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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 10 — May. 19, 2014
  • pp: 11610–11619
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All-fiber fused directional coupler for highly efficient spatial mode conversion

Rand Ismaeel, Timothy Lee, Bernard Oduro, Yongmin Jung, and Gilberto Brambilla  »View Author Affiliations


Optics Express, Vol. 22, Issue 10, pp. 11610-11619 (2014)
http://dx.doi.org/10.1364/OE.22.011610


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Abstract

We model and demonstrate a simple mode selective all-fiber coupler capable of exciting specific higher order modes in two- and few-mode fibres with high efficiency and purity. The coupler is based on inter-modally phase-matching the propagation constants in each arm of the asymmetric fused coupler, formed by dissimilar fibres. At a specific coupler diameter, the launched fundamental LP01 mode is coupled into the higher order mode (LP11, LP21, LP02) in the other arm, over a broadband wave-length range around 1550 nm. Unlike other techniques, the demonstrated coupler is composed of a multimode fiber that is weakly fused with a phase matched conventional single mode telecom fiber (SMF-28). The beating between the supermodes at the coupler waist produces a periodic power transfer between the two arms, and therefore, by monitoring the beating while tapering, it is possible to obtain optimum selection for the desired mode. High coupling efficiencies in excess of 90% for all the higher order modes were recorded over 100 nm spectral range, while insertion losses remain as low as 0.5 dB. Coupling efficiency can be further enhanced by performing slow tapering at high temperature, in order to precisely control the coupler cross-section geometry.

© 2014 Optical Society of America

1. Introduction

2. Working principle

Here, we present a simple, mode selective coupler manufactured via a straightforward two step process. The input light is launched into a telecom fiber (SMF-28) that has been pre-tapered and fused to a step index TMF (core/cladding diameter = 19.7/125μm, NA = 0.12) or FMF (core/cladding diameter =26.1/125μm, NA = 0.12) as shown in Fig. 1. Clearly, the use of a conventional telecom SMF instead of the TMF to support the input LP01 mode is especially convenient for telecommunication systems since it would be compatible for the wide range of devices optimized to interface with telecom fibers. In this paper we use the weakly-fused coupler approximation to model and demonstrate the mode conversion. The coupling region for the weakly fused 2 × 2 coupler is treated as two touching circular cylinders. We chose the weak fusion technique to maintain the geometry of both fibres and therefore achieve accurate index matching. By using this technique, we can directly (and selectively) couple the input LP01 mode from the telecom fiber to the other higher order mode supported in the TMF (or FMF).

Fig. 1 Schematic of the MSC: light is launched in the SMF port; the LP11 is expected to be preferentially excited at the TMF output port, while the uncoupled fundamental LP01 will propagate along the SMF. For the FMF, it is possible to select any of the modes that propagates in the FMF by choosing the SMF pre-tapering diameter.

Tapering the SMF fiber is essential for the mode conversion to occur, since two different fibers (and hence different propagation constants) are used in manufacturing the coupler, Therefore tapering the SMF into a specific diameter will match the propagation constants for the LP01 mode in the SMF with the desired higher order mode in the TMF/FMF. The power distribution at any position z along the coupling region of the coupler, is given by two functions P1(z) and P2(z), which give the power in the SMF and TMF/FMF arm respectively. For our case of asymmetric fibres, where the propagation constant β1 of the fundamental mode in the SMF is phase matched with the β2 of the high-order mode in the multimode fibre, the phase mismatch Δβ = β1β2 is zero and so the power distribution reduces to P1(z) ∝ cos2(κz), P2(z) ∝ sin2(κz) indicating a complete periodic power transfer between the two fibers in the lossless case. κ is the coupling coefficient which depends on the index profile of the fibres and the cross-section geometry (i.e the distance between the cores of the two fibres). The power in each arm would therefore depend on these three factors; the geometry, index and the coupling region length. Although in this work we match the diameters of the fibres composing the coupler close to the ideal value, any change in the size or the surrounding of the fibers would introduce a nonzero phase mismatch Δβ ≠ 0 which limits the fraction of the input power coupled into the higher-order mode. Therefore, we attempted to optimize these three parameters to maximise the possible coupling efficiency. In our modelling, we adapted a three layer system comprised of the coupler cores, cladding and air regions. As the core-cladding interface could still affect the behaviour of the supported modes by providing guidance between the cladding and the external air [21

21. K. Chiang, “Effects of cores in fused tapered single-mode fiber couplers,” Opt. Lett. 12, 431–433 (1987). [CrossRef] [PubMed]

], the overall structure of the couplers should be considered as strongly guiding.

