Abstract

We propose an ultra-broadband mode converter based on the structure of a length-apodized long-period grating, where π-phase shifts are introduced at strategic locations of the grating profile. Using a 3-section length-apodized grating structure, we design and fabricate an LP01-LP11a and an LP01-LP11b mode converter with a sidewall grating and a surface grating formed along a polymer channel waveguide, respectively. The fabricated LP01-LP11a and LP01-LP11b mode converters provide a conversion efficiency higher than 99% over a bandwidth of ~120 nm and ~150 nm, respectively, or a conversion efficiency higher than 90% over a bandwidth of ~180 nm and ~300 nm, respectively. The transmission characteristics of these devices are weakly sensitive to polarization and temperature variations. These mode converters can find applications in ultra-broadband mode-division-multiplexing transmission systems based on few-mode fibers and the design principle can be applied to general grating-based mode-coupling devices for a wide range of applications.

© 2017 Optical Society of America

1. Introduction

Mode-division multiplexing (MDM), which allows different spatial modes of a few-mode fiber (FMF) to transmit independent signal channels, is a promising technology to increase the signal-carrying capacity of an optical fiber. For the MDM technology to be compatible with existing single-mode fiber devices, it is necessary to develop devices that convert the fundamental mode into a high-order mode, or, more generally, between any two modes of an FMF. Such mode converters are also needed for the construction of advanced MDM networks that require routing and switching of modes.

Mode converters have been demonstrated with bulk-optics components [1–4], optical fibers [5–11], and optical waveguides [12–16]. Bulk-optics mode converters [1–4] can be implemented with commercially available components, such as phase plates and liquid-crystal spatial modulators, but they require critical alignment and have a large insertion loss. To overcome the problems with bulk-optics devices, several all-fiber mode converters based on mode-selective directional couplers [5–9] and long-period fiber gratings (LPFGs) [10,11] have been proposed. To further improve the flexibility and the controllability in the design and the production of mode converters, various waveguide-based mode converters have been developed, which include, for example, trenched waveguides [12], Mach-Zehnder interferometers [13], micro-resonators [14], and long-period waveguide gratings (LPWGs) [15,16]. Among these, LPWGs are particularly simple and flexible structures for the design and the implementation of mode converters. For example, surface and sidewall LPWGs can be integrated to realize advanced conversion functions for high-order modes [15].

To exploit the full capacity of MDM, each mode of the FMF should carry dense wavelength-division-multiplexed (DWDM) signals, which suggests that the mode-selective devices used in an MDM system should be able to operate over the entire (C + L)-band (from 1530 nm to 1610 nm). As a conventional long-period grating (LPG) is a strongly wavelength-selective device, all the grating-based mode converters reported so far, whether with fibers [10,11] or waveguides [15,16], have limited bandwidths. The widest bandwidth achieved with an LPFG mode converter is 34 nm [10]. The most commonly used technique to increase the bandwidth of an LPG is to chirp the grating profile [17], but chirping can significantly lower the mode-conversion efficiency and the bandwidth increase is still limited [18]. Another possible technique is to operate an LPG at its turning point along its phase-matching curve [19], but such a turning point may not exist for a given set of guided modes and, even though it exists, the transmission spectrum would be highly sensitive to fabrication errors and environmental disturbances. In this paper, we demonstrate a new technique to largely increase the bandwidth of an LPG mode converter, which is based on applying length apodization to the grating profile.

Length apodization, first proposed in 1998 [20], refers to the use of different section lengths in a multi-section phase-shifted LPG. A detailed theoretical analysis of LPGs shows the possibility of achieving ultra-broadband operation with a length-apodized LPG [21], but no experimental verification has yet been reported. In this study, we present two ultra-broadband length-apodized LPG mode converters: an LP01-LP11a mode converter and an LP01-LP11b mode converter, which are formed with a sidewall grating and a surface grating along a polymer optical waveguide, respectively. The fabricated LP01-LP11a and LP01-LP11b mode converters can operate at a conversion efficiency higher than 99% over a bandwidth of ~120 and ~150 nm, respectively, or at a conversion efficiency higher than 90% over a bandwidth of ~180 and ~300 nm, respectively. The performances of these devices are weakly sensitive to polarization and temperature variations. These mode converters have the largest bandwidths ever reported and could be used in ultra-broadband MDM systems. The design principle can be applied to general grating-based devices for a wide spectrum of applications.

2. Principle and design

Figure 1 shows the variation of the refractive index of a waveguide core along the wave propagation direction (the z direction) for a length-apodized LPG. The grating consists of (M – 1) π-phase shifts, which separate the grating into M sections with different lengths, z1, z2, …, zM. The period and the index-modulation depth (i.e., the grating depth) are the same for all sections, which are denoted as Λ and h, respectively. The total length of the grating is L. The grating is designed to achieve coupling between the LP01 mode and the LP11 mode, where the LP11 mode can be the LP11a mode, whose electric-field distribution changes sign in the horizontal direction, or the LP11b mode, whose electric-field distribution changes sign in the vertical direction. The resonance wavelength of the grating, at which the mode-coupling effect is strongest, is determined by the phase-matching condition [22]:

λ0=(N01N11)Λ,
where N01 and N11 are the effective indices of the LP01 and LP11 modes, respectively. The electric fields of the LP01 and LP11 modes at the output end of the LPG, denoted as E01(L) and E11(L), respectively, can be obtained from the input fields E01(0) and E11(0) by [20,21]
(E01(L)E11(L))=FMF2F1(E01(0)E11(0)),
where Fi (i = 1, 2, …, M) is the transfer matrix for the i-th section:
Fi=(ej(β01δ)zi00ej(β11+δ)zi)×(cos(Qzi)jδQsin(Qzi)κQejφisin(Qzi)κQejφisin(Qzi)cos(Qzi)+jδQsin(Qzi)).
In the above expression, β01 and β11 are the propagation constants of the LP01 and LP11 modes, δ = π/Λ(λ0/λ − 1) is the phase mismatch, which is the deviation of the operation wavelength λ from the resonance wavelength λ0, κ is the coupling coefficient, which is a measure of the spatial overlap between the fields of the two coupled modes in the grating area of the waveguide, Q=(δ2+κ2)1/2, L=i=1i=Mzi, and φi=2πΛj=1j=i1zj+(i1)π is the initial phase of the i-th section.

 figure: Fig. 1

Fig. 1 Profile of a length-apodized LPG.

