Complex mode amplitude measurement for a six-mode optical fiber

Steven Golowich; Nenad Bozinovic; Poul Kristensen; Siddharth Ramachandran

doi:10.1364/OE.21.004931

1. Introduction

Recent years have seen a surge in interest in few-mode optical fibers for many applications, including mode-division optical multiplexing [1] and generation and transmission of optical vortices [2–4]. In contrast with standard graded-index multimode fiber, applications of such fibers depend on properties of individual modes, as opposed to averages over many modes. Therefore, standard mode power distribution measurements, such as those based on encircled flux [5], do not apply, and new methods are required.

In this paper, we demonstrate such a method that is applicable to six-mode fibers that support the pair of fundamental HE₁₁ modes, along with the higher order TM₀₁, TE₀₁, and pair of HE₂₁ modes. We propose a measurement protocol and statistical parameter estimation algorithm that separates both the powers and relative phases of all six modes. The measurements consist of polarization-filtered near-field intensity (NFI) images, which are easily implemented with standard off-shelf components. The complex mode amplitude recovery is, in general, only unique up to a two-fold ambiguity in mode labeling, but under most situations of interest a small amount of prior information allows a unique recovery.

We demonstrate our technique with simulated and measured data from a recently demonstrated six-mode “vortex” fiber [6] that supports the stable propagation of orbital angular momentum (OAM) states. The index profile, radial mode fields, and guided mode spectrum of this fiber is presented in Fig. 1. The feature that allows the stable propagation of OAM in superpositions of the two HE₂₁ modes, as well as that of the azimuthally- and radially-polarized TE₀₁ and TM₀₁ modes, is the enhancement of the effective index splitting among these modes due to the vector term in the wave equation, as first exploited for microbend grating applications [7, 8]. Information about both the magnitude and phase relationships between the modes is valuable for studying superpositions of the orbital angular momentum states [4]. One of the contributions of the current paper is to generalize the analysis method of [4] to recover phase information and to relax the dominant mode assumption made in that paper.

Fig. 1 (a) Measured index profile (red) of the six-mode vortex fiber studied in this paper, along with the calculated radial wave functions F₀₁ (black) and F₁₁ (blue); (b) effective index spectrum of the LP₁₁ modes, with calculated (solid) and measured (dashed) values shown.

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A variety of other methods for modal content analysis of few-mode fibers have been developed in recent years. Using only the geometry of NFI patterns, the recovery of a subset of the complex mode amplitude parameters of few-mode fibers is described in [9]. Modal powers and phases in the scalar approximation are recovered from far-field intensity patterns in [10], although the modes within a degenerate group are not distinguished. A similar method, using NFI patterns, was used to retrieve partial information about the coherent superposition of vector modes in [11]. A full coherent vector reconstruction is given in [12] by use of a specially designed holographic optical element and corresponding reconstruction algorithm. In [13, 14] the mode powers of the scalar modes are recovered by a self-interference method that does not require a priori knowledge of the mode structure. Low-coherence interferometry with an external reference beam, as done in [15], enables a similar, but more versatile, measurement. The modal content of a six-mode fiber, including the vortex content, was analyzed via a different interferometric measurement employing a reference beam in [16]. As part of a MIMO demonstration, the coherent mode content of a six-mode fiber was extracted with a specially designed free-space multiplexer and coherent detection in [1].

The primary features of the method we present that distinguish it from existing methods are its simplicity of implementation combined with the wealth of information it provides. Indeed, in most situations of interest, complete information about the coherent superposition of modes present at the fiber output are recovered from NFI images alone, as filtered through standard quarter wave plates and polarizers. In particular, no special-purpose optical elements nor interferometry with reference beams are required. We discuss our recovery algorithm in Section 2, and show that it retrieves all possible information given the postulated set of measurements. Experimental details are presented in Section 3, and examples on simulated and measured data in Section 4.

2. Algorithms

2.1. Mode field representations

Under monochromatic illumination, the transverse electric field E_t(r, θ, t) at the fiber end-face may be represented as a complex superposition of the six guided mode fields e_i(r, θ)

E_{t} (r, θ, t) = e^{i ω t} \sum_{i = 1}^{6} γ_{i} e_{i} (r, θ),

where the sum is over the mode index i. In the full vectorial treatment of the Maxwell equations, the six fields e_i represent

