Cylindrical vector beam generation in fiber with mode selectivity and wavelength tunability over broadband by acoustic flexural wave

Wending Zhang; Ligang Huang; Keyan Wei; Peng Li; Biqiang Jiang; Dong Mao; Feng Gao; Ting Mei; Guoquan Zhang; Jianlin Zhao

doi:10.1364/OE.24.010376

1. Introduction

Cylindrical vector beams (CVBs), specifically with radial and azimuthal polarizations, have been drawing considerable research attention. Exploitation of their unique properties, such as tight focusing and special polarization distribution [1,2], have been stimulated in a variety of applications, e.g. second harmonic generation (SHG) [3], optical tweezers [4,5], micro/nanofabrication [6], surface plasmon polariton (SPP) excitation [7,8], etc. Recently, there is a burst of enthusiasm on investigation of the CVBs for mode division multiplexing to achieve high-capacity optical communication [9,10]. A series of previously reported the CVBs generation techniques, including both active (intra-cavity) [11–16] and passive (extra-cavity) [17–22], were implemented in free-space optics. The CVBs generated in free space using these techniques, if used in fiber-based optical communication systems [23], would inevitably meet with problems like insertion loss and assembling issue due to the need of light coupling between free space and optical fiber. Therefore, the generation of CVBs in fiber is greatly desired in such circumstance.

The fiber-based generation of CVBs has been studied using eigenmodes in optical fiber [24]. Excitation of the radially polarized TM₀₁ mode in a two-mode optical fiber was demonstrated by coupling with an offset linearly polarized Gaussian beam output from a single-mode fiber [25]. Different combinations of modes from the LP₁₁ group were selectively excited in a two-mode fiber by coupling with a Laguerre-Gaussian beam generated using a holographic mask, and then converted to an azimuthal or radial CVBs with half-wave plates in free space [26]. The vector modes of a two-mode fiber was selectively excited by a linearly polarized fundamental Gaussian laser beam, depending on the input beam polarization, the fiber length, and the launch condition [27]. These techniques, although are fiber-based, depend on the condition of external coupling and thus are technically challenging in aspect of engineering. An in-fiber technique may adopt intermodal coupling within fiber such that the CVBs are generated inside the fiber circuit without critical manipulation of optical beam alignment. Intermodal coupling may be induced by applying external perturbation, preferably in controllable fashion. Long period ber gratings (LPFGs) could be naturally thought about as candidates. LPFGs could be formed by the mechanical means [28], exposing the ber using a ultra-violet (UV) laser [29], or a CO₂ laser [30], and so on, whereas not all types of LPFGs are suitable for generating CVBs unless they have an asymmetric refractive index distribution across the core of fiber [28,30].

The generation of CVBs with high modal purity in fiber was demonstrated using micro-bend LPFG mechanically formed on a so-called vortex fiber which strongly disperses vector modes in the LP₁₁ group [31,32]. Recently, it was successfully demonstrated that an LPFG written by irradiating a two-mode ber from one side with a CO₂ laser is capable of generating the CVBs [30].

In some application like wavelength division multiplexing involving several wavelengths and mode division multiplexing involving several modes, the wavelength tunability and mode selectivity for the CVBs generation are appreciated. Thus in this article, we report a technique for in-fiber generation of different CVBs with wavelength tunability and mode selectivity over broadband. The intermodal coupling caused by the asymmetric refractive index distribution, which is induced by the acoustic flexural wave applied on the unjacketed fiber, is theoretically analyzed. Then the coupling coefficients between the fundamental HE₁₁ mode and the LP₁₁-group higher-order modes are summarized. The phase matching condition for mode conversion is discussed subsequently. Experimental results using the few-mode fiber (Corning SMF-28) for the visible band (532 nm, 633 nm) and the two-mode fiber (OFS) for the optical communication band (1540 nm - 1560 nm) are presented. The output beams show annular intensity patterns in absence of polarizer and characteristic intensity patterns under polarizer, which confirm radial and azimuthal polarizations of the generated CVBs. The utilization efficiency of the laser power reached 35% and a modal purity of ~30 dB was measured using the optical heterodyne interference method.

2. Principle and experimental conguration

As illustrated in Fig. 1 for a few-mode fiber with a step-index profile, the modes in scalar approximation solution, designated as LP₀₁, LP₁₁ and LP₂₁, correspond to the groups of vector modes { ${HE}_{11}^{x}$ , ${HE}_{11}^{y}$ }, { ${TE}_{01}$ , ${HE}_{21}^{even/odd}$ , ${TM}_{01}$ } and { ${HE}_{31}^{even/odd}$ , ${EH}_{11}^{even/odd}$ }, respectively. In the LP₀₁ group, the fundamental ${HE}_{11}^{x}$ and ${HE}_{11}^{y}$ modes are degenerate with their transverse electric fields parallel to the x and y directions, respectively [24]. In the LP₁₁ group, the TE₀₁ and TM₀₁ modes are separated from the strictly degenerate ${HE}_{21}^{even}$ and ${HE}_{21}^{odd}$ modes with tiny differences of modal effective refractive index, and have electric fields with azimuthal and radial polarizations, respectively. Mode conversion via the intermodal coupling between the fundamental modes and the LP₁₁-group higher-order modes may thus be adopted for generating the CVBs in a few-mode fiber.

