Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Realization of linear-mapping between polarization Poincaré sphere and orbital Poincaré sphere based on stress birefringence in the few-mode fiber

Yan Li, Lipeng Feng, Sihan Wu, Chen Yang, Weijun Tong, Wei Li, Jifang Qiu, Xiaobin Hong, Yong Zuo, Hongxiang Guo, and Jian Wu

Open Access Open Access

Abstract

Based on the spatial profiles and polarization states evolution process of the first-order modes resulted from stress-induced birefringence in the few-mode fiber (FMF), we analyze the mapping relationship between the input polarization states represented on polarization PS and the output spatial profiles represented on the orbital PS of the FMF with respect to the magnitude and orientation of birefringence. When the input mode lobe orientation and the phase differences between the four eigenmodes of FMF induced by the stress birefringence satisfy a given condition, the mapping relationship between the input polarization PS and the output orbital PS is linear. Thus, the arbitrary points on the orbit PS can be generated at the output of stressed FMF by controlling the polarization state of the input modes. Then we experimentally verify that, an electrical single-mode polarization controller, a mode converter for converting fundamental mode to higher-order mode, a polarization controller mounting a coil of two-mode fiber and a polarizer can be employed to generate arbitrary first-order spatial modes on the orbital PS by controlling the input single-mode polarization states. The positions on the orbital PS of the generated first-order modes, which are obtained by calculating the three normalized Stokes parameters of output modes, agree well with the simulation ones. The correlation coefficients between the theoretical mode profiles and the experimental ones are higher than 80%. Since the spatial profile evolutions depend on the variations of the input polarization states, a potential advantage of this method is high-speed switching among desired first-order modes by using the commercial devices switching the state of polarization.

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

1. Introduction

Few-mode fibers (FMF) have attracted many research interests in various applications due to the existence of higher-order modes that standard single-mode fibers (SMFs) do not possess [1]. In the SMF, the mode profile of the fundamental mode is symmetrical Gaussian pattern and has two orthogonal linear polarization states. When the SMF is perturbed by stress, the polarization state of the fundamental mode changes due to the stress-induced birefringence and the spatial profile keeps its original distribution. A polarization Poincaré sphere (PS) is generally used to represent the uniform polarization state [2]. In the FMF, the radially asymmetric higher-order modes have four degenerate orthogonal linearly-polarized (LP) modes, including two degenerate spatial orthogonal modes and each has two orthogonal linear polarizations. For the higher-order modes with uniform polarization, an orbital PS can be used to represent the spatial profile [3]. When the FMF is under the circumstances with stress, both spatial profile and polarization state of higher-order mode will evolve because of the birefringence effect. Based on the above phenomena, the mode manipulation by controlling the stress on FMFs has many potential applications in the fabrication of fiber grating [4], fiber optical manipulation [5,6], imaging [7,8], sensing [9,10], mode conversion [11,12] and mode coupling [13,14].

So far, the controllable stress can be induced by entwining a coil of FMF on a paddle-typed polarization controller (PC), pressing the FMF in one direction or corrugated plates, and introducing stress rods into one axis of the FMF cladding. For example, Hong et al. reported a 360° rotation of the mode lobe orientation and polarization direction for first-order LP modes by using a step-index circle-core FMF wound in a three-paddle PC [15] or FMFs with panda-type stress rods [16]. By tuning the orientation of an elliptical core FMF (E-FMF) entwined three-paddle PC, the polarization state of higher-order modes can be controlled without changing the spatial profile of mode [17]. The fiber placed between two corrugated plates can convert fundamental mode to higher-order modes [11,12]. In addition, continuously tunable orbital angular momentum (OAM) can also be generated by pressing the fiber into corrugated plates and metal parallel slabs [18,19]. The fiber under metal parallel slabs can induce the relative phase between two higher-order modes with degenerate spatial orthogonal property. In Ref. [20], our previous work reported a method that can generate tunable OAM modes with FMF wound in a three-paddle PC by continuously changing the angle of the linear polarization state of the input light. This method has a potential advantage of high-speed switching. Up to now, all the reported all-fiber mode manipulation methods are restricted to generate a line on the orbital PS, either a longitude or a latitude. Tunable generation of all the modes on the whole orbit PS is significant to flexibly use high-order modes in various applications, improving the functions of application systems and expanding the advantages of application scenarios based on high-order modes.

In this letter, according to the equivalent Jones vector model based on four linear polarization (LP) mode bases proposed in [20], we further research the evolution process of both the spatial profiles and the polarization states for the first-order modes resulted from stress-induced birefringence in the FMF. The polarization state at the FMF input port is represented by a polarization PS, while the spatial mode with uniform polarization at the FMF output port is represented as the orbital PS. By analyzing the magnitude the stress-induced birefringence with the fixed stress orientation, the mapping relationship between the points on the polarization PS and the orbital PS are obtained. It is found that when the mode lobe orientation of the input LP11 mode is 45° with the stress axis of the FMF and the phase differences between the four eigenmodes induced by the stress birefringence satisfy δa’x’b’y’- δb’x’b’y’=δa’y’b’y’+π, the mapping relationship between the input polarization PS and the output orbital PS is linear. Thus, the arbitrary points on the orbit PS can be generated by controlling the polarization state of the input modes. Since devices switching the polarization state at ultra-high speed are commercially available [21], a potential advantage of this method is high-speed switching among desired first-order modes. Then we experimentally verify that, by propagating through a mode converter, a coil of FMF wound in a PC and a polarizer, the desired first-order modes represented on the orbital PS can be generated by only adjusting the polarization of the input fundamental mode. The experimental results, which are obtained by calculating the three normalized Stokes parameters of orbital PS, agree well with the simulation ones.

