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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 23 — Nov. 5, 2012
  • pp: 25562–25571
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Simple and reliable light launch from a conventional single-mode fiber into a helical-core fiber through an adiabatically tapered splice

Hyuntai Kim, Jongki Kim, Yongmin Jung, Luis Alonso Vazquez-Zuniga, Seung Jong Lee, Geunchang Choi, Kyunghwan Oh, Pu Wang, W. A. Clarkson, and Yoonchan Jeong  »View Author Affiliations


Optics Express, Vol. 20, Issue 23, pp. 25562-25571 (2012)
http://dx.doi.org/10.1364/OE.20.025562


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Abstract

We propose a simple and efficient light launch scheme for a helical-core fiber (HCF) by using an adiabatically tapered splice technique, through which we overcome its inherent difficulty with light launch owing to the large lateral offset and angular tilt of its core. We experimentally demonstrate single-mode excitation in the HCF in this configuration, which yields the coupling efficiency of around −5.9 dB (26%) for a ~1.1-μm light input when the splice joint is tapered down to 30 μm in diameter. To our knowledge, this is the first proof-of-principle report on the fusion-splice coupling between an HCF and a conventional single-mode fiber.

© 2012 OSA

1. Introduction

With recent advances in high-power fiber lasers, power scaling while maintaining excellent beam quality has become an important issue in a variety of large-mode-area (LMA) fiber lasers [1

1. M. E. Fermann, “Single-mode excitation of multimode fibers with ultrashort pulses,” Opt. Lett. 23(1), 52–54 (1998). [CrossRef] [PubMed]

11

11. L. Dong, X. Peng, and J. Li, “Leakage channel optical fibers with large effective area,” J. Opt. Soc. Am. B 24(8), 1689–1697 (2007). [CrossRef]

]. The LMA ensures a large effective mode area, which eventually increases the pump absorption per unit length and reduces the fiber nonlinear response such as self-phase modulation, Brillouin scattering, and Raman scattering. A straightforward design approach to obtain such an LMA is to decrease the numerical aperture (NA) while increasing the core size. However, we cannot decrease it as demanded because too low an NA can give rise to a substantial bend loss even for the fundamental mode, i.e. LP01 mode. Thus, such LMA fibers with an extremely low NA (say 0.01 or 0.02) must be kept straight as rod-type fibers [6

6. J. Limpert, N. Deguil-Robin, I. Manek-Hönninger, F. Salin, F. Röser, A. Liem, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson, and C. Jakobsen, “High-power rod-type photonic crystal fiber laser,” Opt. Express 13(4), 1055–1058 (2005). [CrossRef] [PubMed]

]. Consequently, in order to maintain relatively robust single-mode guidance in LMA fibers with bending flexibility, a moderately low NA (say 0.06 ~0.08) core design is often preferred [4

4. Y. Jeong, J. K. Sahu, D. N. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express 12(25), 6088–6092 (2004). [CrossRef] [PubMed]

, 5

5. Y. Jeong, A. J. Boyland, J. K. Sahu, S.-H. Chung, J. Nilsson, and D. N. Payne, “Multi-kilowatt single-mode ytterbium-doped large-core fiber laser,” J. Opt. Soc. Kor. 13(4), 416–422 (2009). [CrossRef]

], with which a coiled fiber configuration is then utilized in order to suppress unwanted higher-order modes (HOMs) as the curvature losses of the HOMs are significantly higher than that of the fundamental mode [2

2. J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25(7), 442–444 (2000). [CrossRef] [PubMed]

]. By carefully choosing the optimal bend radius, one can therefore achieve an effective single-mode operation through LMA fibers with a moderately low core NA [4

4. Y. Jeong, J. K. Sahu, D. N. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express 12(25), 6088–6092 (2004). [CrossRef] [PubMed]

, 5

5. Y. Jeong, A. J. Boyland, J. K. Sahu, S.-H. Chung, J. Nilsson, and D. N. Payne, “Multi-kilowatt single-mode ytterbium-doped large-core fiber laser,” J. Opt. Soc. Kor. 13(4), 416–422 (2009). [CrossRef]

]. While a typical coiling diameter for single-mode operation in such LMA fibers in conventional fiber geometries is of the order of a few centimeters, fibers of large cores often require even smaller a bending radius in order to achieve a sufficient level of suppression of HOMs. However such a tight bend radius can lead to fracture or degradation of the fiber, including both silica glass and outer polymer material, thereby affecting the long-term reliability of the device. To alleviate such a stress issue incurred in the coiled fiber, helical-core fibers (HCFs) with inbuilt helical-core trajectory within the cladding region have been introduced and investigated [12

12. D. Marcuse, “Radiation loss of a helically deformed optical fiber,” J. Opt. Soc. Am. 66(10), 1025–1031 (1976). [CrossRef]

21

21. R. D. Birch, “Fabrication and characterisation of circularly birefringent helical fibres,” Electron. Lett. 23(1), 50–52 (1987). [CrossRef]

]. In general an HCF can be fabricated from a preform having an off-centered core that is typically obtained through ultrasonic drilling of a hole in a large silica preform and inserting of a high-index core rod into it, with which the helix is readily formed during the fiber drawing process via spinning the preform [17

17. P. Wang, L. J. Cooper, R. B. Williams, J. K. Sahu, and W. A. Clarkson, “Helical-core ytterbium-doped fibre laser,” Electron. Lett. 40(21), 1325–1326 (2004). [CrossRef]

]. The helical core trajectory within the cladding can create a very small bending diameter (a few millimeters) without incurring substantial stress whereby the effective suppression of HOMs in the multimode core can be realized. Early research activities have been devoted to the modeling and analysis of mode area scaling, effective cut-offs of the guided modes, and radiation from pulses in HCFs [12

12. D. Marcuse, “Radiation loss of a helically deformed optical fiber,” J. Opt. Soc. Am. 66(10), 1025–1031 (1976). [CrossRef]

16

16. Z. Jiang and J. R. Marciante, “Mode-area scaling of helical-core, dual-clad fiber lasers and amplifiers using an improved bend-loss model,” J. Opt. Soc. Am. B 23(10), 2051–2058 (2006). [CrossRef]

]. More recently, helical-core fiber lasers have been demonstrated and their mode area scaling with single-mode lasing performance has also been investigated [17

17. P. Wang, L. J. Cooper, R. B. Williams, J. K. Sahu, and W. A. Clarkson, “Helical-core ytterbium-doped fibre laser,” Electron. Lett. 40(21), 1325–1326 (2004). [CrossRef]

, 18

18. P. Wang, L. J. Cooper, J. K. Sahu, and W. A. Clarkson, “Efficient single-mode operation of a cladding-pumped ytterbium-doped helical-core fiber laser,” Opt. Lett. 31(2), 226–228 (2006). [CrossRef] [PubMed]

].