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We assume that only the LP01 mode is launched into the waveguide, i.e., E01(0) ≠ 0 and E11(0) = 0. In that case, the output powers of the two modes are obtained as

|E01(L)|2=cos2[i=1i=M(1)iκzi]|E01(0)|2,
|E11(L)|2=(1cos2[i=1i=M(1)iκzi])|E01(0)|2.
To achieve zero transmission of the LP01 mode at the resonance wavelength, we find from Eq. (4) the following condition:
i=1M(1)izi=±π/2κ,
or

κ=±π/2[i=1M(1)izi]forM >1.

Here we employ a linear length apodization profile to achieve ultra-broadband mode conversion [21], where the lengths of the sections are given by zi = L/M + [(i – 1) – (M − 1)/2]NΛ (for i = 1, 2, .., M) with N (a constant integer) being the difference in the number of periods between adjacent sections. In principle, the bandwidth of the LPG increases with the number of sections, but a larger number of sections requires a larger coupling coefficient [21]. To limit the length of the grating and the magnitude of the coupling coefficient (for practical considerations), we assume three grating sections in our study, i.e., M = 3.

We design two specific ultra-broadband mode converters with channel waveguides. The structures of the two devices are shown in Fig. 2. The structure with a sidewall grating, as shown in Fig. 2(a), is designed for the conversion between the LP01 and the LP11a mode, while the structure with a surface grating, as shown in Fig. 2(b), is designed for the conversion between the LP01 and the LP11b mode. The corrugation duty cycles for both gratings are 50%. As shown in Figs. 1 and 2, a π-phase shift can be introduced between two grating sections by simply adding an extra half grating period between them. To demonstrate the flexibility of the technique, we use different waveguide parameters and different grating profiles in the design of the two mode converters.

 figure: Fig. 2

Fig. 2 Structures of (a) the LP01-LP11a mode converter and (b) the LP01-LP11b mode converter.

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For the mode converter based on the sidewall grating, the refractive indices of the core and the cladding of the waveguide are fixed at nco = 1.5730 and ncl = 1.5595, respectively, which are the refractive indices of the polymer materials used in the fabrication work. The width and the height of the core are wco = 7.0 μm and hco = 3.6 μm, respectively. The waveguide supports only the LP01 and LP11a modes. We calculate these modes with a commercial mode solver (COMSOL) based on the finite-element method. At the wavelength 1570 nm, which is the center wavelength in the (C + L)-band, the effective indices of the LP01 and the LP11a mode are 1.5659 and 1.5606 (for the x-polarization), respectively. According to Eq. (1), the grating period required is Λ = 294 μm.

We fix the total length of the grating to 30 periods, i.e., L = 30Λ = 8.82 mm. With L = 30Λ, the length of the second section is given by z2 = L/M = 10Λ. The lengths of the other two sections depend on the value of N used. For example, with N = 3, 6, and 9, we have z1 = 7Λ and z3 = 13Λ; z1 = 4Λ and z3 = 16Λ; and z1 = Λ and z3 = 19Λ, respectively. Figure 3 shows the normalized transmission spectra of the LP01 mode, i.e., |E01(L)|2/|E01(0)|2, calculated for N = 0, 3, 6, and 9. The coupling coefficient is fixed at κ = 530 m−1, which is the value that satisfies Eq. (7). As shown in Fig. 3, the grating with z1 = 4Λ, z2 = 10Λ, and z3 = 16Λ (i.e., N = 6) provides a −20-dB bandwidth large enough to cover the entire (C + L)-band, or a −10-dB bandwidth as wide as ~200 nm. Figure 4 shows the variation of the normalized transmission spectrum with the coupling coefficient for the grating with N = 6. As shown in Fig. 4, the −20-dB bandwidth of the grating is larger than 80 nm over the range of κ from 500 m−1 to 550 m−1. The −10-dB bandwidth is much less sensitive to the value of κ. In our design, we choose N = 6, i.e., z1 = 4Λ, z2 = 10Λ, and z3 = 16Λ. From the results shown in Fig. 4(b), we choose a coupling coefficient of κ = 504 m−1. Given the mode field distributions and the coupling coefficient, we calculate the sidewall corrugation depth required from the coupled-mode theory [23] and the result is h = 560 nm.

 figure: Fig. 3

Fig. 3 Normalized transmission spectra of the LP01 mode calculated for four 3-section length-apodized LPGs with L = 30Λ, z2 = 10Λ, and κ = 530 m−1.

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 figure: Fig. 4

Fig. 4 (a) Normalized transmission spectra of the LP01 mode calculated for a 3-section length-apodized LPG with z1 = 4Λ, z2 = 10Λ, and z3 = 16Λ at different values of the coupling coefficient κ and (b) variation of the −20-dB bandwidth with the coupling coefficient κ.

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For the mode converter based on the surface grating, the refractive indices of the core and the cladding of the waveguide are fixed at nco = 1.5706 and ncl = 1.5595, respectively. The width and the height of the core and the cladding of the waveguide are wco = 6.0 μm and hco = 7.2 μm, respectively. At the wavelength 1570 nm, the effective indices of the LP01 and the LP11b mode are 1.5661 and 1.5610, respectively. According to Eq. (1), the grating period required is Λ = 310 μm. For the waveguide dimensions chosen, the period obtained is similar to that of the sidewall grating.

We fix the total length of the grating to 24 periods, i.e., L = 24Λ = 7.44 mm. With L = 24Λ, the length of the second section is z2 = 8Λ. Figure 5 shows the normalized transmission spectra of the LP01 mode calculated for N = 1 (z1 = 7Λ and z3 = 9Λ), 3 (z1 = 5Λ and z3 = 11Λ), 5 (z1 = 3Λ and z3 = 13Λ), and 7 (z1 = Λ and z3 = 15Λ). The coupling coefficient is fixed at κ = 630 m−1, which is the value that satisfies Eq. (7). As shown in Fig. 5, the grating with N = 5 provides a −20-dB bandwidth larger than 80 nm, or a −10-dB bandwidth well larger than 200 nm. Figure 6 shows the effects of the value of κ on the normalized transmission spectrum and the bandwidth for the grating with N = 5. The −20-dB bandwidth is larger than 80 nm over the range of the κ value from 590 m−1 to 660 m−1 and the −10-dB bandwidth is larger than 100 nm over an even much wider range of the κ value. In our design, we use N = 5, i.e., z1 = 3Λ, z2 = 8Λ, and z3 = 13Λ. From the results shown in Fig. 6(b), we choose a coupling coefficient of κ = 600 m−1, which corresponds to a surface corrugation depth of h = 640 nm.

 figure: Fig. 5

Fig. 5 Normalized transmission spectra of the LP01 mode calculated for four 3-section length-apodized LPGs with L = 24Λ, z2 = 8Λ, and κ = 630 m−1.

