Abstract

We propose a measurement protocol and parameter estimation algorithm to recover the powers and relative phases of each of the vector modes present at the output of an optical fiber that supports the HE11, TE01, HE21, and TM01 modes. The measurements consist of polarization filtered near-field intensity images that are easily implemented with standard off-shelf components. We demonstrate the accuracy of the method on both simulated and measured data from a recently demonstrated fiber that supports stable orbital angular momentum states.

© 2013 OSA

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 [24]. 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 Et(r, θ, t) at the fiber end-face may be represented as a complex superposition of the six guided mode fields ei(r, θ)

Et(r,θ,t)=eiωti=16γiei(r,θ),
where the sum is over the mode index i. In the full vectorial treatment of the Maxwell equations, the six fields ei represent HE11(e), HE11(o), TE01, TM01, HE21(e), and HE21(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 Fℓm(r), we have a basis set that we refer to as the Vector set
HE11(e)(r,θ)=F01(r)x^HE11(o)(r,θ)=F01(r)y^HE21(e)(r,θ)=(x^cosθy^sinθ)F11(r)HE21(o)(r,θ)=(x^sinθ+y^cosθ)F11(r)TM01(r,θ)=(x^cosθ+y^sinθ)F11(r)TE01(r,θ)=(x^sinθy^cosθ)F11(r).
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 [1820] is given by
V11(HE+)(r,θ)=(HE11(e)+iHE11(o))/2=(x^+iy^)F01/2V11(HE)(r,θ)=(HE11(e)iHE11(o))/2=(x^iy^)F01/2V21(HE+)(r,θ)=(HE21(e)+iHE21(o))/2=eiθ(x^+iy^)F11/2V21(HE)(r,θ)=(HE21(e)iHE21(o))/2=eiθ(x^iy^)F11/2V01(T+)(r,θ)=(TM01iTE01)/2=eiθ(x^+iy^)F11/2V01(T)(r,θ)=(TM01+iTE01)/2=eiθ(x^iy^)F11/2.
The V21(HE±) 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 V01(T±) 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 V11(HE±) 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
LP01(x)(r,θ)=x^F01(r)LP01(y)(r,θ)=y^F01(r)LP11(x+)(r,θ)=x^eiθF11(r)LP11(x)(r,θ)=x^eiθF11(r)LP11(y+)(r,θ)=y^eiθF11(r)LP11(y)(r,θ)=y^eiθF11(r).
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

e1=V11(HE+);e3=V21(HE+);e5=V01(T+);e2=V11(HE);e4=V21(HE);e6=V01(T).
Our ultimate goal is to coherently estimate all of the coefficients
γi=|γi|eiϕi.
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

|P+Et(r,θ)|2=|γ1F01(r)+γ3eiθF11+γ5eiθF11(r)|2=|γ1|2F01(r)2+(|γ3|2+|γ5|2)F11(r)2+2(γ3γ5*)cos(2θ)F11(r)22(γ3γ5*)sin(2θ)F11(r)2+2(γ1(γ3+γ5)*))cos(θ)F01(r)F11(r)+2(γ1(γ3γ5)*))sin(θ)F01(r)F11(r).
The analogous result for a negative helicity filter is
|PEt(r,θ)|2=|γ2|2F01(r)2+(|γ4|2+|γ6|2)F11(r)2+2(γ4γ6*)cos(2θ)F11(r)2+2(γ4γ6*)sin(2θ)F11(r)2+2(γ2(γ6+γ4)*))cos(θ)F01(r)F11(r)+2(γ2(γ6γ4)*))sin(θ)F01(r)F11(r).
A near-field intensity (NFI) image consists of intensity measurements at a number of points {ri, θi}. By linear regression on the two circularly polarized NFI’s, we may estimate the ten coefficients
Ξ1=|γ1|2;Ξ2=|γ2|2;Ξ3+5=|γ3|2+|γ5|2;Ξ4+6=|γ4|2+|γ6|2;Ξ35=γ3γ5*;Ξ46=γ4γ6*;Ξ13+5=γ1(γ3+γ5)*;Ξ26+4=γ2(γ6+γ4)*;Ξ135=γ1(γ3γ5)*;Ξ264=γ2(γ6γ4)*.
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
Γ35(min)=min(|γ3|,|γ5|);Γ46(min)=min(|γ4|,|γ6|);Γ35(max)=max(|γ3|,|γ5|)Γ46(max)=max(|γ4|,|γ6|).
There is an ambiguity in which of Γ35(min), Γ35(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 V21(HE+), then again |γ3| ≫ |γ5|. When the launch is chosen to maximize V21(HE), 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 ϕi in (2.6). Since there will be an ambiguous overall phase, we define ϕ1 = 0. We denote the relative phases

