## Abstract

We propose and experimentally demonstrate a novel scheme to generate LP_{11}/LP_{21} modes with tunable mode lobe orientation (MLO). Wherein, the MLOs have an excellent linear relationship with the linearly-polarized states of input fundamental modes. The proposed scheme is composed of a polarization controller (PC), a mode converter, a mode and polarization controller (PMC) which is twined with the few mode fiber (FMF) and a polarizer. Experimental results show that the deviations of MLOs between generated LP_{11}/LP_{21} modes and simulated ones are less than 3.5 and 8 degrees over C band. Since polarization control up to nanosecond scale is available with GaAs or lithium based electro-optic modulator, the proposed scheme could enable nanosecond time scale MLO control, which would be immensely useful for optical trapping, fiber sensors and optical communications.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Recently, few-mode fibers (FMFs) have shown outstanding performance in many applications with attractive features that standard single mode fibers (SMFs) do not possess. For example, the mode division multiplexing (MDM) technique based on FMFs has been widely studied to solve future capacity crunch arising in SMFs [1–3]. The exploration of FMFs has benefited not only the optical communications but also many other fields like optical sensors. FMF-based sensors have attracted much research interests due to inherently distinctive optical characteristics of higher order modes [4,5]. In the FMFs, although vector modes are the true modes, linearly-polarized (LP* _{lm}*) mode bases are commonly used because LP

*modes are more readily excited and detected than the vector modes. For*

_{lm}*l*≥ 1, the electric field distribution of LP

*mode is divided into several segments along the angular direction. In general, the symmetry optical axis where mode lobes located is mode lobe orientation (MLO) of LP*

_{lm}*(*

_{lm}*l*≥ 1) mode [6]. As the MLOs of LP

*(*

_{lm}*l*≥ 1) modes are changeable, the need for the generation of LP

_{lm}mode with tunable MLO is frequently encountered in both scientific research and engineering applications. For example, Ma

*et al.*demonstrated that the azimuthally and radially polarized beams can be generated by coherent two LP modes whose polarizations and MLOs are orthogonal [7]. The orbital angular momentum (OAM) could also be generated by the superposition of two LP modes of different MLOs with a fixed phase delay [8,9]. Additionally, the LP modes with different MLOs can provide a versatile tool for the applications of optical trapping, optical tweezers and fiber specklegram sensors (FSS) [10–12]. Fiber optical tweezers based on LP

_{11}mode rotation have been used to realize multi-dimensional manipulation of trapped yeast cells [11]. Recently, multiple-input multiple-output (MIMO)-free weakly coupled MDM systems have been introduced to increase the data rates in intra-datacenter networks while reducing the cabling footprint and using the commercial direct detection transceivers [13,14]. In weakly coupled MDM systems, proper MLO of LP

*mode is necessary for mode selective multiplexing and de-multiplexing. Based on the above mentioned applications, the generation of high order modes with the tunable MLO has become a basic necessity, which means that the mode intensity patterns are the same while the symmetry axes are different. For the devices changing the MLO, a planar lightwave circuit (PLC) based mode rotator was fabricated on an asymmetric silica waveguide [15] and an orientation-insensitive azimuthally asymmetric mode rotator (OIAAMR) using chirally-coupled-core fiber [16] was proposed. However, these mode rotators only can realize the conversion between two spatial orthogonal degenerate modes and can’t obtain the tunable MLO for a specific input mode. Recently, the tunable LP*

_{lm}_{11}mode generators based on mechanically induced twist and bending in circle-core FMF [17] and polarization maintaining few-mode fibers (PM-FMFs) [18,19] were reported. Although high efficiency and large bandwidth could be obtained, they require mechanically or manually control the fiber states which are not suitable for high speed, i.e. up to 1GHz, adjustment of MLO.

By means of Jones vector representation, we deduce the transmission matrix of the high-order modes in a coil of paddled circular-core FMF based on LP mode basis. According to the transmission matrix, we propose and demonstrate a novel method to generate LP_{11} and LP_{21} modes with tunable MLO controlled by the polarization of LP_{01} modes in the SMF. The MLOs of LP_{11}/LP_{21} modes rotate 180/90 degrees with the angle of linear polarization varies from 0° to 180°. The angle deviations between experimental MLOs and simulated MLOs of LP_{11}/LP_{21} mode are less than 3.5 and 8 degrees over C band, respectively. Since polarization control up to nanosecond scale is available with GaAs or lithium niobate based electro-optic modulator [20], the proposed method could enable nanosecond time scale MLO control, which would be immensely useful for optical trapping, fiber sensors and optical communications.