In addition, the HCFs exhibit a very high circular birefringence of an order of magnitude greater than those of twisted fibers: Beat lengths down to 3 mm (birefringence: ~2.11 × 10−4) have been reported [19

19. J. Qian and C. D. Hussey, “Circular birefringence in helical-core fibre,” Electron. Lett. 22(10), 515–517 (1986). [CrossRef]

21

21. R. D. Birch, “Fabrication and characterisation of circularly birefringent helical fibres,” Electron. Lett. 23(1), 50–52 (1987). [CrossRef]

]. Consequently, fiber modes in HCFs undergo different phase shifts depending on whether they are clockwise-circularly polarized or counter-clockwise-circularly polarized. Intriguingly, such a high birefringence is dependent only on geometrical effects, thereby being relatively uninfluenced by other external disturbances [21

21. R. D. Birch, “Fabrication and characterisation of circularly birefringent helical fibres,” Electron. Lett. 23(1), 50–52 (1987). [CrossRef]

]. This can find cost-effective applications in electric current or magnetic field sensing based on Faraday effects.

In this paper we, thus, propose a simple and reliable light launch scheme for HCFs via using an adiabatically tapered splice (ATS) technique, which eventually resolves out all the difficulties with HCFs that can be encountered in the free-space launch configuration [17

17. P. Wang, L. J. Cooper, R. B. Williams, J. K. Sahu, and W. A. Clarkson, “Helical-core ytterbium-doped fibre laser,” Electron. Lett. 40(21), 1325–1326 (2004). [CrossRef]

, 18

18. P. Wang, L. J. Cooper, J. K. Sahu, and W. A. Clarkson, “Efficient single-mode operation of a cladding-pumped ytterbium-doped helical-core fiber laser,” Opt. Lett. 31(2), 226–228 (2006). [CrossRef] [PubMed]

]. The ATS technique has previously been used to reduce the splice loss between two dissimilar SMFs [23

23. D. B. Mortimore and J. V. Wright, “Low-loss joints between dissimilar fibres by tapering fusion splices,” Electron. Lett. 22(6), 318–319 (1986). [CrossRef]

], to couple optical power from a single-core fiber to a multi-core fiber [24

24. L. Yuan, Z. Liu, J. Yang, and C. Guan, “Bitapered fiber coupling characteristics between single-mode single-core fiber and single-mode multicore fiber,” Appl. Opt. 47(18), 3307–3312 (2008). [CrossRef] [PubMed]

], to realize low-loss 1 × 2 Y-junctions [25

25. N. Healy, E. McDaid, D. G. Murphy, C. D. Hussey, and T. A. Birks, “Low-loss singlemode fibre 1x2 Y-junction,” Electron. Lett. 42(13), 740–742 (2006). [CrossRef]

], and to selectively excite fundamental mode in multimode fibers [26

26. Y. Jung, Y. Jeong, G. Brambilla, and D. J. Richardson, “Adiabatically tapered splice for selective excitation of the fundamental mode in a multimode fiber,” Opt. Lett. 34(15), 2369–2371 (2009). [CrossRef] [PubMed]

]. However, to the best of our knowledge, there has been no report on the application of the ATS technique to the all-fiber light coupling between an SMF and an HCF that bear a large angular tilt in the core axes as well as having a large lateral offset between two dissimilar cores. Here, we demonstrate that successive core-cladding mode conversion through a tapered splice can lead to a significant improvement in power coupling between a conventional SMF and an HCF.

2. Numerical modeling of ATS between an SMF and an HCF

Figure 1
Fig. 1 Schematic diagram of a typical HCF.
shows the schematic diagram of a typical HCF that has a core offset of R and a helix pitch of D where θ denotes the angle measured between the fiber axis and the helix.

As discussed in Ref [12

12. D. Marcuse, “Radiation loss of a helically deformed optical fiber,” J. Opt. Soc. Am. 66(10), 1025–1031 (1976). [CrossRef]

], the optical modes guided by this helically deformed fiber undergo radiation loss according to the curvature of the waveguiding core, which is given by
2α=1cosθπκ2exp[23(γ3βν2)(Rsin2θ)]eνV2γ32Rsin2θKν1(γa)Kν+1(γa)
(1)
Where
κ=n12k02βν2,
γ=βν2n22k02,
V=k0an12n22.
Note that 2α′ is the effective radiation loss coefficient per unit length of the helical core fiber. (Note the fiber length is measured along the longitudinal axis of the fiber.) In addition, K is the modified Bessel function of the second kind, k0 is the propagation constant in vacuum, βν is the propagation constant of the corresponding core mode, a is the radius of the core, n1 and n2 are the refractive indices of the core and cladding, respectively, and eν is given by 2 for ν = 0 (e.g., LP01 mode) or 1 for ν ≠ 0 (e.g., LP11 mode). The resultant radius of curvature of the helix is given by [12

12. D. Marcuse, “Radiation loss of a helically deformed optical fiber,” J. Opt. Soc. Am. 66(10), 1025–1031 (1976). [CrossRef]

]

R=Rsin2θ=D24π2R.
(2)

It should be noted that while the curvature loss depends on various parameters of the fiber geometry and property, it is also crucially influenced by the mode order as shown in Eq. (1). Figure 2
Fig. 2 Radiation loss rates for LP01 mode (solid line) and LP11 mode (dashed line). HCF details: Core diameter = 10 μm, core NA = 0.14, R = 120 μm, D = 4.3 mm, and outer diameter = 300 μm.
, for example, represents the radiation loss rates for LP01 and LP11 modes, respectively, in terms of the helix pitch D for the other fiber parameters that are given in the figure caption. One can see that there is a range of proper helical pitch over which LP01 mode transmits with a negligible loss while LP11 mode experiences substantially higher loss. In fact LP11 mode is the most difficult higher-order mode to discriminate from the fundamental mode, so that if the effective radius of curvature is made small enough for filtering out LP11 mode, all the higher-order modes can be virtually eliminated out. As a result, single-mode guidance is absolutely possible even if the core is in a multimode form. However, HCFs have an inherent difficulty in terms of light coupling as discussed in [17

