Mode coupling from the fundamental vector mode () to the cylindrical vector beams (CVBs) and optical vortex beams (OVBs) of a few-mode fiber excited by two acoustic flexural waves with orthogonal perturbations is achieved by using a composite acoustic transducer. The mode is converted to TM01 and TE01 modes, which have radial and azimuthal polarizations, by using the lowest-order acoustic flexural modes of and , respectively. Furthermore, mode can also be converted to the ± 1-order OVBs of through the combined acoustic modes of . This technique provides a useful way of generating CVBs and OVBs in optical fiber with conveniently electrically-controlled mode conversion characteristics.
© 2017 Optical Society of America
In recent years, the structured light beams have attracted much attention. Due to possessing the features of the exotic spatial polarization or phase singularities [1,2], potential applications have been explored in areas of the stimulated emission depletion (STED) microscopy [3,4], optical tweezers [5,6], micro/nano fabrication [7,8], surface plasmon polariton (SPP) excitation [9,10], nonlinear optics [11,12], quantum optics [13,14], and high-capacity optical communication [15,16], etc.
To date, many methods, including the spatial light modulators (SLMs) , q-plates , spiral phase plates , plasmonic metasurfaces , and silicon integrated devices , have been adopted to generate the structured light beams in free space. It is especially noteworthy that the most common method is the SLMs, which are used to generate a wide variety of structure light beams with arbitrary polarization and phase distribution in free space. Meanwhile, due to the advantages of long-distance/high-capacity transmission for the optical communication system, the fiber-based generation techniques are also developing rapidly. However, the structured light beams in optical fiber is the vector solution of the eigenvalue equation , thus the types of the structured light beams in optical fiber are not as abundant as in free space.
In an optical fiber, cylindrical vector beams (CVBs) and optical vortex beams (OVBs) are two kinds of typical structured light beams possessing the spatial polarization and phase singularities, respectively. So far, several methods of direct generation in optical fiber have been proposed [23–28], and the CVBs and OVBs have been experimentally generated by exploiting the fiber gratings produced by mechanical micro-bend , laser writing , etc. In these approaches, it is not convenient to actively tune the wavelength of the CVBs and OVBs because the period of the fabricated grating elements are fixed. In contrast, wavelength tunability and mode conversion was achieved in our previous works by exploiting the highly polarization-dependent vector mode coupling characteristic of an acoustically induced fiber grating [29–31]. In the CVBs generator , the output CVBs can be switched between radial and azimuthal polarizations by changing the linear polarization direction of the input mode of HE11 through a half-wave plate. In the OVBs generator [30,31], the OVBs output can be switched between opposite signs of the topological charge by changing the sign of the circular polarization input mode of using a quarter-wave plate.
In this paper, we propose and demonstrate a method to generate the CVBs and OVBs in a few-mode fiber (FMF). The mode conversion is operated via setting radio frequency (RF) driving signals instead of mechanically rotating a waveplate, which eases the operation. In the experiment conducted at 633 nm, the linearly polarized optical mode is converted to the TM01 or TE01 modes for radially or azimuthally polarized CVBs by correspondingly exploiting the acoustic flexural modes of or , whose vibrations are perpendicular to each other. Furthermore, the mode can also be converted to the ± 1-order OVBs of through the combined acoustic modes , for which the acoustic flexural modes of and are set with the same frequency and a ± π/2 phase difference.
Figure 1(a) is the sketch of the composite transducer constructed with two shear mode piezoelectric transducers (PZTs), which are stacked on top of each so that their vibration directions are perpendicular to each other. With an RF driving signal applied to PZT1 or PZT2, a lowest-order acoustic flexural mode or , with vibration along the x- or y-axis, can be excited and then propagates along the unjacketed fiber . The mode fields of and are antisymmetric along their vibration directions, as shown in Fig. 1(b), the corresponding refractive index modulation induced by the and modes are also antisymmetric and can be written as [30,33]Fig. 1(c). The mode coupling coefficient κij between the vector modes i () and j (TE01, , TM01) of the acoustically induced fiber grating can be expressed as 
The mode coupling coefficient κij in Eq. (2) is numerically calculated for the adopted step-index FMF at 633 nm, which supports transmission of fundamental vector modes () and high-order vector modes (TE01, and TM01). The transverse electric field distributions of the vector modes Ei(x, y) and Ej(x, y) are calculated using the finite element method (Comsol). Then, with the grid data of Ei(x, y) and Ej(x, y), the mode coupling coefficient κij is calculated using Eq. (2) with varying θ in Fig. 1(c). Here, the constant N0 in Eq. (1) is set to ~10−5, and the calculation result of κij is shown in Fig. 1(c), which denotes the dependence of vector mode coupling on the polarization direction of the acoustic flexural mode. In the condition of θ = 90°, the can be coupled only to TE01 and via the acoustic flexural mode , as shown the black and blue dots curves in Fig. 1(c). Whereas at θ = 0°, can be converted only to and TM01 by the acoustic flexural mode , as shown the red and green dots curves in Fig. 1(c), respectively.