2. The polarization and orbital Poincaré sphere

The polarization PS represents geometrically the spatially homogeneous polarization states of a beam [2], as shown in Fig. 1(a). The polarization states can be described as the superposition of two orthogonal and linearly polarized components along the x and y axes, as shown in Eq. (1).

$${\textrm E} = {\textrm A}_{\textrm x}{\textrm E}_{\textrm x} + {\textrm A}_{\textrm y}{\textrm E}_{\textrm y}$$
In Eq. (1), Ax and Ay represent the complex amplitudes of the x and y polarizations of a given beam, respectively. The polarization states represented by x and y polarizations can be mapped to the polarization PS’s surface through the Stokes parameters (S1, S2, and S3) in the Cartesian coordinates. The Stokes parameters of polarization PS are expressed in terms of Ax and Ay as:
$$\begin{array}{c} {\textrm S}_{1} = \frac{{{{\textrm I}_{0{^\circ }}} - {{\textrm I}_{90{^\circ }}}}}{{{{\textrm I}_{0{^\circ }}} + {{\textrm I}_{90{^\circ }}}}} = \frac{{\left\langle {{{|{{{\textrm A}_{\textrm x}}} |}^2}} \right\rangle - \left\langle {{{|{{{\textrm A}_{\textrm y}}} |}^2}} \right\rangle }}{{\left\langle {{{|{{{\textrm A}_{\textrm x}}} |}^2}} \right\rangle + \left\langle {{{|{{{\textrm A}_{\textrm y}}} |}^2}} \right\rangle }}\\ {{\textrm S}_2} = \frac{{{{\textrm I}_{45{^\circ }}} - {{\textrm I}_{135{^\circ }}}}}{{{{\textrm I}_{0{^\circ }}} + {{\textrm I}_{90{^\circ }}}}} = \frac{{\left\langle {{{\textrm A}_{\textrm x}}{\textrm A}_{\textrm y}^\ast{+} {{\textrm A}_{\textrm y}}{\textrm A}_{\textrm x}^\ast } \right\rangle }}{{{{\left\langle {{{|{{{\textrm A}_{\textrm x}}} |}^2}} \right\rangle }^2} + \left\langle {{{|{{{\textrm A}_{\textrm y}}} |}^2}} \right\rangle }}\\ {{\textrm S}_3} = \frac{{{{\textrm I}_{\textrm{right}}} - {{\textrm I}_{\textrm{left}}}}}{{{{\textrm I}_{0{^\circ }}} + {{\textrm I}_{90{^\circ }}}}} = \frac{{\left\langle {{\textrm i}({{\textrm A}_{\textrm x}}{\textrm A}_{\textrm y}^\ast{-} {{\textrm A}_{\textrm y}}{\textrm A}_{\textrm x}^\ast )} \right\rangle }}{{{{\left\langle {{{|{{{\textrm A}_{\textrm x}}} |}^2}} \right\rangle }^2} + \left\langle {{{|{{{\textrm A}_{\textrm y}}} |}^2}} \right\rangle }} \end{array}$$
where I0°, I90°, I45°, and I135° are the intensities of the light recorded through various orientations of a linear polarizer. Iright and Ileft are the intensities of the circularly polarized components of the beam. For a completely polarized light beam, the squares of the Stokes parameters add up to unity, i.e.,
$${\textrm S}_1^2 + {\textrm S}_2^2 + {\textrm S}_3^2 = 1$$

 figure: Fig. 1.

Fig. 1. (a) Polarization PS and (b) Orbital sphere of first-order modes. Some commonly encountered polarization states and modes are indicated.

Download Full Size | PDF

Similar to the polarization PS representation, the orbital PS has been proposed as a geometrical construction to represent the spatial profile evolution of higher-order mode [3]. Taking the first-order mode as an example, the orbital PS with unit radius can describe any spatial profile with the same polarization in the mode group, as shown in Fig. 1(b). Those modes on orbital PS carry the same polarization, which can be algebraically described by the Eq. (4) in terms of LP mode bases.

$${\textrm V}_{11} = ({\textrm A}_{\textrm a}\textrm{LP}_{11\textrm{ap}} + {\textrm A}_{\textrm b}\textrm{LP}_{11\textrm{bp}})$$
where LP11ap and LP11bp represent the spatially orthogonal LP11 modes with the same polarization along ‘p’ direction, whose mode orientations are respectively along the horizontal (H) and vertical (V) directions. The ‘Aa’ and ‘Ab’ represent the complex amplitudes of the LP11ap and LP11bp modes, respectively. In analogy to the polarization PS, the normalized Stokes parameters (O1, O2, and O3) of orbital PS are defined as [3]:
$$\begin{array}{c} {{\textrm O}_1} = \frac{{{{\textrm I}_{\textrm{LP}11(0{^\circ )}}} - {{\textrm I}_{\textrm{LP}11(90^\circ )}}}}{{{{\textrm I}_{\textrm{LP}11(0{^\circ )}}} + {{\textrm I}_{\textrm{LP}11(90^\circ )}}}} = \frac{{\left\langle {{{|{{{\textrm A}_{\textrm a}}} |}^2}} \right\rangle - \left\langle {{{|{{{\textrm A}_{\textrm b}}} |}^2}} \right\rangle }}{{\left\langle {{{|{{{\textrm A}_{\textrm a}}} |}^2}} \right\rangle + \left\langle {{{|{{{\textrm A}_{\textrm b}}} |}^2}} \right\rangle }}\\ {{\textrm O}_2} = \frac{{{{\textrm I}_{\textrm{LP}11(45{^\circ )}}} - {{\textrm I}_{\textrm{LP}11(135^\circ )}}}}{{{{\textrm I}_{\textrm{LP}11(45{^\circ )}}} + {{\textrm I}_{\textrm{LP}11(135^\circ )}}}} = \frac{{\left\langle {{{\textrm A}_{\textrm a}}{\textrm A}_{\textrm b}^\ast{+} {{\textrm A}_{\textrm b}}{\textrm A}_{\textrm a}^\ast } \right\rangle }}{{\left\langle {{{|{{{\textrm A}_{\textrm a}}} |}^2}} \right\rangle + \left\langle {{{|{{{\textrm A}_{\textrm b}}} |}^2}} \right\rangle }}\\ {{\textrm O}_3} = \frac{{{{\textrm I}_{{\textrm {OAM}}( + 1)}} - {{\textrm I}_{{\textrm {OAM}}( - 1)}}}}{{{{\textrm I}_{\textrm{LP}11(0{^\circ )}}} + {{\textrm I}_{\textrm{LP}11(90^\circ )}}}} = \frac{{\left\langle {{\textrm i}({{\textrm A}_{\textrm a}}{\textrm A}_{\textrm b}^\ast{-} {{\textrm A}_{\textrm b}}{\textrm A}_{\textrm a}^\ast )} \right\rangle }}{{\left\langle {{{|{{{\textrm A}_{\textrm a}}} |}^2}} \right\rangle + \left\langle {{{|{{{\textrm A}_{\textrm b}}} |}^2}} \right\rangle }} \end{array}$$
where ILP11(α) stands for the intensity of the LP11 mode lobe orientation at angle α with respect of the horizontal direction and IOAM(±1) stands for the intensities of the OAM±1 modes in the beam. The points on the equator of the sphere correspond to the standard LP11 modes with different mode lobe orientations. The angle between mode lobe orientation and the horizontal direction is defined as θ. The north and south poles represent the right- and left-handed helical phase annular beams, which can be taken as OAM+1 and OAM−1, respectively. The modes lied on the upper (lower) hemisphere is neither OAM nor LP mode, whose amplitude distribution is the adhesion of two-mode lobes of LP11 mode and the phase remains the right (left) helical. The degree of adhesion is defined as tan(χ), where -π/4⩽χ⩽π/4, as shown in Fig. 1(b). When the phase is right (left) helical, the χ is negative (positive). Meanwhile, the definitions of the O3 and the average OAM value (Lave) are the same as in [20]. Therefore, the points on every longitude of the sphere are precisely the tunable OAM modes with Lave between −1 and 1. The relationship between θ, χ and Stokes parameters can be represented as Eq. (6) according to the deduction in Ref. [2].
$$\begin{array}{c} {{\textrm O}_1} = \cos (2\chi )\cos (2\theta ) \\ {{\textrm O}_2} = \cos (2\chi )\sin (2\theta ) \\ {{\textrm O}_3} = \sin (2\chi ) \end{array}$$