17. P. Wang, L. J. Cooper, R. B. Williams, J. K. Sahu, and W. A. Clarkson, “Helical-core ytterbium-doped fibre laser,” Electron. Lett. 40(21), 1325–1326 (2004). [CrossRef]

, 18

18. P. Wang, L. J. Cooper, J. K. Sahu, and W. A. Clarkson, “Efficient single-mode operation of a cladding-pumped ytterbium-doped helical-core fiber laser,” Opt. Lett. 31(2), 226–228 (2006). [CrossRef] [PubMed]

], and thus, a simple and efficient scheme for stable light coupling is desirable.

Here we exploit an ATS configuration to provide an efficient light coupling between a conventional SMF and an HCF. The ATS can be realized by fusion-splicing an SMF to an HCF and adiabatically tapering the splice joint. The optical field evolution along the splice taper is shown in Fig. 3
Fig. 3 Schematic of the proposed light launch scheme into a helical core fiber via an ATS. Through the successive mode conversion along the adiabatic taper [LP01core(SMF)LP01taper(SMF)LP01taper(HCF)LP01core(HCF)], the coupling efficiency between the conventional SMF and the HCF can be significantly improved. The images in the lower part represent some typical mode field patterns along the ATS sections.
. Under adiabatic mode transition through the tapered fiber, the fundamental core-mode of the input SMF fiber [LP01core(SMF)] is continuously mode-converted into a guided cladding-mode [LP01taper(SMF)] in the down-taper region [24

24. L. Yuan, Z. Liu, J. Yang, and C. Guan, “Bitapered fiber coupling characteristics between single-mode single-core fiber and single-mode multicore fiber,” Appl. Opt. 47(18), 3307–3312 (2008). [CrossRef] [PubMed]

26

26. Y. Jung, Y. Jeong, G. Brambilla, and D. J. Richardson, “Adiabatically tapered splice for selective excitation of the fundamental mode in a multimode fiber,” Opt. Lett. 34(15), 2369–2371 (2009). [CrossRef] [PubMed]

]. When LP01taper(SMF) mode passes through the splice joint, there occurs a mode transition into a guide cladding-mode in the up-taper region of the HCF [LP01taper(SMF)LP01taper(HCF)] with a negligible transition loss if the mode fields are well matched [24

24. L. Yuan, Z. Liu, J. Yang, and C. Guan, “Bitapered fiber coupling characteristics between single-mode single-core fiber and single-mode multicore fiber,” Appl. Opt. 47(18), 3307–3312 (2008). [CrossRef] [PubMed]

26

26. Y. Jung, Y. Jeong, G. Brambilla, and D. J. Richardson, “Adiabatically tapered splice for selective excitation of the fundamental mode in a multimode fiber,” Opt. Lett. 34(15), 2369–2371 (2009). [CrossRef] [PubMed]

]. Then LP01taper(HCF) mode is again adiabatically converted into the fundamental core-mode of the HCF [LP01core(HCF)] in the up-taper region. It should be noted that the guided modes in the waist region are mainly confined by the cladding/air boundary so that the original core has a negligible effect on their guidance as long as the waist is sufficiently thin [26

26. Y. Jung, Y. Jeong, G. Brambilla, and D. J. Richardson, “Adiabatically tapered splice for selective excitation of the fundamental mode in a multimode fiber,” Opt. Lett. 34(15), 2369–2371 (2009). [CrossRef] [PubMed]

]. If the waist were not thin enough, the mode overlap between LP01taper(SMF) and LP01taper(HCF) would not be high, thereby giving rise to high transition or coupling loss [22

22. K. Shiraishi, T. Yanagi, and S. Kawakami, “Light-propagation characteristics in thermally diffused expanded core fibers,” J. Lightwave Technol. 11(10), 1584–1591 (1993). [CrossRef]

]. In addition, it is noteworthy that while the cladding/air boundary may also guide HOMs, their contributions to power coupling between the SMF and the HCF through the ATS section must be extremely limited because such HOMs would end up having enormous loss in the un-tapered section of the HCF as discussed in Fig. 2. Therefore, one can simply ignore the contributions from HOMs.

In general, if two dissimilar optical fibers are joined together, the resultant coupling efficiency is determined by their normalized mode overlap [22

22. K. Shiraishi, T. Yanagi, and S. Kawakami, “Light-propagation characteristics in thermally diffused expanded core fibers,” J. Lightwave Technol. 11(10), 1584–1591 (1993). [CrossRef]

], e.g. LP01taper(SMF) and LP01taper(HCF) in our case, which can be given by
η=|E1E2cos(βνθx)dxdy|2|E1|2dxdy|E2|2dxdy,
(3)
where Ei(x,y) denotes the electric field in the transverse plane of each fiber and θ is the angle between the two fibers if their longitudinal fiber axes are not parallel.

Let us suppose that an SMF and an HCF are simply spliced in a normal condition, that is, the occasion before tapering the spliced joint. Then, it is clear that the large lateral offset and angular tilt between the SMF mode and the HCF mode would lead to a very small overlap integral value. However, if we taper the spliced joint down, the guidance by the core becomes loose, thereby leading to a significant increase in the mode overlap. Figure 4
Fig. 4 Electric-field intensities of LP01taper(SMF) and LP01taper(HCF) modes with respect to the fiber dimension calculated. OD: outer diameter.
shows how the field intensities of LP01taper(SMF) and LP01taper(HCF) modes evolve as we scale down the fiber dimensions, i.e., we taper the fibers down. The mode fields were obtained via a numerical method based on the finite-element method (FEM: COMSOL Multiphysics®). In particular, one can see that the mode overlap substantially improves as the fibers are tapered down to below 30 μm in the given fiber parameters. Furthermore, it should be noted that as we taper down the SMF-HCF splice joint, the effective angular tilt between the SMF mode and the HCF modes substantially decreases, also contributing to the increase in the mode overlap. Consequently, the ATS technique should work for obtaining an efficient mode coupling between a conventional SMF and an HCF.