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 figure: Fig. 6

Fig. 6 (a) Normalized transmission spectra of the LP01 mode calculated for a 3-section length-apodized LPG with z1 = 3Λ, z2 = 8Λ, and z3 = 13Λ at different values of the coupling coefficient κ and (b) variation of the −20-dB bandwidth with the coupling coefficient κ.

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As shown by the above examples, there are many possible designs for the achievement of ultra-broadband operation. While the condition given by Eq. (7) provides a suitable value for the coupling coefficient required, it is not necessarily the optimal value and, as shown in Figs. 4(b) and 6(b), a large range of the coupling coefficient is allowed for good performance, which suggests much relaxed tolerances in the control of the refractive-index modulation in the fabrication process. Our results presented in Figs. 3-6 ignore the dispersion effects in the waveguide and, as a result, the transmission spectra obtained are symmetrical with respect to the resonance wavelength. The results are universal and material-independent. Because the core and the cladding material have similar dispersion properties, material dispersion should not affect much the resonance wavelength and the coupling coefficient. By including the dispersion effects in the calculation, we obtain slightly asymmetrical rejection bands with somewhat larger bandwidths. Because the refractive-index difference between the core and the cladding is small, the resonance wavelengths of the two mode converters are only weakly sensitive to the polarization. The difference in the resonance wavelength between the two polarizations, which is much smaller than the bandwidth of the device, does not significantly affect the performance of the device.

3. Device fabrication

We followed the design parameters as closely as possible in the fabrication of the two mode converters. The polymer materials used were EpoClad and EpoCore (Micro Resist Technology), which could be mixed in different proportions to control the refractive indices. The refractive indices of the core and cladding materials in thin-film form were measured at the wavelength 1538 nm with a commercial prism coupler system (Metricon 2010).

To fabricate the LP01-LP11a mode converter, we first spin-coated a thick film of lower cladding onto a silicon (Si) substrate and then an EpoCore film onto the lower cladding to the desired thickness. The waveguide core was formed by the photolithography process with a mask that contained the designed sidewall-corrugated pattern and a biconical taper near one end of the core. The taper, which was used to filter out the LP11a mode to facilitate the launching of a pure LP01 mode into the mode converter, had a linear profile with a width of 3.0 μm at its waist and a total length of 3.0 mm. Finally, we spin-coated a thick (>10 μm) upper cladding layer (EpoClad) onto the core. The total length of the device was ~20 mm. Figure 7(a) shows a microscopic image of a sidewall corrugation (before the application of the upper cladding) and Fig. 7(b) shows an end face of a fabricated device.

 figure: Fig. 7

Fig. 7 Images (a) of sidewall corrugation and (b) an end face of a typical fabricated LP01-LP11a mode converter.

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The fabrication of the LP01-LP11b mode converter was similar to that of the LP01-LP11a mode converter, except that an additional mask was required to define the surface grating and, hence, more steps were involved. After the formation of the waveguide core, the core was etched into a grating with the inductive coupling plasma (ICP) etching process. To filter out the LP11b mode, a vertical taper with gradual reduction of the core height was formed near one end of the core. To create such a vertical taper, two stacks of staggered 3-mm thick glass slides were placed on the core with a horizontal gap of 2 mm at the top and 5 mm at the bottom and the gap area was subject to ICP etching for about 50 minutes. Because of the shadows of the glass slides, the etching depth increased gradually from the center of the gap towards the edges of the glass slides, which thus resulted in a biconical taper with a varying core height. The core height at the waist of the taper was ~3 μm. Figure 8(a) shows an image of a surface corrugation (before the application of the upper cladding) and Fig. 8(b) shows an end face of a typical fabricated device. The total length of the device was ~17 mm.

 figure: Fig. 8

Fig. 8 Images (a) of surface corrugation and (b) an end face of a typical fabricated LP01-LP11b mode converter.

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4. Measurements and discussions

We first measured the characteristics of the LP01-LP11a mode converter. We launched the LP01 mode into the waveguide with a lensed single-mode fiber (SMF) and a broadband source (SuperK COMPACT KOHERAS) and detected only the LP01 mode at the output end with another lensed SMF and an optical spectrum analyzer (AQ6370, Yokogawa). We could adjust the polarization state of the input light with a polarization controller placed at the input end and select the polarization to be detected with a polarizer placed at the output end. Figure 9 shows the transmission spectra measured at different operating temperatures for the x and y polarizations, where strong rejection bands can be seen. The transmission spectra shown in Fig. 9 were normalized with respect to the output spectrum of a reference waveguide of the same dimensions formed on the same substrate. As shown in Fig. 9, the results are similar for both polarizations. The −10-dB bandwidth is larger than ~180 nm, which is weakly sensitive to temperature variations. The −20-dB bandwidth increases from ~80 nm to ~120 nm as the temperature increases from 20 °C to 34 °C, and does not change much as the temperature increases further to 47 °C.

 figure: Fig. 9

Fig. 9 Normalized transmission spectra of the LP01-LP11a mode converter measured at different operating temperatures for (a) the x-polarization and (b) the y-polarization.

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To confirm that the rejection bands shown in Fig. 9 were due to the coupling from the LP01 mode to the LP11a mode, we launched the LP01 mode into the mode converter with a tunable laser (Agilent 8164B) and captured the near-field images from the output end with an infrared camera (Hamamatsu, Model C2741-03). Figure 10 shows the output near-field images taken at different wavelengths in the C-band, where clean LP11a mode patterns can be seen for both polarizations. These results confirm the function of LP01-LP11a mode conversion. The excess losses induced by the grating and the taper in the mode converter, measured by comparing the output power of the device with those of reference waveguides of the same dimensions, were ~0.8 dB and ~0.13 dB, respectively. The propagation loss of the device, measured by the cutback method on a reference waveguide, was ~2 dB/cm.

 figure: Fig. 10

Fig. 10 Output near-field images taken at different wavelengths for the LP01-LP11a mode converter, when the LP01 mode was launched into the device.