3. Modelling

Firstly, the mode effective indices were calculated at different fiber diameters for step index profiles in order to find the optimum diameter at which the LP01 modes are phase matched to the desired higher order mode.

The phase matching curves in Figs. 2(b) and 2(d) indicate that the SMF should be pre-tapered to a diameter of 79 μm to phase match to the LP11 mode in the TMF and pre-tapered to 78 μm, 45 μm and 33 μm to phase match the LP11, LP21 and the LP02 in the FMF respectively. After fabrication, the index profiles of both fibers (which was measured using a S-14 profilometer) was imported into the simulation software COMSOL to solve for the modes numerically and thus confirm the phase matching condition between both modes Figs. 2(a) and 2(c).

Fig. 2 (a) The electric field component (x direction) of the phase matched modes LP01 in the SMF port and the LP11 in the TMF port.(b) the phase matching graphs for the LP01 in the SMF and the LP11 in the TMF. (c) The electric field component of the LP01 in the SMF and the LP11, LP21, LP02 in the FMF port. (d) The effective index matching graph for the LP01 in the SMF and the desired mode in the FMF (all the modelling was performed using geometrical parameters obtained experimentally, The dotted line represents the dimensions chosen for the fabricated coupler.)

The results from both methods confirm that the ratio between the SMF to the TMF diameters should be around 0.63 as shown in Fig. 2(a). Since the scalar LP11 mode is actually composed of multiple vectorial modes (TE01, TM01 and and the two hybrid HE21 modes), the difficulty in selecting one specific vectorial mode could be a source of loss. To minimise this issue, a long tapering length was used to separate the modes that have different group velocity. The optimum coupler length was optimized experimentally, until the maximum power transfer was obtained. The same technique was exploited to efficiently couple the modes in the FMF.

In order to determine the possibility of obtaining high conversion efficiency, the TMF coupler cross-section was measured by scanning electron microscopy as shown in Fig. 3(a), which confirms that the coupler cross-section is weakly fused as required for the expected efficient power transfer between the two coupler arms. Figure 3(b), the result of the modelling, shows that it is possible to transfer most of the power from the phase matched SMF to the TMF in the form of the LP11 mode. The analytical results shows that a maximum transfer would accrue around a coupler diameter of 10 μm and coupler length of 25 μm, which are the parameters of giving the first peak of the beating oscillations.

Fig. 3 (a) Scanning electron microscope image of the weakly fused coupler cross section. (b) Simulated resulted for specific coupler diameter, most of the power is expected to be transferred to the output of the coupler as LP11.

To determine the dependency of the coupling efficiency between the LP01 and the LP11 on the change in the tapering diameter, we solved the following system of coupled equations:
ddz(A1(z)A2(z))=(iΔβ2iκiκiΔβ2)(A1(z)A2(z))
(1)
where z represents the distance along the coupler length, A1, A2 are the slowly-varying field amplitudes in the two fibres forming the coupler, κ is the coupling coefficient, which is approximated as half the difference between the propagation constants of the symmetrical (even) and antisymmetrical (odd) modes on the composite waveguide. The value of κ was calculated to be 5130 m−1 assuming that the coupler cross-section is composed of two touching circular cylinders (weakly fused asymmetric coupler) [26

26. H. Chang, T. Lin, and T. Wu, “Accurate coupling coefficients for fiber couplers with weakly fused cross sections,” Appl. Opt. 34, 6168–6171 (1995). [CrossRef] [PubMed]

].