H E_{11}^{(e)}

,

H E_{11}^{(o)}

, TE₀₁, TM₀₁,

H E_{21}^{(e)}

, and

H E_{21}^{(o)}

. The representation (2.1) also applies to unchirped pulses as long as the coherent relationship between the modes is maintained, which will be true if the temporal pulse width is larger than the group delay differences of the fiber modes. For a weakly guided fiber, it is a very good approximation to express these vector mode fields as linear combinations of the LP₀₁ and LP₁₁ degenerate sets of solutions to the scalar wave equation [17]. Denoting the radial wave functions of the LP states by F_ℓm(r), we have a basis set that we refer to as the Vector set

\begin{array}{l} H E_{11}^{(e)} (r, θ) = F_{01} (r) \hat{x} \\ H E_{11}^{(o)} (r, θ) = F_{01} (r) \hat{y} \\ H E_{21}^{(e)} (r, θ) = (\hat{x} cos θ - \hat{y} sin θ) F_{11} (r) \\ H E_{21}^{(o)} (r, θ) = (\hat{x} sin θ + \hat{y} cos θ) F_{11} (r) \\ T M_{01} (r, θ) = (\hat{x} cos θ + \hat{y} sin θ) F_{11} (r) \\ T E_{01} (r, θ) = (\hat{x} sin θ - \hat{y} cos θ) F_{11} (r) . \end{array}

The spatial polarization patterns of these modes are depicted in Fig. 2. Although the six fields represented by (2.2) are physically the ones that correspond to the guided modes of the ideal fiber, they do not form the most useful basis for analysis due to their spatially varying polarization patterns. Instead, we introduce two new basis sets that span the same subspace. The Vortex basis set [18–20] is given by

\begin{array}{l} V_{11}^{(H E +)} (r, θ) = (H E_{11}^{(e)} + i H E_{11}^{(o)}) / \sqrt{2} = (\hat{x} + i \hat{y}) F_{01} / \sqrt{2} \\ V_{11}^{(H E -)} (r, θ) = (H E_{11}^{(e)} - i H E_{11}^{(o)}) / \sqrt{2} = (\hat{x} - i \hat{y}) F_{01} / \sqrt{2} \\ V_{21}^{(H E +)} (r, θ) = (H E_{21}^{(e)} + i H E_{21}^{(o)}) / \sqrt{2} = e^{i θ} (\hat{x} + i \hat{y}) F_{11} / \sqrt{2} \\ V_{21}^{(H E -)} (r, θ) = (H E_{21}^{(e)} - i H E_{21}^{(o)}) / \sqrt{2} = e^{- i θ} (\hat{x} - i \hat{y}) F_{11} / \sqrt{2} \\ V_{01}^{(T +)} (r, θ) = (T M_{01} - i T E_{01}) / \sqrt{2} = e^{- i θ} (\hat{x} + i \hat{y}) F_{11} / \sqrt{2} \\ V_{01}^{(T -)} (r, θ) = (T M_{01} + i T E_{01}) / \sqrt{2} = e^{i θ} (\hat{x} - i \hat{y}) F_{11} / \sqrt{2} . \end{array}

The

V_{21}^{(H E \pm)}

are the OAM states; each carries ħ/photon each of orbital and spin angular momentum, which are aligned, for a total angular momentum of 2ħ/photon. The

V_{01}^{(T \pm)}

elements are not true fiber modes, as the TE and TM modes are separated in effective index. However, they are valid elements of a basis that may be used to expand the transverse electric fields at a given point in the fiber, which is the only way in which we will use them. Also, the

V_{11}^{(H E \pm)}

elements are not vortex states, as they do not vanish on the axis; we label them with the symbol V for consistency of notation. We note that, unlike the Vector basis, the polarization of each element of the Vortex set does not depend on position; all are uniformly circularly polarized. The same is true of the LP basis set, given by

\begin{array}{l} L P_{01}^{(x)} (r, θ) = \hat{x} F_{01} (r) \\ L P_{01}^{(y)} (r, θ) = \hat{y} F_{01} (r) \\ L P_{11}^{(x +)} (r, θ) = \hat{x} e^{i θ} F_{11} (r) \\ L P_{11}^{(x -)} (r, θ) = \hat{x} e^{- i θ} F_{11} (r) \\ L P_{11}^{(y +)} (r, θ) = \hat{y} e^{i θ} F_{11} (r) \\ L P_{11}^{(y -)} (r, θ) = \hat{y} e^{- i θ} F_{11} (r) . \end{array}

In Section 2.2 we will demonstrate how to recover the powers in each of the vortex states with a pair of intensity measurements. In Section 2.3, we will verify that, under most situations of interest, a single additional intensity measurement is sufficient to recover all of the six complex mode amplitudes, up to an overall phase.

Fig. 2 Spatial polarization patterns of the Vector basis set.

