1. Introduction

Optical vortex, with ability to carry orbital angular momentum (OAM) [1], has recently attracted great research interest in a wide range of applications, such as optical micromanipulation [2], quantum optics [3], nonlinear optics [4], optical communications [5, 6], etc.

Optical vortex in free space has typically been generated using cylindrical lens mode converters [7], q-plates [8], spiral phase plates [9], computer-generated holograms [10], metamaterials-based phase plates [11], subwavelength gratings [12], EIT-based light pulse storage and retrieval process [13], etc. Meanwhile, due to the advantages of long-distance/large-capacity transmission for optical communication systems, the fiber-based generation techniques are also developing rapidly. Several direct methods to generate optical vortex in fiber have been proposed [14–18], and the ± 1-order optical vortex have been experimentally generated based on the fiber gratings only with asymmetric refractive index modulation due to mechanical microbend [16,17], laser writing [18], etc. In these approaches, it is difficult to actively tune the wavelength of the optical vortex because the grating period of the fabricated fiber grating elements is fixed. Dashti et al. demonstrated that the OAM of the ± 1-order acoustic vortex can be transferred to a circularly polarized fundamental optical mode, therefore, to form a stable ± 1-order optical vortex in the fiber carrying OAM [19]. In their method, active wavelength tunability with two sets of coworking acoustic transducers and radio frequency (RF) drivers is possible but practically complicated.

In this Letter, we proposed a method to actualize the optical vortex generation with dynamic wavelength tunability based on an acoustically-induced fiber grating (AIFG) driven by an RF signal. The AIFG could selectively convert the left/right handed circular polarization fundamental mode to the + 1/-1 order optical vortex in a two-mode fiber. The optical vortex was experimentally generated within the wavelength range 1540 nm - 1560 nm by tuning the frequency of RF driving signal. A uniform mode conversion efficiency ∼95% was kept in the whole wavelength tuning range. The topological charge of the generated optical vortex was verified using the coaxial interference pattern between the optical vortex and a Gaussian-reference beam.

2. Principle and experimental configuration

In an unjacketed fiber with cylindrical symmetry, the lowest-order acoustic flexural mode F₁₁ [20], with its vibration along the x-axis, can be excited and then propagates along the unjacketed fiber, as shown in Fig. 1(a). The F₁₁ mode is antisymmetric with respect to its vibration direction, as shown in Fig. 1(b), the corresponding refractive index modulation induced by the F₁₁ mode on the cross section of the unjacketed fiber is also antisymmetric and can be expressed as [21,22]

 

Fig. 1 (a) Example F₁₁ mode of the acoustic flexural wave propagating along the unjacketed fiber. (b-c) Field patterns of F₁₁ and HE₁₁ modes on the cross-section of fiber, respectively. The arrows indicate the vibration directions of the F₁₁ mode (b) and the polarization directions of the transverse electric field of the HE₁₁ mode (c), which are all linearly polarized. (d) Sketch of the mode field overlap between the linearly-polarized F₁₁ and HE₁₁ modes on the cross section of fiber, ϕ denotes the crossing angle between the two modes. (e₁-e₄) Coupling coefficient κ_ij versus ϕ for each of the first four high-order vector modes (TE₀₁, ${\text{HE}}_{21}^{even/odd}$, TM₀₁) with respect to different linear polarizations of F₁₁ and HE₁₁ modes.

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The mode coupling coefficient κ_ij in Eq. (2) was numerically calculated for the adopted step-index two-mode fiber (TMF, OFS), which is optimized to stably support the transmission of fundamental vector modes (${\text{HE}}_{11}^{x/y}$) and four high-order vector modes (TE₀₁, ${\text{HE}}_{21}^{even/odd}$and TM₀₁). The transverse electric field distributions of the vector modes E_i(x, y) and E_j(x, y) were calculated using the finite element method (Comsol). Then, with the grid data of E_i(x, y) and E_j(x, y), the mode coupling efficiency κ_ij was calculated using Eq. (2) when ϕ varied, and the polarization direction of E_i(x, y) is illustrated in Fig. 1(d). Here, the constant N₀ in Eq. (1) was set to ∼10⁻⁵, and the calculation result of κ_ij is respectively shown in Figs. 1(e₁-e₄), which denote the dependence of mode coupling on the polarization direction. At ϕ = 90^o, ${\text{HE}}_{11}^y$can be coupled only to TE₀₁ and${\text{HE}}_{21}^{odd}$, as shown in Figs. 1(e₁) and 1(e₃), respectively. Whereas at ϕ = 0^o, ${\text{HE}}_{11}^x$ can be converted only to ${\text{HE}}_{21}^{even}$and TM₀₁, as shown in Figs. 1(e₂) and 1(e₄), respectively.

