Modal decomposition for few mode fibers using the fractional Fourier system

Wei Yan; Xiaojun Xu; Jianguo Wang

doi:10.1364/OE.27.013871

1. Introduction

Modal decomposition (MD) for characterizing the fiber laser beams is an emerging research field [1,2]. This technique is demonstrated to decompose the total field propagating in an optical fiber into sum of individual transverse modes and has become an indispensable diagnostic tool for the development of novel high power fiber laser systems [3], beam quality measurements [4], fiber to fiber coupling processing [5], wavefront reconstruction [6], and adaptive mode control [7], etc.

Recent years, a plethora of MD techniques have been developed in literature [8–18]. Some of the most popular ones are the spatially and spectrally (S²) resolved imaging technique [13], the correlation filter method (CFM) [14], and the numerical analysis method (NAM) [15–18], etc. Each of these MD techniques is based on unique procedures, using numerical or experimental approaches (or both) to access modal properties, and individually possesses outstanding merits. The S² imaging technique is possible to extract the modal fields from the measurement data instead of being reliant upon theoretical calculated fields as they have been used for the CFM and NAM. So this technique is in particular suitable for characterizing novel or unknown fiber architectures where the actual mode set is unknown. The CFM highlights the promising ability of real-time decomposition of multi-mode beams and has been widely employed in various applications [1,2,4–6]. The NAM stands out by their remarkably low experimental effort [16].

However, the above mentioned techniques still have some limitations. The S² imaging technique requires the most experimental effort (the broad-band source or tunable laser, optical spectrum analyzer, and two-axis mechanical shifter) as well as the highest measurement time consumption [19]. The resolution power of CFM for measuring the low higher-order mode contents is limited by the sensitivity of the CCD sensor at the measurement wavelength and the ambient noise. The center piece of NAM is the optimization algorithms, such as Gerchberg-Saxton (GS) algorithm [15], Stochastic Parallel Gradient Descent (SPGD) algorithm [17], hybrid genetic algorithm [18], and so on. These optimization algorithms easily suffer from initial value sensitivity and local minima problems.

In this paper, a novel MD method that makes use of the fractional Fourier system is presented. The role of the fractional Fourier system played in MD is to recover the phase of modal field. The most attractable advantages of PR algorithm based on the fractional Fourier system are non-iterative, non-interferometric, and experimentally simple. However, the existing PR algorithm doesn’t cover the effect of the optical vortex [20–22] while the higher-order fiber mode possesses optical vortices intrinsically [14]. Thus, we extend the existing PR algorithm to account for the effect of optical vortex by employing the hidden phase algorithm developed by Fried in adaptive optics area [23]. The extended PR algorithm allows the phase of the modal field to be reconstructed to high fidelity even when the optical vortices exist. Combing the reconstructed phase with the measured NF intensity, a preliminary modal field can be established. Based upon the obtained modal field, the power ratio and relative phase difference of each transverse mode could be determined unambiguously. The core idea behind the proposed method is very similar to the NAM, but the PR algorithm developed in this paper enables the phase to be reconstructed in a non-iterative manner. So the current approach possesses the merits of NAM: low experiment effort and high processing speed, but frees suffering from the challenge of it. The validity and reliability are demonstrated by several numerical examples including noisy signals with different SNR levels.

The outline of this article is as follows. Section 2 and 3 provide the fundamentals on the modal decomposition and PR algorithm using the fractional Fourier system, respectively. Thereafter, the efficacy of the proposed approach is presented in Section 4. Finally, some brief concluding remarks are given in Section 5.

2. Modal decomposition for step-index fiber (SIF)

Modal fields in weakly guiding SIF can be described to good approximation by linearly polarized (LP) modes [24]. The transverse structure of this mode set {ψ_i(x, y)} can be derived from the scalar Helmholtz equation. The ith LP mode structure can be factorized using

ψ_{i} (r, ϕ) = R_{n}^{m} (r) Φ_{n} (ϕ)

where n and m are integers

R_{n}^{m} (r) = c {\begin{matrix} \frac{J_{n} (U_{m} r / r_{0})}{J_{n} (U_{m})} (0 \leq r < r_{0}) \\ \frac{K_{n} (W_{m} r / r_{0})}{K_{n} (W_{m})} (r \geq r_{0}) \end{matrix} and Φ_{n} (ϕ) = {\begin{array}{l} cos (n ϕ) for even modes \\ sin (n ϕ) for odd modes \end{array}

respectively. The symbol r₀ denotes the fiber core radius and c is chosen so that ψ_i fulfills the normalization condition. J_n means the nth Bessel function of first kind, whereas K_n means the nth modified Bessel function of second kind. The value U_m and W_m can be calculated from the characteristic equations