## 2. Theoretical analysis

The basic concept of the proposed method that can generate LP* _{lm}* modes with tunable MLO is illustrated in Fig. 1. The method is composed of an SMF-based polarization controller (PC), a mode converter (MC), and a paddle-type polarization mode controller (PMC) which is twined with the FMF. The SMF-based PC is responsible for adjusting polarization state of LP

_{01}mode [21,22]. The MC should be polarization-insensitive and used to convert LP

_{01}mode to LP

*mode (*

_{lm}*l$\ge $*1) with same polarization state. In the circular-core FMF, the

*l*-order mode can be described by four orthogonal LP

*mode bases, as shown in Eq. (1).*

_{lm}_{ax}, E

_{ay}, E

_{bx}and E

_{by}represent the amplitudes of the four orthogonal modes, respectively. LP

_{lm}_{ax}, LP

_{lm}_{ay}, LP

_{lm}_{bx}and LP

_{lm}_{by}are the four orthogonal mode bases along horizontal and vertical (H and V) axes, which can be expressed by Eq. (2).

*l*and

*m*denote transverse and radial indices, F

*(r) represents the radial field distribution and $\Phi $ is the azimuthal coordinate. $\stackrel{\rightharpoonup}{x}$and $\stackrel{\rightharpoonup}{y}$ indicate polarization directions along H and V axes, respectively; a and b indicate the MLO along H and V axes, respectively. Figure 1(b) shows the mode bases of*

_{lm}*l*= 1 and

*l*= 2 which are along the H-V coordinate systems. Analogizing to the Jones vector representation, the

*l*-order mode can be represented in terms of the four mode bases, as shown in Eq. (3)

when the output mode of the MC has the characteristics that MLO is parallel with H axis (LP_{lm}_{a}) and polarization state is linear (oriented at θ with H axis), the Jones vectors of the mode can be expressed as Eq. (4).

Then, E_{in}* ^{l}* will be transmitted into the PMC, which is produced by entwining an FMF around the paddles of a commercial PC. For a normal circle-core FMF, since effective refractive index (RI) differences between any two modes of mode bases are less than 10

^{−7}, the directions of the mode bases vary at random [23]. When the FMF is twined around the paddles of the PC, due to modal birefringences induced by the fiber bending, effective RIs of mode bases will be separated. Meanwhile, the mode and polarization directions of the aforementioned mode bases will be forced to follow the stress direction and the orthogonal direction, which are defined as the slow and fast axes (S and F axes) of the PMC, as shown in the left view of the PMC in Fig. 1(c). The mode bases along S and F axes are described as LP

_{lm}_{a’x’}, LP

_{lm}_{a’y’}, LP

_{lm}_{b’x’}and LP

_{lm}_{b’y’}. x’ and y’ indicate polarization directions along S and F axes, respectively. a’ and b’ indicate the MLOs along S and F axes, respectively. The effective RI differences between LP

_{lm}_{a’x’}and LP

_{lm}_{a’y’}, LP

_{lm}_{b’x’}and LP

_{lm}_{a’y’}, LP

_{lm}_{b’y’}and LP

_{lm}_{a’y’}can be described by Δn

_{eff(a’x’a’y’)}, Δn

_{eff(b’x’a’y’)}, Δn

_{eff(b’y’a’y’)}, which are related to optical fiber parameters and bending radius. The corresponding phase differences between the mode bases can be calculated by δ = 2πLΔn

_{eff}/λ, where λ is the operating wavelength and L is the length of the FMF. As the stress direction rotates with the paddle, mode and polarization directions of mode bases also rotate with the paddle. If the paddle rotates α degrees with respect to the H axis, the four new mode bases along S and F axes will also rotate α degrees correspondingly, as shown in Fig. 1(c).

