1. Introduction

The data transmission volume in classical and quantum communication has been increasing dramatically over the past several years, thereby driving the need for higher-capacity channels as well as higher bit rates. Different methods have been used to increase the amount of bits of information per pulse sent from sender (Alice) to receiver (Bob) in free space and fiber optics communications.

For example in free space Quantum Key Distribution (QKD) protocols, high dimensional encoding using spatial degrees of freedom of light, e.g., orbital angular momentum (OAM), can increase the secure key generation rate of the protocol compared to the case that the polarization of light is using to encode information [1–5]. OAM degree of freedom is corresponding to the helical wave-front of the photon which is inherently unbounded and is in infinite-dimensional Hilbert space, so the photon can be used as a qudit to increase the amount of data transfer by transmitting one photon from 1 bit of information to $log_2 d$ bits of information, where d is the dimension of the qudit [6]. Using qudit instead of qubit provides many benefits in QKD protocols such as increasing the key rate as well as improving noise resistance [7]. Moreover, arbitrary superposition of polarization and spatial modes which is called structured light, can be used to realize higher dimensional states of photon [7–9]. The structured light modes are the eigenmodes of the free space propagation so they are invariant upon free-space propagation.

The other classical and quantum channel is optical fiber. Since a large amount of data exchange and QKD protocols are carried out through optical fibers, using structured light in optical fiber for encoding in higher dimension could be helpful. In this method instead of single mode fiber (SMF) which can only carries one spatial mode (with Gaussian profile) with two orthogonal polarization (HE11x, HE11y), multi mode or few mode fibers with the ability to transfer more spatial modes should be used. These eigenmodes of the fiber could be applied to encode data but due to some imperfections and environmental effects the eigenmodes of the fiber cannot maintain through the fiber and couple to each other [8].

In transmitting one bit of information using two polarization modes in SMF fiber, the state of polarization and accordingly the information is changed in the fiber due to the various disturbances which exist unavoidably in fiber resulting from asymmetries in the fiber geometry or from bending, stress and other environmental factors. It corresponds to a unitary transformation and given that this transformation is almost stable, Alice and Bob can compensate for it by using polarization controllers based on feedback control [10,11]. In multimode fibers, spatial and polarization modes change along the length of the fiber due to disturbances, so to compensate for these changes, both the spatial and polarization degrees of freedom of photon must be controlled at the same time. So far, the production and control of polarization and spatial degree of freedom of light in optical fibers have been done separately [12–15]. In this paper we propose a structure to control spatial and polarization degree of freedom of light simultaneously. This device which we name it $SPC$ (Spatial and Polarization Controller) can be used to compensate the environmental effects and disturbance on fiber modes in quantum and classical channels. Using this $SPC$ we can implement classical and quantum communication protocols in higher dimension in optical fibers which give us a higher bit rate. This paper has been organized as follows: in Section 2, the theoretical model of coupling between fiber modes is described. Also the important fiber perturbations factors and their effect on coupling between modes are studied. The structure of our proposed SPC and its calculation are studied in Section 3, and, finally, we conclude this paper in Section 4.