U_{m} \frac{J_{n + 1} (U_{m})}{J_{n} (U_{m})} = W_{m} \frac{K_{n + 1} (W_{m})}{K_{n} (W_{m})} and U_{m}^{2} + W_{m}^{2} = V^{2}

where

U_{m} = \frac{2 π a}{λ} \sqrt{n_{co}^{2} - n_{eff}^{2}} and W_{m} = \frac{2 π a}{λ} \sqrt{n_{eff}^{2} - n_{cl}^{2}}

with

V = \frac{2 π a}{λ} \sqrt{n_{co}^{2} - n_{cl}^{2}}

being the well-known V-parameter of the fiber. λ is the light wavelength. n_co is the refractive index of the fiber core, while n_cl is the refractive index of the fiber cladding and n_eff corresponds to an effective index.

An arbitrary fully coherent modal field that is normalized to unit power can be described as the coherent superposition of LP modes.

ψ (x, y) = | ψ (x, y) | exp [i ϕ (x, y)] = \sum_{n = 1}^{N} c_{n} ψ_{n} (x, y)

where ψ(x, y) denotes the modal field and ϕ(x, y) is the phase of it. Due to the orthogonal property of the LP modes, the modal coefficient c_n can be given by

c_{n} = ρ_{n} exp (i ϕ_{n}) = \iint ψ_{n}^{*} (x, y) ψ (x, y) d x d y

where

ρ_{n}^{2}

is the modal weight, which fulfills the relation

\sum_{n = 1}^{N} {| c_{n} |}^{2} = \sum_{n = 1}^{N} ρ_{n}^{2} = 1

, ϕ_n is the inter-modal phase, and

ψ_{n}^{*}

is the complex conjugate of the ψ_n. The integrals, unless specified, run from −∞ to +∞.

The difficulty along with Eq. (6) to evaluate modal coefficients stems from the fact that the modal field ψ(x, y) is generally immeasurable except its amplitude. Consequently the PR is indispensable. The PR algorithm employed in this paper can recover the phase from the FrFT power spectra measurement and the detail will be illustrated in the next section.

3. PR algorithm using the fractional Fourier system

Nowadays, The FrFT plays an important role in many branches of optics especially in optical information processing [25]. Its application in optics started with the seminal work of Mendlovic and Ozaktas [26]. There are many ways to define the FrFT, while in the present article the two dimensional FrFT is defined as [27]

ψ_{α_{x}, α_{y}} (u, v) = \iint K^{α_{x}} (x, u) K^{α_{y}} (y, v) ψ (x, y) d x d y

where ψ_{α_x,α_y} is the FrFT of ψ(x, y) with two independent transformation parameters α_q, α_q ∈ [0, 2π], q is a placeholder for x and y, K^α_q is the FrFT transformation kernel which is given by

K^{α_{q}} (q, p) = \frac{exp {\frac{i π}{σ^{2} sin α_{q}} [(q^{2} + p^{2}) cos α_{q} - 2 q p]}}{σ \sqrt{i sin a_{q}}}

where p is a variable for u and v, σ is the scale parameter with the dimension of length. If α_x = α_y = 0, we have the identity transformation ψ_0,0(x, y) = ψ(x, y). When

α_{x} = α_{y} = \frac{π}{2}

or

α_{x} = α_{y} = - \frac{π}{2}

, we have the ordinary Fourier transform or inverse Fourier transform of the modal field, respectively. The properties and numerical simulation method of the FrFT are detailed in [27]. We will not elaborate these topics in the present paper.