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We next characterized the performance of the LP01-LP11b mode converter. Figure 11 shows the normalized transmission spectra measured at different operating temperatures for the two polarizations. The −10-dB bandwidth and the −20-dB bandwidth are larger than ~300 nm and ~150 nm, respectively, which cover the entire (C + L)-band and beyond. As shown in Fig. 11, the transmission characteristics are only weakly sensitive to the polarization of light and temperature variations. Figure 12 shows the output near-field images taken at different wavelengths for both polarizations when the LP01 mode was launched into the device. The clean LP11b mode patterns obtained confirms the function of LP01-LP11b mode conversion. The slight asymmetry of the mode field in the vertical direction was due to the rather thin upper cladding used in this device. The excess losses induced by the grating and the vertical taper in the mode converter were estimated to be 1.2 dB and 1.0 dB, respectively. The waveguide loss was ~2 dB/cm.

 figure: Fig. 11

Fig. 11 Normalized transmission spectra of the LP01-LP11b mode converter measured at different operating temperatures for (a) the x-polarization and (b) the y-polarization.

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 figure: Fig. 12

Fig. 12 Output near-field images taken at different wavelengths for the LP01-LP11b mode converter, when the LP01 mode was launched into the device.

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For both devices, the agreements between the experimental results and the theoretical results are good, considering the uncertainties in the control of the waveguide and grating parameters in the fabrication process and the assumptions made in the theory. According to our calculation, the −20-dB bandwidths of the mode converters vary a few nanometers when the core dimensions and the corrugation depths deviate from the design values by ± 0.3 μm and ± 30 nm, respectively, which are roughly the tolerances in the control of these parameters in our fabrication process.

5. Conclusion

We have proposed a technique of realizing ultra-broadband mode converters for MDM applications based on length-apodized LPGs. To demonstrate the flexibility and the effectiveness of this technique, we have designed and fabricated LP01-LP11a and LP01-LP11b mode converters with 3-section length-apodized sidewall and surface grating structures formed along polymer channel waveguides. The fabricated LP01-LP11a and LP01-LP11b mode converters can operate at a conversion efficiency higher than 99% over a bandwidth of ~120 nm and ~150 nm, respectively, or a conversion efficiency higher than 90% over a bandwidth of ~180 nm and ~300 nm, respectively. The performance of these devices is weakly sensitive to polarization and temperature variations. These mode converters have the largest bandwidths ever demonstrated and can find immediate applications in MDM systems. The design principle can be applied to general grating-based mode-coupling devices, whether with fibers or waveguides, for a wide range of applications.

Funding

National Natural Science Foundation of China (61377057, 61177054, U1533121); National Postdoctoral Program for Innovative Talents, China (BX201600027); Fundamental Research Funds for the Central Universities, China (ZYGX2015J006); Open Fund of State Key Laboratory of Integrated Optoelectronics of Jilin University, China (IOSKL2014KF07); Open Fund of State Key Laboratory of IPOC of BUPT, China (IPOC2016B007); The 111 Project of UESTC, China (B14039).

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References

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  1. S. Randel, R. Ryf, A. Sierra, P. J. Winzer, A. H. Gnauck, C. A. Bolle, R. J. Essiambre, D. W. Peckham, A. McCurdy, and R. Lingle., “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt. Express 19(17), 16697–16707 (2011).
    [Crossref] [PubMed]
  2. R. Ryf, M. A. Mestre, S. Randel, C. Schmidt, A. H. Gnauck, R.-J. Essiambre, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, X. Jiang, D. W. Peckham, A. McCurdy, and R. Lingle., “Mode-multiplexed transmission over a 209-km DGD-compensated hybrid few-mode fiber span,” IEEE Photonics Technol. Lett. 24(21), 1965–1968 (2012).
    [Crossref]
  3. C. Koebele, M. Salsi, D. Sperti, P. Tran, P. Brindel, H. Mardoyan, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, F. Cerou, and G. Charlet, “Two mode transmission at 2×100 Gb/s, over 40 km-long prototype few-mode fiber, using LCOS-based programmable mode multiplexer and demultiplexer,” Opt. Express 19(17), 16593–16600 (2011).
    [Crossref] [PubMed]
  4. J. von Hoyningen-Huene, R. Ryf, and P. Winzer, “LCoS-based mode shaper for few-mode fiber,” Opt. Express 21(15), 18097–18110 (2013).
    [Crossref] [PubMed]
  5. S. Choi, K. Oh, W. Shin, C. S. Park, U. C. Paek, K. J. Park, Y. C. Chung, G. Y. Kim, and Y. G. Lee, “Novel Mode Converter Based on Hollow Optical Fiber for Gigabit LAN Communication,” IEEE Photonics Technol. Lett. 14(2), 248–250 (2002).
    [Crossref]
  6. A. Witkowska, S. G. Leon-Saval, A. Pham, and T. A. Birks, “All-fiber LP11 mode convertors,” Opt. Lett. 33(4), 306–308 (2008).
    [Crossref] [PubMed]
  7. G. Pelegrina-Bonilla, K. Hausmann, H. Sayinc, U. Morgner, J. Neumann, and D. Kracht, “Analysis of the modal evolution in fused-type mode-selective fiber couplers,” Opt. Express 23(18), 22977–22990 (2015).
    [Crossref] [PubMed]
  8. K. J. Park, K. Y. Song, Y. K. Kim, J. H. Lee, and B. Y. Kim, “Broadband mode division multiplexer using all-fiber mode selective couplers,” Opt. Express 24(4), 3543–3549 (2016).
    [Crossref] [PubMed]
  9. A. B. Taher, P. Di Bin, F. Bahloul, E. Tartaret-Josnière, M. Jossent, S. Février, and R. Attia, “Adiabatically tapered microstructured mode converter for selective excitation of the fundamental mode in a few mode fiber,” Opt. Express 24(2), 1376–1385 (2016).
    [Crossref] [PubMed]
  10. J. Dong and K. S. Chiang, “Temperature-insensitive mode converters with CO2-laser written long-period fiber gratings,” IEEE Photonics Technol. Lett. 27(9), 1006–1009 (2015).
    [Crossref]
  11. Y. Zhao, Y. Liu, L. Zhang, C. Zhang, J. Wen, and T. Wang, “Mode converter based on the long-period fiber gratings written in the two-mode fiber,” Opt. Express 24(6), 6186–6195 (2016).
    [Crossref] [PubMed]
  12. K. Saitoh, T. Uematsu, N. Hanzawa, Y. Ishizaka, K. Masumoto, T. Sakamoto, T. Matsui, K. Tsujikawa, and F. Yamamoto, “PLC-based LP11 mode rotator for mode-division multiplexing transmission,” Opt. Express 22(16), 19117–19130 (2014).
    [Crossref] [PubMed]
  13. Y. Huang, G. Xu, and S. Ho, “An ultracompact optical mode order converter,” IEEE Photonics Technol. Lett. 18(21), 2281–2283 (2006).
    [Crossref]
  14. Y. Yu, M. Ye, and S. Fu, “On-chip polarization controlled mode converter with capability of WDM operation,” IEEE Photonics Technol. Lett. 27(18), 1957–1960 (2015).
    [Crossref]
  15. W. Jin and K. S. Chiang, “Mode converters based on cascaded long-period waveguide gratings,” Opt. Lett. 41(13), 3130–3133 (2016).
    [Crossref] [PubMed]
  16. Y. Yang, K. Chen, W. Jin, and K. S. Chiang, “Widely wavelength-tunable mode converter based on polymer waveguide grating,” IEEE Photonics Technol. Lett. 27(18), 1985–1988 (2015).
    [Crossref]
  17. Q. Liu and K. S. Chiang, “Design of long-period waveguide grating filter by control of waveguide cladding profile,” J. Lightwave Technol. 24(9), 3540–3546 (2006).
    [Crossref]
  18. T. Liu, I. B. Djordjevic, Z. Song, Y. Chen, R. Zhang, K. Zhang, W. Zhao, and B. Li, “Broadband wavelength converters with flattop responses based on cascaded second-harmonic generation and difference frequency generation in Bessel-chirped gratings,” Opt. Express 24(10), 10946–10955 (2016).
    [Crossref] [PubMed]
  19. Q. Liu, K. S. Chiang, and K. P. Lor, “Dual resonance in a long-period waveguide grating,” Appl. Phys. B 86(1), 147–150 (2007).
    [Crossref]
  20. H. Ke, K. S. Chiang, and J. H. Peng, “Analysis of phased-shifted long-period fiber gratings,” IEEE Photonics Technol. Lett. 10(11), 1596–1598 (1998).
    [Crossref]
  21. F. Y. M. Chan and K. S. Chiang, “Analysis of apodized phase-shifted long-period fiber gratings,” Opt. Commun. 244(1-6), 233–243 (2005).
    [Crossref]
  22. T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
    [Crossref]
  23. Q. Liu, K. S. Chiang, and V. Rastogi, “Analysis of corrugated long-period gratings in slab waveguides and their polarization dependence,” J. Lightwave Technol. 21(12), 3399–3405 (2003).
    [Crossref]