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2.2. Vortex mode power estimation

For reasons that will soon be apparent, the Vortex basis is most convenient for analysis. Referring to the expansion (2.1), we define

\begin{array}{l} e_{1} = V_{11}^{(H E +)}; & e_{3} = V_{21}^{(H E +)}; & e_{5} = V_{01}^{(T +)}; \\ e_{2} = V_{11}^{(H E -)}; & e_{4} = V_{21}^{(H E -)}; & e_{6} = V_{01}^{(T -)} . \end{array}

Our ultimate goal is to coherently estimate all of the coefficients

γ_{i} = | γ_{i} | e^{i ϕ_{i}} .

From inspection of (2.3) it is evident that a spatially uniform circular polarizer of positive helicity will project onto the three “+” vortex states. We now show that the resulting intensity image gives enough information to estimate the powers in each of the three states, up to a possible ambiguity in the labeling of the modes, as well as some information about their relative phases. Analogous information is available from the negative helicity projection, so all six vortex mode powers can be recovered from these two intensity measurements. The relative phases may be estimated with additional information from intensity measurements after linear polarizers, as described in Section 2.3. We will initially assume perfect measurements and knowledge of the radial mode fields; practicalities due to various sources of noise and bias will be addressed in Section 3.

The intensity of the transverse electric field at the fiber end-face (2.1), after passing through a positive helicity polarizer P₊, is given by

\begin{array}{l} {| P_{+} E_{t} (r, θ) |}^{2} & = & {| γ_{1} F_{01} (r) + γ_{3} e^{i θ} F_{11} + γ_{5} e^{- i θ} F_{11} (r) |}^{2} \\ = & {| γ_{1} |}^{2} F_{01} {(r)}^{2} \\ + ({| γ_{3} |}^{2} + {| γ_{5} |}^{2}) F_{11} {(r)}^{2} \\ + 2 ℜ (γ_{3} γ_{5}^{*}) cos (2 θ) F_{11} {(r)}^{2} \\ - 2 ℑ (γ_{3} γ_{5}^{*}) sin (2 θ) F_{11} {(r)}^{2} \\ + 2 ℜ (γ_{1} {(γ_{3} + γ_{5})}^{*})) cos (θ) F_{01} (r) F_{11} (r) \\ + 2 ℑ (γ_{1} {(γ_{3} - γ_{5})}^{*})) sin (θ) F_{01} (r) F_{11} (r) . \end{array}

The analogous result for a negative helicity filter is

\begin{array}{l} {| P_{-} E_{t} (r, θ) |}^{2} & = & {| γ_{2} |}^{2} F_{01} {(r)}^{2} \\ + ({| γ_{4} |}^{2} + {| γ_{6} |}^{2}) F_{11} {(r)}^{2} \\ + 2 ℜ (γ_{4} γ_{6}^{*}) cos (2 θ) F_{11} {(r)}^{2} \\ + 2 ℑ (γ_{4} γ_{6}^{*}) sin (2 θ) F_{11} {(r)}^{2} \\ + 2 ℜ (γ_{2} {(γ_{6} + γ_{4})}^{*})) cos (θ) F_{01} (r) F_{11} (r) \\ + 2 ℑ (γ_{2} {(γ_{6} - γ_{4})}^{*})) sin (θ) F_{01} (r) F_{11} (r) . \end{array}

A near-field intensity (NFI) image consists of intensity measurements at a number of points {r_i, θ_i}. By linear regression on the two circularly polarized NFI’s, we may estimate the ten coefficients

\begin{array}{l} Ξ_{1} = {| γ_{1} |}^{2}; & Ξ_{2} = {| γ_{2} |}^{2}; \\ Ξ_{3 + 5} = {| γ_{3} |}^{2} + {| γ_{5} |}^{2}; & Ξ_{4 + 6} = {| γ_{4} |}^{2} + {| γ_{6} |}^{2}; \\ Ξ_{35} = γ_{3} γ_{5}^{*}; & Ξ_{46} = γ_{4} γ_{6}^{*}; \\ Ξ_{13 + 5} = ℜ γ_{1} {(γ_{3} + γ_{5})}^{*}; & Ξ_{26 + 4} = ℜ γ_{2} {(γ_{6} + γ_{4})}^{*}; \\ Ξ_{13 - 5} = ℑ γ_{1} {(γ_{3} - γ_{5})}^{*}; & Ξ_{26 - 4} = ℑ γ_{2} {(γ_{6} - γ_{4})}^{*} . \end{array}