Using the phase space formalism, Bastiaans and Wolf derived the relationship between the phase gradient and angular derivative of the FrFT power spectra [20]

g_{x} (x, y) = \frac{\partial ϕ (x, y)}{\partial x} = \frac{π}{I_{0, 0} (x, y)} \iint {\frac{\partial I_{α_{x}, 0} (x^{'}, y^{'})}{\partial α_{x}} |}_{α_{x} = 0} sgn (x - x^{'}) δ (y - y^{'}) d x^{'} d y^{'}

g_{y} (x, y) = \frac{\partial ϕ (x, y)}{\partial y} = \frac{π}{I_{0, 0} (x, y)} \iint {\frac{\partial I_{0, α_{y}} (x^{'}, y^{'})}{\partial α_{y}} |}_{α_{y} = 0} sgn (y - y^{'}) δ (x - x^{'}) d x^{'} d y^{'}

where g(x, y) = [g_x(x, y), g_y(x, y)]^T is the phase gradient field, ^T denotes transpose, sgn labels the sign function, δ is the Dirac delta function, I_{α_x,α_y} = |ψ_{α_x,α_y} (x, y)|² is the FrFT power spectra, and I_0,0(x, y) = |ψ(x, y)|² is the NF intensity.

If two close fractional Fourier power spectra separated by small angle ε are measured, the phase gradient can be approximated to 2nd order of accuracy

g_{x} (x, y) \approx \frac{π}{I_{0, 0} (x, y)} \iint \frac{I_{ε, 0} (x^{'}, y^{'}) - I_{- ε, 0} (x^{'}, y^{'})}{2 ε} sgn (x - x^{'}) δ (y - y^{'}) d x^{'} d y^{'}

g_{y} (x, y) \approx \frac{π}{I_{0, 0} (x, y)} \iint \frac{I_{0, ε} (x^{'}, y^{'}) - I_{0, - ε} (x^{'}, y^{'})}{2 ε} sgn (y - y^{'}) δ (x - x^{'}) d x^{'} d y^{'}

In the practical situation, there are tradeoffs among the angle separation ε, the accuracy of the reconstruction result, and noise considerations. Small ε avoids the nonlinearity error, however, small ε also leads to noisy phase reconstructions. Utilizing the numerical technique to solve the Eqs. (11) and (12), the phase gradient could be obtained.

In [20–22], the authors employed some kind of numerical integral techniques to recover the phase from the phase gradient. While these kind of numerical integral techniques all depend on the integrand being free of singularities. In [14], Kaiser et al show that the higher-order transverse modes possess optical vortices intrinsically. An optical vortex (also known as branch point, phase singularity or screw dislocation) is one type of optical singularity that possesses a spiral wavefront pattern [29]. The spiral wavefront rotates around a point where the light intensity is zero and phase is undefined. According to the direction of rotation, the sign of the optical vortex could be positive or negative. Consequently, the above mentioned numerical integral techniques can not produce results that properly match the actual phase. The least-squares (LS) wavefront reconstruction technique is another standard way to recover the phase from the phase gradient field and has been widely utilized in adaptive optics systems [28]. Unfortunately, the LS techniques still does not work well in the presence of optical vortex [23].

We extend the existing PR algorithm to account for the effect of the optical vortex by employing the hidden phase theorem. This theorem is developed by Fried in adaptive optics area where the optical vortices is caused by the strong atmospheric turbulence [23]. According to the hidden phase theorem, if the modal field possesses optical vortices, the gradient field can be decomposed into two terms: the gradient of the scalar potential and the curl of the Hertz potential [23].

g (x, y) = \nabla s (x, y) + \nabla \times H (x, y)

where ∇ is the gradient operator, ∇× defines the curl of the vector, s is the scalar potential, and H is the Hertz potential which is the contribution of the optical vortices and is along the z direction as [0, 0, h(x, y)]^T. h(x, y) is known as the Hertz function.

Under this circumstance, the reconstructed phase ϕ_rc could be written as

ϕ_{rc} (x, y) = ϕ_{ls} (x, y) + ϕ_{hid} (x, y)

The first term on the right-hand side of Eq. (14) is the LS phase. In this paper, we will make use of Zernike modal reconstructor to obtain this term. Recalling the fact that the wavefront can be expanded by the Zernike polynomials [28], the LS phase can be described as

ϕ_{ls} (x, y) = \sum_{p = 1}^{P} a_{p} Z_{p} (x, y)

where a_p is the expansion coefficient of the pth Zernike polynomial Z_p.

