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The reflectogram of a fiber grating is used to characterize vector modes of an optical fiber supporting orbital angular momentum states. All modes, with a minimal effective index separation around 10−4, are simultaneously measured. OAM states are reflected by the FBG, along with a charge inversion, at the center wavelength of the Bragg reflection peak of the corresponding fiber vector mode.
© 2014 Optical Society of America
1. Introduction
Orbital angular momentum (OAM) of light is represented by an electrical field with a helical phase proportional to exp(ilϕ) where l is an integer and ϕ is the azimuthal angle [1]. The unique property of OAM light beams is that an infinite number of orthogonal OAM states can be supported in principle [2]. It is therefore possible to use them as carriers to transport data in optical communications in addition to wavelength division multiplexing (WDM). In the past decade, OAM-based optical free space communications has been extensively studied. The reported data-transferring capacity rapidly increased from 2 Tbit/s to 100 Tbit/s [3–5]. Photonic integrated techniques were also implemented to achieve more compact systems [6–8]. After initial studies in free space communication links, OAM modes are now being considered for spatial multiplexing in optical fibers [9, 10]. Terabit-scale data-transferring capacity in specially designed optical “vortex” fibers has been demonstrated by multiplexing two OAM modes over 10 wavelengths, which is a huge step forward in efforts to develop this technology as an efficient means to scale the capacity of current fiber optic networks [11].
OAM beams are generally considered to be unstable in conventional optical few mode fibers because of the presence of mode coupling. Therefore, specialty fibers that support OAM states are a critical technology for future OAM-WDM systems and several designs have been proposed [10, 12–14]. One key parameter in the performance of such OAM guiding fibers is the separation of the effective refractive indices, ∆neff, of the fiber vector modes. Large ∆neff values alleviate the problem of mode coupling and enable the stable copropagation of multiple OAM states. Efficient techniques that accurately and rapidly measure the vector modal effective indices are highly desirable for the characterization of OAM fibers and subsequent optimization of their designs. Savolainen et al. measured the difference in modal effective indices by stretching the fiber and monitoring the interference pattern [15]. This method works well for few mode fibers in which only few modes propagate. When the number of propagating modes increases, the analysis becomes rather complicated. The index separations can also be measured by using mechanically-induced micro-bend long period fiber gratings (LPGs) [16]. ∆neff is deduced from the center wavelength of the mode conversion peak. However, a number of micro-bend LPGs with different periods are required for the measurement in the wavelength range of interest, e.g. the C-band, due to the large separation of the mode conversion peaks (around 100 nanometers), making this technique practically inconvenient. Fiber Bragg gratings (FBGs) can spectrally filter fiber modes over a much smaller wavelength region. FBGs have been successfully used to characterize and monitor fiber modes, but most of them are in conventional multi-mode fibers and for measuring linearly polarized modes [17–20]. Until now FBGs written in few-mode fibers supporting OAM modes have never been reported to the best of our knowledge.
In this paper we demonstrate a FBG-based technique for characterizing vector modes in OAM fibers. The OAM fiber under test was designed to suppress OAM mode coupling and was fabricated in our laboratory [12]. Weak FBGs were photoinduced in the OAM fibers and all the fiber vector modes were simultaneously measured by analyzing the FBG reflectogram. The measured effective index separations are in good agreement with numerical simulations of the fiber refractive index profile. Furthermore, Bragg reflections of OAM states by the FBG are also demonstrated and analyzed.
2. OAM fiber with an inverse-parabolic graded-index profile
The OAM fiber under test presents an inverse-parabolic graded refractive index profile (Fig. 1 (a)) written as
where n1 and n2 are the refractive indices at the center core (r = 0) and in the cladding respectively, N is the curvature parameter, and [12]. Inverse-parabolic profiles are obtained with N<0. The designed index profile are plotted in Fig. 1(a) (dashed blue line) with rcore = 3 μm, N = −4, n1 = 1.4539, n2 = 1.444 and a maximal value of nmax = 1.4928 at the core-cladding interface.
Fig. 1 a) The designed and fabricated index profile of the OAM fiber; b) calculated effective indices of the fiber vector modes based on the fabricated profile.