Figure 5
Fig. 5 Theoretical coupling efficiency between LP01taper(SMF) mode and LP01taper(HCF) modes based on the normalized mode overlap.
shows the theoretical coupling efficiency between LP01taper(SMF) and LP01taper(HCF) modes obtained by numerically calculating the normalized mode overlap given by Eq. (3) based on the results from Fig. 4 for various outer diameter of the SMF-HCF tapered splice. Here we assumed that the fiber parameters of the SMF and HCF are those of a standard SMF and the HCF defined in Fig. 2, respectively.

3. Experimental result

In our experiments we investigated an in-house fabricated, undoped (passive) HCF having a core diameter of 10 μm with an NA of 0.14 and a cladding diameter of 300 μm. The HCF was coated with a standard high-index polymer. It should be noted that the V parameter of the core is ~4.0 at ~1.1 μm, which indicates that the core supports multiple modes if the helical geometry is not taken into account. The core is laterally offset from the center of the fiber cross-section by 120 μm (R) and the helix pitch (D) is 4.3 mm. It is noteworthy that the fiber parameters for the numerical results shown in Figs. 3-5 are primarily based on this HCF. According to Eq. (1), this HCF would lead to 2.2-dB/m radiation loss for LP01 and 3.5 × 103-dB/m loss for LP11 mode, respectively. This indicates that the transmission of higher-order modes in the given HCF, including LP11 mode, is forbidden in practice. We experimentally characterized the radiation loss by the cutback method and found that the loss of our HCF is ~2.5 dB/m at ~1.1 μm, which is in good agreement with the theoretical value estimated by Eq. (1) as shown in Fig. 2. We attribute the small discrepancy to the excessive core background loss due to the impurities or imperfection of the waveguide. Although it would have been better to choose the helix pitch of around 5 mm in order for reducing the loss of the fundamental mode as negligible as possible, we note that the proof-of-principle demonstration of the light launch through the ATS technique is absolutely possible even with the HCF in the current form.

Here is the procedure to realize ATS between an SMF and our in-house fabricated HCF: At first, to match the cladding diameter of the HCF to the input standard SMF (Corning SMF-28) of 125 μm in outer diameter (OD), a 40-mm length of the HCF was pre-tapered from 300 μm to 125 μm in terms of the cladding diameter and, then, spliced it to the input SMF. After fusion-splicing the two fibers, the splice point was carefully positioned at the center of a tapering rig and adiabatically tapered using the modified “flame-brushing” technique [26

26. Y. Jung, Y. Jeong, G. Brambilla, and D. J. Richardson, “Adiabatically tapered splice for selective excitation of the fundamental mode in a multimode fiber,” Opt. Lett. 34(15), 2369–2371 (2009). [CrossRef] [PubMed]

28

28. Y. Jung, G. Brambilla, and D. J. Richardson, “Broadband single-mode operation of standard optical fibers by using a sub-wavelength optical wire filter,” Opt. Express 16(19), 14661–14667 (2008). [CrossRef] [PubMed]

]. We investigated two cases of the resultant splice taper, one of 40 μm and the other of 30 μm in terms of the taper waist diameter. Both of them had a uniform waist section of 20 mm embedded by up- and down-taper sections of 30 mm. It is noteworthy that the coupling efficiency would crucially depend on the positional accuracy of the splice point that should locate at the waist of the uniform taper as well as on the adiabaticity of the taper profile [26

26. Y. Jung, Y. Jeong, G. Brambilla, and D. J. Richardson, “Adiabatically tapered splice for selective excitation of the fundamental mode in a multimode fiber,” Opt. Lett. 34(15), 2369–2371 (2009). [CrossRef] [PubMed]

, 29

29. F. Gonthier, A. Hénault, S. Lacroix, R. J. Black, and J. Bures, “Mode coupling in nonuniform fibers: comparison between coupled-mode theory and finite-difference beam-propagation method simulations,” J. Opt. Soc. Am. B 8(2), 416–421 (1991). [CrossRef]

]. Based on the mass conservation consideration, the uniform taper waist in case of the diameter of 30 μm was stretched from the solid volume with a length of 1.1 mm and diameter of 125 μm. The accurate positioning of the splice point in the center of the hot zone (with a margin of the error below 1.1 mm) is crucial. We utilized the top-mounted microscope to minimize the alignment error. Figure 6
Fig. 6 Side views of the initial, pre-tapered, and splice-tapered HCF.
shows the side views of the initial, pre-tapered, and splice-tapered HCF. One can clearly see the helical trajectory of the core around the axis of the fiber in the initial HCF, and it becomes elongated as the fiber is tapered down although it becomes very hard to identify the location of the helix when it was tapered down to 30 μm in OD. It is also noteworthy that in this case the core diameter of 10 μm of the initial HCF was scaled down to 4.2 μm and 1 μm in the pre-tapered and splice-tapered sections, respectively. In addition, the ATS section was fixed on a plate to be kept straight in order to avoid any additional bending loss and unwanted higher-order mode excitation in the tapered section.

To quantify the light coupling efficiency to the HCF, the input SMF in Fig. 3 was fed by a single-mode ytterbium-doped fiber laser (YDFL) operating at ~1.1 μm that was pigtailed by a passive single-mode fiber (Corning HI1060). It should be noted that the standard SMF (Corning SMF-28) does not operate in pure single-mode at ~1.1 μm so that fractions of the excited modes in the SMF-28 through the HI1060 could be in higher-order modes. However, such fractions of higher-order modes are immediately filtered out through the down-taper section of the SMF-28 thanks to the much smaller core size in the tapered section, and in result their contribution to the coupling into the HCF is negligible. We estimate that the insertion loss of the fundamental mode from the HI1060 to SMF-28 was ~0.74 dB by the cutback analysis on a fusion-spliced assembly of HI1060−SMF-28−HI1060. In addition, the coupling efficiency between the input SMF and the HCF through the tapered splice was determined by the ratio between the input and output powers through the SMF-HCF tapered splice assembly that was pigtailed by a ~60-cm-long HCF where the background propagation loss of the HCF itself was excluded.