According to Eqs. (13)–(15), the gradient of the LS phase can be described as

\begin{array}{l} \frac{\partial ϕ_{ls} (x, y)}{\partial x} = \frac{\partial s (x, y)}{\partial x} = \sum_{p = 1}^{P} a_{p} \frac{\partial Z_{p} (x, y)}{\partial x} \\ \frac{\partial ϕ_{ls} (x, y)}{\partial y} = \frac{\partial s (x, y)}{\partial y} = \sum_{p = 1}^{P} a_{p} \frac{\partial Z_{p} (x, y)}{\partial y} \end{array}

It is possible to formulate a matrix version of Eq. (16) as

[\partial s] = [\partial Z] [a_{p}]

where [∂s] = [∂_xs, ∂_ys]^T, [∂Z] = [∂_xZ, ∂_yZ]^T, and [a_p] is the expansion coefficient column. Because the Hertz potential has no contribution to LS phase [23], the [a_p] can be estimated through a LS manner as

[a_{p}] = {[\partial Z]}^{+} g

where [∂Z]⁺ = ([∂Z]^T [∂Z])⁻¹[∂Z]^T is the pseudo-inverse of [∂Z] called Zernike modal reconstructor. Once the g is obtained, the coefficient column as well as the LS phase can also be determined straightforwardly.

The second term on the right-hand side of Eq. (14) is the hidden phase which is given by [23]

ϕ_{hid} (x, y) = Im {log [\frac{\prod_{n = 1}^{N} (x - x_{n}) + i (y - y_{n})}{\prod_{m = 1}^{M} (x - x_{m}^{'}) + i (y - y_{m}^{'})}]}

where the coordinate (x_n, y_n) represents the nth of N positive optical vortices location and (x′_m, y′_m) represents the mth of M negative optical vortices location. It is easy to find that the hidden phase is directly related to the locations and signs of the optical vortices. However, these two properties of the optical vortex are difficult to estimate in practical situation unless several complex algorithms are applied [30]. In this article, these two properties will be determined in a straightforward way.

The Hertz function h(x, y) that satisfies to Eq. (13) can be written as [23]

h (x, y) = log [\frac{\prod_{m = 1}^{M} | \sqrt{{(x - x_{m}^{'})}^{2} + {(y - y_{m}^{'})}^{2}} |}{\prod_{n = 1}^{N} | \sqrt{{(x - x_{n})}^{2} + {(y - y_{n})}^{2}} |}]

According to Eq. (20), it is evident that that the presence of optical vortices could be visualized as the peaks and valleys of the Hertz function where the peaks correspond to the positive optical vortices and the valleys correspond to the negative ones. Once the Hertz function is obtained, the locations and signs of the optical vortices will also be determined straightforwardly. We will show in the following that the Hertz function can be obtained from the phase gradient measurement.

The Eq. (13) can be expanded and expressed in matrix form as

[\begin{matrix} g_{x} (x, y) \\ g_{y} (x, y) \end{matrix}] = [\begin{matrix} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} \\ \frac{\partial}{\partial y} & - \frac{\partial}{\partial x} \end{matrix}] [\begin{matrix} s (x, y) \\ h (x, y) \end{matrix}]

It is evident that the Fourier transform of Eq. (21) is

[\begin{matrix} {\bar{g}}_{x} (k_{x}, k_{y}) \\ {\bar{g}}_{y} (k_{x}, k_{y}) \end{matrix}] = [\begin{matrix} i 2 π k_{x} & i 2 π k_{y} \\ i 2 π k_{y} & - i 2 π k_{x} \end{matrix}] [\begin{matrix} \bar{s} (k_{x}, k_{y}) \\ \bar{h} (k_{x}, k_{y}) \end{matrix}]

where ḡ(k_x, k_y) = [ḡ_x(k_x, k_y), ḡ_y(k_x, k_y)]^T is the Fourier transform of the gradient field g, k_x and k_y denote the spatial frequency, s̄ (k_x, k_y) and h̄ (k_x, k_y) are the Fourier transform of the scalar potential s(x, y) and Hertz function h(x, y), respectively. The Matrix inversion, together with the inverse Fourier transform, can yield

h (x, y) = \int d k_{x} \int d k_{y} \frac{k_{y} {\bar{g}}_{x} (k_{x}, k_{y}) - k_{x} {\bar{g}}_{y} (k_{x}, k_{y})}{i 2 π (k_{x}^{2} + k_{y}^{2})} exp [i 2 π (k_{x} x + k_{y} y)]

Eq. (23) shows that the Hertz function can also be expressed in the Fourier transform terms of the gradient field g. As we will show in the next section, by using Eqs. (11) and (12), plenty of gradient data could be obtained based on which not only will the vortex structure of the gradient field be manifested clearly, but also could the Hertz function be obtained using Eq. (23). Thenceforth the locations of optical vortices will be recognized in a very convenient way.