We fabricated the fiber preform via modified chemical vapor deposition (MCVD) and drew it to a 7 km total fiber length. Figure 1(a) (solid red line) shows the measured refractive index profile of the preform. The maximum values of the refractive index and the core-cladding contrast are 1.4871 and 0.0426 respectively. We also measured the fabricated fiber directly using a visible-light refractometer, and the measurement was in agreement with the profile measured on the preform but with lower spatial resolution. A finite-element method (FEM) was used to calculate the effective indices of all the vector modes based on the measured index profile. The calculated results are plotted in Fig. 1(b) indicating that this fiber supports 10 vector modes: HE11 even & odd, TE01, HE21 even & odd, TM01, EH11 even & odd, and HE31 even & odd. Through coherent combinations of the vector modes, this fiber theoretically supports 6 OAM states that include 3 sets of 2 degenerated modes with ± s (circular) polarizations and the associated ± l topological charge: , and . The radial index, m, is 1 for all OAM states. Information about the fiber design and fabrication will be reported in detail elsewhere.
3. Measurement of the effective index difference between the vector modes
The OAM fiber used in the experiment was firstly deuterium loaded for 6 weeks (2000 psi) to increase its photosensitivity. A FBG was then written on the fiber using a frequency doubled Argon ion laser (244 nm) and a uniform phase mask with a period ΛPM = 1066.1 nm. The FBG is 20 mm long with a uniform profile. The UV beam is slightly defocused on the fiber core using a cylindrical lens (35 mm focal length). In order to obtain a weak FBG, the incident UV power before the phase mask is kept as low as 19 mW and the scan speed is 1.4 mm/s. The photo-induced index change is estimated to be around 1 × 10−4, which would result in self-coupling Bragg reflection peaks with bandwidths of about 0.1 nm.
According to the coupled mode theory, a fiber mode j with an effective index neff,j is reflected by the fiber grating and coupled to a counter-propagating fiber mode k with an effective index neff,k in the vicinity of wavelength:
j = k corresponds to Bragg modal self-reflection, while j≠k is the cross-coupling between different fiber modes. In the experiment, cross-coupling between modes occurs because of asymmetry of the photo-induced index modification in the transverse fiber plane due to stronger exposure on one side of the fiber.The reflectogram of the FBG is measured using the set up illustrated in Fig. 2. An optical vector analyzer (OVA, Luna technologies) is connected to a polarization controller, which is used for OVA calibration, and a length of single mode fiber. In order to excite all the fiber vector modes simultaneously, the single mode fiber and the OAM fiber are purposely misaligned.
Figure 3(a) shows the measured reflectogram from 1545 nm to 1570 nm. Figures 3(b), 3(c) and 3(d) zoom on the self-coupling Bragg reflections of mode groups #1 (HE11), #2 ({TE01, HE21, TM01}) and #3 ({EH11, HE31}) respectively. Figures 3(e), 3(f) and 3(g) zoom on the cross-coupling peaks between modes. Figures 3(b)-3(g) have wavelength spans of 2 nm. The cross-coupling peaks are narrower because the coupling coefficient (overlap) is weaker [21]. With this weak grating, we observe that the Bragg reflection peaks of all the fiber vector modes are clearly separated and identifiable. The modal effective indices can be directly calculated based on the center wavelength of the self-coupling Bragg reflection peaks and Eq. (2):
The effective index of mode j, neff,j , can also be deduced from the cross-coupling peaks (Figs. 3(e) and 3(f)) with respect to the Bragg reflection peak of the fundamental mode HE11:where λj,HE11 is the center wavelength of the cross-coupling peak between the fundamental mode HE11 and mode j, where λHE11 is the Bragg wavelength of the HE11 mode. This is useful if the bandwidths of the Bragg reflection peaks are so large that they spectrally superimpose. In our case, this provides a second method of verifying the differential effective indices.Fig. 3 a) Reflectogram of the FBG; b) zoom-in on mode group #1; c) zoom-in on mode group #2; d) zoom-in on mode group #3; e) zoom-in on the cross-coupling peak between mode group #1 and #2; f) zoom-in on the cross-coupling peak between mode group #1 and #3; g) zoom-in on the cross-coupling peak between mode group #2 and #3. The symbol “←→” in the figure denotes “cross-coupling”.