Before tapering the splicing joint (just when the SMF and the HCF were directly fusion-spliced), we observed that there was simply no light coming out through the HCF (higher than 40 dB loss, i.e., below 0.01% coupling efficiency), which is due obviously to the large lateral offset and angular tilt of the fiber cores of the SMF and the HCF. It should be noted that the HCF is coated with a standard high-index polymer so that it does not guide cladding modes at all. However, once the splice joint was tapered down as described above, the coupling efficiency dramatically increased to −8.4 dB (14%) and −5.9 dB (26%) for the uniform taper waists of 40 and 30 μm, respectively. This highlights that the splice loss before and after the splice-taper process was improved by more than 34 dB. Table 1

Table 1. Estimated overall coupling efficiencies based on the experiment and theory.

table-icon
View This Table
summarizes the estimated overall coupling efficiencies based on the experiment and theory (See Fig. 5).

Based on the theoretical estimation shown in Table 1 where we assume an ideal adiabatic transition through the tapered splice, the transition losses across the tapered splice of SMF and HCF are given by −2.8 dB and −0.42 dB for the uniform taper waists of 40 and 30 μm, respectively. While there is a discrepancy between the experimental and theoretical estimates, one can see that the discrepancy is only an offset of ~5.5 dB for the both cases of 40 and 30 μm in terms of the uniform waist diameter. This implies that there must be an excessive loss mechanism regardless of the dimension of the uniform taper waist. We attribute this excessive loss to the limited length of the tapered splice, comprising a uniform waist section of 20 mm and two tapered sections of 30 mm. (There was a technical limitation in our tapering rig available at the time of the experiment in terms of the maximum tapering length.) It should be noted that a nearly negligible coupling loss of below 0.1 dB was successfully demonstrated for a SMF-MMF ATS when the tapered splice section was twice as long as the current one [26

26. Y. Jung, Y. Jeong, G. Brambilla, and D. J. Richardson, “Adiabatically tapered splice for selective excitation of the fundamental mode in a multimode fiber,” Opt. Lett. 34(15), 2369–2371 (2009). [CrossRef] [PubMed]

]. Except for the dimension of the uniform waist, the adiabaticity of the transitional sections of up- and down-taper also plays a crucial role to minimize the excessive loss. A numerical analysis based on the beam-propagation method (BPM) may be very useful for analyzing the cause of the excessive loss [24

24. L. Yuan, Z. Liu, J. Yang, and C. Guan, “Bitapered fiber coupling characteristics between single-mode single-core fiber and single-mode multicore fiber,” Appl. Opt. 47(18), 3307–3312 (2008). [CrossRef] [PubMed]

, 29

29. F. Gonthier, A. Hénault, S. Lacroix, R. J. Black, and J. Bures, “Mode coupling in nonuniform fibers: comparison between coupled-mode theory and finite-difference beam-propagation method simulations,” J. Opt. Soc. Am. B 8(2), 416–421 (1991). [CrossRef]

], which is outside the scope of this paper. However, we emphasize that apart from the excessive loss offset, the tendencies of the coupling loss depending on the uniform waist diameter are in good agreement with the numerical estimation discussed in Fig. 5. While there is still room for improvement in terms of the coupling efficiency, our experimental result confirms that the large splice loss between the SMF and the HCF can be substantially reduced by the ATS method.

To visualize the light coupled from the SMF to the HCF via the ATS, the cross-sectional and side views are shown in Fig. 7
Fig. 7 The light emission from the HCF via the SMF-HCF ATS visualized in the cross-sectional (a) and side views (b).
, which were taken with the aid of a microscope and a CCD camera. One can clearly see the light emission from the flat-cleaved, off-centered core: Because of the angular tilt between the flat-cleaved fiber facet and the helix of the core, the light ray emitting from the core are exiting with a noticeably large angle as shown in Fig. 7(b), which is in good agreement with a simple estimation via Snell’s law, i.e., θexit = sin−1(n1 ⋅ sinθ) = 14.5°, where θ ~2πR/D is the trajectory angle of the helix [12

12. D. Marcuse, “Radiation loss of a helically deformed optical fiber,” J. Opt. Soc. Am. 66(10), 1025–1031 (1976). [CrossRef]

].

In addition, to confirm the robust single-mode operation of our SMF-HCF ATS assembly, the output field pattern through the 60-cm-long HCF was imaged with a microscope objective lens and a CCD camera. In general, when coherent light is guided in a multimode fiber, severe speckle patterns must be observed or move around if external perturbations are applied. This is due to the intermodal interference between the guided modes travelling within the fiber [30

30. Y. Jeong and B. Lee, “Effect of a random pattern through a multimode-fiber bundle on angular and spatial selectivity in volume holograms: experiments and theory,” Appl. Opt. 41(20), 4085–4091 (2002). [CrossRef] [PubMed]

]. It is noteworthy that, as shown in Fig. 8
Fig. 8 Measured output field intensity profile from the end of the HCF.
, a clean and stationary, fundamental-order beam of a Gaussian shape was detected, and it remained completely unchanged even with external perturbations, such as bending or squeezing the HCF section. This is clear evidence that there was only one mode supported through the SMF-HCF ATS assembly and the pigtail HCF. This should be understood in twofold: One is that the tapered splice selectively excites the fundamental mode only as it was fed by a single-mode input. The other is that the HCF effectively supports only the fundamental mode imposing substantially high curvature loss to higher-order modes. Previously it was reported that a YDFL based on an active HCF could operate in near single-mode where the core diameter, NA, and helix pitch were 30 μm, 0.087, ~9 mm, respectively, which gives rise to the V parameter of 7.45 [18

18. P. Wang, L. J. Cooper, J. K. Sahu, and W. A. Clarkson, “Efficient single-mode operation of a cladding-pumped ytterbium-doped helical-core fiber laser,” Opt. Lett. 31(2), 226–228 (2006). [CrossRef] [PubMed]

]. Considering that the current HCF used in our experiment has the V parameter of 4.0 that is significantly lower than 7.45, we presume that the current HCF effectively supports only the fundamental mode. In addition, while the HCF used in our experiment had a rather moderate core dimension, the ATS technique is readily applicable to HCFs having even larger core dimensions, such as the large-core HCFs demonstrated in [17

17. P. Wang, L. J. Cooper, R. B. Williams, J. K. Sahu, and W. A. Clarkson, “Helical-core ytterbium-doped fibre laser,” Electron. Lett. 40(21), 1325–1326 (2004). [CrossRef]

, 18

18. P. Wang, L. J. Cooper, J. K. Sahu, and W. A. Clarkson, “Efficient single-mode operation of a cladding-pumped ytterbium-doped helical-core fiber laser,” Opt. Lett. 31(2), 226–228 (2006). [CrossRef] [PubMed]

].

