Kano, "In vivo fluorescence confocal microendoscopy," in Biomedical Imaging, 2002. Proceedings. 2002 IEEE International Symposium on (2002), pp. 277-280. [CrossRef]
  3. Y. S. Sabharwal, A. R. Rouse, L. Donaldson, M. F. Hopkins, and A. F. Gmitro, "Slit-Scanning Confocal Microendoscope for High-Resolution In Vivo Imaging," Appl. Opt. 38, 7133-7144 (1999). [CrossRef]
  4. J. Knittel, L. Schnieder, G. Buess, B. Messerschmidt, and T. Possner, "Endoscope-compatible confocal microscope using a gradient index-lens system," Opt. Commun. 188, 267-273 (2001). [CrossRef]
  5. V. Dubaj, A. Mazzolini, A. Wood, and M. Harris, "Optic fibre bundle contact imaging probe employing a laser scanning confocal microscope," J. Microsc. 207, 108-117 (2002). [CrossRef] [PubMed]
  6. W. Göbel, J. N. D. Kerr, A. Nimmerjahn, and F. Helmchen, "Miniaturized two-photon microscope based on a flexible coherent fiber bundle and a gradient-index lens objective," Opt. Lett. 29, 2521-2523 (2004). [CrossRef] [PubMed]
  7. K.-B. Sung, R. Richards-Kortum, M. Follen, A. Malpica, C. Liang, and M. R. Descour, "Fiber optic confocal reflectance microscopy: a new real-time technique to view nuclear morphology in cervical squamous epithelium in vivo," Opt. Express 11, 3171 (2003). [CrossRef] [PubMed]
  8. B. Flusberg, E. Cocker, W. Piyawattanametha, J. C. Jung, E. Cheung, and M. J. Schnitzer, "Fiber-optic Fluorenscence Imaging," Nature Methods 2, 941 (2005). [CrossRef] [PubMed]
  9. C. Amatore, A. Chovin, P. Garrigue, L. Servant, N. Sojic, S. Szunerits, and L. Thouin, "Remote Fluorescence Imaging of Dynamic Concentration Profiles with Micrometer Resolution Using a Coherent Optical Fiber Bundle," Anal. Chem. 76, 7202-7210 (2004). [CrossRef] [PubMed]
  10. T. Xie, D. Mukai, S. Guo, M. Brenner, and Z. Chen, "Fiber-optic-bundle-based optical coherence tomography," Opt. Lett. 30, 1803-1805 (2005). [CrossRef] [PubMed]
  11. K. L. Reichenbach and C. Xu, "Numerical analysis of light propagation in image fibers or coherent fiber bundles," Opt. Express 15, 2151 (2007). [CrossRef] [PubMed]
  12. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. d. Sterke, and L. C. Botten, "Multipole method for microstructured optical fibers. I. Formulation," J. Opt. Soc. Am. B 19, 2322-2330 (2002). [CrossRef]
  13. A. Snyder, and J. Love, Optical Waveguide Theory (Kluwer, London, 1983).
  14. A. W. Snyder, "Coupled-Mode Theory for Optical Fibers," J. Opt. Soc. Am. 62, 1267 (1972). [CrossRef]
  15. S.-L. Chuang, "A coupled-mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation," J. Lightwave Technol. 5, 174-183 (1987). [CrossRef]
  16. J. Fini, "Perturbative numerical modeling of microstructure fibers," Opt. Express 12, 4535-4545 (2004). [CrossRef] [PubMed]
  17. K. Saitoh, Y. Sato, and M. Koshiba, "Coupling characteristics of dual-core photonic crystal fiber couplers," Opt. Express 11, 3188-3195 (2003). [CrossRef] [PubMed]
  18. E. Marcatili, "Improved coupled-mode equations for dielectric guides," IEEE J. Quantum Electron. 22, 988-993 (1986). [CrossRef]
  19. K. L. Reichenbach, and C. Xu, "Independent core propagation in two-core photonic crystal fibers resulting from structural nonuniformities," Opt. Express 13, 10336 (2005). [CrossRef] [PubMed]
  20. X. Feng, T. Monro, P. Petropoulos, V. Finazzi, and D. Hewak, "Solid microstructured optical fiber," Opt. Express 11, 2225-2230 (2003). [CrossRef] [PubMed]
  21. E. Beaurepaire, M. Oheim, and J. Mertz, "Ultra-deep twophoton fluorescence excitation in turbid media," Opt. Commun. 188,25-29 (2001). [CrossRef]

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