Not surprisingly, the measured cross-coupling peaks between mode group #2 and #3 (Fig. 2 (g)) is not as clear and more difficult to interpret due to the larger number of vector modes involved and the low excitation power in these high order modes.
Both the calculated and measured effective indices of the vector modes are listed in Table 1. Determination of the absolute neff values is difficult because of the uncertainty in the fiber refractive index profile both before and after exposure (as well as the applied strain on the FBG). However, differential effective indices Δneff are not ambiguous, and can be obtained from measurements of either the Bragg reflection peaks or the cross-coupling peaks. The measured index differences calculated from the Bragg reflection peaks are 2.9 × 10−4 (TE01 and HE21), 3.69 × 10−4 (HE21 and TM01) and 1.57 × 10−4 (EH11 and HE31). These compare well with the indices Δneff. while they are slightly larger than the FEM calculations [12]: 2.12 × 10−4 (TE01 and HE21), 2.05 × 10−4 (HE21 and TM01) and 1.70 × 10−4 (EH11 and HE31), this discrepancy may originate from longitudinal variations in the refractive index profile of the fiber due to manufacturing imperfections and/or transverse refractive index modifications induced by FBG writing. We note in Table 1 that the ∆neff retrieved from the cross-coupling peaks are very similar to those retrieved from the Bragg reflection peaks.
Table 1. Comparison between the calculated and measured modal effective index values.
4. Reflection of OAM states by FBGs
Another experiment was implemented to confirm that the FBG can selectively reflect and filter OAM beams (the upper part of Fig. 4). Light from a tunable laser (TL) was converted from a fiber mode to a free-space Gaussian beam by a collimator with an output beam diameter of 2 mm. Then it was split into two branches by a polarization beam splitter (PBS1), vertically-polarized in the upper branch and horizontally-polarized in the lower branch. A polarization controller (PC) was inserted between the TL and the collimator to adjust the power at the outputs of PBS1. In the upper branch, a vertically-polarized OAM beam was first generated by a spatial light modulator (SLM). A beam expander (BE), which consists of two lenses with different focal lengths, expanded the beam size from 2 mm to 4 mm. After transmitting through another PBS (PBS2) and a quarter-wave plate (QWP), the OAM beam was converted to circular polarization and was then coupled into the OAM fiber via Lens1 (focal length: 6.24 mm, N.A.: 0.4). The fiber end for coupling was anti-reflection coated and the other end was immersed in the index matching oil to eliminate the reflection by the fiber facet. The beam reflected by the FBG was converted into horizontal polarization by QWP and PBS2 and, was combined and interfered with the beam from the lower branch which contains a mirror (M3) and a beam splitter (BS). Lens2 (focal length: 50cm) was used to adjust the size of the reference beam. A CCD camera was used to record the interference patterns. Two flip mirrors (FM1 and FM2) and a half-wave plate (HWP) were inserted in the setup for measuring the profiles and the interference patterns of the incident beams. As shown in the lower part of Fig. 4, the fiber under test is 73 cm long in total and the 2 cm long FBG is placed 47.5 cm away from the input coupling end.
Fig. 4 Upper part: experimental setup for detecting the Bragg reflections of the FBG. TL: tunable laser; PC: polarization controller; PBS: polarization beam splitter; M: mirror; SLM: spatial light modulator; FM: flip mirror; BE: beam expander; QWP: quarter-wave plate; BS: beam splitter; HWP: half-wave plate; IM: index matching oil. Lower part: the physical sizes of the OAM fiber and FBG under test.
We sent the fundamental mode (HE11) and ± 1st/ ± 2nd order OAM beams with different polarization states into the fiber and swept the laser wavelength. The coupling losses for the fundamental mode, ± 1st and ± 2nd order OAM beams are −4.52 dB, −2.69 dB and −5.69 dB respectively. The fundamental mode and ± 1st/ ± 2nd order OAM beams are strongly reflected at the wavelengths that correspond to the Bragg reflections of HE11, HE21, EH11 and HE31 modes respectively. The beam profiles and interference patterns of the incident and reflected beams are displayed in Table 2, Table 3, Table 4, and Table 5. The experimental results show that the OAM modes supported by HE modes in fiber have circular polarizations that rotate in the same directions as those of their helical phases, while the OAM modes supported by EH modes are in the opposite directions. For example, + 1st order/left-circular polarized AND −1st order/right-circular polarized OAM beams cannot be reflected in the vicinity of 1559.84 nm that correspond to the Bragg reflection of the HE21 mode. This result confirms the theoretical prediction [12].