2016 (5)

2015 (4)

Y. Yang, K. Chen, W. Jin, and K. S. Chiang, “Widely wavelength-tunable mode converter based on polymer waveguide grating,” IEEE Photonics Technol. Lett. 27(18), 1985–1988 (2015).
[Crossref]

G. Pelegrina-Bonilla, K. Hausmann, H. Sayinc, U. Morgner, J. Neumann, and D. Kracht, “Analysis of the modal evolution in fused-type mode-selective fiber couplers,” Opt. Express 23(18), 22977–22990 (2015).
[Crossref] [PubMed]

Y. Yu, M. Ye, and S. Fu, “On-chip polarization controlled mode converter with capability of WDM operation,” IEEE Photonics Technol. Lett. 27(18), 1957–1960 (2015).
[Crossref]

J. Dong and K. S. Chiang, “Temperature-insensitive mode converters with CO2-laser written long-period fiber gratings,” IEEE Photonics Technol. Lett. 27(9), 1006–1009 (2015).
[Crossref]

2014 (1)

2013 (1)

2012 (1)

R. Ryf, M. A. Mestre, S. Randel, C. Schmidt, A. H. Gnauck, R.-J. Essiambre, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, X. Jiang, D. W. Peckham, A. McCurdy, and R. Lingle., “Mode-multiplexed transmission over a 209-km DGD-compensated hybrid few-mode fiber span,” IEEE Photonics Technol. Lett. 24(21), 1965–1968 (2012).
[Crossref]

2011 (2)

2008 (1)

2007 (1)

Q. Liu, K. S. Chiang, and K. P. Lor, “Dual resonance in a long-period waveguide grating,” Appl. Phys. B 86(1), 147–150 (2007).
[Crossref]

2006 (2)

Q. Liu and K. S. Chiang, “Design of long-period waveguide grating filter by control of waveguide cladding profile,” J. Lightwave Technol. 24(9), 3540–3546 (2006).
[Crossref]

Y. Huang, G. Xu, and S. Ho, “An ultracompact optical mode order converter,” IEEE Photonics Technol. Lett. 18(21), 2281–2283 (2006).
[Crossref]

2005 (1)

F. Y. M. Chan and K. S. Chiang, “Analysis of apodized phase-shifted long-period fiber gratings,” Opt. Commun. 244(1-6), 233–243 (2005).
[Crossref]

2003 (1)

2002 (1)

S. Choi, K. Oh, W. Shin, C. S. Park, U. C. Paek, K. J. Park, Y. C. Chung, G. Y. Kim, and Y. G. Lee, “Novel Mode Converter Based on Hollow Optical Fiber for Gigabit LAN Communication,” IEEE Photonics Technol. Lett. 14(2), 248–250 (2002).
[Crossref]

1998 (1)

H. Ke, K. S. Chiang, and J. H. Peng, “Analysis of phased-shifted long-period fiber gratings,” IEEE Photonics Technol. Lett. 10(11), 1596–1598 (1998).
[Crossref]

1997 (1)

T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[Crossref]

Astruc, M.

Attia, R.

Bahloul, F.

Bigo, S.

Birks, T. A.

Bolle, C. A.

Boutin, A.

Brindel, P.

Cerou, F.

Chan, F. Y. M.

F. Y. M. Chan and K. S. Chiang, “Analysis of apodized phase-shifted long-period fiber gratings,” Opt. Commun. 244(1-6), 233–243 (2005).
[Crossref]

Charlet, G.