Therefore, the output mode of the PMC can be obtained through below three procedures. Firstly, the mode bases of the E_{in}* ^{l}* mode along the S and F axes can be obtained by rotating the mode bases along the H-V coordinate system counterclockwise by an angle α. The corresponding rotation matrix R(α) can be written as Eq. (5), which is interpreted by taking the rotation from LP

_{11ax}to mode bases along S and F axes as an example, as shown in Fig. 2. For example, the rotation factor between LP

_{11ax}and LP

_{11a’x}is cos(α), which is called mode rotation factor. If the

*l*> 1, the mode rotation factor is cos(

*l*α); The rotation factor between LP

_{11a’x}and LP

_{11a’x’}is cos(α), which is called polarization rotation factor. If the

*l*> 1, the mode rotation factor still is cos (α); Thus, the rotation factor between LP

_{11ax}and LP

_{11a’x’}is cos(α)cos(α). If the

*l*> 1, the rotation factor between LP

_{lm}_{ax}and LP

_{lm}_{a’x’}is cos(

*l*α)cos(α) and the rest elements of the rotation matrix can be obtained by using the same way

Secondly, after the mode bases along S and F axes passing through the PMC, the phase differences between mode bases along S and F axes (δ_{a’x’a’y’}, δ_{b’x’a’y’} and δ_{b’y’a’y’}) will be introduced. The effect of the phase differences can be described by a phase retarding matrix r().

Finally, if we want to describe the output mode in terms of the mode bases along H and V axes, the mode bases of undergoing retarding matrix need to be rotated clockwise by an angle α. In other words, the mode bases along S and F axes need to be rotated back to the mode bases along H and V axes. Thus, the Jones matrix of the PMC is written as:

In a special case, δ_{a’x’a’y’} = π and δ_{b’x’a’y’} = δ_{b’y’a’y’} = 0, when the PMC’s paddle rotates 90/2*l* degrees, the output mode of the PMCis given by:

The four mode components of Jones vector E_{out}* ^{l}* are LP

_{11ax}, LP

_{11bx}, LP

_{11ay}and LP

_{11by}, respectively. Thus, the x and y polarizations of the mode E

_{out}

*can be expressed as:*

^{l}At the end of the generator, the output mode E_{out}* ^{l}* is injected to a polarizer. The Jones matrix for a general linear polarizer is given by

If the angle 𝛽 = 𝛼 + π/4, the distribution of output mode P_{out}* ^{l}* can be written as:

It is seen from the Eq. (13) that the MLO of output mode is [θ + (2*l*-1)π/4*l*]/*l* which varies with polarization angle θ. The MLO of output mode and linear-polarized state of input mode have an excellent linear relationship. Figure 3 shows the simulation results of tunable LP_{11}/LP_{21} mode generators, which illustrate that the MLO rotate 180 or 90 degrees when the input polarization varies from 0° to 180° under the conditions of β = π/2 or 3π/8 for LP_{11} or LP_{21} modes, respectively.

## 3. Experimental setup and results

In this work, we verify the generation of LP_{11}/LP_{21} modes with tunable MLO by using an electric PC, a homemade six-mode selective photonic lantern (MSPL), a PMC and a polarizer. The experimental setup for the mode generation and detection is sketched in Fig. 4. The optical source is a tunable laser whose operation wavelength is initially set at 1550 nm for the subsequent characterizations. The electric PC transfers arbitrary elliptical polarization state to desired linear polarization in the SMF. The output light of the electric PC is coupled into two different ports (port3 and port4) of the MSPL to efficiently excite LP_{11a} and LP_{21a} modes [24], respectively. After mode conversion, LP_{11a}/ LP_{21a} modes are launched into a section of FMF which the core and outer trench radii, the refractive index of the core, cladding and outer trench are 10.7μm, 1.81μm, 1.448, 1.444 and 1.441, respectively. In order to make sure that the PMC can introduce phase differences of δ_{a’x’a’y’} = π and δ_{b’x’a’y’} = δ_{b’y’a’y’} = 0, we use COMSOL software to simulate the bending TMF with the same radius as the commercial PC paddle size (Ф = 27mm). Figure 5(a) shows the calculated phase differences between the four mode bases along S and F axes for *l* = 1 and *l* = 2 with the variation of loops numbers. According to the simulated results, with 8 fiber loops mounted, the phase differences of the LP_{11} mode bases along S and F axes (δ_{11a’x’a’y’}, δ_{11b’x’a’y’} and δ_{11b’y’a’y’}) are 1.2π, 80.2π and 81.8π, which are approximately equal to π, 80π and 82π, respectively. At the same time, the phase differences of the LP_{21} mode bases along S and F axes (δ_{21a’x’a’y’}, δ_{21b’x’a’y’} and δ_{21b’y’a’y’}) are 1.1π, 1.95π and 2.01π, which are approximately equal to 1π, 2π and 2π, respectively. Those phase differences keep basically stable over the C-band, as shown in Fig. 5(b). Thus, we twin the FMF into a commercial PC whose paddle size is 27mm by loops of 3, 8 and 3. The 3 loops of the first and third paddles are for compensating the unavoidable perturbations of the superfluous FMF. By rotating three paddles, we are able to realize 45°/22.5° angles between the MLOs of LP_{11a}/LP_{21a} modes and the PMC’s stress orientation, respectively. The FMF’s output mode is collimated by using an objective lens. The polarizer with scale after the objective lens is used to obtain a polarization component of the output mode. The pass axes of the polarizers are set as 90/67.5 degrees for the LP_{11}/LP_{21} generators, respectively. Then, the beam intensity distributions are imaged by using a CCD camera, respectively. Maintaining the states of PMC, LP_{11}/LP_{21} modes with tunable MLO can be observed just by adjusting the input electric PC.