Table 2. Profile and interference pattern of the incident and reflected beams. The wavelength is located in the Bragg wavelength of HE11 mode.
Table 3. Profiles and interference patterns of the incident and reflected beams. The wavelength is located in the Bragg wavelength of HE21 mode.
Table 4. Profiles and interference patterns of the incident and reflected beams. The wavelength is located in the Bragg wavelength of EH11 mode.
Table 5. Profiles and interference patterns of the incident and reflected beams. The wavelength is located in the Bragg wavelength of HE31 mode.
Another interesting phenomenon that can be clearly seen from Table 2 is that the detected OAM states invert their topological charges compared to the incident beams. Note that mirrors and lens can invert the topological charge of an incident OAM beam [22, 23]. In our experimental setup, the fiber was placed slightly before the focus of Lens1 so that Lens1 did not affect the topological charge and, PBS2 inverts the topological charge when it reflects an OAM beam, as does M3. The OAM beam reflected by the FBG out from the fiber experienced two charge inversions in the free space optical path thus keeping the same topological charge when it reached the CCD camera. Therefore, it can be concluded that the FBG does indeed invert the topological charge of an OAM mode upon reflection. This phenomenon can be explained as follows. The OAM states supported by the fiber are a linear combination of the vector modes. When reflecting the OAM states, the FBG induces the same phase shift to the degenerated even and odd vector modes, e.g. even and odd HE21 modes, which construct the OAM state. However, the relative phase between them is shifted by 180° because they propagate in the opposite direction with respect to the incident light. Thus the FBG inverts both the topological charge and polarization states of OAM beams, as do mirrors in free space propagation.
5. Conclusion
A fiber Bragg grating (FBG) was written in an optical fiber supporting OAM modes. The reflectogram of the FBG was then exploited to measure the effective index separations of the vector modes. We demonstrate that both the Bragg reflection peaks and the cross-coupling peaks can be used for deducing the modal effective indices of vector modes. Although it is difficult to obtain the accurate absolute values of the effective indices, the measured modal index splitting is found to be reliable and coincides well with the modal calculations of the fabricated fiber profile. Furthermore, the fundamental mode, ± 1st and ± 2nd order OAM states were successfully filtered and reflected by the FBG near the Bragg wavelengths corresponding to HE11, HE21, EH11 and HE31 modes respectively. Both topological charge and circular polarization inversions are observed when an OAM is reflected by the FBG. These results indicate that weak photoinduced gratings are a convenient tool for characterizing the vector modes and the corresponding OAM states in fibers. We believe that the FBG can have applications in the manipulation and control of the OAM states of light in optical fibers.
Acknowledgement
This work was supported by the Canada Research Chair in Advanced photonics technologies for emerging communication strategies (APTECS), by the Canada Excellence Research Chair in Photonic innovations and the Natural sciences and engineering research council of Canada (NSERC). The authors would like to acknowledge Charles Brunet and Pierre-André Bélanger for insightful discussions.