Chen, K.

Y. Yang, K. Chen, W. Jin, and K. S. Chiang, “Widely wavelength-tunable mode converter based on polymer waveguide grating,” IEEE Photonics Technol. Lett. 27(18), 1985–1988 (2015).
[Crossref]

Chen, Y.

Chiang, K. S.

W. Jin and K. S. Chiang, “Mode converters based on cascaded long-period waveguide gratings,” Opt. Lett. 41(13), 3130–3133 (2016).
[Crossref] [PubMed]

Y. Yang, K. Chen, W. Jin, and K. S. Chiang, “Widely wavelength-tunable mode converter based on polymer waveguide grating,” IEEE Photonics Technol. Lett. 27(18), 1985–1988 (2015).
[Crossref]

J. Dong and K. S. Chiang, “Temperature-insensitive mode converters with CO2-laser written long-period fiber gratings,” IEEE Photonics Technol. Lett. 27(9), 1006–1009 (2015).
[Crossref]

Q. Liu, K. S. Chiang, and K. P. Lor, “Dual resonance in a long-period waveguide grating,” Appl. Phys. B 86(1), 147–150 (2007).
[Crossref]

Q. Liu and K. S. Chiang, “Design of long-period waveguide grating filter by control of waveguide cladding profile,” J. Lightwave Technol. 24(9), 3540–3546 (2006).
[Crossref]

F. Y. M. Chan and K. S. Chiang, “Analysis of apodized phase-shifted long-period fiber gratings,” Opt. Commun. 244(1-6), 233–243 (2005).
[Crossref]

Q. Liu, K. S. Chiang, and V. Rastogi, “Analysis of corrugated long-period gratings in slab waveguides and their polarization dependence,” J. Lightwave Technol. 21(12), 3399–3405 (2003).
[Crossref]

H. Ke, K. S. Chiang, and J. H. Peng, “Analysis of phased-shifted long-period fiber gratings,” IEEE Photonics Technol. Lett. 10(11), 1596–1598 (1998).
[Crossref]

Choi, S.

S. Choi, K. Oh, W. Shin, C. S. Park, U. C. Paek, K. J. Park, Y. C. Chung, G. Y. Kim, and Y. G. Lee, “Novel Mode Converter Based on Hollow Optical Fiber for Gigabit LAN Communication,” IEEE Photonics Technol. Lett. 14(2), 248–250 (2002).
[Crossref]

Chung, Y. C.

S. Choi, K. Oh, W. Shin, C. S. Park, U. C. Paek, K. J. Park, Y. C. Chung, G. Y. Kim, and Y. G. Lee, “Novel Mode Converter Based on Hollow Optical Fiber for Gigabit LAN Communication,” IEEE Photonics Technol. Lett. 14(2), 248–250 (2002).
[Crossref]

Delbue, R.

R. Ryf, M. A. Mestre, S. Randel, C. Schmidt, A. H. Gnauck, R.-J. Essiambre, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, X. Jiang, D. W. Peckham, A. McCurdy, and R. Lingle., “Mode-multiplexed transmission over a 209-km DGD-compensated hybrid few-mode fiber span,” IEEE Photonics Technol. Lett. 24(21), 1965–1968 (2012).
[Crossref]

Di Bin, P.

Djordjevic, I. B.

Dong, J.

J. Dong and K. S. Chiang, “Temperature-insensitive mode converters with CO2-laser written long-period fiber gratings,” IEEE Photonics Technol. Lett. 27(9), 1006–1009 (2015).
[Crossref]

Erdogan, T.

T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[Crossref]

Essiambre, R. J.

Essiambre, R.-J.

R. Ryf, M. A. Mestre, S. Randel, C. Schmidt, A. H. Gnauck, R.-J. Essiambre, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, X. Jiang, D. W. Peckham, A. McCurdy, and R. Lingle., “Mode-multiplexed transmission over a 209-km DGD-compensated hybrid few-mode fiber span,” IEEE Photonics Technol. Lett. 24(21), 1965–1968 (2012).
[Crossref]

Février, S.

Fu, S.

Y. Yu, M. Ye, and S. Fu, “On-chip polarization controlled mode converter with capability of WDM operation,” IEEE Photonics Technol. Lett. 27(18), 1957–1960 (2015).
[Crossref]

Gnauck, A. H.

R. Ryf, M. A. Mestre, S. Randel, C. Schmidt, A. H. Gnauck, R.-J. Essiambre, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, X. Jiang, D. W. Peckham, A. McCurdy, and R. Lingle., “Mode-multiplexed transmission over a 209-km DGD-compensated hybrid few-mode fiber span,” IEEE Photonics Technol. Lett. 24(21), 1965–1968 (2012).
[Crossref]

S. Randel, R. Ryf, A. Sierra, P. J. Winzer, A. H. Gnauck, C. A. Bolle, R. J. Essiambre, D. W. Peckham, A. McCurdy, and R. Lingle., “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt. Express 19(17), 16697–16707 (2011).
[Crossref] [PubMed]

Hanzawa, N.

Hausmann, K.

Ho, S.

Y. Huang, G. Xu, and S. Ho, “An ultracompact optical mode order converter,” IEEE Photonics Technol. Lett. 18(21), 2281–2283 (2006).
[Crossref]

Huang, Y.

Y. Huang, G. Xu, and S. Ho, “An ultracompact optical mode order converter,” IEEE Photonics Technol. Lett. 18(21), 2281–2283 (2006).
[Crossref]

Ishizaka, Y.

Jiang, X.

R. Ryf, M. A. Mestre, S. Randel, C. Schmidt, A. H. Gnauck, R.-J. Essiambre, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, X. Jiang, D. W. Peckham, A. McCurdy, and R. Lingle., “Mode-multiplexed transmission over a 209-km DGD-compensated hybrid few-mode fiber span,” IEEE Photonics Technol. Lett. 24(21), 1965–1968 (2012).
[Crossref]

Jin, W.

W. Jin and K. S. Chiang, “Mode converters based on cascaded long-period waveguide gratings,” Opt. Lett. 41(13), 3130–3133 (2016).
[Crossref] [PubMed]

Y. Yang, K. Chen, W. Jin, and K. S. Chiang, “Widely wavelength-tunable mode converter based on polymer waveguide grating,” IEEE Photonics Technol. Lett. 27(18), 1985–1988 (2015).
[Crossref]

Jossent, M.