Fig. 1 Schematic of the index profile of a few-mode fiber denoted with modal indices and intensity patterns for (a) the scalar modes and (b) corresponding groups of the vector modes. Field directions are denoted on the intensity patterns of vector modes.

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An acoustic flexural wave propagating in the z-direction with vibration along the x-axis in an unjacketed fiber may induce the asymmetric refractive index modulation with respect to the vibration direction and consequently cause the intermodal coupling, i.e. to transfer energy from one mode to another. The induced asymmetric refractive index modulation at the cross section of the fiber can be expressed as [33,34]

Δ n (r, ϕ) = N_{0} r \cos (ϕ),

where r and ϕ are the radial and azimuthal variables in cylindrical coordinates, respectively, and ϕ = 0 is set for the acoustic flexural wave vibration along the x-axis. N₀ = n₀(1 + χ)K²u₀, where n₀ is the refractive index of the core, χ = −0.22 is the elasto-optical coefficient of silica, K and u₀ are the wavevector and amplitude of the acoustic flexural wave, respectively. For the acoustically-induced fiber grating, the coupling coefcient κ_pm between a fundamental p mode and a higher-order m mode can be expressed as [35,36]

κ_{p m} = \frac{π}{λ} \sqrt{\frac{ε_{0}}{μ_{0}}} n_{0} \iint E_{t}^{p} \cdot Δ n (r, ϕ) E_{t}^{m} r d r d ϕ .

In weakly guidance approximation, the transverse components of the electric fields of the fundamental and higher-order vector modes can be expressed as [24]

E_{t}^{p} = u_{p} F_{01} (r),

and

E_{t}^{m} = Φ_{m} (ϕ) F_{11} (r),

where

F_{ℓ 1} (r), (ℓ = 0, 1)

is the radial function of the corresponding scalar mode

{LP}_{ℓ 1}, (ℓ = 0, 1)

, u_p =

\hat{x}

,

\hat{y}

is the unit vector in x- or y-directions for p =

{HE}_{11}^{x}

,

{HE}_{11}^{y}

, and

Φ_{m} (ϕ)

is the field direction function given in Table 1. We may rewrite the Eq. (2) as

κ^{p m} = C κ_{r} κ_{ϕ}^{p m}

such that

C = (π / λ) \sqrt{ε_{0} / μ_{0}} n_{0} N_{0},

κ_{r} = \int F_{01} (r) F_{11} (r) r^{2} d r,

and

κ_{ϕ}^{p m} = \int_{0}^{2 π} u_{p} \cdot Φ_{m} (ϕ) \cos ϕ d ϕ,

where the term cosϕ in Eq. (7) originates from the acoustic flexural wave vibration along the x-axis as expressed in Eq. (1). Thus the azimuthal component

κ_{ϕ}^{p m}

reveals allowed or forbidden the intermodal coupling between p and m modes, as shown in Table 1, which tells that the mode for an output CVB is selected via polarization control on the input light.

Table 1. $κ_{ϕ}^{p m}$ for Coupling between Fundamental Modes and LP₁₁-group Higher-order Modes due to the Acoustic Flexural Wave Vibrating along the x-axis

View Table

To guarantee the success of the mode conversion, the phase matching condition L_B = Λ should be satisfied at the same time. L_B = λ/Δn_pm is the beatlength of the two coupled modes, where λ is the resonance wavelength and Δn_pm is the modal refractive index difference. $Λ = \sqrt{π R C_{e x t} / f}$ is the period of the acoustic flexural wave propagating along the unjacketed fiber [33], where R is the ber radius and C_ext = 5760 m/s is the phase velocity of the acoustic wave in silica. The plots of the beatlengths for the LP₁₁-group higher-order modes converted from a fundamental mode are shown in Fig. 2(a), and the required frequencies f of the RF driving signal applied on the acoustic transducer are shown in Fig. 2(b) for the SMF-28 fiber based on the phase matching condition of the acoustically-induced ber grating

f = π R C_{ext} {(Δ n_{p m} / λ)}^{2} .