Fig. 4 (a) Coupling efficiency versus the deviation from the ideal SMF phase matching diameter at the coupler waist, r1 is the ideal SMF diameter and r2 represents smaller or larger diameter than the ideal diameter(b) The electric field amplitude ratio at the fibre centres, |E2|/|E1|, versus the ratio between both fibres propagation constants β1/β2.

In the case of strong fusion, the effects of the unequal propagation constants on the field distribution is illustrated in Fig. 4(b) which gives the electric field amplitude ratio, E2/E1, versus the ratio between the two unequal propagation constants, where E1 and E2 are the electric field amplitudes at the centres of each fiber composing the coupler, respectively. It is seen that |E2|/|E1| depends strongly on β1/β2 and achieving high coupling efficiency above 90% requires the difference between β1 and β2 to be within a strict tolerance of < 0.5%.

4. SMF-TMF coupler experiment

Fig. 5 (a) Power transfer between the two arms of the coupler against pulling length: a periodic power transfer between the output ports is observed. (b) Output spectra of the two arms of the coupler. Far field images of the MSC output at λ ≈1550 nm from the (c) TMF port and (d) SMF port.

In this experiment, tapering was stopped at the point of maximum power exchange, i.e. when the output of the TMF was maximized. When tapering continued we noticed a sinusoidal power transfer between the two output ports Fig. 5(a), which occurs as a result of the beating between the two phase matched modes, while tapering continues even further, the frequency of the beating would decrease, which would decrease the wavelength range at the region for which the LP11 operates, therefore, it is essential to stop tapering at the point at which the first power transfer maxima is observed, in order to obtain the broadband operation range for the selected mode. From the figure above, The insertion loss in the TMF, calculated (from the power meter reading), as the difference between the TMF output at time t (where t represents the time when pulling was stopped) and the SMF output at t = 0 (when pulling started), is estimated to be smaller than 0.4 dB and can be attributed to coupling to other high order modes, or index mismatch.

5. SMF-FMF coupler experiment

To generate the higher order modes (LP11, LP21, LP02), we pre-tapered the SMF to diameters of 79 μm,45 μm and 33 μm respectively. The pre-tapered SMF fiber was then fused to the FMF until maximum power transfer between the two output ports was observed, with the length of the fused region selected to be 15 mm for the LP11 and 35 mm for the rest of the modes. Longer fusion lengths were required for the LP21 and LP02 modes to overcome the coupling losses resulting from the bigger geometrical difference between the cores of both fibers at the coupler waist region. The results obtained from the SMF-FMF coupler are shown in Fig. 6. Different pulling lengths were required for each coupler, and tapering was occasionally only stopped after several full power transfer cycles in order to ensure that the two fibres were indeed fused together. Figures 6(a), 6(c), and 6(e) show that the total insertion losses for all the three couplers were less than 0.5dB.

Fig. 6 Power transfer between the two arms of the coupler against pulling length for the (a) LP11, (c) LP21, (e) LP02 modes. Output spectra of the two arms of the coupler for the (b) LP11, (d) LP21, (f) LP02 modes.

All the three higher order modes supported by the FMF were excited with high coupling efficiencies of 91%,95% and 92% for the LP11, LP21 and LP02 respectively as shown in Figs. 6(b), 6(d), and 6(f). The purity of the modes was verified by the tight bend approach, as well as the far field image field intensity, to be in excess of 92% in all cases, and the performance of the coupler was tested over different wavelengths, at which they all showed coupling efficiencies above 90% as depicted in Fig. 7.

Fig. 7 CCD images of the LP11, LP21 and LP02 modes excited in the FMF at different launching wavelengths.

6. Conclusion

In summary, we have modelled and experimentally demonstrated a high coupling efficiency between the fundamental mode and the selected higher order mode in a mode selective coupler composed of a SMF-28 telecom fiber and a TMF or FMF, at telecom wavelengths around 1.55 μm. The phase matching diameter was calculated for a three layer system so that both of the fundamental mode and the selected higher order mode would have equal propagation constant, therefore helping to maximise the power transfer which could be obtained. Meanwhile, low insertion losses and high coupling efficiency were ensured by using a variable speed/temperature tapering process to maintain the shape of the waist of the coupler.