3. The mapping relationship between the polarization PS and the Orbit PS

In the circular-core FMF, the first-order modes can be described by the superposition of four orthogonal LP11 modes and can be expressed as:

$${\textrm E} = {{\textrm E}_{\textrm{ax}}}\textrm{LP}_{11\textrm{ax}} + {\textrm E}_{\textrm{bx}}\textrm{LP}_{11\textrm{bx}} + {\textrm E}_{\textrm{ay}}\textrm{LP}_{11\textrm{ay}} + {\textrm E}_{\textrm{by}}\textrm{LP}_{11\textrm{by}}$$
where Eax, Ebx, Eay and Eby represent the amplitudes of the four orthogonal LP11 modes (LP11ax, LP11bx, LP11ay, and LP11by), as shown in Fig. 2(a). The four LP modes have the following electric field distributions [22]:
$$\begin{array}{l} \textrm{LP}_{11\textrm{ax}} = {\textrm F}_{11}({\textrm r})\cos (\Phi \overrightarrow {\textrm x} )\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\textrm{LP}_{11\textrm{bx}} = {\textrm F}_{11}({\textrm r})\sin (\Phi \overrightarrow {\textrm x} )\\ \textrm{LP}_{11\textrm{ay}} = {\textrm F}_{11}({\textrm r})\cos (\Phi \overrightarrow {\textrm y} )\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\textrm{LP}_{11\textrm{by}} = {\textrm F}_{11}({\textrm r})\sin (\Phi \overrightarrow {\textrm y} ) \end{array}$$
where F11(r) is the radial field distribution of the corresponding scalar mode solution; r and Φ are the radial and azimuthal coordinates ; ${\vec{\textrm x}}$ and ${\vec{\textrm y}}$ indicate linear polarization states along the H direction and V direction, respectively; ‘a’ and ‘b’ indicate the mode orientations along the H direction and V direction, respectively.

 figure: Fig. 2.

Fig. 2. (a) Mode profiles of LP11 mode along H and V axes; (b) The front view of the PMC and the PMC’s middle line; (c) The left view, obtained by looking to the middle line from the left and the mode profiles of LP11 mode along S and F axes.

Download Full Size | PDF

Utilizing the Jones vector representation, the first-order mode can be represented as a 4×1 matrix, as shown in Eq. (9). The corresponding four bases of the matrix are LP11ax, LP11bx, LP11ay, and LP11by, respectively.

$${\textrm E} = {[{\textrm E}_\textrm{ax}\;\;\;{\textrm E}_\textrm{bx}\;\;{\textrm E}_\textrm{ay}\;\;{\textrm E}_\textrm{by}]^{\textrm T}}$$
Since the polarization state of LP11a mode is homogeneous, the LP11a mode with different polarization states can be represented by Eq. (10):
$${\textrm E}_\textrm{in} = {[{\textrm A}_{\textrm x}\;\;\;0\;\;{\textrm A}_{\textrm y}\;\;0]^\textrm{T}}$$
To introduce the stress birefringence, an FMF is entwined around the paddles of a commercial PC can act as a polarization and mode controller (PMC). The Fig. 2(b) shows the front view of the PMC. When the LP11a passes through the PMC, the polarization and mode profile will be simultaneously changed. Due to the fiber bending in the PMC, the mode lobe and polarization orientations of four LP eigenmodes in the PMC will be along the stress and the orthogonal directions (S and F directions). The new four eigenmodes along S and F axes are defined as LP11a’x’, LP11b’x’, LP11a’y’ and LP11b’y’, as shown in Fig. 2(c). x’ and y’ indicate polarization directions along S and F axes, respectively. a’ and b’ indicate the mode directions along S and F axes, respectively. If the paddle of the PMC rotates β degrees with respect to the horizontal direction, the four eigenmodes will rotate β degrees correspondingly. Meanwhile, the phase differences between eigenmodes, which are introduced by fiber bending, can be calculated via δ=2πLΔneff/λ, where λ is the operating wavelength, L is the length of the FMF and Δneff is the effective refractive index difference accordingly. The phase differences between LP11a’x’ and LP11b’y’, LP11b’x’ and LP11b’y’, LP11a’y’ and LP11b’y’, are represented as δa’x’b’y’, δb’x’b’y’, δa’y’b’y’, respectively. For convenience, δ1, δ2, and δ3 stand for δa’x’b’y’, δb’x’b’y’ and δa’y’b’y’ in the equations, respectively.