All of the coefficients in (2.9) are real except for Ξ₃₅ and Ξ₄₆, of which both the real and imaginary parts may be independently recovered. We observe that the fundamental mode powers |γ₁|² and |γ₂|² are directly observed, while those of the LP₁₁ states must be calculated from |Ξ₃₊₅|, |Ξ₄₊₆|, |Ξ₃₅|, and |Ξ₄₆|. From these four quantities we may compute

\begin{matrix} Γ_{35}^{(min)} = min (| γ_{3} |, | γ_{5} |); & Γ_{46}^{(min)} = min (| γ_{4} |, | γ_{6} |); \\ Γ_{35}^{(max)} = max (| γ_{3} |, | γ_{5} |) & Γ_{46}^{(max)} = max (| γ_{4} |, | γ_{6} |) . \end{matrix}

There is an ambiguity in which of

Γ_{35}^{(min)}

,

Γ_{35}^{(max)}

is assigned to |γ₃|, |γ₅|, and similarly for the |γ₄|, |γ₆| pair. In many cases of interest, this ambiguity may be removed with prior knowledge of the mode powers. For instance, if most power is localized in one of the transverse modes, we expect that |γ₅| ≫ |γ₃| and |γ₄| ≫ |γ₆|. If most power is localized in one of the HE₂₁ modes, we expect that |γ₃| ≫ |γ₅| and |γ₄| ≫ |γ₆|. If the power is localized in

V_{21}^{(H E +)}

, then again |γ₃| ≫ |γ₅|. When the launch is chosen to maximize

V_{21}^{(H E -)}

, and the P₊ projection is taken (or vice versa), then we encounter the ambiguity of how to assign the values of |γ₄| and γ₆. In this case, we remark that all six mode powers are estimated (in the vortex basis); but the complex amplitudes are not, and the powers of two of the six states are ambiguous. In most cases of practical interest, this ambiguity is resolved with the full complex mode amplitude estimation, as discussed in Section 2.3.

2.3. Complex amplitude estimation

We now proceed to estimate the relative phases ϕ_i in (2.6). Since there will be an ambiguous overall phase, we define ϕ₁ = 0. We denote the relative phases

δ_{a b} = ϕ_{a} - ϕ_{b} .

We assume that the analysis of Section 2.2 has been performed, so estimates of the coefficients (2.9) are available. We will, for now, ignore the power labeling ambiguity referenced at the end of Section 2.2 and assume that all powers

{{| γ_{i} |}^{2}}_{i = 1}^{6}

have been correctly estimated; we will return to this point after discussing phase estimation. From the coefficients Ξ₃₅ and and Ξ₄₆ we may read off δ₃₅ and δ₄₆. To go further we need more information.

The next step is to project the field pattern onto two orthogonal LP subspaces via orthogonal linear polarizers, which yields the NFI images

\begin{array}{l} 2 {| P_{\hat{x}} E_{t} |}^{2} & = & {| γ_{1} + γ_{2} |}^{2} F_{01} {(r)}^{2} \\ + ({| γ_{3} + γ_{6} |}^{2} + {| γ_{5} + γ_{4} |}^{2}) F_{11} {(r)}^{2} \\ + 2 cos (2 θ) ℜ (γ_{3} + γ_{6}) {(γ_{5} + γ_{4})}^{*} F_{11} {(r)}^{2} \\ - 2 sin (2 θ) ℑ (γ_{3} + γ_{6}) {(γ_{5} + γ_{4})}^{*} F_{11} {(r)}^{2} \\ + 2 cos (θ) ℜ (γ_{1} + γ_{2}) {(γ_{3} + γ_{6} + γ_{5} + γ_{4})}^{*} F_{01} (r) F_{11} (r) \\ + 2 sin (θ) ℑ (γ_{1} + γ_{2}) {(γ_{3} + γ_{6} - γ_{5} - γ_{4})}^{*} F_{01} (r) F_{11} (r) \end{array}

and

\begin{array}{l} 2 {| P_{\hat{y}} E_{t} |}^{2} & = & {| γ_{1} - γ_{2} |}^{2} F_{01} {(r)}^{2} \\ + ({| γ_{3} - γ_{6} |}^{2} + {| γ_{5} - γ_{4} |}^{2}) F_{11} {(r)}^{2} \\ + 2 cos (2 θ) ℜ (γ_{3} - γ_{6}) {(γ_{5} - γ_{4})}^{*} F_{11} {(r)}^{2} \\ - 2 sin (2 θ) ℑ (γ_{3} - γ_{6}) {(γ_{5} - γ_{4})}^{*} F_{11} {(r)}^{2} \\ + 2 cos (θ) ℜ (γ_{1} - γ_{2}) {(γ_{3} - γ_{6} + γ_{5} - γ_{4})}^{*} F_{01} (r) F_{11} (r) \\ + 2 sin (θ) ℑ (γ_{1} - γ_{2}) {(γ_{3} - γ_{6} - γ_{5} + γ_{4})}^{*} F_{01} (r) F_{11} (r) . \end{array}