δab=ϕaϕb.
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 {|γi|2}i=16 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

2|Px^Et|2=|γ1+γ2|2F01(r)2+(|γ3+γ6|2+|γ5+γ4|2)F11(r)2+2cos(2θ)(γ3+γ6)(γ5+γ4)*F11(r)22sin(2θ)(γ3+γ6)(γ5+γ4)*F11(r)2+2cos(θ)(γ1+γ2)(γ3+γ6+γ5+γ4)*F01(r)F11(r)+2sin(θ)(γ1+γ2)(γ3+γ6γ5γ4)*F01(r)F11(r)
and
2|Py^Et|2=|γ1γ2|2F01(r)2+(|γ3γ6|2+|γ5γ4|2)F11(r)2+2cos(2θ)(γ3γ6)(γ5γ4)*F11(r)22sin(2θ)(γ3γ6)(γ5γ4)*F11(r)2+2cos(θ)(γ1γ2)(γ3γ6+γ5γ4)*F01(r)F11(r)+2sin(θ)(γ1γ2)(γ3γ6γ5+γ4)*F01(r)F11(r).
From (2.12) and (2.13) ten linear regression coefficients may be estimates, that we label
Ξ12(x)=|γ1+γ2|2;Ξ12(y)=|γ1γ2|2;Ξ36+54(x)=|γ3+γ6|2+|γ5+γ4|2;Ξ36+54(y)=|γ3γ6|2+|γ5γ4|2;Ξ3654(x)=(γ3+γ6)(γ5+γ4)*;Ξ3654(y)(γ3γ6)(γ5γ4)*;Ξ1236+54(x)=(γ1+γ2)(γ3+γ6+γ5+γ4)*;Ξ1236+54(y)=(γ1γ2)(γ3γ6+γ5γ4)*;Ξ123654(x)=(γ1+γ2)(γ3+γ6γ5γ4)*;Ξ123654(y)=(γ1γ2)(γ3γ6γ5+γ4)*.
The coefficients Ξ3654(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=Ξ3645(x)Ξ35Ξ46*=|γ3||γ4|eiδ43+|γ5||γ6|ei(δ64+δ35)eiδ43.
The only unknown is δ43, which we find as the solution to
(zz)=(|γ5||γ6|cos(δ64+δ35)=|γ3||γ4||γ5||γ6|sin(δ64+δ35)|γ5||γ6|sin(δ64+δ35)|γ5||γ6|cos(δ64+δ35)|γ3||γ4|)(cosδ43sinδ43).
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,
(Ξ13+5/|γ1|Ξ135/|γ1|)=(|γ5|cosδ53+|γ3||γ5|sinδ53|γ5|sinδ53|γ5|cosδ53|γ3|)(cosϕ3sinϕ3),
which yields the phase ϕ3, and, immediately, ϕ4, ϕ5, and ϕ6. Similarly,
(Ξ26+4/|γ2|Ξ264/|γ2|)=(|γ6|cosδ64+|γ4||γ6|sinδ64|γ6|sinδ64|γ6|cosδ64|γ4|)(cosδ24sinδ24),
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 γγ̃, with

γ˜2=γ2*;γ˜3=γ5*;γ˜5=γ3*;γ˜4=γ6*;γ˜6=γ4*.
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 Γ46(min) and Γ46(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 ϕi 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 V21(HE+) or V21(HE) 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 V21(HE+), 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 rr0, 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

γ^=argminγp,x|Ip(x)PpEt(x,γ)|2,
where p ∈ {+, −, x̂, ŷ} are the polarization states on which we project, x is the pixel index, Ip are the measured polarization filtered images, Pp are the polarization projections, and Et 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 V21(HE+), 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 V21(HE+) 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 V21(HE+) 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 V21(HE) 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 HE1,1 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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2. C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Vortex-preserving weakly guiding anisotropic twisted fibres,” J. Opt. A-Pure Appl. Op. 6(5), S162–S165 (2004). [CrossRef]  