2. Theoretical model

2.1 Coupling between fiber modes

The modes of the step-index optical fibers are obtained by solving the Maxwell’s equations considering the boundary conditions. These eigenmodes named cylindrical vector (CV) modes. When the difference between the refractive indices of core and clad is very small (for common fiber $\Delta n \simeq 1\%$.), the weekly-guiding approximation can be used. In this case, the dispersion equations of a number of fiber modes become the same. The solution of the dispersion equation is the propagation constant of different modes. A group of modes that have equal dispersion equations and therefore equal propagation constants in the above approximation are called degenerate modes. CV modes are categorized into different azimuthal orders ($l$). The zeroth azimuthal order eigenmodes are two degenerate fundamental modes $HE_{1,m}^{odd}$, $HE_{1,m}^{even}$ which have the same propagation constants. Here, $m$ is the radial order, denoting the number of radial nodes of the mode. The first azimuthal order eigenmodes are $TM_{0,1}$, $TE_{0,1}, HE_{2,1}^{even}, HE_{2,1}^{odd}$ and the $l$th ($l>1$) azimuthal order eigenmodes are $EH_{l-1,m}^{even}$, $EH_{l-1,m}^{odd}$, $HE_{l+1,m}^{even}$, and $HE_{l+1,m}^{odd}$[16]. In weakly-guiding approximation regime, the eigenmodes of each subspace of $l$ are considered degenerated with the same propagation constant, so they form a degenerate subspace and any linear combination of them could be considered as a fiber eigenmode [17]. Linearly polarized (LP) modes and Orbital Angular Momentum (OAM) modes are two groups of fiber mode bases. Any electric field in subspace $l \geq 1$ inside the fiber can be decomposed into four LP or CV or OAM mode bases with different complex amplitudes. Due to the some environmental disturbance effect on the fiber (that will be mentioned in the following), the eigenmodes of the fiber change and couple to each other along the length of the fiber. The coupling equation between these modes are as follows [8]:

Here $a(z)$ is a vector that shows the amplitude of each eigenmodes of the fiber, $\beta$ is the diagonal matrix of propagation constants, $K$ is the coupling matrix, and $z$ is the length of the disturbance in the direction of fiber $(z)$.

In an isotropic material(before perturbation), the coupling matrix elements is obtained from the equation [8]:

Here, $a$ is the radius of the core, $d$ is the diameter of the clad, $\beta$ is propagation constant, $J_l$ and $K_l$ are the Bessel functions of the first kind and modified Bessel functions of the second kind, respectively, and $A$ is normalization coefficient. $u$ is equal to $a\sqrt {k^2n^2_{core}-\beta ^2}$ and $w$ is equal to $a\sqrt {\beta ^2- k^2n^2_{clad}}$ where $k$ is the wave number, and, $n_{core}$ and $n_{clad}$ are the refractive indices of core and clad, respectively. The $z$ component of the fields is the linear combination of $\cos (l \pm 1)\theta$ or $\sin (l \pm 1)\theta$ [16]. Any arbitrary state in a degenerate subspace of the first order azimuthal modes is a linear superposition of these four bases.

Figure 1 shows the intensity patterns of the first azimuthal order of LP modes $LP_{1,1}$.

 

Fig. 1. Four orthogonal bases $\hat {x} LP_{11}^{\cos }, \hat {x} LP_{11}^{\sin },\hat {y} LP_{11}^{\cos }$ and $\hat {y} LP_{11}^{\sin }$ in first azimuthal order degenerate subspace of an ideal optical fiber.

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In principle, any perturbation can produce two types of coupling: the first is coupling between non-degenerate modes and the second is coupling between degenerate modes. The exchange energy coefficient($C_{ex}$) between two non-degenerate modes is:

The coupling matrix between degenerate modes ($l> 0$) is a $4 \times 4$ matrix. The operator that relates the states of polarization and spatial profile before and after the coupling is a unitary operator in $SU(4)$ group (Special Unitary matrix Group), which is written in the form $U=e^{-i\big (\beta +K\big )z}$. The $SU(4)$ group has $15$ generators, so each member of this group is characterized by $15$ independent parameters. However, we will show that important fiber perturbations produce operators that are in $SU(2) \otimes SU(2)$ subgroup of $SU(4)$ group and therefore they are described with a maximum of $6$ parameters, so we can compensate for it by controlling only 6 parameters. Important fiber perturbations are: 1. Birefringence; 2. Core ellipticity; 3. Tortion (or twist); 4. Fiber bending; 5. Magnetic field; and 6. Change in temperature [19–21]. In the following, we will give a brief explanation of the mentioned cases.