From (2.12) and (2.13) ten linear regression coefficients may be estimates, that we label

\begin{array}{l} Ξ_{12}^{(x)} = {| γ_{1} + γ_{2} |}^{2}; & Ξ_{12}^{(y)} = {| γ_{1} - γ_{2} |}^{2}; \\ Ξ_{36 + 54}^{(x)} = {| γ_{3} + γ_{6} |}^{2} + {| γ_{5} + γ_{4} |}^{2}; & Ξ_{36 + 54}^{(y)} = {| γ_{3} - γ_{6} |}^{2} + {| γ_{5} - γ_{4} |}^{2}; \\ Ξ_{3654}^{(x)} = (γ_{3} + γ_{6}) {(γ_{5} + γ_{4})}^{*}; & Ξ_{3654}^{(y)} (γ_{3} - γ_{6}) {(γ_{5} - γ_{4})}^{*}; \\ Ξ_{1236 + 54}^{(x)} = ℜ (γ_{1} + γ_{2}) {(γ_{3} + γ_{6} + γ_{5} + γ_{4})}^{*}; & Ξ_{1236 + 54}^{(y)} = ℜ (γ_{1} - γ_{2}) {(γ_{3} - γ_{6} + γ_{5} - γ_{4})}^{*}; \\ Ξ_{1236 - 54}^{(x)} = ℑ (γ_{1} + γ_{2}) {(γ_{3} + γ_{6} - γ_{5} - γ_{4})}^{*}; & Ξ_{1236 - 54}^{(y)} = ℑ (γ_{1} - γ_{2}) {(γ_{3} - γ_{6} - γ_{5} + γ_{4})}^{*} . \end{array}

The coefficients

Ξ_{3654}^{(x, y)}

are complex valued, while the rest of those in (2.14) are real valued.

In order to recover the phases in (2.6) we form

z = Ξ_{3645}^{(x)} - Ξ_{35} - Ξ_{46}^{*} = | γ_{3} | | γ_{4} | e^{- i δ_{43}} + | γ_{5} | | γ_{6} | e^{i (δ_{64} + δ_{35})} e^{i δ_{43}} .

The only unknown is δ₄₃, which we find as the solution to

(\begin{matrix} ℜ z \\ ℑ z \end{matrix}) = (\begin{matrix} | γ_{5} | | γ_{6} | cos (δ_{64} + δ_{35}) = | γ_{3} | | γ_{4} | & - | γ_{5} | | γ_{6} | sin (δ_{64} + δ_{35}) \\ | γ_{5} | | γ_{6} | sin (δ_{64} + δ_{35}) & | γ_{5} | | γ_{6} | cos (δ_{64} + δ_{35}) - | γ_{3} | | γ_{4} | \end{matrix}) (\begin{matrix} cos δ_{43} \\ sin δ_{43} \end{matrix}) .

At this point we have the relative phases of γ₃, γ₄, γ₅, and γ₆, but not their absolute phases (i.e. relative to γ₁), nor that of γ₂. We obtain these from Ξ₁₃₊₅, Ξ₁₃₋₅, Ξ₂₆₊₄ and Ξ₂₆₋₄. From the definitions,

(\begin{matrix} Ξ_{13 + 5} / | γ_{1} | \\ Ξ_{13 - 5} / | γ_{1} | \end{matrix}) = (\begin{matrix} | γ_{5} | cos δ_{53} + | γ_{3} | & - | γ_{5} | sin δ_{53} \\ | γ_{5} | sin δ_{53} & | γ_{5} | cos δ_{53} - | γ_{3} | \end{matrix}) (\begin{matrix} cos ϕ_{3} \\ sin ϕ_{3} \end{matrix}),

which yields the phase ϕ₃, and, immediately, ϕ₄, ϕ₅, and ϕ₆. Similarly,

(\begin{matrix} Ξ_{26 + 4} / | γ_{2} | \\ Ξ_{26 - 4} / | γ_{2} | \end{matrix}) = (\begin{matrix} | γ_{6} | cos δ_{64} + | γ_{4} | & | γ_{6} | sin δ_{64} \\ - | γ_{6} | sin δ_{64} & | γ_{6} | cos δ_{64} - | γ_{4} | \end{matrix}) (\begin{matrix} cos δ_{24} \\ sin δ_{24} \end{matrix}),

which yields δ₂₄, and therefore ϕ₂, as ϕ₄ is already known.