References and links
1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]
2. A. E. Willner, J. Wang, and H. Huang, “Applied physics. A Different Angle on Light Communications,” Science 337(6095), 655–656 (2012). [CrossRef] [PubMed]
3. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef] [PubMed]
4. J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). [CrossRef]
5. H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I. Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. Lavery, M. J. Padgett, M. Tur, and A. E. Willner, “100 Tbit/s free-space data link enabled by three-dimensional multiplexing of orbital angular momentum, polarization, and wavelength,” Opt. Lett. 39(2), 197–200 (2014). [CrossRef] [PubMed]
6. N. Fontaine, C. Doerr, and L. Buhl, “Efficient Multiplexing and Demultiplexing of Free-space Orbital Angular Momentum using Photonic Integrated Circuits,” in Optical Fiber Communication Conference, OSA Technical Digest (Optical Society of America, 2012), paper OTu1I.2. [CrossRef]
7. J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493(7431), 195–199 (2013). [CrossRef] [PubMed]
8. B. Guan, R. P. Scott, C. Qin, N. K. Fontaine, T. Su, C. Ferrari, M. Cappuzzo, F. Klemens, B. Keller, M. Earnshaw, and S. J. Yoo, “Free-space coherent optical communication with orbital angular, momentum multiplexing/demultiplexing using a hybrid 3D photonic integrated circuit,” Opt. Express 22(1), 145–156 (2014). [CrossRef] [PubMed]
9. N. Bozinovic, S. Golowich, P. Kristensen, and S. Ramachandran, “Control of orbital angular momentum of light with optical fibers,” Opt. Lett. 37(13), 2451–2453 (2012). [CrossRef] [PubMed]
10. P. Gregg, P. Kristensen, S. Golowich, J. Olsen, P. Steinvurzel, and S. Ramachandran, “Stable Transmission of 12 OAM States in Air-Core Fiber,” in CLEO: 2013, OSA Technical Digest (online) (Optical Society of America, 2013), paper CTu2K.2.
11. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-Scale Orbital Angular Momentum Mode Division Multiplexing in Fibers,” Science 340(6140), 1545–1548 (2013). [CrossRef] [PubMed]
12. B. Ung, L. Wang, C. Brunet, P. Vaity, C. Jin, L. A. Rusch, Y. Messaddeq, and S. LaRochelle, “Inverse-parabolic graded-index profile for transmission of cylindrical vector modes in optical fibers,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Tu3K.4. [CrossRef]
13. C. Brunet, B. Ung, Y. Messaddeq, S. LaRochelle, E. Bernier, and L. Rusch, “Design of an Optical Fiber Supporting 16 OAM Modes”, in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.24. [CrossRef]
14. S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett. 34(16), 2525–2527 (2009). [CrossRef] [PubMed]
15. J. M. Savolainen, L. Grüner-Nielsen, P. Kristensen, and P. Balling, “Measurement of effective refractive-index differences in a few-mode fiber by axial fiber stretching,” Opt. Express 20(17), 18646–18651 (2012). [CrossRef] [PubMed]
16. S. Ramachandran, S. Golowich, M. F. Yan, E. Monberg, F. V. Dimarcello, J. Fleming, S. Ghalmi, and P. Wisk, “Lifting polarization degeneracy of modes by fiber design: a platform for polarization-insensitive microbend fiber gratings,” Opt. Lett. 30(21), 2864–2866 (2005). [CrossRef] [PubMed]
17. F. Stutzki, C. Jauregui, J. Limpert, and A. Tünnermann, “Real-time characterisation of modal content in monolithic few-mode fibre lasers,” Electron. Lett. 47(4), 274–275 (2011). [CrossRef]
18. F. Stutzki, C. Jauregui, C. Voigtländer, J. U. Thomas, J. Limpert, S. Nolte, and A. Tünnermann, “Passively stabilized 215 W monolithic CW LMA-fiber laser with innovative transversal mode filter,” Proc. SPIE 7580(75801K), 75801K (2010). [CrossRef]
19. J. M. O. Daniel, J. S. P. Chan, J. W. Kim, J. K. Sahu, M. Ibsen, and W. A. Clarkson, “Novel technique for mode selection in a multimode fiber laser,” Opt. Express 19(13), 12434–12439 (2011). [CrossRef] [PubMed]
20. C. Jocher, C. Jauregui, C. Voigtländer, F. Stutzki, S. Nolte, J. Limpert, and A. Tünnermann, “Fiber based polarization filter for radially and azimuthally polarized light,” Opt. Express 19(20), 19582–19590 (2011). [CrossRef] [PubMed]
21. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997). [CrossRef]
22. S. Roychowdhury, V. K. Jaiswal, and R. P. Singh, “Implementing controlled NOT gate with optical vortex,” Opt. Commun. 236(4-6), 419–424 (2004). [CrossRef]
23. G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87(2), 023902 (2001). [CrossRef] [PubMed]