Acknowledgments

G. Brambilla gratefully acknowledges the Royal Society (London) for his University Research fellowship. The authors acknowledge partial support of this research by the European Communities 7th Framework Programme under grant agreement 258033 (MODE-GAP) for the fabrication of few mode fibre. The authors thank the EPSRC Centre for Innovative Manufacturing in Photonics for support.

References and links

1.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013). [CrossRef]

2.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012). [CrossRef]

3.

S. Leon-Saval, T. Birks, J. Bland-Hawthorn, and M. Englund, “Multimode fiber devices with single-mode performance,” Opt. Lett. 30, 2545–2547 (2005). [CrossRef] [PubMed]

4.

S. Leon-Saval, A. Argyros, and J. Bland-Hawthorn, “Photonic lanterns: a study of light propagation in multimode to single-mode converters,” Opt. Express 18, 8430–8439 (2010). [CrossRef] [PubMed]

5.

S. Leon-Saval, N. Fontaine, J. Salazar-Gil, B. Ercan, R. Ryf, and J. Bland-Hawthorn, “Mode-selective photonic lanterns for space-division multiplexing,” Opt. Express 22, 1036–1044 (2014). [CrossRef] [PubMed]

6.

N. K. Fontaine, S. G. Leon-Saval, R. Ryf, J. R. S. Gil, B. Ercan, and J. Bland-Hawthorn, “Mode-selective dissimilar fiber photonic-lantern spatial multiplexers for few-mode fiber,” Optical Communication (ECOC 2013), 39th European Conference and Exhibition, pp. 1, 3, 22–26 Sept. 2013

7.

A. M. Vengsarkar, J. A. Greene, and K. A. Murphy, “Photoinduced refractive-index changes in two-mode elliptical-core fibers: sensing applications,” Opt. Lett. 16, 1541–1543 (1991). [CrossRef] [PubMed]

8.

C. D. Poole, C. D. Townsend, and K. T. Nelson, “Helical-grating two-mode fiber spatial-mode coupler,” J. Light-waveTechnol. 9, 598–604 (1991). [CrossRef]

9.

H. G. Park and B. Y. Kim, “Intermodal coupler using permanently photoinduced grating in two-modeoptical fiber,” Electron. Lett. 25, 797–799 (1989). [CrossRef]

10.

F. A. Castro, S. R. M. Carneiro, O. Lisboa, and S. L. A. Carrara, “Two-mode optical fiber accelerometer,” Opt. Lett. 17, 1474–1475 (1992). [CrossRef] [PubMed]

11.

S. Y. Haung, J. N. Blake, and B. Y. Kim, “Perturbation effects on mode propagation in highly elliptical core two-mode fibers,” J. Lightwave Technol. 8, 23–33 (1990). [CrossRef]

12.

J. N. Blake, S. U. Huang, B. Y. Kim, and H. J. Shaw, “Strain effects on highly elliptical core two-mode fibers,” Opt. Lett. 12, 732–734 (1987). [CrossRef] [PubMed]

13.

W. V. Sorin, B. Y. Kim, and H. J. Shaw, “Highly selective evanescent modal filter for two-mode optical fibers,” Opt. Lett. 11, 581–583 (1986). [CrossRef] [PubMed]

14.

R. C. Youngquist, J. L. Brooks, and H. J. Shaw, “Two-mode fiber modal coupler,” Opt. Lett. 9, 177–179 (1984). [CrossRef] [PubMed]

15.

M. Vaziri and C. Chen, “An etched two-mode fiber modal coupling element,” J. Lightwave Technol. 15, 474–481 (1997). [CrossRef]

16.

Y. Shou, J. Bures, S. Lacroix, and X. Daxhelet, “Mode separation in fused fiber coupler made of two-mode fibers,” Opt. Fiber Technol. 5, 92–104 (1999). [CrossRef]

17.

K. Y. Song, I. K. Hwang, S. H. Yun, and B. Y. Kim, “High performance fused-type mode selective coupler for two-mode fiber devices,” in Opt. Fibre Commun. Conf., Baltimore, 7–10 Mar (2000).

18.