2.2 Important fiber perturbations

2.2.1 Stress birefringence

Applying an external force on the fiber leads to two phenomena: stress-birefringence and fiber deformation. The presence of birefringence in the optical fiber leads to a change in the polarization of the light passing through the fiber while maintaining its spatial profile. On the other hand, changing the shape of the cross section of the fiber changes the spatial profile of the mode passing through it while changing its polarization too. The overall effect comes from the sum of these two effects [22]. In this subsection, we will talk about stress-birefringence and postpone the explanation of fiber deformation to the next part.

If the external force that acts on the fiber is in $y$ direction in the transverse plane, then the birefringence, defined as the difference in the refractive index between two directions $x$ and $y$ occurs. The permittivity disorder $\tilde {\epsilon }$ is written as follows: [23,24]

For a fiber with the parameters listed in Table 1, $m_1=0.52P(\text {cm}^{-1})$ and $m_2=-0.10P(\text {cm}^{-1})$ for $LP_{11}$ modes. $m_3$ which is caused by the coupling of $z$ components, is obtained for different $l$s from the order of $10^{-5}P(\text {cm}^{-1})$. As you can see, $m_1$ and $m_2$ are much greater than $m_3$, and coupling of $z$ components can be ignored.

2.2.2 Core ellipticity

Deformation in the shape of transverse plane of fiber can change the fiber modes. To calculate the coupling coefficients from Eq. (2), we need to change the limits of the integrals based on the deformation of the fiber, but for a very small deformation, instead of changing the limits of the integral, a scalar $\tilde {\epsilon }$ dependent on $\theta$ can be used [25].

If the shape of transverse plane is an ellipse whose major and minor diameter are in the direction of the $x$ and $y$ axes, then we have $\tilde {\epsilon _e}$ as follows [19,22,25]

By putting this $\tilde {\epsilon }$ in Eq. (2), the coupling matrix in $l>0$ degenerate subspace can be written as:

Here $p_1= \frac {\beta \eta (n_{core}^2 -n_{clad}^2)}{n_{core}^2} \times \frac {u^2}{u^2+w^2}$ and $p_2= \frac {- \eta (n_{core}^2 -n_{clad}^2)}{2n_{core}^2 \beta \frac {a^2}{u^2}} \times \frac {-J_{l-1}(u)J_{l+1}(u) }{J_l^2(u)-J_{l+1}(u)J_{l-1}(u)}$. The core ellipticity changes the polarization of all modes while only affecting the intensity pattern of $LP_{1m}$ which can be used to create OAM modes with $l=1$ in optical fiber [12,13]. The first and second terms of Eq. (9) affect the polarization and OAM, respectively. Since these two terms are commuted by each other, the operator resulting from core deformation can be written as a tensor product of two operators, which one affects the polarization and the other affects the OAM.

2.2.3 Torsion (or twist)

If $\tau$ is torsion per unit of length, then the resulting $\tilde {\epsilon }$ is written as follows [19,23]:

Here $n_{av}$ is the mean refractive index of core and clad, and $g=p_{12}-p_{11}$ is an elasto-optic coefficient. The coupling matrix between degenerate modes with $l>0$ is calculated as follows [26]:

Similar to core ellipticity, torsion affects both polarization and OAM but, the resulting operator is in the form of a tensor product of two polarization and OAM operators.

As a general rule, silica fibers can tolerate $\tau$ less than three turns in one meter then, $\frac {g \tau }{2}\leq 1.42 (\text {cm}^{-1})$ [27].

2.2.4 Fiber bending

The bending of the fiber affects the intensity, polarization and OAM of its modes. When the fiber bends, some photons escape and the light intensity inside the fiber decreases. The greater the curvature of the bend, the greater the decrease in intensity. Therefore, the effect of bending on the intensity is not reversible, while the rotation of the other two degrees of freedom can be compensated. The effect of fiber bending on the remaining light in the fiber can be described by the coupling equation.