Inspection of (2.9) and (2.14) reveals that all of the regression coefficients are invariant under the mapping γ → γ̃, with

{\tilde{γ}}_{2} = γ_{2}^{*}; {\tilde{γ}}_{3} = γ_{5}^{*}; {\tilde{γ}}_{5} = γ_{3}^{*}; {\tilde{γ}}_{4} = γ_{6}^{*}; {\tilde{γ}}_{6} = γ_{4}^{*} .

This invariance is the origin of the ambiguity discussed at the end of Section 2.2. Furthermore, the ambiguity under (2.19) persists even when all four NFI images are included. Therefore, some ancillary information is necessary for a unique complex mode amplitude reconstruction. We argue now that only two bits of information is sufficient: knowledge that one of |γ₃| > |γ₅| or |γ₄| > |γ₆|, and which pair of modes satisfies the inequality. Fortunately, in most cases of practical interest this ancillary information is available.

Suppose we are given that |γ₃| > |γ₅| (the other case works analogously). Then the assignment of $Γ_{46}^{(min)}$ and $Γ_{46}^{(max)}$ to |γ₄| and |γ₆|, as done at the end of Section 2.2, is ambiguous. We may proceed through the regression analysis of the current section to estimate the phases ϕ_i under both hypotheses |γ₄| ≷ |γ₆|. In light of (2.19), because the values of |γ₃| and |γ₅| are, by assumption, correct, only the correct assignment of powers to |γ₄| and |γ₆| will result in values of all regression coefficients that are consistent with those observed. Therefore, a consistency check may be performed by comparing the estimated value of, say, Ξ̂₃₆₊₅₄ = |γ̂₃ +γ̂₆|² + |γ̂₅ + γ̂₄|² with the observed value. The hypothesized assignment of powers to |γ₄| and |γ₆| that yields the best match should be chosen. By including more regression coefficients in the consistency metric a more robust choice can be made. We remark that only the coefficients $Ξ_{*}^{(x)}$ from the x̂ linearly polarized image were employed to uniquely reconstruct the complex mode amplitudes. The ŷ polarized image is, in principle, unnecessary, but its inclusion in the non-linear regression discussed in Section 3 will result in better estimates.

Finally, we comment on the conditions under which the ambiguity (2.19) can be avoided. Typically, we are interested in putting most power into either the HE₂₁ or the TE/TM₀₁ pairs, and it is exactly these situations in which the ambiguity does not arise. If it is one of the transverse modes that is preferentially excited, then either |γ₅| > |γ₃| or |γ₆| > |γ₄|. The propagation constants of the transverse modes are split both from one another and from the HE₂₁ pair, so there is no ambiguity as to which mode is present. The HE₂₁ pair, by contrast, is degenerate, so mode coupling may, in principle, spoil the state that is prepared at the input end of the fiber. However, when most power is contained in the degenerate pair, then one of the vortex states $V_{21}^{(H E +)}$ or $V_{21}^{(H E -)}$ must contain at least nearly half of the total power, and we may determine which one is dominant by observing the circularly polarized NFI images. If the dominant state is $V_{21}^{(H E +)}$ , say, then we have |γ₃| > |γ₅|.

3. Implementation

Two experimental setups that have been used to obtain the measurements necessary for the recovery algorithms are presented in Fig. 3. In both, the vortex states were excited by the methods described in [4]. The output of a narrowband tunable continuous-wave (CW) laser is coupled into a standard single-mode fiber (SMF), with its polarization state determined by a polarization controller before coupling into the vortex fiber. A microbend grating converts power in the fundamental mode into a vortex state that is determined by the input polarization state. In the experiments we discuss, the conversion efficiency is better than 25 dB at the resonant wavelength of the grating. The output of the vortex fiber is imaged onto a camera. The appropriate polarization projections may be obtained, in series, with appropriately oriented polarizers and quarter-wave plates. In long fibers, temperature fluctuations in the vortex fiber can change the phase relationships between the modes over the time it takes to obtain the images in series. To avoid this problem, the scheme shown in Fig. 3(b) was implemented to capture both linear and circular polarization projections simultaneously. The results presented below use data from this latter setup.

Fig. 3 Experimental setups. (a) A vortex state is excited by a polarization controller (Pol-Con) and microbend grating, and NFI images are taken in series by an InGaAs short-wave infrared (SWIR) camera after passing through a quarter-wave plate (QWP) and/or linear polarizer. (b) Setup for simultaneous imaging of all four polarization filtered images. Components are non-polarizing beam splitters (NPBS), polarizing beam displacing prisms (PBDP), and mirrors (M). (c) Sample image from setup (b).