K. Y. Song, I. K. Hwang, S. H. Yun, and B. Y. Kim, “High performance fused-type mode-selective coupler using elliptical core two-mode fiber at 1550nm,” IEEE Photon. Technol. Lett. 14, 501–503 (2002). [CrossRef]

19.

Y. Jung, R. Chen, R. Ismaeel, G. Brambilla, S. Alam, I. Giles, and D. Richardson, “Dual mode fused optical fiber couplers suitable for mode division multiplexed transmission,” Opt. Express 21, 24326–24331 (2013). [CrossRef] [PubMed]

20.

S. Yerolatsitis, I. Gris-Sánchez, and T. Birks, “Adiabatically-tapered fiber mode multiplexers,” Opt. Express 22, 608–617 (2014). [CrossRef] [PubMed]

21.

K. Chiang, “Effects of cores in fused tapered single-mode fiber couplers,” Opt. Lett. 12, 431–433 (1987). [CrossRef] [PubMed]

22.

G. Brambilla, “Optical fiber nanowires and microwires: a review,” J. Opt. 12, 043001 (2010). [CrossRef]

23.

A. W. Snyder and X. H. Zheng, “Fused couplers of arbitrary cross-section,” Electron. Lett. 21, 1079–1080 (1985). [CrossRef]

24.

S. Lacroix, F. Gonthier, and J. Bures, “Modeling of symmetric 2× 2 fused-fiber couplers,” Appl. Opt. 33, 8361–8369 (1994). [CrossRef] [PubMed]

25.

F. P. Payne, C. D. Hussey, and M. S. Yataki, “Polarisation analysis of strongly fused and weakly fused tapered couplers,” Electron. Lett. 21, 561–563 (1985). [CrossRef]

26.

H. Chang, T. Lin, and T. Wu, “Accurate coupling coefficients for fiber couplers with weakly fused cross sections,” Appl. Opt. 34, 6168–6171 (1995). [CrossRef] [PubMed]

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(060.1810) Fiber optics and optical communications : Buffers, couplers, routers, switches, and multiplexers

ToC Category:
Fiber Optics

History
Original Manuscript: February 27, 2014
Revised Manuscript: April 7, 2014
Manuscript Accepted: April 23, 2014
Published: May 6, 2014

Citation
Rand Ismaeel, Timothy Lee, Bernard Oduro, Yongmin Jung, and Gilberto Brambilla, "All-fiber fused directional coupler for highly efficient spatial mode conversion," Opt. Express 22, 11610-11619 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-10-11610