Ke, H.

H. Ke, K. S. Chiang, and J. H. Peng, “Analysis of phased-shifted long-period fiber gratings,” IEEE Photonics Technol. Lett. 10(11), 1596–1598 (1998).
[Crossref]

Kim, B. Y.

Kim, G. Y.

S. Choi, K. Oh, W. Shin, C. S. Park, U. C. Paek, K. J. Park, Y. C. Chung, G. Y. Kim, and Y. G. Lee, “Novel Mode Converter Based on Hollow Optical Fiber for Gigabit LAN Communication,” IEEE Photonics Technol. Lett. 14(2), 248–250 (2002).
[Crossref]

Kim, Y. K.

Koebele, C.

Kracht, D.

Lee, J. H.

Lee, Y. G.

S. Choi, K. Oh, W. Shin, C. S. Park, U. C. Paek, K. J. Park, Y. C. Chung, G. Y. Kim, and Y. G. Lee, “Novel Mode Converter Based on Hollow Optical Fiber for Gigabit LAN Communication,” IEEE Photonics Technol. Lett. 14(2), 248–250 (2002).
[Crossref]

Leon-Saval, S. G.

Li, B.

Lingle, R.

R. Ryf, M. A. Mestre, S. Randel, C. Schmidt, A. H. Gnauck, R.-J. Essiambre, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, X. Jiang, D. W. Peckham, A. McCurdy, and R. Lingle., “Mode-multiplexed transmission over a 209-km DGD-compensated hybrid few-mode fiber span,” IEEE Photonics Technol. Lett. 24(21), 1965–1968 (2012).
[Crossref]

S. Randel, R. Ryf, A. Sierra, P. J. Winzer, A. H. Gnauck, C. A. Bolle, R. J. Essiambre, D. W. Peckham, A. McCurdy, and R. Lingle., “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt. Express 19(17), 16697–16707 (2011).
[Crossref] [PubMed]

Liu, Q.

Liu, T.

Liu, Y.

Lor, K. P.

Q. Liu, K. S. Chiang, and K. P. Lor, “Dual resonance in a long-period waveguide grating,” Appl. Phys. B 86(1), 147–150 (2007).
[Crossref]

Mardoyan, H.

Masumoto, K.

Matsui, T.

McCurdy, A.

R. Ryf, M. A. Mestre, S. Randel, C. Schmidt, A. H. Gnauck, R.-J. Essiambre, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, X. Jiang, D. W. Peckham, A. McCurdy, and R. Lingle., “Mode-multiplexed transmission over a 209-km DGD-compensated hybrid few-mode fiber span,” IEEE Photonics Technol. Lett. 24(21), 1965–1968 (2012).
[Crossref]

S. Randel, R. Ryf, A. Sierra, P. J. Winzer, A. H. Gnauck, C. A. Bolle, R. J. Essiambre, D. W. Peckham, A. McCurdy, and R. Lingle., “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt. Express 19(17), 16697–16707 (2011).
[Crossref] [PubMed]

Mestre, M. A.

R. Ryf, M. A. Mestre, S. Randel, C. Schmidt, A. H. Gnauck, R.-J. Essiambre, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, X. Jiang, D. W. Peckham, A. McCurdy, and R. Lingle., “Mode-multiplexed transmission over a 209-km DGD-compensated hybrid few-mode fiber span,” IEEE Photonics Technol. Lett. 24(21), 1965–1968 (2012).
[Crossref]

Morgner, U.

Neumann, J.

Oh, K.

S. Choi, K. Oh, W. Shin, C. S. Park, U. C. Paek, K. J. Park, Y. C. Chung, G. Y. Kim, and Y. G. Lee, “Novel Mode Converter Based on Hollow Optical Fiber for Gigabit LAN Communication,” IEEE Photonics Technol. Lett. 14(2), 248–250 (2002).
[Crossref]

Paek, U. C.

S. Choi, K. Oh, W. Shin, C. S. Park, U. C. Paek, K. J. Park, Y. C. Chung, G. Y. Kim, and Y. G. Lee, “Novel Mode Converter Based on Hollow Optical Fiber for Gigabit LAN Communication,” IEEE Photonics Technol. Lett. 14(2), 248–250 (2002).
[Crossref]

Park, C. S.

S. Choi, K. Oh, W. Shin, C. S. Park, U. C. Paek, K. J. Park, Y. C. Chung, G. Y. Kim, and Y. G. Lee, “Novel Mode Converter Based on Hollow Optical Fiber for Gigabit LAN Communication,” IEEE Photonics Technol. Lett. 14(2), 248–250 (2002).
[Crossref]

Park, K. J.

K. J. Park, K. Y. Song, Y. K. Kim, J. H. Lee, and B. Y. Kim, “Broadband mode division multiplexer using all-fiber mode selective couplers,” Opt. Express 24(4), 3543–3549 (2016).
[Crossref] [PubMed]

S. Choi, K. Oh, W. Shin, C. S. Park, U. C. Paek, K. J. Park, Y. C. Chung, G. Y. Kim, and Y. G. Lee, “Novel Mode Converter Based on Hollow Optical Fiber for Gigabit LAN Communication,” IEEE Photonics Technol. Lett. 14(2), 248–250 (2002).
[Crossref]

Peckham, D. W.

R. Ryf, M. A. Mestre, S. Randel, C. Schmidt, A. H. Gnauck, R.-J. Essiambre, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, X. Jiang, D. W. Peckham, A. McCurdy, and R. Lingle., “Mode-multiplexed transmission over a 209-km DGD-compensated hybrid few-mode fiber span,” IEEE Photonics Technol. Lett. 24(21), 1965–1968 (2012).
[Crossref]

S. Randel, R. Ryf, A. Sierra, P. J. Winzer, A. H. Gnauck, C. A. Bolle, R. J. Essiambre, D. W. Peckham, A. McCurdy, and R. Lingle., “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt. Express 19(17), 16697–16707 (2011).
[Crossref] [PubMed]

Pelegrina-Bonilla, G.

Peng, J. H.

H. Ke, K. S. Chiang, and J. H. Peng, “Analysis of phased-shifted long-period fiber gratings,” IEEE Photonics Technol. Lett. 10(11), 1596–1598 (1998).
[Crossref]

Pham, A.

Provost, L.

Pupalaikis, P.