According to the Eq. (5) in the Ref. [20], when the paddle rotates 45 degrees with respect with the H-V axis, the input LP11a mode described in Eq. (10) will evolve to Eout, which is described by Eq. (11).

$$\begin{aligned} {\textrm E}_{\textrm{out}} &= {\textrm R}( - \beta )\left[ {\begin{array}{{cccc}} {{\delta_1}}&0&0&0\\ 0&{{\delta_2}}&0&0\\ 0&0&{{\delta_3}}&0\\ 0&0&0&1 \end{array}} \right]{\textrm R}(\beta ){\textrm E}_{\textrm{in}} = {\textrm R}( - \frac{\pi }{4})\left[ {\begin{array}{{cccc}} {{\delta_1}}&0&0&0\\ 0&{{\delta_2}}&0&0\\ 0&0&{{\delta_3}}&0\\ 0&0&0&1 \end{array}} \right]{\textrm R}(\frac{\pi }{4}){\textrm E}_{\textrm{in}}\\ &= \frac{1}{4}\left[ {\begin{array}{{c}} {{\textrm A}_{\textrm x}({{\textrm e}^{{\textrm i}{\delta_1}}} + {{\textrm e}^{{\textrm i}{\delta_2}}} + {{\textrm e}^{{\textrm i}{\delta_3}}} + 1) + {\textrm A}_{\textrm y}({\textrm e}^{{\textrm i}{\delta_1}} - {\textrm e}^{{\textrm i}{\delta_2}} + {\textrm e}^{{\textrm i}{\delta_3}} - 1)}\\ {{\textrm A}_{\textrm x}({\textrm e}^{{\textrm i}{\delta_1}} + {\textrm e}^{{\textrm i}{\delta_2}} - {\textrm e}^{{\textrm i}{\delta_3}} - 1) + {\textrm A}_{\textrm y}({\textrm e}^{{\textrm i}{\delta_1}} - {\textrm e}^{{\textrm i}{\delta_2}} + {\textrm e}^{{\textrm i}{\delta_3}} + 1)}\\ {\textrm A}_{\textrm x}({\textrm e}^{{\textrm i}{\delta_1}} - {\textrm e}^{{\textrm i}{\delta_2}} + {\textrm e}^{{\textrm i}{\delta_3}} - 1) + {\textrm A}_{\textrm y}({\textrm e}^{{\textrm i}{\delta_1}} + {\textrm e}^{{\textrm i}{\delta_2}} + {\textrm e}^{{\textrm i}{\delta_3}} + 1)\\ {\textrm A}_{\textrm x}({\textrm e}^{{\textrm i}{\delta_1}} - {\textrm e}^{{\textrm i}{\delta_2}} - {\textrm e}^{{\textrm i}{\delta_3}} + 1) + {\textrm A}_{\textrm y}({\textrm e}^{{\textrm i}{\delta_1}} + {\textrm e}^{{\textrm i}{\delta_2}} - {\textrm e}^{{\textrm i}{\delta_3}} - 1) \end{array}} \right] \end{aligned}$$
where the R(β) and R(-β) are the rotation matrix and reverse rotation matrix, respectively.
$${\mathbf R}({\beta }) = \left[ {\begin{array}{{cccc}} {\cos \beta \cos \beta }&{\cos \beta \sin \beta }&{\cos \beta \sin \beta }&{\sin \beta \sin \beta }\\ { - \cos \beta \sin \beta }&{\cos \beta \cos \beta }&{ - \sin \beta \sin \beta }&{\cos \beta \sin \beta }\\ { - \cos \beta \sin \beta }&{ - \sin \beta \sin \beta }&{\cos \beta \cos \beta }&{\cos \beta \sin \beta }\\ {\sin \beta \sin \beta }&{ - \cos \beta \sin \beta }&{ - \cos \beta \sin \beta }&{\cos \beta \cos \beta } \end{array}} \right]$$

The spatial profiles of x and y polarization components for the output mode Eout can be represented on the orbit PS, and can be described with 1 × 2 Jones vector Ex and Ey as follows:

$${\textrm E}_{\textrm x} = \frac{1}{4}\left[ \begin{array}{{c}} {\textrm A}_{\textrm x}({\textrm e}^{{\textrm i}{\delta_1}} + {\textrm e}^{{\textrm i}{\delta_2}} + {\textrm e}^{{\textrm i}{\delta_3}} + 1) + {\textrm A}_{\textrm y}({\textrm e}^{{\textrm i}{\delta_1}} - {\textrm e}^{{\textrm i}{\delta_2}} + {\textrm e}^{{\textrm i}{\delta_3}} - 1)\\ {\textrm A}_{\textrm x}({\textrm e}^{{\textrm i}{\delta_1}} + {\textrm e}^{{\textrm i}{\delta_2}} - {\textrm e}^{{\textrm i}{\delta_3}} - 1) + {\textrm A}_{\textrm y}({\textrm e}^{{\textrm i}{\delta_1}} - {\textrm e}^{{\textrm i}{\delta_2}} + {\textrm e}^{{\textrm i}{\delta_3}} + 1) \end{array} \right] = \left[ {\begin{array}{{c}} {{\textrm{A}_\textrm{a}}}\\ {{\textrm{A}_\textrm{b}}} \end{array}} \right]$$
$${\textrm{E}_\textrm{y}} = \frac{1}{4}\left[ {\begin{array}{{c}} {{\textrm{A}_\textrm{x}}({\textrm{e}^{\textrm{i}{\delta_1}}} - {\textrm{e}^{\textrm{i}{\delta_2}}} + {\textrm{e}^{\textrm{i}{\delta_3}}} - 1) + {\textrm{A}_\textrm{y}}({\textrm{e}^{\textrm{i}{\delta_1}}} + {\textrm{e}^{\textrm{i}{\delta_2}}} + {\textrm{e}^{\textrm{i}{\delta_3}}} + 1)}\\ {{\textrm{A}_\textrm{x}}({\textrm{e}^{\textrm{i}{\delta_1}}} - {\textrm{e}^{\textrm{i}{\delta_2}}} - {\textrm{e}^{\textrm{i}{\delta_3}}} + 1) + {\textrm{A}_\textrm{y}}({\textrm{e}^{\textrm{i}{\delta_1}}} + {\textrm{e}^{\textrm{i}{\delta_2}}} - {\textrm{e}^{\textrm{i}{\delta_3}}} - 1)} \end{array}} \right] = \left[ {\begin{array}{{c}} {{\textrm{A}_\textrm{a}}}\\ {{\textrm{A}_\textrm{b}}} \end{array}} \right]$$

For the spatial profiles with x polarization, according to Eq. (2), Eq. (5) and the relationship described by (13), we can deduce the mapping relationships between the Stokes parameters (S1, S2 and S3) of input mode on the polarization PS and the ones (O1, O2 and O3) of output mode on the orbital PS as shown in Eq. (15).