4. Conclusions

We have presented a simple and reliable light launch scheme for the HCFs by using an ATS technique. A successive mode conversion [LP01core(SMF)LP01taper(SMF)LP01taper(HCF)LP01core(HCF)] along the ATS section, the coupling efficiency between the SMF and the HCF with a large core offset and angular tilt was significantly improved to −5.9 dB (26%). We believe this coupling efficiency can further be improved to better than, perhaps, −0.1 dB (98%) if we can elongate the tapered splice to be twice as long as the current form. (See Table 1.) This all-fiber, fusion splicing technique with HCFs offers enormous advantages and potentials over the conventional methods based on bulky and delicate free-space optics [17

17. P. Wang, L. J. Cooper, R. B. Williams, J. K. Sahu, and W. A. Clarkson, “Helical-core ytterbium-doped fibre laser,” Electron. Lett. 40(21), 1325–1326 (2004). [CrossRef]

, 18

18. P. Wang, L. J. Cooper, J. K. Sahu, and W. A. Clarkson, “Efficient single-mode operation of a cladding-pumped ytterbium-doped helical-core fiber laser,” Opt. Lett. 31(2), 226–228 (2006). [CrossRef] [PubMed]

]. The method proposed and demonstrated here is definitely a route forward to realize align-free, all-fiberized systems consisting of HCFs. For example, this scheme can readily be used for building an all-fiberized master-oscillator power amplifier systems [3

3. Y. Jeong, J. K. Sahu, D. B. S. Soh, C. A. Codemard, and J. Nilsson, “High-power tunable single-frequency single-mode erbium:ytterbium codoped large-core fiber master-oscillator power amplifier source,” Opt. Lett. 30(22), 2997–2999 (2005). [CrossRef] [PubMed]

] comprised with a master-oscillator laser based on a conventional SMF and a booster power amplifier based on an HCF. In this case side-pumping schemes [31

31. R. Herda, A. Liem, B. Schnabel, A. Drauschke, H.-J. Fuchs, E.-B. Kley, H. Zellmer, and A. Tuennermann, “Efficient side-pumping of fibre lasers using binary gold diffraction gratings,” Electron. Lett. 39(3), 276–277 (2003). [CrossRef]

, 32

32. S.-U. Alam, J. Nilsson, P. W. Turner, M. Ibsen, and A. B. Grudinin, “Low cost multi-port reconfigurable erbium doped cladding pumped fibre amplifier,” in Proc. ECOC 2000, München, Germany, 4–7 Sep., 2000, 5.4.3.

] may be preferred to simple end-pumping schemes since pump beams can bypass the thin ATS sections. We expect that the proposed ATS scheme can also be applied to the HCF-HCF splicing or be directly applicable to other operational regimes and will provide a new degree of freedom in the application of HCFs to high performance fiber lasers and optical fiber sensors.

Acknowledgment

This work was supported in part by Research Settlement Fund for the new faculty of SNU and the Ministry of Knowledge Economy (Project No.10040429).

References and links

1.

M. E. Fermann, “Single-mode excitation of multimode fibers with ultrashort pulses,” Opt. Lett. 23(1), 52–54 (1998). [CrossRef] [PubMed]

2.

J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25(7), 442–444 (2000). [CrossRef] [PubMed]

3.

Y. Jeong, J. K. Sahu, D. B. S. Soh, C. A. Codemard, and J. Nilsson, “High-power tunable single-frequency single-mode erbium:ytterbium codoped large-core fiber master-oscillator power amplifier source,” Opt. Lett. 30(22), 2997–2999 (2005). [CrossRef] [PubMed]

4.

Y. Jeong, J. K. Sahu, D. N. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express 12(25), 6088–6092 (2004). [CrossRef] [PubMed]

5.

Y. Jeong, A. J. Boyland, J. K. Sahu, S.-H. Chung, J. Nilsson, and D. N. Payne, “Multi-kilowatt single-mode ytterbium-doped large-core fiber laser,” J. Opt. Soc. Kor. 13(4), 416–422 (2009). [CrossRef]

6.

J. Limpert, N. Deguil-Robin, I. Manek-Hönninger, F. Salin, F. Röser, A. Liem, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson, and C. Jakobsen, “High-power rod-type photonic crystal fiber laser,” Opt. Express 13(4), 1055–1058 (2005). [CrossRef] [PubMed]

7.

C.-H. Liu, G. Chang, N. Litchinitser, and A. Galvanauskas, “Effectively single-mode chirally-coupled core fiber,” in Proc. ASSP 2007, Vancouver, Canada, 28–31 Jan., 2007, ME2.

8.

A. E. Siegman, “Gain-guided, index-antiguide fiber lasers,” J. Opt. Soc. Am. B 24(8), 1677–1682 (2007). [CrossRef]

9.

D. A. Rockwell, V. V. Shkunov, and J. R. Marciante, “Semi-guiding high-aspect-ratio core (SHARC) fiber providing single-mode operation and an ultra-large core area in a compact coilable package,” Opt. Express 19(15), 14746–14762 (2011). [CrossRef] [PubMed]

10.

F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett. 36(5), 689–691 (2011). [CrossRef] [PubMed]

11.

L. Dong, X. Peng, and J. Li, “Leakage channel optical fibers with large effective area,” J. Opt. Soc. Am. B 24(8), 1689–1697 (2007). [CrossRef]

12.

D. Marcuse, “Radiation loss of a helically deformed optical fiber,” J. Opt. Soc. Am. 66(10), 1025–1031 (1976). [CrossRef]

13.

J. Qian, “Coupled-mode theory for helical fibres,” IEE Proceedings J. 135, 178–182 (1988).

14.

A. Altintas and J. D. Love, “Effective cut-offs for modes on helical fibers,” Opt. Quantum Electron. 22(3), 213–226 (1990). [CrossRef]

15.