To allow the mode conversion from ${\text{HE}}_{11}^{x/y}$ to ${\text{HE}}_{21}^{even/odd}$ while prevent the generation of TE₀₁ and TM₀₁ modes, the phase matching condition [24]

 

Fig. 2 Beatlength (L_B) between ${\text{HE}}_{11}^{x/y}$and TE₀₁, ${\text{HE}}_{21}^{even/odd}$, TM₀₁ modes of the TMF. Relationship between the calculated grating period (Λ) of the AIFG and frequency of RF driving signal applied to the acoustic transducer.

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The experimental configuration for the actively wavelength tunable all-fiber optical vortex generation and examination is sketched in Fig. 3. The output beam from a tunable laser was amplified by an erbium doped fiber amplifier (EDFA) and then divided into two paths by a 3-dB coupler. One path was adopted for generating the optical vortex ${\text{VM}}_{11}^{\pm }$ and the other was used as a reference beam to interfere with the generated${\text{VM}}_{11}^{\pm }$ mode. For the path of ${\text{VM}}_{11}^{\pm }$generation, the beam was firstly coupled into a section of a single-mode fiber (SMF, Corning SMF-28) and the intensity was controlled by a tunable attenuator. The linear polarization characteristic of the beam was further purified by a polarizer, and then the linearly polarized beam was converted to a circularly polarized mode CP ^± by a polarization controller (PC). The SMF was directly spliced to the TMF, the fusion splice between the two kinds of fiber was pretty smooth to guarantee high coupling efficiency. Moreover, to further eliminate the effects of the unwanted high-order vector modes (TE₀₁, ${\text{HE}}_{21}^{even/odd}$, TM₀₁) before the AIFG, a mode tripper (MS), which was made of 5 turns of TMF wound on a 12 mm diameter rod [26], was used to ensure a pure CP ^± mode launching. The diameter of the TMF for forming the AIFG was etched down to 40 µm by hydrofluoric (HF) acid in order to adjust the resonance wavelength based on the phase matching condition Eq. (3) and to enhance the overlap between the acoustic and optical waves [21], thus increasing the acousto-optic coupling efficiency of the AIFG within the 50 mm long etched segment. One end of the unjacketed fiber was glued with epoxy to the tip of the horn-like acoustic transducer and the other end was fixed on an optical fiber clamp.

 

Fig. 3 Experimental configuration of the actively wavelength tunable all-fiber optical vortex generation and examination. EDFA: Erbium doped fiber amplifier. SMF: single mode fiber. PC: polarization controller. MS: mode stripper. TMF: two mode fiber. MO: micro-objective. NPBS: Non polarizing beam splitter prism. CCD: charge coupled device.

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By tuning the power and frequency of the RF driving signal applied to the acoustic transducer and adjusting the input polarization state through controlling the PC, the input CP ^± mode was converted to the ${\text{VM}}_{11}^{\pm }$ mode when the phase matching condition was satisfied. Subsequently, the TMF output terminal was collimated using a 40 × micro-objective (MO) and the mode intensity image was recorded using an infrared charge coupled device (IR CCD). Because the frequency of the ${\text{VM}}_{11}^{\pm }$ mode was downshifted from that of the CP ^± mode by an amount equal to the frequency of the acoustic flexural wave [24,27], a phase modulator was used to adjust the reference beam at the acoustic frequency to create a sideband of the same frequency as that of the ${\text{VM}}_{11}^{\pm }$ mode [19]. The interference pattern formed by the lower sideband reference and the ${\text{VM}}_{11}^{\pm }$ mode was captured by the IR CCD. The topological charge number of the ${\text{VM}}_{11}^{\pm }$ mode could be identified from the interference pattern [17, 19].