Bending in $xz$ plane makes two types of stress: longitudinal stress and compressive stress which make $\tilde {\epsilon }_z$ and $\tilde {\epsilon }_x$, respectively [19]:

Because Young’s modules is too large (in order of GPa), $\kappa$ is very small and coupling between non-degenerate modes can be neglected. The coupling matrix is obtained as follows:

As can be seen from Eq. (15) Bending just changes polarization and it has no effect on OAM. $K_{\text {bend}}$ is of order $10^{-4}\kappa ^2 (\text {cm}^{-1})$, which is very insignificant.

2.2.5 Magnetic field

Another unwanted disturbance that can change the state of light inside the fiber is the presence of an external magnetic field around the fiber. Although this effect is very small, it still rotates the polarization and OAM with an operator from $SU(2) \otimes SU(2)$ subspace.

If $B$ is the magnetic field component along the fiber axis, then the resulting $\tilde {\epsilon _M}$ is written as follows [19]:

$V$ is the Verdet constant and for silica in 1550nm is about $0.6(\text {Rad}\text {T}^{-1}\text {m}^{-1})$. The coupling matrix is obtained as follows:

Because usually $B$ is very small (for example, the Earth’s magnetic field is about $50\mu$T), so $K_{M}$ is negligible. However, the effect of the external magnetic field on the fiber modes can be described with an operator from group $SU(2) \otimes SU(2)$

2.2.6 Change in temperature

The effect of temperature changes on the cylindrical fiber is only the uniform expansion or contraction of the fiber and as a result the uniform change of its refractive index. In this way, temperature changes apply the same overall phase to all fiber modes.

In all above cases coupling matrix is in the form $K=K_{\pi } \otimes I_l +I_{\pi } \otimes K_{l}$. Because of $[K_{\pi } \otimes I_l ,I_{\pi }\otimes K_{l}]=0$, therefore according to Eq. (1), the operator that convert the input state to the output state will be obtained as:

So, to control the input state in degenerate subspaces of fiber, we must have a device that can produce any productive operator in that subspaces.

After all, it is necessary to pay attention to the fact that in all above cases, The perturbation direction was in the $y$ axes of fiber. If the perturbation direction is at angle $\phi$ with $y$ axes, then the coupling matrix rotates as follows [19].

3. All-fiber proposed system for spatial and polarization controller

As seen in the last section for any state in degenerate 4-dimensional subspace of $l=1$, all perturbation environmental effects and asymmetries in fiber could be considered as a two productive unitary operators in two subspaces of polarization degree of freedom and spatial degree of freedom ($SU(2) \otimes SU(2)$). Different types of polarization controllers (PCs) have been proposed before which are based on squeezing [11]. But non of them could control the spatial modes of light simultaneously. The aim of this paper is to control this two degrees of freedom simultaneously in order to compensate all perturbation effects on input modes through the fiber. Our proposed structure which is based on squeezing the fiber in different directions is shown in Fig. 2.

 

Fig. 2. Structure of all-fiber system designed for control of polarization and spatial profile of $LP_{1m}$ modes. Two pairs of triple mechanical squeezers used to control the polarization are placed next to each other. One of them affects the gray fiber and the other affects the yellow one. Gray and yellow fiber are different in mechanical, optical or geometrical properties.

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Compared to the polarization controller (based on mechanical pressure) which consists of three squeezers in angles $\frac {\pi }{2},\frac {\pi }{4}$ and $\frac {\pi }{2}$ respect to axis $x$ [11], our proposed structure is composed of six squeezers in angles $\frac {\pi }{2},\frac {\pi }{4},\frac {\pi }{2},\frac {\pi }{2},\frac {\pi }{4}$ and $\frac {\pi }{2}$ respect to axis $x$, so that it can perform spatial mode control at the same time as polarization control. It should be noted that the difference in color in different parts of the fiber means the difference either in core geometry, optomechanical ($P_{11} \& P_{12}$) and optical ($n_{core}$) properties or $\Delta n$ (difference between refractive index of core and clad).