Furthermore, to guarantee the success of the vector mode conversion, the phase matching condition of the acoustically induced fiber grating 36], where R is the fiber radius and Cext = 5760 m/s is the phase velocity of the acoustic wave in silica. Based on Eq. (3) and the acoustic wave dispersion equation, the phase matching condition can be given by 
According to Eq. (4), the relationship between the resonance wavelength λ and the acoustic frequency f is plotted in Fig. 1(d). As seen in Fig. 1(d), the RF driving frequencies are different for different output vector modes due to the effective refractive index differences between the high-order vector modes. The frequency difference isFig. 1(d), the generation of TE01, , and TM01 modes at λ = 633 nm requires a setting of f = 0.8227, 0.8270 and 0.8289 MHz, respectively. Thus the mode can be converted to TE01 mode via the acoustic flexural mode with f = 0.8227 MHz. By changing as mode with f = 0.8289 MHz, the mode can be converted to TM01 mode. Furthermore, when two RF driving signals, with same frequency of f = 0.8270 MHz and a ± π/2 phase shift, are simultaneously applied to PZT1 and PZT2, and mode can be generated and also has a ± π/2 phase shift between them. Thus the modes can be converted to and modes through and modes, respectively. Meanwhile, there is also a ± π/2 phase difference between and modes, thus the combined result is the ± 1-order OVBs of . Note that the frequency difference of ~kHz can be easily controlled with the RF driving source. Therefore, this approach is highly advantageous with the electronically-controlled tunability for the phase matching in generating different types of CVBs and OVBs at the same wavelength.
3. Experimental results and discussions
The experimental configuration of the CVBs and OVBs generation is depicted in Fig. 2. The light beam emitted from a 20 mW laser at λ = 633 nm is polarized to horizontal linear polarization by the polarizer (P1) with polarization orientation adjusted by the half-wave plate (HWP). The linearly polarized light with power of 17 mW is coupled into the FMF by the micro-objective (MO1). Moreover, to further eliminate the effects of the unwanted high-order vector modes before the two acoustically induced fiber gratings, a mode tripper (MS), which is made of 7 turns of FMF wound on a 5 mm diameter rod, is used to ensure a pure mode launching. When the light propagates through the mode stripper (MS) , there is only the linearly polarized mode with a power of 7 mW left in the fiber core. The unjacketed optical fiber used in the experiment has a core with radius ρco = 4.5 μm, a cladding with radius ρco = 62.5 µm, and a step index of ∆ = 0.32%. The diameter of the optical fiber for forming the acoustically induced grating was etched down to 22 µm by the hydroﬂuoric (HF) acid in order to adjust the resonant wavelength according to the phase matching condition and enhance the overlap between the acoustic and optical waves, thus increasing the acousto-optic coupling efficiency of the two acoustically induced fiber gratings within the 40 mm long etched segment . One end of the unjacketed fiber was glued with epoxy to the tip of the horn acoustic transducer, and the other end was fixed on the optical fiber clamps.
By tuning the frequency of RF driving signal applied to the acoustic transducer and adjusting the input polarization state through the half-wave plate (HWP), the input mode is converted to the CVBs or OVBs when the phase matching condition in Eq. (3) is satisfied. The FMF output terminal is collimated using a 40 micro-objective (MO2) and the CVBs or OVBs mode intensity patterns are recorded using a charge coupled device (CCD1).
For generating the radial polarization vector mode of TM01, an acoustic flexural wave is generated on the PZT1 being actuated by an RF driving source with frequency of f = 0.8289 MHz, and amplified at the tip of the horn-like transducer. The propagating acoustic flexural wave with vibration along the x-axis produce a dynamic micro-bend grating with Λ = 693 μm. The linear polarization mode is then converted to the TM01 mode along the acousto-optic coupling region. By adjusting the RF driving frequency as f = 0.8227 MHz and applied to the PZT2, an acoustic flexural wave is generated and propagated with vibration along the y-axis. Thus the dynamic micro-bend grating with Λ = 695.6 μm is induced and converted the mode to TE01 mode. To examine the modal field patterns of the generated CVBs, the beams are projected on the CCD1 covered by a linear polarizer after the NPBS2 . The intensity patterns at various polarizations are shown in Figs. 3(a) and 3(b).