Combing the reconstructed phase ϕ_rc with the NF intensity I_0,0, a preliminary modal field can be established by $ψ_{rc}^{(p)} (x, y) = \sqrt{I_{0, 0} (x, y)} exp (i ϕ_{rc} (x, y))$ . Based upon Eq. (6), the modal weights $ρ_{n}^{2}$ and modal phases ϕ_n could be obtained. The estimated modal weights may have a sub-normalized energy, i.e. $\sum_{n} ρ_{n}^{2} < 1$ . We compensated for this by renormalizing at the end of MD.

In order to filter out any information that cannot be represented as a superposition of the fiber modes, the modal field is reconstructed again from the obtained modal coefficients using Eq. (5). To quantify the accuracy of the final reconstructed modal field, the correlation coefficient between it and the original field ψ will be defined as:

C = | \frac{\iint ψ (x, y) ψ_{rc}^{*} (x, y) d x d y}{\sqrt{\iint ψ (x, y) ψ^{*} (x, y) d x d y \iint ψ_{rc} (x, y) ψ_{rc}^{*} (x, y) d x d y}} |

where ψ_rc denotes the final reconstructed modal field. It is noted that in the case of perfect agreement between ψ_rc and ψ, the correlation coefficient will be 1.

4. Numerical examples

In the following, we will demonstrate the validity and reliability of the MD method derived above through numerical simulations. As a way of example, an arbitrary linearly polarized modal field of the few mode step-index fiber (core diameter = 20μm, V-parameter = 4.7) operated at λ = 1μm will be considered and about 6 LP modes (LP₀₁, LP₀₂, LP_11e, LP_11o, LP_21e, and LP_21o) could be guided in it. Two different modal combinations are demonstrated in Table 1 and the numerical dimension of the modal field is 128 × 128. Last but not least, The first 70 Zernike polynomial terms are used to establish the Zernike modal reconstructor. Fig. 1(a) shows an example experimental setup that measures the FrFT power spectra and implements the MD. The FrFT system performs the transformation depending on the input fractional orders and the output FrFT power spectra are registered by a CCD camera which is connected to a PC. The customized software can output the modal contents based on the input FrFT power spectra. We accomplish this experimental setup using the numerical simulation method and the flowchart of the numerical simulation code is illustrated in Fig. 1(b). It should be noted that we do not demonstrate the detailed 2D FrFT optical setup in Fig. 1(a) because that isn’t the emphasis of the present work. But the reader can consult [31] where a flexible optical system able to perform the 2D FrFT is given in.

Table 1. Two Different Modal Combinations.

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Fig. 1 (a) schematic for the modal decomposition using the fractional Fourier systems; (b) flowchart of the numerical simulation code.

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4.1. Example1

In the first numerical example, the input modal field is generated following the modal combination 1 and no noise effect is considered. Choosing ε = 0.3 rad and measuring five FrFT power spectra: I_0,0, I_ε,0, I_−ε,0, I_0,ε, and I_0,−ε, the two dimensional x, y component of the phase gradient g can be obtained using Eqs. (11) and (12). Figs. 2(a) and 2(b) depict these two components respectively. Due to the high density of gradient data, only a zoomed portion of whole phase gradient vector is illustrated in Fig. 2(c).

Fig. 2 The two dimensional phase gradient and phase obtained using the FrFT power spectra. The input modal field is generated according to the modal combination 1 and no noise effect is considered. (a), (b) the x and y component of the phase gradient g, respectively; (c) a zoomed portion of the phase gradient vector; (d) the reconstructed phase obtained by Zernike modal reconstructor.

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In this example, the modal field does not possess any optical vortex. This conclusion is evident from the sketch in Fig. 2(c) that the phase gradient vector shows no spiral pattern. The reason may be that the fundamental mode dominates the modal combination. Under this circumstance, the Zernike modal reconstructor can furnish the required phase. Fig. 2(d) illustrates the reconstructed phase. As the Zernike polynomials are orthogonal over the unit disk, only the phase over the whole unit disk can be obtained.