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Once the requisite images are obtained, the analyses of Section 2 can, in principle, be implemented directly via standard linear regression algorithms. However, in practice there are additional steps that must be taken to obtain reliable results. The first issue involves the radial mode fields F₀₁(r) and F₁₁(r). These can, in principle, be obtained from a mode solver applied to a measured radial index profile of the fiber. In practice, we found that these calculated mode fields did not agree with the NFI measurements, rendering the regression results unstable. We attribute the differences to a combination of inaccuracies in the fiber profile measurement, uncertainties in translating the measured index profile to wavelength used in the experiments, aberrations and vignetting in the optical train in front of the camera, scattering at the fiber end-face; and camera calibration error. We did not attempt to unravel the relative importance of these possible causes, as the problem is easily circumvented.

There are two possible resolutions. The approach taken in [4] was to confine attention to a narrow annulus of pixels r ≃ r₀, for which F₀₁(r₀) ≃ F₁₁(r₀). The information at other radii is not used. When it is assumed that most power in the fiber resides in the HE₂₁ mode pair, then it is possible to uniquely recover all mode powers (although not their relative phases). In order to relax the assumption of a dominant HE₂₁ mode, as we do in this paper, we must also consider the power in pixels near the fiber center r ≃ 0, and have knowledge of the ratio F₀₁(r₀)/F₀₁(0). This ratio is easily measured by imaging the vortex fiber output with the microbend grating removed. In this paper we take a slightly different approach, in that we use the entire image but with measured, as opposed to calculated, radial mode fields. We therefore require NFI images of one of the LP₁₁ states without the fundamental mode present. Since the microbend grating can transfer power from the fundamental to the LP₁₁ mode with better than 25 dB efficiency, such images are easy to obtain. With the two images in hand, the mode field center and radial mode field are obtained with a non-linear fit.

Once the preprocessing steps are completed, the complex mode amplitude recovery algorithm consists of two additional steps. The first is the linear regression analysis of Section 2. This analysis requires only three NFI images: those that project onto the two circularly polarized subspaces, and one of the two linearly polarized images. The results are estimates of all mode powers and relative phases, up to the two-fold ambiguity that was previously discussed. The second step is to use the results of the linear regression as a starting point for a non-linear regression to refine the complex mode amplitude estimates. This non-linear regression uses all four available images: both, instead of just one, linear projections, along with the two circularly polarized projections. The non-linear programming problem is

\hat{γ} = \underset{γ}{argmin} \sum_{p, x} {| I_{p} (x) - P_{p} E_{t} (x, γ) |}^{2},

where p ∈ {+, −, x̂, ŷ} are the polarization states on which we project, x is the pixel index, I_p are the measured polarization filtered images, P_p are the polarization projections, and E_t is the model transverse electric field given in Eq. (2.1), with basis set given by (2.5). In addition to providing a more efficient estimator, and using the additional data contained in the extra linearly polarized projection, an additional advantage of the non-linear technique is that it enforces physically meaningful values of the parameters, which need not be the case in the linear regression approach. We have observed that the results of the non-linear regression can, rarely, violate the assumptions used to disambiguate the linear parameter estimates with respect to the invariance (2.19). The disambiguation is therefore repeated after the non-linear regression.

Finally, we comment on sources of error in the complex mode amplitude recovery. In Section 4.1 we show results for simulated images that include sources of noise similar to those in the experiments. With a single camera shot for each image, we find that the mode powers and phases are estimated to within a few percent at worst. There are also several sources of bias that must be minimized. The beam splitters employed may exhibit polarization dependent loss and birefringence, which will tend to underestimate mode purity. The preparation of the fiber end-face can distort the beams; while we measure the radial wavefunction, we assume the angular dependence is that of a flat interface. Camera miscalibration will bias the results, and dark counts can impose a floor on sensitivity.

4. Results

4.1. Simulations

In this section we present the results of the analysis of Sections 2.2 and 2.3 applied to two simulated data sets, one in which most power is concentrated in the the positive helicity vortex state $V_{21}^{(H E +)}$ , the other in which the TM₀₁ state dominates. In both cases the “nuisance” mode powers are more than 20 dB lower than the dominant mode. In each case shot noise, dark noise, and quantization noise was simulated, with a twelve-bit camera with mean dark count of 250 assumed. The four polarization filtered images were assumed to have equal exposure time, set so that the brightest pixel across all images would have a count of 3200 in the absence of dark current. These assumptions correspond to the experimental conditions in Section 4.2. The images, including all noise sources, for the vortex mode are presented in Fig. 4, and those for the TM₀₁ are shown in Fig. 5.