The length of each squeezer is about 10 (cm). The force applied uniformly along the length of the squeezer is less than 120 (N), which is equivalent to $P=12 \text {N/cm}$. This weight should be selected with an accuracy of 1 grams in order to control the polarization with good accuracy. Silica optical fiber can tolerant strain less than $10^{-3}$ [27]. A pressure of $P=12$(N/cm) produces a strain of $10^{-4}$ in the optical fiber, which is smaller than its tolerance limit.

The main challenge is to choose two different types of fiber with different $q$ values which depends on core geometry, optomechanical ($P_{11} \& P_{12}$) and optical ($n_{core}$) properties or $\Delta n$. One way is to use a same size silica fiber with $\Delta n \simeq 0.336\%$ which has already been designed and used to produce OAM in fiber [12]. Another suggestion for the second type of fibers is to use elliptical few mode fiber (e-FMF) with initial $\frac {A_1}{B_1} =1.6$ [14]. We can use common circular fiber and an e-FMF for grey and yellow parts, respectively. e-FMF with length $L$ creates a constant phase difference between orthogonal spatial modes, which must be taken into account, but the phase difference caused by the external pressure is related to the changes of form factor.

To better explain the function of this structure it is good to consider the effect of pressure in a certain direction on the polarization and spatial degrees of freedom of light. Consider the pressure is applied in direction $y$. Pressure causes both birefringence and core ellipticity. Imoto et al. have shown when the strain of the fiber is very small, the effect of the core deformation and the change of its refractive index can be added together. The $\tilde {\epsilon }$ caused by the pressure in $y$ direction on the fiber can be written as follows [22]:

So the parameters $p_1$ and $p_2$ in coupling matrix (Eq. (9)) are obtained. The operator of a squeezer at direction $y$ with length $L$ can be written in the form below:

The effect of pressure on SOP and SOS can be shown on polarization Poincaré sphere and orbital Poincaré sphere, respectively (see Fig. 3) [28]. $S_1, S_2$ and $S_3$ are stokes parameters of SOP, similarly, $O_1, O_2$ and $O_3$ can be defined as stokes parameters of SOS.

 

Fig. 3. Rotation of SOP and SOS which is made by squeezing the fiber in direction $y$. We assume the start points are on $S_2$ and $O_2$ axes of polarization and orbital angular momentum Poincaré sphere, respectively. If SOP rotates $\alpha$ degrees around $S_1$ then SOS rotates $q\alpha$ degree around $O_1$.

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The effect of applying pressure on direction $y$ is to rotate $\alpha$ the input SOP and rotate $q \alpha$ the input SOS around the axis $S_1$ of polarization Poincaré sphere and $O_1$ of orbital Poincaré spheres, respectively. It is clear that with a single pressure, angels of rotation of SOP and SOS are dependent to each other. A pair of squeezers that acts on fibers with different $q$ values can be used to make them independent.

If two successive vertical squeezers (two first squeezers in Fig. 4) with different $q$ values rotate the polarization by $\alpha _1$ and $\alpha _5$, respectively, then the following operator is applied to the state in the fiber:

 

Fig. 4. The theoretical equivalent of Fig. 2. Three pairs of squeezers press the fiber with angles of $\frac {\pi }{2}$, $\frac {\pi }{4}$, and $\frac {\pi }{2}$ respect to the axis $x$. Here, the difference in the color of the fiber means the difference in its properties.

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By defining parameter $\beta _1$ as the rotation angle of SOP, and parameters $\beta _2$ as the rotation angle of SOS as follows:

So, if $q \neq q{\prime }$ we can rotate SOP and SOS on polarization and orbital angular momentum Poincaré sphere independently with two squeezers act in the same direction.

According to Eq. (19), if the direction of force is at angle $\phi$ with $y$ axis then $\sigma _1$ must be replaced with $\sigma _{\phi }$ in Eq. (23).