Combining the reconstructed phase with the measured NF intensity I_0,0, the preliminary modal field is obtained. And then the complete MD is performed. In order to make sure that the energy conservation is compromised, the modal weights are renormalized after the MD. The revised modal weights and phases are compared with the actual results in Figs. 3(a) and 3(b) respectively. We can see that the agreement is very high.

Fig. 3 The comparison between the MD results and the actual results. (a) modal weights; (b) modal phases.

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As described in section 3, the modal field is reconstructed again from the obtained modal weights and phases using Eq. (5). This procedure is very necessary since it can filter out any information that can not be represented as the superposition of the transverse modes. Fig. 4 depicts the comparison between the reconstructed NF intensity and phase with the original ones. We can see that the reconstructed NF intensity and phase of modal field are visually almost the same as the original ones. The correlation coefficient between the reconstructed modal field and the original one is 0.9998. The relative root-mean-square (RMS) error of the reconstructed NF intensity and phase are 8.65 × 10⁻³, 4.27 × 10⁻², respectively.

Fig. 4 The comparison between the reconstructed results and the original ones. (a) the original NF intensity; (b) the reconstructed NF intensity; (c) the discrepancy between (a) and (b); (d) the original phase of modal field; (e) the reconstructed phase; (f) the discrepancy between (d) and (e).

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4.2. Example2

In the second numerical example, the input modal field is generated following the modal combination 2 and also no noise effect is considered. In this combination, the power ratio of the fundamental mode is decreased while the corresponding higher-order mode contents are increased. The values of ε are the same as example 1. The two dimensional x, y component of the phase gradient g are demonstrated in Figs. 5(a) and 5(b), respectively. Because of the same reason as example 1, only a zoomed portion of the gradient vector is shown in Fig. 5(c) where two spiral patterns are clearly evident, which means that the modal field possesses two optical vortices. The counter-clockwise one marked by plus corresponds to the positive optical vortex and the clockwise one marked by minus corresponds to the negative optical vortex.

Fig. 5 The two dimensional phase gradient obtained using the FrFT power spectra. The input modal field is generated according to the modal combination 2 and no noise effect is considered. (a), (b) the x and y component of the phase gradient g, respectively; (c) a zoomed portion of the phase gradient vector. Evidently, it has two spiral patterns where plus marks the positive one and minus marks the negative one.

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The Hertz function obtained from Eq. (23) is given in Fig. 6(a). It is apparent that the positive and negative optical vortices are visualized as peaks and valleys of the Hertz function which permits the locations and signs of optical vortex to be determined in a very convenient way. Figs. 6(b) and 6(c) provide the LS phase and hidden phase respectively. The LS phase is reconstructed by the Zernike modal reconstructor too and the hidden phase is obtained by Eq. (19) where a 2π phase discontinuity is quite obvious, which cannot be reconstructed by the Zernike modal reconstructor.

Fig. 6 (a) the Hertz function which obviously demonstrate a peak and a valley indicating the positive and negative optical vortex, respectively; (b) the LS phase reconstructed by Zernike modal reconstructor; (c) the hidden phase obtained by Eq. (19).

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Combining the LS phase and the hidden phase with the measured NF intensity I_0,0, the preliminary modal field is obtained. The further process is the same as example 1. The complete MD is performed. The obtained modal weights and phases are compared with the actual results in Figs. 7(a) and 7(b) respectively. The agreement is also quite high.

Fig. 7 The comparison between the MD results and actual results. (a) modal weights; (b) modal phases.

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The modal field is reconstructed again from the obtained modal weights and phases. Fig. 8 demonstrates the comparison between the reconstructed NF intensity and phase with the original ones. We can see again that the reconstructed NF intensity and phase are visually almost the same as the original ones. The correlation coefficient (CC), the relative RMS error of the reconstructed NF intensity (RMS_I) and phase (RMS_P) are shown in Table 2. In comparison, The results where the hidden phase algorithm is not employed also illustrated in it. From Table 2, we can conclude that the extended PR algorithm outperforms the existing one in our numerical example.

Fig. 8 The comparison between the reconstructed results and the original ones. (a) the original NF intensity; (b) the reconstructed NF intensity; (c) the discrepancy between (a) and (b); (d) the original phase of modal field; (e) the reconstructed phase; (f) the discrepancy between (d) and (e).

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Table 2. The Comparison of the Correlation Coefficient and the Relative RMS Errors between the Extended PR Algorithm and the Existing One.