For generating the ± 1-order OVBs, two RF driving signal, with frequency f = 0.8270 MHz and a ± π/2 phase shift, are applied to the PZT1 and PZT2, simultaneously. Thus the two acoustically induced fiber gratings, which have the same period of Λ = 693.8 μm and a ± π/2 phase shift, are generated by PZT1 and PZT2, with vibration direction along the x and y-axis. The mode is coupled to and modes through the two acoustically induced gratings induced by and modes, respectively. It is noted that there is also a ± π/2 phase shift between and mode due to the two acoustic flexural modes with a ± π/2 phase difference. Thus the combined results is ± 1-order OVBs of . As shown in Figs. 4(a1) and 4(b1), the images taken by the charge-coupled device (CCD1) displayed annular intensity patterns for the ± 1-order OVBs output [23,26]. Furthermore, the helical phase exp( ± iℓθ) of the ± 1-order OVBs signifying OAM = ± ħ per photon is verified by examining its interference pattern with a linearly polarized Gaussian beam. Note that the frequency of the ± 1-order OVBs is down-shifted from that of the mode by an amount equal to the frequency of the acoustic flexural wave, the reference light could not be directly taken from the laser. Instead, part of the vortex mode output from the FMF is reflected using a non-polarization beam splitting prism (NPBS1), and then coupled into a segment of 630 HP fiber to convert the ± 1-order OVBs back to a fundamental mode because the 630 HP is a single-mode fiber in the wavelength range of 600–770 nm. The fundamental mode output from 630 HP fiber is aligned to a parallel light by MO4 and converted to a linearly polarized Gaussian beam using P2. A polarizer (P3) after the NPBS2 is used to convert the circular polarization of the ± 1-order OVBs to linear polarization. Then the reference light and the linearly polarized optical vortex mode are combined through NPBS3, and the interference pattern is recorded by CCD2. As the signature of the OVBs with a topological charge, the forklike fringes are observed in off-axial interference, as shown in Figs. 4(a2) and 4(b2). As another signature of the ± 1-order OVBs in coaxial interference, the spiral patterns are observed, as shown in Figs. 4(a3) and 4(b3), with opposite chirality for the left- and right-handed circular polarization, respectively.
Furthermore, the mode conversion efficiency of the acoustically induced fiber grating is also measured by using the difference of the critical bend loss between the mode and the high-order vector modes . At the output terminal of the FMF, the other mode stripper, is made to filter out the generated CVBs and OVBs, and leave only the mode in the core of FMF. Take the case of TM01 mode, without RF driving signal loaded on the PZT1, the acoustically induced grating is not generated in the FMF, thus the mode can output from the terminal of the FMF without loss. With the RF driving voltage increases, an acoustically induced grating with vibration along the x-axis is produced in FMF and its refractive index modulation depth is gradually increasing. Thus, more and more mode is converted to the mode in the acousto-optic interaction region. After the two vector modes passing through the mode stripper, the generated TM01 mode is filtered out, only the mode is left in the FMF and received by the CCD1. In the condition of the critical coupling, the mode is almost coupled to TM01 mode the by acoustically induced fiber grating. Therefore, no mode is received by the CCD1. Figure 5(a) are the modes intensity patterns of mode with different RF driving voltage applied on PZT1. Figure 5(b) are the corresponding horizontal intensity profiles of the modes through the center of the Fig. 5(a), respectively. Note that the intensity of the modes decrease from 255 and 14, with increasing the RF driving voltage from Vpp = 0 V to Vpp = 8 V, respectively. Therefore, the mode conversion efficiency of the acoustically induced fiber grating can be deduced as ~94.5%.
In summary, we propose and demonstrate a method to generate the CVBs and OVBs in an FMF based on two acoustically induced gratings with orthogonal vibration directions. The mode is converted to TM01 and TE01 modes, which have radial and azimuthal polarizations, by exploiting the lowest-order acoustic flexural modes of and , respectively. Furthermore, the mode can also be converted to the ± 1-order OVBs by using the combined acoustic modes of . The experiment was conducted at 633 nm and the mode conversion efficiency of 94.5% was reached. This work provides a useful way for generating CVBs and OVBs in FMF with conveniently electric-control tunable mode conversion characteristics.
This work is financially supported by the Natural Science Foundation of China (NSFC) (11404263, 61675169, 11634010, 61377055, 61675171, 61405161, and 11574161).
The authors would like to thank Wei Gao in Xi’an Institute of Optics and Precision Mechanics of Chinese Academy of Science for the helpful discussions.