$$\begin{array}{c} {\textrm{O}_1} = \frac{{[\cos ({\delta _1} - {\delta _3}) + \cos ({\delta _2})] + [\cos ({\delta _1}) + \cos ({\delta _2} - {\delta _3})]{\textrm{S}_1} + [\cos ({\delta _1} - {\delta _3}) - \cos ({\delta _2})]{\textrm{S}_2} + [\sin ({\delta _2} - {\delta _3}) - \sin ({\delta _1})]{\textrm{S}_3}}}{{2 + [\cos ({\delta _1} - {\delta _2}) + \cos ({\delta _3})]{\textrm{S}_1} + [\sin ({\delta _1} - {\delta _2}) + \sin ({\delta _3})]{\textrm{S}_3}}} \\ {\textrm{O}_2} = \frac{{[\cos ({\delta _1} - {\delta _2}) - \cos ({\delta _3})]{\textrm{S}_1} + (\sin ({\delta _2} - {\delta _1}) + \cos ({\delta _3})){\textrm{S}_3}}}{{2 + (\cos ({\delta _1} - {\delta _2}) + \cos ({\delta _3})){\textrm{S}_1} + (\sin ({\delta _1} - {\delta _2}) + \sin ({\delta _3})){\textrm{S}_3}}} \\ {\textrm{O}_3} = \frac{{ - [\sin ({\delta _3} - {\delta _2}) - \sin ({\delta _2})] - [\sin ({\delta _3} - {\delta _2}) - \sin ({\delta _1})]{\textrm{S}_1} - [\sin ({\delta _3} - {\delta _1}) + \sin ({\delta _2})]{\textrm{S}_2} + [\cos ({\delta _1}) - \sin ({\delta _2} - {\delta _3})]{\textrm{S}_3}}}{{2 + (\cos ({\delta _1} - {\delta _2}) + \cos ({\delta _3})){\textrm{S}_1} + (\sin ({\delta _1} - {\delta _2}) + \sin ({\delta _3})){\textrm{S}_3}}}\end{array}$$

It is found that the mapping relationship between the input mode on the polarization PS and output mode on the orbital PS is related with the phase differences δ1, δ2, and δ3. For better understanding, Fig. 3 shows the above-mentioned mapping relationship with phase differences δ1=π/4, δ2=−π/2 and δ3 varies from −π to π. The polarization states of the input modes are depicted by the red points on the polarization PS, as shown in Fig. 3(a). The corresponding orbital PSs of the output modes with respect to different phase differences are shown in the Figs. 3(b)–3(i). The colors of the dots represent the normalized intensities of the modes, which are normalized by the input power. The red dots represent that the intensities of modes are 1, while the blue dots represent that the intensities of modes are 0.

 figure: Fig. 3.

Fig. 3. The mapping relationship between polarization PS and orbital PS with the phase differences δ1=π/4, δ2=−π/2 and δ3 varies from –π to π. (a) The polarization state of the input modes. (b)-(i) The corresponding orbital PSs of the output modes with respect to different phase differences.

Download Full Size | PDF

To realize the tunable generation of arbitrary modes on the orbit PS by controlling the polarization state of the input mode, we need to find the suitable phase differences for making sure the mapping relationship between the polarization PS and the orbit PS is linear. According to Eq. (14), when the coefficients of S1 and S3 in the denominator are zero, e.g. the phase differences are δ1-δ2=δ3+π, the mapping from Oi to Si (i = 1, 2, 3) is linear, as shown in Eq. (16). Under this condition, the location of the points on the output orbit PS can be obtained by counterclockwise rotating the points on the input polarization PS around the y, x, and z axis consequently. The corresponding rotation radians are δ3, (δ1-δ3) and $\pi /2$ respectively.

$$\begin{array}{l} {\textrm{O}_1} = \sin ({\delta _3})\sin ({\delta _2}){\textrm{S}_1} - \cos ({\delta _2}){\textrm{S}_2} + \sin ({\delta _2})\cos ({\delta _3}){\textrm{S}_3}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;{\textrm{O}_2} ={-} \cos ({\delta _3}){\textrm{S}_1} + \sin ({\delta _3}){\textrm{S}_3}\\ {\textrm{O}_3} ={-} \cos ({\delta _2})\sin ({\delta _3}){\textrm{S}_1} - \sin ({\delta _2}){\textrm{S}_2} - \cos ({\delta _2})\cos ({\delta _3}){\textrm{S}_3} \end{array}$$
For simplification, we set δ1$= 2n\pi + \pi $, δ2$= ({2n\pi + \pi /2} )$ and δ3$={-} (2n\pi + \pi /2$), and the relationship between Oi and Si (i = 1,2,3) can be simplified to O1=−S1, O2=S3, O3=S2. It indicates that the orbital PS can be obtained by rotating the input polarization PS 90 degrees counterclockwise around the y-axis, x-axis and z-axis consequently. We plot three circles on two PSs to explain the relationship between them. Assumed that input polarizations are the states of the three special circles (blue, red and green circles) on the polarization PS (Fig. 4(a)), the ideal output mode will be the point of three circles with the same color on the orbital PS (Fig. 4(b)).

 figure: Fig. 4.

Fig. 4. (a) The input polarizations (blue, red and green circles) on the polarization PS. (b) The ideal output modes on the orbital PS.

Download Full Size | PDF

4. Experiment verification of arbitrary spatial modes generation

The experimental setup for the generation and detection of the points on the orbital PS is sketched in Fig. 5. The output light of the laser is divided into two paths by a 1:1 optical coupler. The bottom branch is used to generate first-order modes, while the top branch is used as a reference Gaussian beam to verify the phase structure of the generated mode. A tunable attenuator in the reference branch is to make the powers of the reference beam and the generated beam in the same order of magnitude, which can obtain clear interferograms. In the bottom branch, an electrical PC is inserted ahead of a photonic lantern to control the SOPs of the input light. The photonic lantern realizes the conversion of LP01 mode to LP11a mode [23]. Then, the converted LP11a mode is launched into a step-index TMF with a core diameter of 16µm and a 0.5% relative index difference between the core and cladding. The V-number is 4.091, which can support LP01 and LP11 modes. The MFDs of LP01 and LP11 modes are respectively are 13.684µm and 17.741µm. In order to make sure that the TMF can introduce the phase differences of δ1$= 2n\pi + \pi $, δ2$= 2n\pi + \pi /2$ and δ3$={-} ({2n\pi + \pi /2} )$, we use the Finite Element Method to simulate the effective refractive indices (RIs) of four LP11 modes. When the bending radius of TMF is 29 mm, the phase differences of δ1, δ2 and δ3 at 1550 nm with one loop are −0.974${\pi }$, 0.551${\pi }$ and −2.424${\pi }$, respectively. As the bending radius is same as the paddle size of a commercial polarization controller (PC), the phase differences can be achieved by entwining the TMF into a commercial PC with loops of 2, 1 and 2. The first paddle of PC is used to compensate for the phase error induced by the imperfect fabrication of photonic lantern. The second one of PC is used to complete the mapping from polarization PS to orbital PS. The last one is used to compensate for the phase error induced by the TMF tail. In the experiment, the total insertion loss is 2.08 dB. The powers of generated modes fluctuate within 0.4 dB. The losses introduced by photonic lantern and PMC are respectively are 1.8 dB and 0.28 dB.