The insert losses for the generators of LP_{11}/LP_{21} modes with tunable MLO are about 7.1 dB and 7.5 dB in the C-band, when the polarization state is varied from 0° to 180°. The mode dependent losses are lower than 1.5dB at the wavelength of 1540nm, 1550nm and 1560nm. The MSPL introduces 3.5 dB and 3.8 dB losses for LP_{11} and LP_{21} mode generators, respectively. The polarizer can introduces 3.5dB loss, which includes 3dB intrinsic loss, since that only one polarization component is received. The link loss introduced by other components is probably resulted from the splicing between MSPL and FMF.

The experimental results, shown in Figs. 6(a) and 6(b), illustrate that LP_{11}/LP_{21} modes rotate 180/90 degrees when the polarization angle θ varies from 0° to 180° at 1540nm, 1550nm and 1560nm, respectively. The red arrows represent the MLOs of experimental results. The MLOs vary with the input linear-polarized angles are shown as the dotted lines of Figs. 7(a) and 7(b). These results coincide with the simulated ones, sketched in Figs. 3(a) and 3(b). The deviations between experimental MLOs and simulated MLOs of LP_{11}/LP_{21} modes are individually measured at 1540nm, 1550nm and 1560nm, as shown in the solid lines of Figs. 7(a) and 7(b). For tunable LP_{11} and LP_{21} mode generators, the deviations of MLOs are below 3.5° and 8°, respectively, indicating that the proposed mode generators are wavelength-insensitive. Compared with the tunable LP_{21} mode generator, the fluctuations of MLOs in the tunable LP_{11} mode generator are a little bigger. It might be the reason that the phase differences δ_{a’x’a’y’}, δ_{b’x’a’y’}, and δ_{b’y’a’y’} of LP_{21} mode bases are closer to the ideal values than that of the LP_{11} mode bases in the experiment. Thus, we could improve the performance of tunable LP_{11} mode generator by choosing more appropriate bending radius or the numbers of loops. Mode correlation coefficients between the captured mode profiles and the corresponding ideal LP mode profiles with specific MLO are calculated. The correlation coefficients of the LP_{11} modes related to the input polarization angle are shown in Fig. 8(a), and are higher than 76%, 87% and 74% at the wavelength of 1540nm, 1550nm and 1560nm. The correlation coefficients of LP_{21} modes are shown in Fig. 8(b) for the polarization angle form 0° to 180°, and are higher than 70%, 75% and 56% at the wavelength of 1540nm, 1550nm and 1560nm. The imperfect fabrication including the uneven heating of the tube, the twisting of optical fiber in the tube and some uncertain reasons leads to wavelength-dependence of the home-made MSPL, which induces degradations of correlation coefficients at 1560nm.

## 4. Conclusions

In summary, a novel method to generate LP_{11} and LP_{21} modes with tunable MLO is proposed and experimentally verified. We have theoretically deduced transmission matrix of the high-order modes in a coil of paddled circular-core FMF based on LP mode basis. The continuously tunable MLOs can be obtained by changing the input linear polarizations, which are with 3.5°/8° precision for tunable LP_{11} and LP_{21} mode generators at the wavelength of 1540nm, 1550nm and 1560nm, respectively. The experimental characterization agrees well with the theoretical calculation. The mode dependent losses are lower than 0.8dB of proposed generators, when the operation wavelength is varied from 1540nm to 1560nm.

## Funding

National Natural Science Foundation of China (NSFC) (61875019, 61675034, 61875020, 61571067); The Fund of State Key Laboratory of IPOC (BUPT); The Fundamental Research Funds for the Central Universities.

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