As seen in Fig. 2(b), the RF driving frequencies are different for different output modes due to the effective refractive index differences between the higher-order modes. The frequency difference is

Δ f_{m m'} = \frac{2 π R C_{e x t}}{λ^{2}} Δ n_{p m} Δ n_{m m'},

where

Δ n_{m m'}

is the effective refractive index difference between higher-order modes. Therefore, the mode selection at the same wavelength is achievable via tuning the frequency of the RF driving signal. In the example shown in Fig. 2(b), the generation of TE₀₁ and TM₀₁ modes at λ = 633 nm requires a setting of f = 0.8227 MHz and 0.8289 MHz, respectively. The frequency difference Δf = 6.2 kHz can be easily discriminated by an electrical function generator. Compared to the mechanical micro-bend [28,31] and CO₂-laser writing technology [30], this approach is highly advantageous with the tunability for phase matching in generating different types of CVBs at the same wavelength or varying wavelengths of CVBs over a broad band.

Fig. 2 (a) Beatlength and (b) acoustic wave frequency as functions of the resonance wavelength for the SMF-28 fiber (Corning). Insets show the zoomed-in curves at λ = 532 and 633 nm.

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Figure 3(a) shows the process flow for generating the CVBs. The beam is firstly purified as an HE₁₁ mode by stripping out all higher-order modes after polarization control that defines the mode as either ${HE}_{11}^{x}$ or ${HE}_{11}^{y}$ . Mode conversion is then performed to obtain a desired CVB in a segment of unjacketed ber (50 mm long) with one end glued with epoxy to the tip of the horn-like acoustic transducer and the other end xed on a ber clamp. The acoustic wave was generated and then amplied at the tip of the horn-like acoustic transducer by applying an RF driving signal on the piezoelectric transducer. To obtain the desired CVB, the mode conversion is selected via polarization control, i.e. ${HE}_{11}^{x}$ →TM₀₁ and ${HE}_{11}^{y}$ →TE₀₁, and the phase matching condition is satisfied via frequency tuning on the RF driver.

Fig. 3 (a) The process flow for generating CVBs. (b) Experimental setup for optical communication band based on the Two-Mode Step-Index Fiber (OFS). PC: polarization controller; MO: micro-objective; MS: mode stripper; GT: Glan-Taylor prism polarizer; SMF: single-mode ber; TMF: two-mode fiber; CCD: charge coupled device.

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3. Experimental results and discussions

Two types of optical fibers were adopted in the experiment, the SMF-28 fiber (Corning) and the Two-Mode Step-Index Fiber (OFS). The SMF-28 fiber, with a core radius of ρ_co = 4.5 µm, a cladding radius of ρ_cl = 62.5 µm and a step index of ∆ = 0.32%, were used as a few-mode fiber in the visible band. Experiment for the SMF-28 fiber was done with wavelength at 633 nm and 532 nm to acquire the required RF driving frequencies for the desired CVB modes. Beatlength and acoustic frequency as functions of the resonance wavelength as shown in Fig. 2 are deduced from the experimental results. To examine the modal field of the CVBs, the beam was projected on a CCD covered by a linear polarizer through a micro-objective in free space. The intensity patterns at various polarizations are shown in Figs. 4(a)-4(d) for the SMF-28 fiber [31]. The Two-Mode Step-Index Fiber, with a core radius of ρ_co = 10 µm, a cladding radius of ρ_cl = 62.5 µm, and a step index of ∆ = 0.32%, is suitable for generating CVBs in the optical communication band. Figure 3(b) shows the experimental setup for generation of CVBs in optical communication band with all-fiber devices. Experiment was done at λ = 1540 nm, 1545 nm, 1550 nm, 1555 nm and 1560 nm to acquire the required RF driving frequencies for desired modes of the CVBs, Theoretic calculation on the beatlength and the acoustic frequency as functions of the resonance wavelength has also been done based on Eq. (8) and the results are shown in Fig. 5 with experimental measurement denoted with solid dots. As the input power to the polarization controller (PC) was set at 6.8 mW, an output power of 2.4 mW was received for the CVBs, so the power utilization efficiency reached 35%. The system lost 1.4 mW from the PC and 3.0 mW from the mode stripper (MS) [37]. The modal intensity patterns at various polarizations at λ = 1550 nm are shown in Figs. 4(e) and 4(f). In experiment, the generated CVBs show annular intensity patterns in absence of polarizer, whereas splits appear in the patterns after adding a polarizer and rotate for rotating polarizer.

Fig. 4 Images taken by the CCD in absence of polarizer (a₁-f₁) and in presence of polarizer at different polarization orientations [(a₂ - a₅) to (f₂ - f₅)]. (a, c, e) are for TM₀₁ mode; (b, d, f) are for TE₀₁ mode; (a, b), (c, d), and (e, f) are for λ = 633 nm, 532 nm, and 1550 nm, respectively. The image sizes are 1.8 mm × 1.8 mm (a-d) and 2.25 mm × 2.25 mm (e, f), respectively.