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References

  1. D. J. Richardson, J. M. Fini, L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013). [CrossRef]
  2. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012). [CrossRef]
  3. S. Leon-Saval, T. Birks, J. Bland-Hawthorn, M. Englund, “Multimode fiber devices with single-mode performance,” Opt. Lett. 30, 2545–2547 (2005). [CrossRef] [PubMed]
  4. S. Leon-Saval, A. Argyros, J. Bland-Hawthorn, “Photonic lanterns: a study of light propagation in multimode to single-mode converters,” Opt. Express 18, 8430–8439 (2010). [CrossRef] [PubMed]
  5. S. Leon-Saval, N. Fontaine, J. Salazar-Gil, B. Ercan, R. Ryf, J. Bland-Hawthorn, “Mode-selective photonic lanterns for space-division multiplexing,” Opt. Express 22, 1036–1044 (2014). [CrossRef] [PubMed]
  6. N. K. Fontaine, S. G. Leon-Saval, R. Ryf, J. R. S. Gil, B. Ercan, J. Bland-Hawthorn, “Mode-selective dissimilar fiber photonic-lantern spatial multiplexers for few-mode fiber,” Optical Communication (ECOC 2013), 39th European Conference and Exhibition, pp. 1, 3, 22–26 Sept. 2013
  7. A. M. Vengsarkar, J. A. Greene, K. A. Murphy, “Photoinduced refractive-index changes in two-mode elliptical-core fibers: sensing applications,” Opt. Lett. 16, 1541–1543 (1991). [CrossRef] [PubMed]
  8. C. D. Poole, C. D. Townsend, K. T. Nelson, “Helical-grating two-mode fiber spatial-mode coupler,” J. Light-waveTechnol. 9, 598–604 (1991). [CrossRef]
  9. H. G. Park, B. Y. Kim, “Intermodal coupler using permanently photoinduced grating in two-modeoptical fiber,” Electron. Lett. 25, 797–799 (1989). [CrossRef]
  10. F. A. Castro, S. R. M. Carneiro, O. Lisboa, S. L. A. Carrara, “Two-mode optical fiber accelerometer,” Opt. Lett. 17, 1474–1475 (1992). [CrossRef] [PubMed]
  11. S. Y. Haung, J. N. Blake, B. Y. Kim, “Perturbation effects on mode propagation in highly elliptical core two-mode fibers,” J. Lightwave Technol. 8, 23–33 (1990). [CrossRef]
  12. J. N. Blake, S. U. Huang, B. Y. Kim, H. J. Shaw, “Strain effects on highly elliptical core two-mode fibers,” Opt. Lett. 12, 732–734 (1987). [CrossRef] [PubMed]
  13. W. V. Sorin, B. Y. Kim, H. J. Shaw, “Highly selective evanescent modal filter for two-mode optical fibers,” Opt. Lett. 11, 581–583 (1986). [CrossRef] [PubMed]
  14. R. C. Youngquist, J. L. Brooks, H. J. Shaw, “Two-mode fiber modal coupler,” Opt. Lett. 9, 177–179 (1984). [CrossRef] [PubMed]
  15. M. Vaziri, C. Chen, “An etched two-mode fiber modal coupling element,” J. Lightwave Technol. 15, 474–481 (1997). [CrossRef]
  16. Y. Shou, J. Bures, S. Lacroix, X. Daxhelet, “Mode separation in fused fiber coupler made of two-mode fibers,” Opt. Fiber Technol. 5, 92–104 (1999). [CrossRef]
  17. K. Y. Song, I. K. Hwang, S. H. Yun, B. Y. Kim, “High performance fused-type mode selective coupler for two-mode fiber devices,” in Opt. Fibre Commun. Conf., Baltimore, 7–10 Mar (2000).
  18. K. Y. Song, I. K. Hwang, S. H. Yun, B. Y. Kim, “High performance fused-type mode-selective coupler using elliptical core two-mode fiber at 1550nm,” IEEE Photon. Technol. Lett. 14, 501–503 (2002). [CrossRef]
  19. Y. Jung, R. Chen, R. Ismaeel, G. Brambilla, S. Alam, I. Giles, D. Richardson, “Dual mode fused optical fiber couplers suitable for mode division multiplexed transmission,” Opt. Express 21, 24326–24331 (2013). [CrossRef] [PubMed]
  20. S. Yerolatsitis, I. Gris-Sánchez, T. Birks, “Adiabatically-tapered fiber mode multiplexers,” Opt. Express 22, 608–617 (2014). [CrossRef] [PubMed]
  21. K. Chiang, “Effects of cores in fused tapered single-mode fiber couplers,” Opt. Lett. 12, 431–433 (1987). [CrossRef] [PubMed]
  22. G. Brambilla, “Optical fiber nanowires and microwires: a review,” J. Opt. 12, 043001 (2010). [CrossRef]
  23. A. W. Snyder, X. H. Zheng, “Fused couplers of arbitrary cross-section,” Electron. Lett. 21, 1079–1080 (1985). [CrossRef]
  24. S. Lacroix, F. Gonthier, J. Bures, “Modeling of symmetric 2× 2 fused-fiber couplers,” Appl. Opt. 33, 8361–8369 (1994). [CrossRef] [PubMed]
  25. F. P. Payne, C. D. Hussey, M. S. Yataki, “Polarisation analysis of strongly fused and weakly fused tapered couplers,” Electron. Lett. 21, 561–563 (1985). [CrossRef]
  26. H. Chang, T. Lin, T. Wu, “Accurate coupling coefficients for fiber couplers with weakly fused cross sections,” Appl. Opt. 34, 6168–6171 (1995). [CrossRef] [PubMed]

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