 figure: Fig. 5.

Fig. 5. The experimental setup for the generation and detection of the points on the orbital PS.

Download Full Size | PDF

The output beam from the TMF is collimated by an objective lens into space and then passes through a linear polarizer to obtain the x polarization. The output modes are recorded by an infrared camera (CCD), when the input polarizations are adjusted according to the dots (blue, red and green) on the polarization PS in the Fig. 4(a). The first rows of Figs. 6(a)–6(c) show the SOPs of the input beam corresponding to the dots of three circles on the polarization PS, respectively. The second rows of Figs. 6(a)–6(c) show the theoretical spatial profiles, which correspond to the dots on the orbital PS in the Fig. 4(b). The third rows of Figs. 6(a)–6(c) show the experimental spatial profiles corresponding to different input polarizations. The fourth rows of Figs. 6(a)–6(c) show the interferograms, which are recorded by the CCD. The experimental results, shown in the third row of Fig. 6(a), illustrate that the recorded spatial profiles keep the two lobes with different angles when the input light SOPs are elliptical polarization states with different ellipticities in the same direction. The experimental results, shown in the third row of Fig. 6(b), illustrate that the recorded spatial profiles evolve from LP11 mode to OAM+1, to orthogonal rotated LP11 mode, to OAM−1 and then back to initial LP11 mode. The experimental results, shown in the third row of Fig. 6(c), illustrate that both the mode lobe orientations and the adhesion degrees of recorded spatial modes evolve with the input polarizations. From the view of mode evolution, the experimental results agree well with the theoretical ones.

 figure: Fig. 6.

Fig. 6. The polarizations of input, intensity distributions of the theoretical and experimental results and the interferograms of experimental results. (a) The modes on the red circle of orbital PS. (b) The modes on the green circle of orbital PS and (c) The modes on the blue circle of orbital PS. The first rows are the SOPs of the input light. The second and third rows are, respectively, the theoretical and experimental results. The fourth rows are the corresponding interferograms of experimental results.

Download Full Size | PDF

For further verification, we point out the positions of the experiment modes on the orbital PS by using Eq. (6) with extracting the mode lobe orientation and average OAM value of modes, as shown in Fig. 7. The solid lines in Fig. 7 are the paths where simulation results are located on, while the dots represent the positions where experiment results lie. The deviations between the experimental results and simulation ones are mainly because of the inaccuracy locations of both input polarizations in the experiment and the image center in Fourier series expansion [20].

 figure: Fig. 7.

Fig. 7. Experimental orbital Poincaré sphere. The dots on the sphere are the positions where experimental higher-order modes are located.

Download Full Size | PDF

To evaluate the profile properties of output beams, we calculate the correlation coefficients between the theoretical mode profiles and the experimental ones [15]. The mode profile similarities will be higher than 80%, as shown in Fig. 8.

 figure: Fig. 8.

Fig. 8. The correlation coefficients between simulation mode profiles and the experimental ones.

Download Full Size | PDF

5. Conclusion

In conclusion, we research the mapping relationship between polarization PS and orbital PS when the phase differences between eigenmodes in the FMF with the stress changes. Based on the theoretical analysis, the phase difference condition satisfying the linear relationship between two PS is found. Then, we propose a new method to generate all points of the orbital PS with the combination of a two-mode fiber entwining around the PC and a polarizer by only changing the polarization states of the input light. With the setup, the corresponding polarization can be converted to the desirable modes. The experimental results show well agreement with the theoretical ones, which is proved by measuring the Stokes parameters of the tested modes. It is believed that the scheme shall have potential applications in encoding information and quantum computation.

Funding

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

Disclosures

The authors declare no conflicts of interest.

References

1. R. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010). [CrossRef]  

2. A. Kumar and A. K. Ghatak, Polarization of light with applications in optical fibers, vol. 246 (SPIE press, 2011).

3. M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24(7), 430–432 (1999). [CrossRef]  

4. 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]  

5. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235(4795), 1517–1520 (1987). [CrossRef]  

6. Y. Zhang, L. Zhao, Y. Chen, Z. Liu, Y. Zhang, E. Zhao, J. Yang, and L. Yuan, “Single optical tweezers based on elliptical core fiber,” Opt. Commun. 365, 103–107 (2016). [CrossRef]  

7. M. Gu, H. Kang, and X. Li, “Breaking the diffraction-limited resolution barrier in fiber-optical two-photon fluorescence endoscopy by an azimuthally-polarized beam,” Sci. Rep. 4(1), 3627 (2015). [CrossRef]  

8. L. Yan, P. Gregg, E. Karimi, A. Rubano, L. Marrucci, R. Boyd, and S. Ramachandran, “Q-plate enabled spectrally diverse orbital-angular-momentum conversion for stimulated emission depletion microscopy,” Optica 2(10), 900–903 (2015). [CrossRef]  

9. Y. Fan, G. Wu, W. Wei, Y. Yuan, F. Lin, and X. Wu, “Fiber-optic bend sensor using LP21 mode operation,” Opt. Express 20(24), 26127–26134 (2012). [CrossRef]  

10. Y. Zhao, C. Wang, G. Yin, B. Jiang, K. Zhou, C. Mou, Y. Liu, L. Zhang, and T. Wang, “Simultaneous directional curvature and temperature sensor based on a tilted few-mode fiber Bragg grating,” Appl. Opt. 57(7), 1671–1678 (2018). [CrossRef]  