3. Experimental results and discussions

Upon tuning of the optical wavelength, the frequency of RF driving signal was adjusted accordingly to satisfy the phase matching condition. As a proof of principle, Figs. 4 (a₁-e₁) and (a₃-e₃) depict the near-field intensity distributions of the ${\text{VM}}_{11}^{\pm }$ modes at λ = 1540 nm,1545 nm, 1550 nm, 1555 nm and 1560 nm, while the frequency of RF driving signal was correspondingly set to be f = 0.2972 MHz, 0.2983 MHz, 0.2994 MHz, 0.3004 MHz and 0.3015 MHz, respectively. The corresponding grating period of the AIFG was calculated to be about Λ = 1103.5 µm, 1101.5 µm, 1099.4 µm, 1097.6 µm and 1095.6 µm, respectively. The${\text{VM}}_{11}^{\pm }$modes were obtained in the wavelength range from 1540 nm to 1560 nm and exhibited the annular shapes with null intensity in the center as the characteristic of the first-order vortex. Moreover, the spiral images, which are the signature of the ${\text{VM}}_{11}^{\pm }$ modes, were experimentally recorded using coaxial interference between the ${\text{VM}}_{11}^{\pm }$modes and the Gaussian-reference beams, as shown in Figs. 4(a₂-e₂) and 4(a₄-e₄). The rotating spiral interference patterns at λ = 1550 nm are shown in Visualization 1 and Visualization 2, respectively.

 

Fig. 4 (a₁-e₁) and (a₃-e₃) are the mode intensity patterns of the ${\text{VM}}_{11}^{\pm }$ modes at different resonance wavelengths of λ = 1540 nm, 1545 nm, 1550 nm, 1555 nm and 1560 nm, respectively. The annular shape with null intensity in the center is characteristic of these modes. (a₂-e₂) and (a₄-e₄) are images of the ${\text{VM}}_{11}^{\pm }$modes output when coaxially interfered with Gaussian-reference beams at above resonance wavelengths. The rotating spiral interference patterns at λ = 1550 nm are shown in Visualization 1 and Visualization 2, respectively.

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Furthermore, the transmission spectra of the AIFG were measured to deduce the mode conversion efficiency of ${\mathrm{CP}}^{\pm }\to {\text{VM}}_{11}^{\pm }$ [16]. Spontaneous emission spectrum of EDFA was used as the broadband light source at the input, while the output terminal of the AIFG was directly spliced to a segment of SMF to prevent resultant ${\text{VM}}_{11}^{\pm }$ modes from coupling into the SMF, and the output spectra were measured by an optical spectrum analyzer (OSA). Figure 5 (a) depicts the transmission spectra of the AIFG with the same frequencies and powers of RF driving signals as those used in Fig. 4. We obtained ~13 dB (95%) of mode conversion efficiency at five resonance wavelengths. With increasing the frequency of the RF driving signal, the resonance wavelength shifted toward long wavelength with a spectral tunability slope of 4.65 nm/kHz, as shown in Fig. 5(b).

 

Fig. 5 Transmission spectra of the AIFG used to deduce the mode conversion efficiency. (a) Mode conversion efficiency were all kept ~13 dB with its resonance wavelength at λ = 1540 nm, 1545 nm, 1550 nm, 1555 nm and 1560 nm, respectively, when the RF driving frequency was set at f = 0.2972 MHz, 0.2983 MHz, 0.2994 MHz, 0.3004 MHz and 0.3015 MHz, respectively. (b) The resonance wavelength versus the RF driving frequency. The resonance wavelength tunability slope was measured to be 4.65 nm/kHz.

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The switching time between the modes is mainly determined by the transit time of the acoustic flexural wave propagating through the acousto-optic coupling region of the AIFG [28, 29], and can be expressed as

4. Conclusions

In conclusion, we developed a method for optical vortex generation in TMF with the wavelength tunability. An AIFG driven by an RF source successfully converted the circularly-polarized fundamental mode to the first-order optical vortex, while actively wavelength tunability was demonstrated in the wavelength range 1540 nm - 1560 nm by varying RF frequency and with a spectral tuning rate of 4.65 nm/kHz. The mode conversion efficiency was kept at ~95% in the whole wavelength tuning range. The topological charge was chosen by controlling the polarization state of the input light, and verified by the spiral pattern using coaxial interference between the ${\text{VM}}_{11}^{\pm }$ mode and a Gaussian-reference beam. The demonstrated compact all-fiber device with capability of active wavelength tunability is advantageous for practical applications.