References and links
1. Q. W. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009). [CrossRef]
2. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011). [CrossRef]
5. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011). [CrossRef]
7. K. Lou, S. X. Qian, X. L. Wang, Y. Li, B. Gu, C. Tu, and H. T. Wang, “Two-dimensional microstructures induced by femtosecond vector light fields on silicon,” Opt. Express 20(1), 120–127 (2012). [CrossRef] [PubMed]
8. K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity to nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013). [CrossRef] [PubMed]
10. L. Du, D. Y. Lei, G. Yuan, H. Fang, X. Zhang, Q. Wang, D. Tang, C. Min, S. A. Maier, and X. Yuan, “Mapping plasmonic near-field profiles and interferences by surface-enhanced Raman scattering,” Sci. Rep. 3, 3064 (2013). [CrossRef] [PubMed]
13. J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010). [CrossRef] [PubMed]
15. 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]
16. J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). [CrossRef]
18. D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincar sphere beams from a laser,” Nat. Photonics 10(5), 327–332 (2016). [CrossRef]
19. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5-6), 321–327 (1994). [CrossRef]
20. J. X. Li, S. Q. Chen, H. F. Yang, J. J. Li, P. Yu, H. Cheng, C. Z. Gu, H. T. Chen, and J. G. Tian, “Simultaneous control of light polarization and phase distributions using plasmonic metasurfaces,” Adv. Funct. Mater. 25(5), 704–710 (2015). [CrossRef]
21. X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338(6105), 363–366 (2012). [CrossRef] [PubMed]
22. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).
23. S. Ramachandran and P. Kristensen, “Optical vortices in fiber,” Nanophotonics 2(5-6), 455–474 (2013). [CrossRef]
24. Y. Yan, L. Zhang, J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, A. E. Willner, and S. J. Dolinar, “Fiber structure to convert a Gaussian beam to higher-order optical orbital angular momentum modes,” Opt. Lett. 37(16), 3294–3296 (2012). [CrossRef] [PubMed]
25. 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]
29. W. Zhang, L. Huang, K. Wei, P. Li, B. Jiang, D. Mao, F. Gao, T. Mei, G. Zhang, and J. Zhao, “Cylindrical vector beam generation in fiber with mode selectivity and wavelength tunability over broadband by acoustic flexural wave,” Opt. Express 24(10), 10376–10384 (2016). [CrossRef] [PubMed]
30. W. Zhang, K. Wei, L. Huang, D. Mao, B. Jiang, F. Gao, G. Zhang, T. Mei, and J. Zhao, “Optical vortex generation with wavelength tunability based on an acoustically-induced fiber grating,” Opt. Express 24(17), 19278–19285 (2016). [CrossRef] [PubMed]
31. W. Zhang, L. Huang, K. Wei, P. Li, B. Jiang, D. Mao, F. Gao, T. Mei, G. Zhang, and J. Zhao, “High-order optical vortex generation in a few-mode fiber via cascaded acoustically driven vector mode conversion,” Opt. Lett. 41(21), 5082–5085 (2016). [CrossRef] [PubMed]
33. T. A. Birks, P. St. J. Russell, and D. O. Culverhouse, “The acousto-optic effect in single-mode fiber tapers and couplers,” J. Lightwave Technol. 14(11), 2519–2529 (1996). [CrossRef]
34. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14(8), 1760–1773 (1997). [CrossRef]
35. W. Zhang, W. Gao, L. Huang, D. Mao, B. Jiang, F. Gao, D. Yang, G. Zhang, J. Xu, and J. Zhao, “Optical heterodyne micro-vibration measurement based on all-fiber acousto-optic frequency shifter,” Opt. Express 23(13), 17576–17583 (2015). [CrossRef] [PubMed]
36. T. A. Birks, P. S. J. Russell, and D. O. Culverhouse, “The acousto-optic effect in single–mode fiber tapers and couplers,” J. Lightwave Technol. 14(11), 2519–2529 (1996). [CrossRef]
38. P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of orbital angular momentum transfer between acoustic and optical vortices in optical fiber,” Phys. Rev. Lett. 96(4), 043604 (2006). [CrossRef] [PubMed]
39. W. Zhang, L. Huang, F. Gao, F. Bo, L. Xuan, G. Zhang, and J. Xu, “Tunable add/drop channel coupler based on an acousto-optic tunable filter and a tapered fiber,” Opt. Lett. 37(7), 1241–1243 (2012). [CrossRef] [PubMed]
40. J. L. 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]
41. K. S. Hong, H. C. Park, B. Y. Kim, I. K. Hwang, W. Jin, J. Ju, and D. I. Yeom, “1000 nm tunable acousto-optic filter based on photonic crystal fiber,” Appl. Phys. Lett. 92(3), 031110 (2008). [CrossRef]