R. Ryf, M. A. Mestre, S. Randel, C. Schmidt, A. H. Gnauck, R.-J. Essiambre, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, X. Jiang, D. W. Peckham, A. McCurdy, and R. Lingle., “Mode-multiplexed transmission over a 209-km DGD-compensated hybrid few-mode fiber span,” IEEE Photonics Technol. Lett. 24(21), 1965–1968 (2012).
[Crossref]

Randel, S.

R. Ryf, M. A. Mestre, S. Randel, C. Schmidt, A. H. Gnauck, R.-J. Essiambre, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, X. Jiang, D. W. Peckham, A. McCurdy, and R. Lingle., “Mode-multiplexed transmission over a 209-km DGD-compensated hybrid few-mode fiber span,” IEEE Photonics Technol. Lett. 24(21), 1965–1968 (2012).
[Crossref]

S. Randel, R. Ryf, A. Sierra, P. J. Winzer, A. H. Gnauck, C. A. Bolle, R. J. Essiambre, D. W. Peckham, A. McCurdy, and R. Lingle., “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt. Express 19(17), 16697–16707 (2011).
[Crossref] [PubMed]

Rastogi, V.

Ryf, R.

J. von Hoyningen-Huene, R. Ryf, and P. Winzer, “LCoS-based mode shaper for few-mode fiber,” Opt. Express 21(15), 18097–18110 (2013).
[Crossref] [PubMed]

R. Ryf, M. A. Mestre, S. Randel, C. Schmidt, A. H. Gnauck, R.-J. Essiambre, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, X. Jiang, D. W. Peckham, A. McCurdy, and R. Lingle., “Mode-multiplexed transmission over a 209-km DGD-compensated hybrid few-mode fiber span,” IEEE Photonics Technol. Lett. 24(21), 1965–1968 (2012).
[Crossref]

S. Randel, R. Ryf, A. Sierra, P. J. Winzer, A. H. Gnauck, C. A. Bolle, R. J. Essiambre, D. W. Peckham, A. McCurdy, and R. Lingle., “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt. Express 19(17), 16697–16707 (2011).
[Crossref] [PubMed]

Saitoh, K.

Sakamoto, T.

Salsi, M.

Sayinc, H.

Schmidt, C.

R. Ryf, M. A. Mestre, S. Randel, C. Schmidt, A. H. Gnauck, R.-J. Essiambre, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, X. Jiang, D. W. Peckham, A. McCurdy, and R. Lingle., “Mode-multiplexed transmission over a 209-km DGD-compensated hybrid few-mode fiber span,” IEEE Photonics Technol. Lett. 24(21), 1965–1968 (2012).
[Crossref]

Shin, W.

S. Choi, K. Oh, W. Shin, C. S. Park, U. C. Paek, K. J. Park, Y. C. Chung, G. Y. Kim, and Y. G. Lee, “Novel Mode Converter Based on Hollow Optical Fiber for Gigabit LAN Communication,” IEEE Photonics Technol. Lett. 14(2), 248–250 (2002).
[Crossref]

Sierra, A.

Sillard, P.

Song, K. Y.

Song, Z.

Sperti, D.

Sun, Y.

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Opt. Lett. (2)

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Figures (12)

Fig. 1
Fig. 1 Profile of a length-apodized LPG.
Fig. 2
Fig. 2 Structures of (a) the LP01-LP11a mode converter and (b) the LP01-LP11b mode converter.
Fig. 3
Fig. 3 Normalized transmission spectra of the LP01 mode calculated for four 3-section length-apodized LPGs with L = 30Λ, z2 = 10Λ, and κ = 530 m−1.
Fig. 4
Fig. 4 (a) Normalized transmission spectra of the LP01 mode calculated for a 3-section length-apodized LPG with z1 = 4Λ, z2 = 10Λ, and z3 = 16Λ at different values of the coupling coefficient κ and (b) variation of the −20-dB bandwidth with the coupling coefficient κ.
Fig. 5
Fig. 5 Normalized transmission spectra of the LP01 mode calculated for four 3-section length-apodized LPGs with L = 24Λ, z2 = 8Λ, and κ = 630 m−1.
Fig. 6
Fig. 6 (a) Normalized transmission spectra of the LP01 mode calculated for a 3-section length-apodized LPG with z1 = 3Λ, z2 = 8Λ, and z3 = 13Λ at different values of the coupling coefficient κ and (b) variation of the −20-dB bandwidth with the coupling coefficient κ.
Fig. 7
Fig. 7 Images (a) of sidewall corrugation and (b) an end face of a typical fabricated LP01-LP11a mode converter.
Fig. 8
Fig. 8 Images (a) of surface corrugation and (b) an end face of a typical fabricated LP01-LP11b mode converter.
Fig. 9
Fig. 9 Normalized transmission spectra of the LP01-LP11a mode converter measured at different operating temperatures for (a) the x-polarization and (b) the y-polarization.
Fig. 10
Fig. 10 Output near-field images taken at different wavelengths for the LP01-LP11a mode converter, when the LP01 mode was launched into the device.
Fig. 11
Fig. 11 Normalized transmission spectra of the LP01-LP11b mode converter measured at different operating temperatures for (a) the x-polarization and (b) the y-polarization.
Fig. 12
Fig. 12 Output near-field images taken at different wavelengths for the LP01-LP11b mode converter, when the LP01 mode was launched into the device.

Equations (7)

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λ 0 = ( N 01 N 11 ) Λ ,
( E 01 ( L ) E 11 ( L ) ) = F M F 2 F 1 ( E 01 ( 0 ) E 11 ( 0 ) ) ,
F i = ( e j ( β 01 δ ) z i 0 0 e j ( β 11 + δ ) z i ) × ( cos ( Q z i ) j δ Q sin ( Q z i ) κ Q e j φ i sin ( Q z i ) κ Q e j φ i sin ( Q z i ) cos ( Q z i ) + j δ Q sin ( Q z i ) ) .
| E 01 ( L ) | 2 = cos 2 [ i = 1 i = M ( 1 ) i κ z i ] | E 01 ( 0 ) | 2 ,
| E 11 ( L ) | 2 = ( 1 cos 2 [ i = 1 i = M ( 1 ) i κ z i ] ) | E 01 ( 0 ) | 2 .
i = 1 M ( 1 ) i z i = ± π / 2 κ ,
κ = ± π / 2 [ i = 1 M ( 1 ) i z i ] for M   > 1 .

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