For $\phi =\frac {\pi }{4}$ we have $\sigma _{\phi }=\sigma _2=\left (\begin {array}{cc} 0 & 1\\ 1 & 0 \end {array} \right )$. A full rotation on Poincaré sphere can be achieved by three Euler rotations around $S_1,S_2$ and $S_1$ directions. Euler rotations can be made by placing three pairs of squeezers in the angles $\frac {\pi }{2}, \frac {\pi }{4}$ and $\frac {\pi }{2}$ respect to axis $x$ (Fig. 4). As shown in Appendix 2, these three pairs of squeezers are equivalent to what is shown in Fig. 2 these six squeezers make six linearly independent equations and six parameters $\alpha _i$ will be obtained independently. Therefore, we can make any desired rotation of SOP and SOS with six squeezers in Fig. 2. Since the environmental effects on fiber produce operators that are in the $SU(2) \otimes SU(2)$ subgroup of the $SU(4)$ group and do not entangle two subspaces of polarization and spatial modes, therefore by using this spatial and polarization controller (SPC) we can compensate all environmental effects and transmit data on eigenmodes of fiber.

One application of SPC could be in the implementation of a QKD protocol in higher dimensions in optical fiber. For example in BB84 which is the most famous QKD protocol that has been widely used till now, Alice and Bob use two mutually unbiased bases for encoding and decoding information [29]. In most implementation of BB84 in optical fiber, the value of qubits are coded on polarization or phase of photons. Because the state of polarization is arbitrarily changed in optical fiber, the polarization controller must be used at Bob’s site to bring it back. To increase the amount of data transmitted by one photon, the value of qudits in 4 dimensional space can be coded on 4 dimensional subspace of fiber modes in $l=1$. Alice must encode her qudit randomly on two sets of mutually unbiased bases. One set of basis could be LP modes in subspace $l=1$ of fiber:

3.1 Calculation

By the parameters of Table 1 for $\lambda =1550$ nm in circular fiber, $q$ becomes approximately $0.086$ which means that, because of an external pressure, period of SOP rotation is about 11 times less than the period of SOS rotation. By these parameters $\alpha \simeq 0.31 L P \text {(Rad)}$ and $q\alpha \simeq 0.027 L P \text {(Rad)}$. The phase difference between $x$ and $y$ polarization and, $\cos$ and $\sin$ spatial profile versus an external force is shown in Fig. 5.

 

Fig. 5. Phase difference between two orthogonal polarization (blue line) and two orthogonal spatial $LP_{11}$ modes (red line) versus force that affects fiber.

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Table 1. Parameters of a Two Mode Circular Silica Fiber

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In Fig. 5, the red and blue graphs intersect at 11 points (green points) in each period of rotation of SOS where the amount of rotation of SOP and SOS are equal. In point $m$, $(LP)_m=9.33(m-1)$(N). The intensity patterns of input mode $(\hat {x}+\hat {y})(\frac {LP_{11}^{\cos }+LP_{11}^{\sin }}{2})$ after passing a squeezer in $y$ direction with different forces are shown in Fig. 6. Also, the SOP of these states are shown in Fig. 7.

 

Fig. 6. Intensity pattern of input mode $(\hat {x}+\hat {y})(\frac {LP_{11}^{\cos }+LP_{11}^{\sin }}{2})$ before and after passing a squeezer in $y$ direction with different forces.

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Fig. 7. SOP states of input mode $(\hat {x}+\hat {y})(\frac {LP_{11}^{\cos }+LP_{11}^{\sin }}{2})$ after passing a squeezer in $y$ direction with different forces. The forces associated with points 1 to 11 which are calculated in Fig. 6.

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Because $q$ is small, therefore the polarization adjustment requires about 11 times more precision. For example, if we want to adjust the polarization with an accuracy of about one degree, $L \Delta P$ must be less than $0.06$ (N), but for the accuracy of one degree in OAM, an error of $L \Delta P=0.65$(N) is acceptable.

4. Conclusion

In this paper, we have shown that the important fiber perturbations change the SOP and SOS independently. We proposed a device that can control these two degrees of freedom simultaneously. The proposed device also can be used to generate different modes in few mode fibers. This device can have many application in classical communication as well as quantum communication and QKD.