D. B. S. Soh, J. Nilsson, J. K. Sahu, and L. J. Cooper, “Geometrical factor modification of helical-core fiber radiation loss formula,” Opt. Commun. 222(1-6), 235–242 (2003). [CrossRef]

16.

Z. Jiang and J. R. Marciante, “Mode-area scaling of helical-core, dual-clad fiber lasers and amplifiers using an improved bend-loss model,” J. Opt. Soc. Am. B 23(10), 2051–2058 (2006). [CrossRef]

17.

P. Wang, L. J. Cooper, R. B. Williams, J. K. Sahu, and W. A. Clarkson, “Helical-core ytterbium-doped fibre laser,” Electron. Lett. 40(21), 1325–1326 (2004). [CrossRef]

18.

P. Wang, L. J. Cooper, J. K. Sahu, and W. A. Clarkson, “Efficient single-mode operation of a cladding-pumped ytterbium-doped helical-core fiber laser,” Opt. Lett. 31(2), 226–228 (2006). [CrossRef] [PubMed]

19.

J. Qian and C. D. Hussey, “Circular birefringence in helical-core fibre,” Electron. Lett. 22(10), 515–517 (1986). [CrossRef]

20.

A. Argyros, J. Pla, F. Ladouceur, and L. Poladian, “Circular and elliptical birefringence in spun microstructured optical fibres,” Opt. Express 17(18), 15983–15990 (2009). [CrossRef] [PubMed]

21.

R. D. Birch, “Fabrication and characterisation of circularly birefringent helical fibres,” Electron. Lett. 23(1), 50–52 (1987). [CrossRef]

22.

K. Shiraishi, T. Yanagi, and S. Kawakami, “Light-propagation characteristics in thermally diffused expanded core fibers,” J. Lightwave Technol. 11(10), 1584–1591 (1993). [CrossRef]

23.

D. B. Mortimore and J. V. Wright, “Low-loss joints between dissimilar fibres by tapering fusion splices,” Electron. Lett. 22(6), 318–319 (1986). [CrossRef]

24.

L. Yuan, Z. Liu, J. Yang, and C. Guan, “Bitapered fiber coupling characteristics between single-mode single-core fiber and single-mode multicore fiber,” Appl. Opt. 47(18), 3307–3312 (2008). [CrossRef] [PubMed]

25.

N. Healy, E. McDaid, D. G. Murphy, C. D. Hussey, and T. A. Birks, “Low-loss singlemode fibre 1x2 Y-junction,” Electron. Lett. 42(13), 740–742 (2006). [CrossRef]

26.

Y. Jung, Y. Jeong, G. Brambilla, and D. J. Richardson, “Adiabatically tapered splice for selective excitation of the fundamental mode in a multimode fiber,” Opt. Lett. 34(15), 2369–2371 (2009). [CrossRef] [PubMed]

27.

G. Brambilla, Y. Jung, and F. Renna, “Optical fiber microwires and nanowires manufactured by modified flame brushing technique: properties and applications,” Front. Optoelectron. China 3(1), 61–66 (2010). [CrossRef]

28.

Y. Jung, G. Brambilla, and D. J. Richardson, “Broadband single-mode operation of standard optical fibers by using a sub-wavelength optical wire filter,” Opt. Express 16(19), 14661–14667 (2008). [CrossRef] [PubMed]

29.

F. Gonthier, A. Hénault, S. Lacroix, R. J. Black, and J. Bures, “Mode coupling in nonuniform fibers: comparison between coupled-mode theory and finite-difference beam-propagation method simulations,” J. Opt. Soc. Am. B 8(2), 416–421 (1991). [CrossRef]

30.

Y. Jeong and B. Lee, “Effect of a random pattern through a multimode-fiber bundle on angular and spatial selectivity in volume holograms: experiments and theory,” Appl. Opt. 41(20), 4085–4091 (2002). [CrossRef] [PubMed]

31.

R. Herda, A. Liem, B. Schnabel, A. Drauschke, H.-J. Fuchs, E.-B. Kley, H. Zellmer, and A. Tuennermann, “Efficient side-pumping of fibre lasers using binary gold diffraction gratings,” Electron. Lett. 39(3), 276–277 (2003). [CrossRef]

32.

S.-U. Alam, J. Nilsson, P. W. Turner, M. Ibsen, and A. B. Grudinin, “Low cost multi-port reconfigurable erbium doped cladding pumped fibre amplifier,” in Proc. ECOC 2000, München, Germany, 4–7 Sep., 2000, 5.4.3.

OCIS Codes
(060.2340) Fiber optics and optical communications : Fiber optics components
(060.2430) Fiber optics and optical communications : Fibers, single-mode
(230.1150) Optical devices : All-optical devices
(060.3510) Fiber optics and optical communications : Lasers, fiber

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: August 28, 2012
Revised Manuscript: October 12, 2012
Manuscript Accepted: October 16, 2012
Published: October 25, 2012

Citation
Hyuntai Kim, Jongki Kim, Yongmin Jung, Luis Alonso Vazquez-Zuniga, Seung Jong Lee, Geunchang Choi, Kyunghwan Oh, Pu Wang, W. A. Clarkson, and Yoonchan Jeong, "Simple and reliable light launch from a conventional single-mode fiber into a helical-core fiber through an adiabatically tapered splice," Opt. Express 20, 25562-25571 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-23-25562