Fig. 4 Simulated polarization filtered NFI images for the example with dominant $V_{21}^{(H E +)}$ state.

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Fig. 5 Simulated polarization filtered NFI images for the example with dominant TM₀₁ state.

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The linear regression analysis was performed in each case for 100 noise realizations; the resulting mode power and phase estimates are presented in Figs. 6 and 7. We observe that the reconstructions are quite stable in both cases, with some variability present in a few of the parameters.

Fig. 6 Parameter estimates for simulated example with dominant $V_{21}^{(H E +)}$ state. (a) Mode power estimates; (b) phase estimates. In both, the green dots represent the estimates obtained for each of 100 noise realizations.

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Fig. 7 Parameter estimates for simulated example with dominant TM₀₁ state. (a) Mode power estimates; (b) phase estimates. In both, the green dots represent the estimates obtained for each of 100 noise realizations.

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4.2. Experimental

The measurement setup of Fig. 3 was applied to a short length (3 m) of the fiber shown in Fig. 1. In the top row of Fig. 8 we show the three measured polarization-filtered images necessary to carry out the analysis, applied to a fiber in which the $V_{21}^{(H E -)}$ was preferentially excited. The fitted values obtained after the complex mode amplitude recovery are shown in the bottom row of Fig. 8. The close agreement between the two rows is evidence for the accuracy of the reconstructed amplitudes, since, as described in Section 2, the model is identifiable under these launch conditions.

Fig. 8 Measured and fitted values of polarization filtered near-field intensity images.

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Figure 9(a) shows the powers estimated in each of the three pairs of modes HE₁₁, HE₂₁, and TE/TM₀₁, at a number of wavelengths surrounding the microbend grating resonance. A quantitative measure of the accuracy of the reconstruction can be obtained by looking at the power in the fundamental mode. An independent measurement of this quantity was obtained by coupling the vortex fiber into a single mode fiber, thereby stripping the power in the higher order vortex modes. The comparison beween this measurement and that estimated from the NFI images is shown in Fig. 9(b), with excellent agreement. During this measurement the two OAM states that comprise HE₂₁ were excited with roughly equal power and stable relative phase, as shown in Fig. 10. We note that the variance of the phase estimate increases dramatically at the extreme ends of the wavelength range, and some apparent errors in mode assignment are visible in the power estimates in this region as well. Such estimation error is expected in these regions, as the assumptions that allow us to resolve the two-fold ambiguity (2.19) do not hold as the OAM-state powers tend to zero.

Fig. 9 (a) Power fraction in the fundamental, transverse, and HE₂₁ pairs; (b) Power estimated in the HE_1,1 modes by regression analysis (red crosses) and by coupling into a SMF (blue curve).

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Fig. 10 (a) Estimated power fraction of each of the two OAM states; (b) Estimated relative phase of the two OAM states.

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5. Conclusion

We proposed a novel technique for recovering the mode powers and relative phases at the output of any six-mode optical fiber that supports the HE₁₁, TE₀₁, HE₂₁, and TM₀₁ modes. The method relies only on polarization filtered NFI measurements that are easily implemented with standard off-the-shelf components. The reconstruction algorithm is based on a statistical regression analysis and is unique up to a two-fold ambiguity that is resolvable in most cases of practical interest. We demonstrated the method on simulated and measured data from a recently developed fiber that supports stable OAM propagation. The experiments we have described employed a short length of fiber, but the method is equally applicable to longer fiber lengths, and such studies will be presented elsewhere. The analysis of fibers supporting larger numbers of modes is complicated by the high dimensionality of the corresponding vector spaces, but we believe that generalizations of our method can give useful information about these fibers as well. In addition to probing the properties of few-mode fibers themselves, this method is also valuable for characterizing launching devices.

Acknowledgments

This work is sponsored by the Defense Advanced Research Projects Agency Information in a Photon (InPho) program under Air Force Contract FA8721-05-C-0002 and grant HR0011-11-1-0004. Opinions, interpretations, conclusions, and recommendations are those of the authors and not necessarily endorsed by the United States Government.

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Complex mode amplitude measurement for a six-mode optical fiber

Abstract

1. Introduction

2. Algorithms

2.1. Mode field representations

2.2. Vortex mode power estimation

2.3. Complex amplitude estimation

3. Implementation

4. Results

4.1. Simulations

4.2. Experimental

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (10)

Equations (20)

Optics Express