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4.3. Example3

In this example, the proposed MD method is tested under several different SNR levels. When the FrFT power spectra are recorded by CCD-based optical imaging systems, the noise is generally inevitable. In the present paper, we mainly incorporate the photon noise and readout noise (RON) into the FrFT power spectra. The photon noise can be well-approximated by the Poisson random numbers, while the RON can be well-approximated by the zero-mean Gaussian random numbers with standard deviation (STD) σ_r. The input modal field is generated following the modal combination 1. The values ε are the same as in example 1. But the five FrFT power spectra: I_0,0, I_ε,0, I_−ε,0, I_0,ε, and I_0,−ε, are contaminated by RON with σ_r = 3e⁻ and photon noise. For each SNR level, the numerical simulation is repeated 100 times, thereafter several important statistical data for evaluation the method, such as: the average correlation coefficient (ACC), the average relative RMS error of reconstructed NF intensity (ARMS_I), and their STDs (STD_C and STD_I, respectively) are recorded. All these simulation results are given in Table 3.

Table 3. Summary of the Numerical Simulation Results under Different SNR Level.

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From the results given in Table 3, we can conclude that the proposed MD method can work quite well for the SNR higher than 20dB and the uncertainty of the results are also very small. Below this value, the degradation of the reconstructed result is significant. In the future, we will explore the more effective noise suppression algorithm to improve the ability of the proposed MD technique in the presence of noise.

5. Conclusion

To conclude, we have introduced a novel MD technique which can process the modal content of few mode fibers very conveniently and quickly. The core of this method is the extended PR algorithm using the fractional Fourier system. The numerical simulation results show that the discussed MD method produces quite good result no matter whether the optical vortex exists or not. The noise sensitivity for the proposed technique is also analyzed. It is shown that the present MD method can still work well when the SNR level is as low as 20dB in our numerical example. The main advantages of the proposed method are its non-iterative, non-interferometric manner, and low experimental effort. This technique can be employed in multiple areas to improve the MD ability.

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Combinations	Parameters	LP₀₁	LP₀₂	LP_11e	LP_11o	LP_21e	LP_21o
1	$ρ_{n}^{2}$	0.6	0.01	0.2	0.1	0.03	0.06
1	ϕ_n	0	0.678	−0.611	−0.840	−1.258	2.195
2	$ρ_{n}^{2}$	0.3	0.1	0.2	0.2	0.1	0.1
2	ϕ_n	0	−1.047	0.785	−0.785	1.047	0.785

Algorithm	CC	RMS_I	RMS_P
LS phase+hidden phase	0.9991	2.88 × 10⁻²	0.14
LS phase only	0.98	0.1	0.46

SNR (dB)	ACC	STD_C	ARMS_I	STD_I
10	0.48	0.19	0.70	0.17
20	0.971	7.29 ×10⁻²	8.75×10⁻²	0.10
30	0.9991	7.52 ×10⁻⁴	1.38×10⁻²	4.27×10⁻³
40	0.9997	4.85 ×10⁻⁵	8.74×10⁻³	4.50×10⁻⁴
50	0.9998	4.37 ×10⁻⁶	8.65×10⁻³	3.99×10⁻⁵
60	0.9998	8.15 ×10⁻⁷	8.65×10⁻³	4.13×10⁻⁶
70	0.9998	3.82 ×10⁻⁸	8.65×10⁻³	4.24×10⁻⁷

Combinations	Parameters	LP₀₁	LP₀₂	LP_11e	LP_11o	LP_21e	LP_21o
1	$ρ_{n}^{2}$	0.6	0.01	0.2	0.1	0.03	0.06
1	ϕ_n	0	0.678	−0.611	−0.840	−1.258	2.195
2	$ρ_{n}^{2}$	0.3	0.1	0.2	0.2	0.1	0.1
2	ϕ_n	0	−1.047	0.785	−0.785	1.047	0.785

Algorithm	CC	RMS_I	RMS_P
LS phase+hidden phase	0.9991	2.88 × 10⁻²	0.14
LS phase only	0.98	0.1	0.46

Modal decomposition for few mode fibers using the fractional Fourier system

Abstract

1. Introduction

2. Modal decomposition for step-index fiber (SIF)

3. PR algorithm using the fractional Fourier system

4. Numerical examples

4.1. Example1

4.2. Example2

4.3. Example3

5. Conclusion

References

Cited By

Figures (8)

Tables (3)

Equations (24)

Optics Express