11. Y. Zhao, Y. Liu, C. Zhang, L. Zhang, G. Zheng, C. Mou, J. Wen, and T. Wang, “All-fiber mode converter based on long-period fiber gratings written in few-mode fiber,” Opt. Lett. 42(22), 4708–4711 (2017). [CrossRef]  

12. I. Giles, A. Obeysekara, R. Chen, D. Giles, F. Poletti, and D. Richardson, “Fiber LPG mode converters and mode selection technique for multimode SDM,” IEEE Photonics Technol. Lett. 24(21), 1922–1925 (2012). [CrossRef]  

13. M. M. Ali, Y. Jung, K. S. Lim, M. R. Islam, S. U. Alam, D. J. Richardson, and H. Ahmad, “Characterization of mode coupling in few-mode FBG with selective mode excitation,” IEEE Photonics Technol. Lett. 27(16), 1713–1716 (2015). [CrossRef]  

14. C. Schulze, R. Brüning, S. Schröter, and M. Duparré, “Mode coupling in few-mode fibers induced by mechanicalstress,” J. Lightwave Technol. 33(21), 4488–4496 (2015). [CrossRef]  

15. Z. Hong, S. Fu, D. Yu, M. Tang, and D. Liu, “All-fiber tunable LP11 mode rotator with 360° range,” IEEE Photonics J. 8(5), 1–7 (2016). [CrossRef]  

16. Q. Mo, Z. Hong, D. Yu, S. Fu, L. Wang, K. Oh, M. Tang, and D. Liu, “All-fiber spatial rotation manipulation for radially asymmetric modes,” Sci. Rep. 7(1), 2539 (2017). [CrossRef]  

17. J. Cui, Z. Hong, S. Fu, B. Chen, M. Tang, and D. Liu, “All-fiber polarization manipulation for high-order LP modes with mode profile maintenance,” Opt. Express 25(15), 18197–18204 (2017). [CrossRef]  

18. S. Li, Q. Mo, X. Hu, C. Du, and J. Wang, “Controllable all-fiber orbital angular momentum mode converter,” Opt. Lett. 40(18), 4376–4379 (2015). [CrossRef]  

19. Y. Jiang, G. Ren, Y. Lian, B. Zhu, W. Jin, and S. Jian, “Tunable orbital angular momentum generation in optical fibers,” Opt. Lett. 41(15), 3535–3538 (2016). [CrossRef]  

20. S. Wu, Y. Li, L. Feng, X. Zeng, W. Li, J. Qiu, Y. Zuo, X. Hong, H. Yu, R. Chen, I. P. Giles, and J. Wu, “Continuously tunable orbital angular momentum generation controlled by input linear polarization,” Opt. Lett. 43(9), 2130–2133 (2018). [CrossRef]  

21. For instance, http://versawave.com/products/polarizationmodulators/.

22. R. D. Niederriter, M. E. Siemens, and J. T. Gopinath, “Simultaneous control of orbital angular momentum and beamprofile in two-mode polarization-maintaining fiber,” Opt. Lett. 41(24), 5736–5739 (2016). [CrossRef]  

23. S. G. Leon-Saval, N. K. Fontaine, J. R. Salazar-Gil, B. Ercan, R. Ryf, and J. Bland-Hawthorn, “Mode-selective photonic lanterns for space division multiplexing,” Opt. Express 22(1), 1036–1044 (2014). [CrossRef]  

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)
Fig. 1.
Fig. 1. (a) Polarization PS and (b) Orbital sphere of first-order modes. Some commonly encountered polarization states and modes are indicated.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. (a) Mode profiles of LP11 mode along H and V axes; (b) The front view of the PMC and the PMC’s middle line; (c) The left view, obtained by looking to the middle line from the left and the mode profiles of LP11 mode along S and F axes.

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. The mapping relationship between polarization PS and orbital PS with the phase differences δ1=π/4, δ2=−π/2 and δ3 varies from –π to π. (a) The polarization state of the input modes. (b)-(i) The corresponding orbital PSs of the output modes with respect to different phase differences.

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. (a) The input polarizations (blue, red and green circles) on the polarization PS. (b) The ideal output modes on the orbital PS.

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. The experimental setup for the generation and detection of the points on the orbital PS.

View in Article | Download Full Size | PDF

Fig. 6.
Fig. 6. The polarizations of input, intensity distributions of the theoretical and experimental results and the interferograms of experimental results. (a) The modes on the red circle of orbital PS. (b) The modes on the green circle of orbital PS and (c) The modes on the blue circle of orbital PS. The first rows are the SOPs of the input light. The second and third rows are, respectively, the theoretical and experimental results. The fourth rows are the corresponding interferograms of experimental results.

View in Article | Download Full Size | PDF

Fig. 7.
Fig. 7. Experimental orbital Poincaré sphere. The dots on the sphere are the positions where experimental higher-order modes are located.

View in Article | Download Full Size | PDF

Fig. 8.
Fig. 8. The correlation coefficients between simulation mode profiles and the experimental ones.