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Fig. 5 Beatlengths and RF driving frequencies as functions of the resonance wavelength deduced from experimental data denoted as solid dots.

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Subsequently, the modal purity was measured for the generated CVBs using the optical heterodyne interference method [38]. When the ${HE}_{11}^{x}$ .( ${HE}_{11}^{y}$ ) mode is converted to the TM₀₁ (TE₀₁) mode in presence of the acoustic vibration, the frequency of the obtained mode after conversion will be downshifted by an amount equal to the acoustic frequency f. Thus the signal waveform for interference of the output waves can be expressed as

s (t) = β \sqrt{I_{1} I_{2}} \cos (2 π f t),

where I₁ and I₂ are the respective powers of the

{HE}_{11}^{x}

.(

{HE}_{11}^{y}

) and TM₀₁ (TE₀₁) modes, and β is the photo-electric conversion coefcient of the photodetector. Taking the example at λ = 633 nm in Fig. 4(a), the power of

{HE}_{11}^{x}

and TM₀₁ modes became equalized so that the visibility of the heterodyne interference signal reached the maximum by tuning RF driving voltage to 5.1 V, as shown the blue curve in Fig. 6. By further increasing the RF driving voltage to 7.4 V, the visibility of heterodyne interference signal reached the minimum, as shown the red curve in Fig. 6, when the power of

{HE}_{11}^{x}

was converted to that of TM₀₁ at maximum limit. The time waveforms of the detected heterodyne interference signals in these two extreme cases presented a maximum and a minimum peak-to-peak amplitudes (V_p−p) of 1 V (blue curve) and 0.12 V (red curve), respectively. Assuming a unity total power, i.e. I₁ + I₂ = 1, the former was for the case I₁:I₂ = 50%:50% and the later for the case of most complete conversion which derived I₁:I₂ = 0.1%:99.9% or a modal purity of ~30 dB. The same modal purity was experimentally reached for the converted TE₀₁ mode as well, while similar results were obtained in the optical communication band.

Fig. 6 Time waveforms of the interference signal with different power ratios between ${HE}_{11}^{x}$ and TM₀₁ at 633 nm.

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4. Conclusion

Intermodal coupling due to the asymmetrical refractive index modulation induced by applying acoustic flexural wave on fiber can be utilized to achieve CVBs with TM₀₁ and TE₀₁ modal fields, which have the characteristics of radial and azimuthal polarizations. Mode conversion of either ${HE}_{11}^{x}$ →TM₀₁ or ${HE}_{11}^{y}$ →TE₀₁ for acoustic vibration along the x-axis is chosen via polarization control at the input. The phase matching condition allowing the mode conversion is satisfied by tuning the acoustic wave frequency. A power utilization efficiency of 35% and a modal purity of ~30 dB were achieved for generated CVBs experimentally. This approach is advantageous with tunability for phase matching in generating different types of CVBs at the same wavelength over a broad wavelength range and implementable in fiber circuit.

Acknowledgments

This work is financially supported by the 973 Programs (2012CB921900, 2013CB328702), the NSFC (11404263, 61377055, 61405161, and 11174153, 11574161), the Fundamental Research Funds for the Central Universities (3102015ZY060).

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$u_{p}$ $κ_{ϕ}^{p m}$		TE₀₁	${HE}_{21}^{even}$	${HE}_{21}^{odd}$	TM₀₁
$u_{p}$ $κ_{ϕ}^{p m}$		$\hat{x} \sin ϕ - \hat{y} \cos ϕ$	$\hat{x} \cos ϕ - \hat{y} \sin ϕ$	$\hat{x} \sin ϕ + \hat{y} \cos ϕ$	$\hat{x} \cos ϕ + \hat{y} \sin ϕ$
${HE}_{11}^{x}$	$\hat{x}$	$\int_{0}^{2 π} \sin ϕ \cos ϕ d ϕ = 0$	$\int_{0}^{2 π} \cos^{2} ϕ d ϕ = π$	$\int_{0}^{2 π} \sin ϕ \cos ϕ d ϕ = 0$	$\int_{0}^{2 π} \cos^{2} ϕ d ϕ = π$
${HE}_{11}^{y}$	$\hat{y}$	$- \int_{0}^{2 π} \cos^{2} ϕ d ϕ = - π$	$- \int_{0}^{2 π} \sin ϕ \cos ϕ d ϕ = 0$	$\int_{0}^{2 π} \cos^{2} ϕ d ϕ = π$	$\int_{0}^{2 π} \sin ϕ \cos ϕ d ϕ = 0$

Cylindrical vector beam generation in fiber with mode selectivity and wavelength tunability over broadband by acoustic flexural wave

Abstract

1. Introduction

2. Principle and experimental conguration

3. Experimental results and discussions

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Tables (1)

Equations (10)

Optics Express