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References

  1. M. E. Fermann, “Single-mode excitation of multimode fibers with ultrashort pulses,” Opt. Lett.23(1), 52–54 (1998). [CrossRef] [PubMed]
  2. J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett.25(7), 442–444 (2000). [CrossRef] [PubMed]
  3. Y. Jeong, J. K. Sahu, D. B. S. Soh, C. A. Codemard, and J. Nilsson, “High-power tunable single-frequency single-mode erbium:ytterbium codoped large-core fiber master-oscillator power amplifier source,” Opt. Lett.30(22), 2997–2999 (2005). [CrossRef] [PubMed]
  4. Y. Jeong, J. K. Sahu, D. N. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express12(25), 6088–6092 (2004). [CrossRef] [PubMed]
  5. Y. Jeong, A. J. Boyland, J. K. Sahu, S.-H. Chung, J. Nilsson, and D. N. Payne, “Multi-kilowatt single-mode ytterbium-doped large-core fiber laser,” J. Opt. Soc. Kor.13(4), 416–422 (2009). [CrossRef]
  6. J. Limpert, N. Deguil-Robin, I. Manek-Hönninger, F. Salin, F. Röser, A. Liem, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson, and C. Jakobsen, “High-power rod-type photonic crystal fiber laser,” Opt. Express13(4), 1055–1058 (2005). [CrossRef] [PubMed]
  7. C.-H. Liu, G. Chang, N. Litchinitser, and A. Galvanauskas, “Effectively single-mode chirally-coupled core fiber,” in Proc. ASSP 2007, Vancouver, Canada, 28–31 Jan., 2007, ME2.
  8. A. E. Siegman, “Gain-guided, index-antiguide fiber lasers,” J. Opt. Soc. Am. B24(8), 1677–1682 (2007). [CrossRef]
  9. D. A. Rockwell, V. V. Shkunov, and J. R. Marciante, “Semi-guiding high-aspect-ratio core (SHARC) fiber providing single-mode operation and an ultra-large core area in a compact coilable package,” Opt. Express19(15), 14746–14762 (2011). [CrossRef] [PubMed]
  10. F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett.36(5), 689–691 (2011). [CrossRef] [PubMed]
  11. L. Dong, X. Peng, and J. Li, “Leakage channel optical fibers with large effective area,” J. Opt. Soc. Am. B24(8), 1689–1697 (2007). [CrossRef]
  12. D. Marcuse, “Radiation loss of a helically deformed optical fiber,” J. Opt. Soc. Am.66(10), 1025–1031 (1976). [CrossRef]
  13. J. Qian, “Coupled-mode theory for helical fibres,” IEE Proceedings J.135, 178–182 (1988).
  14. A. Altintas and J. D. Love, “Effective cut-offs for modes on helical fibers,” Opt. Quantum Electron.22(3), 213–226 (1990). [CrossRef]
  15. D. B. S. Soh, J. Nilsson, J. K. Sahu, and L. J. Cooper, “Geometrical factor modification of helical-core fiber radiation loss formula,” Opt. Commun.222(1-6), 235–242 (2003). [CrossRef]
  16. Z. Jiang and J. R. Marciante, “Mode-area scaling of helical-core, dual-clad fiber lasers and amplifiers using an improved bend-loss model,” J. Opt. Soc. Am. B23(10), 2051–2058 (2006). [CrossRef]
  17. P. Wang, L. J. Cooper, R. B. Williams, J. K. Sahu, and W. A. Clarkson, “Helical-core ytterbium-doped fibre laser,” Electron. Lett.40(21), 1325–1326 (2004). [CrossRef]
  18. P. Wang, L. J. Cooper, J. K. Sahu, and W. A. Clarkson, “Efficient single-mode operation of a cladding-pumped ytterbium-doped helical-core fiber laser,” Opt. Lett.31(2), 226–228 (2006). [CrossRef] [PubMed]
  19. J. Qian and C. D. Hussey, “Circular birefringence in helical-core fibre,” Electron. Lett.22(10), 515–517 (1986). [CrossRef]
  20. A. Argyros, J. Pla, F. Ladouceur, and L. Poladian, “Circular and elliptical birefringence in spun microstructured optical fibres,” Opt. Express17(18), 15983–15990 (2009). [CrossRef] [PubMed]
  21. R. D. Birch, “Fabrication and characterisation of circularly birefringent helical fibres,” Electron. Lett.23(1), 50–52 (1987). [CrossRef]
  22. K. Shiraishi, T. Yanagi, and S. Kawakami, “Light-propagation characteristics in thermally diffused expanded core fibers,” J. Lightwave Technol.11(10), 1584–1591 (1993). [CrossRef]
  23. D. B. Mortimore and J. V. Wright, “Low-loss joints between dissimilar fibres by tapering fusion splices,” Electron. Lett.22(6), 318–319 (1986). [CrossRef]
  24. L. Yuan, Z. Liu, J. Yang, and C. Guan, “Bitapered fiber coupling characteristics between single-mode single-core fiber and single-mode multicore fiber,” Appl. Opt.47(18), 3307–3312 (2008). [CrossRef] [PubMed]
  25. N. Healy, E. McDaid, D. G. Murphy, C. D. Hussey, and T. A. Birks, “Low-loss singlemode fibre 1x2 Y-junction,” Electron. Lett.42(13), 740–742 (2006). [CrossRef]
  26. Y. Jung, Y. Jeong, G. Brambilla, and D. J. Richardson, “Adiabatically tapered splice for selective excitation of the fundamental mode in a multimode fiber,” Opt. Lett.34(15), 2369–2371 (2009). [CrossRef] [PubMed]
  27. G. Brambilla, Y. Jung, and F. Renna, “Optical fiber microwires and nanowires manufactured by modified flame brushing technique: properties and applications,” Front. Optoelectron. China3(1), 61–66 (2010). [CrossRef]
  28. Y. Jung, G. Brambilla, and D. J. Richardson, “Broadband single-mode operation of standard optical fibers by using a sub-wavelength optical wire filter,” Opt. Express16(19), 14661–14667 (2008). [CrossRef] [PubMed]
  29. F. Gonthier, A. Hénault, S. Lacroix, R. J. Black, and J. Bures, “Mode coupling in nonuniform fibers: comparison between coupled-mode theory and finite-difference beam-propagation method simulations,” J. Opt. Soc. Am. B8(2), 416–421 (1991). [CrossRef]
  30. Y. Jeong and B. Lee, “Effect of a random pattern through a multimode-fiber bundle on angular and spatial selectivity in volume holograms: experiments and theory,” Appl. Opt.41(20), 4085–4091 (2002). [CrossRef] [PubMed]
  31. R. Herda, A. Liem, B. Schnabel, A. Drauschke, H.-J. Fuchs, E.-B. Kley, H. Zellmer, and A. Tuennermann, “Efficient side-pumping of fibre lasers using binary gold diffraction gratings,” Electron. Lett.39(3), 276–277 (2003). [CrossRef]
  32. S.-U. Alam, J. Nilsson, P. W. Turner, M. Ibsen, and A. B. Grudinin, “Low cost multi-port reconfigurable erbium doped cladding pumped fibre amplifier,” in Proc. ECOC 2000, München, Germany, 4–7 Sep., 2000, 5.4.3.

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