View in Article | Download Full Size | PDF

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

E = A x E x + A y E y
S 1 = I 0 ∘ − I 90 ∘ I 0 ∘ + I 90 ∘ = ⟨ | A x | 2 ⟩ − ⟨ | A y | 2 ⟩ ⟨ | A x | 2 ⟩ + ⟨ | A y | 2 ⟩ S 2 = I 45 ∘ − I 135 ∘ I 0 ∘ + I 90 ∘ = ⟨ A x A y ∗ + A y A x ∗ ⟩ ⟨ | A x | 2 ⟩ 2 + ⟨ | A y | 2 ⟩ S 3 = I right − I left I 0 ∘ + I 90 ∘ = ⟨ i ( A x A y ∗ − A y A x ∗ ) ⟩ ⟨ | A x | 2 ⟩ 2 + ⟨ | A y | 2 ⟩
S 1 2 + S 2 2 + S 3 2 = 1
V 11 = ( A a LP 11 ap + A b LP 11 bp )
O 1 = I LP 11 ( 0 ∘ ) − I LP 11 ( 90 ∘ ) I LP 11 ( 0 ∘ ) + I LP 11 ( 90 ∘ ) = ⟨ | A a | 2 ⟩ − ⟨ | A b | 2 ⟩ ⟨ | A a | 2 ⟩ + ⟨ | A b | 2 ⟩ O 2 = I LP 11 ( 45 ∘ ) − I LP 11 ( 135 ∘ ) I LP 11 ( 45 ∘ ) + I LP 11 ( 135 ∘ ) = ⟨ A a A b ∗ + A b A a ∗ ⟩ ⟨ | A a | 2 ⟩ + ⟨ | A b | 2 ⟩ O 3 = I OAM ( + 1 ) − I OAM ( − 1 ) I LP 11 ( 0 ∘ ) + I LP 11 ( 90 ∘ ) = ⟨ i ( A a A b ∗ − A b A a ∗ ) ⟩ ⟨ | A a | 2 ⟩ + ⟨ | A b | 2 ⟩
O 1 = cos ⁡ ( 2 χ ) cos ⁡ ( 2 θ ) O 2 = cos ⁡ ( 2 χ ) sin ⁡ ( 2 θ ) O 3 = sin ⁡ ( 2 χ )
E = E ax LP 11 ax + E bx LP 11 bx + E ay LP 11 ay + E by LP 11 by
LP 11 ax = F 11 ( r ) cos ⁡ ( Φ x → ) LP 11 bx = F 11 ( r ) sin ⁡ ( Φ x → ) LP 11 ay = F 11 ( r ) cos ⁡ ( Φ y → ) LP 11 by = F 11 ( r ) sin ⁡ ( Φ y → )
E = [ E ax E bx E ay E by ] T
E in = [ A x 0 A y 0 ] T
E out = R ( − β ) [ δ 1 0 0 0 0 δ 2 0 0 0 0 δ 3 0 0 0 0 1 ] R ( β ) E in = R ( − π 4 ) [ δ 1 0 0 0 0 δ 2 0 0 0 0 δ 3 0 0 0 0 1 ] R ( π 4 ) E in = 1 4 [ A x ( e i δ 1 + e i δ 2 + e i δ 3 + 1 ) + A y ( e i δ 1 − e i δ 2 + e i δ 3 − 1 ) A x ( e i δ 1 + e i δ 2 − e i δ 3 − 1 ) + A y ( e i δ 1 − e i δ 2 + e i δ 3 + 1 ) A x ( e i δ 1 − e i δ 2 + e i δ 3 − 1 ) + A y ( e i δ 1 + e i δ 2 + e i δ 3 + 1 ) A x ( e i δ 1 − e i δ 2 − e i δ 3 + 1 ) + A y ( e i δ 1 + e i δ 2 − e i δ 3 − 1 ) ]
R ( β ) = [ cos ⁡ β cos ⁡ β cos ⁡ β sin ⁡ β cos ⁡ β sin ⁡ β sin ⁡ β sin ⁡ β − cos ⁡ β sin ⁡ β cos ⁡ β cos ⁡ β − sin ⁡ β sin ⁡ β cos ⁡ β sin ⁡ β − cos ⁡ β sin ⁡ β − sin ⁡ β sin ⁡ β cos ⁡ β cos ⁡ β cos ⁡ β sin ⁡ β sin ⁡ β sin ⁡ β − cos ⁡ β sin ⁡ β − cos ⁡ β sin ⁡ β cos ⁡ β cos ⁡ β ]
E x = 1 4 [ A x ( e i δ 1 + e i δ 2 + e i δ 3 + 1 ) + A y ( e i δ 1 − e i δ 2 + e i δ 3 − 1 ) A x ( e i δ 1 + e i δ 2 − e i δ 3 − 1 ) + A y ( e i δ 1 − e i δ 2 + e i δ 3 + 1 ) ] = [ A a A b ]
E y = 1 4 [ A x ( e i δ 1 − e i δ 2 + e i δ 3 − 1 ) + A y ( e i δ 1 + e i δ 2 + e i δ 3 + 1 ) A x ( e i δ 1 − e i δ 2 − e i δ 3 + 1 ) + A y ( e i δ 1 + e i δ 2 − e i δ 3 − 1 ) ] = [ A a A b ]
O 1 = [ cos ⁡ ( δ 1 − δ 3 ) + cos ⁡ ( δ 2 ) ] + [ cos ⁡ ( δ 1 ) + cos ⁡ ( δ 2 − δ 3 ) ] S 1 + [ cos ⁡ ( δ 1 − δ 3 ) − cos ⁡ ( δ 2 ) ] S 2 + [ sin ⁡ ( δ 2 − δ 3 ) − sin ⁡ ( δ 1 ) ] S 3 2 + [ cos ⁡ ( δ 1 − δ 2 ) + cos ⁡ ( δ 3 ) ] S 1 + [ sin ⁡ ( δ 1 − δ 2 ) + sin ⁡ ( δ 3 ) ] S 3 O 2 = [ cos ⁡ ( δ 1 − δ 2 ) − cos ⁡ ( δ 3 ) ] S 1 + ( sin ⁡ ( δ 2 − δ 1 ) + cos ⁡ ( δ 3 ) ) S 3 2 + ( cos ⁡ ( δ 1 − δ 2 ) + cos ⁡ ( δ 3 ) ) S 1 + ( sin ⁡ ( δ 1 − δ 2 ) + sin ⁡ ( δ 3 ) ) S 3 O 3 = − [ sin ⁡ ( δ 3 − δ 2 ) − sin ⁡ ( δ 2 ) ] − [ sin ⁡ ( δ 3 − δ 2 ) − sin ⁡ ( δ 1 ) ] S 1 − [ sin ⁡ ( δ 3 − δ 1 ) + sin ⁡ ( δ 2 ) ] S 2 + [ cos ⁡ ( δ 1 ) − sin ⁡ ( δ 2 − δ 3 ) ] S 3 2 + ( cos ⁡ ( δ 1 − δ 2 ) + cos ⁡ ( δ 3 ) ) S 1 + ( sin ⁡ ( δ 1 − δ 2 ) + sin ⁡ ( δ 3 ) ) S 3
O 1 = sin ⁡ ( δ 3 ) sin ⁡ ( δ 2 ) S 1 − cos ⁡ ( δ 2 ) S 2 + sin ⁡ ( δ 2 ) cos ⁡ ( δ 3 ) S 3 O 2 = − cos ⁡ ( δ 3 ) S 1 + sin ⁡ ( δ 3 ) S 3 O 3 = − cos ⁡ ( δ 2 ) sin ⁡ ( δ 3 ) S 1 − sin ⁡ ( δ 2 ) S 2 − cos ⁡ ( δ 2 ) cos ⁡ ( δ 3 ) S 3
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use