3. N. K. Viswanathan and V. V. G. Inavalli, “Generation of optical vector beams using a two-mode fiber,” Opt. Lett. 34(8), 1189–1191 (2009). [CrossRef]   [PubMed]  

4. N. Bozinovic, S. Golowich, P. Kristensen, and S. Ramachandran, “Control of orbital angular momentum of light with optical fibers,” Opt. Lett. 37(13), 2451–2453 (2012). [CrossRef]   [PubMed]  

5. “Launched Power Distribution Measurement Procedure for Graded-Index Multimode Fiber Transmitters (FOTP-203),” ANSI/TIA/EIA-455-203-2001, (2001).

6. S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett. 34(16), 2525–2527 (2009). [CrossRef]   [PubMed]  

7. S. Golowich and S. Ramachandran, “Impact of fiber design on polarization dependence in microbend gratings,” Opt. Express 13(18), 6870–6877 (2005). [CrossRef]   [PubMed]  

8. 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]   [PubMed]  

9. A. Volyar, “Algebra and geometry of optical vortices: measuring of a mode spectrum in optical fibres,” in First International Conference on Advanced Optoelectronics and Lasers, 2003. Proceedings of CAOL 2003 , 1, 30–35 (2003).

10. M. Skorobogatiy, C. Anastassiou, S. G. Johnson, O. Weisberg, T. D. Engeness, S. A. Jacobs, R. U. Ahmad, and Y. Fink, “Quantitative characterization of higher-order mode converters in weakly multimoded fibers,” Opt. Express 11(22), 2838–2847 (2003). [CrossRef]   [PubMed]  

11. B. A. Alvi, M. Asif, and A. Israr, “Measurement of complex mode amplitudes in multimode fiber,” in SPIE OPTO , 82840K–82840K (2012). [CrossRef]  

12. T. Kaiser, D. Flamm, S. Schrter, and M. Duparr, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express 17(11), 9347–9356 (2009). [CrossRef]   [PubMed]  

13. J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16(10), 7233–7243 (2008). [CrossRef]   [PubMed]  

14. J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 61–70 (2009). [CrossRef]  

15. D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (C2) imaging of fiber and waveguide modes,” Opt. Express 19(14), 13,008–13,019 (2011). [CrossRef]  

16. V. Inavalli and N. Viswanathan, “Switchable vector vortex beam generation using an optical fiber,” Opt. Commun. 283(6), 861–864 (2010). [CrossRef]  

17. A. Snyder and J. Love, Optical waveguide theory, vol. 190 (Springer, 1983).

18. J. J. Kapany and N. S. Burke, Optical Waveguides (Academic Press, 1972).

19. A. V. Volyar and T. A. Fadeeva, “Vortex nature of optical fiber modes: I. structure of the natural modes,” Tech. Phys. Lett. 22, 330–332 (1996).

20. R. J. Black and L. Gagnon, Optical Waveguide Modes: Polarization, Coupling and Symmetry, 1st ed. (McGraw-Hill Professional, 2010).

References

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  1. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-Division Multiplexing Over 96 km of Few-Mode Fiber Using Coherent 6 × 6 MIMO Processing,” J. Lightwave Technol. 30(4), 521–531 (2012).
    [Crossref]
  2. C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Vortex-preserving weakly guiding anisotropic twisted fibres,” J. Opt. A-Pure Appl. Op. 6(5), S162–S165 (2004).
    [Crossref]
  3. N. K. Viswanathan and V. V. G. Inavalli, “Generation of optical vector beams using a two-mode fiber,” Opt. Lett. 34(8), 1189–1191 (2009).
    [Crossref] [PubMed]
  4. N. Bozinovic, S. Golowich, P. Kristensen, and S. Ramachandran, “Control of orbital angular momentum of light with optical fibers,” Opt. Lett. 37(13), 2451–2453 (2012).
    [Crossref] [PubMed]
  5. “Launched Power Distribution Measurement Procedure for Graded-Index Multimode Fiber Transmitters (FOTP-203),” ANSI/TIA/EIA-455-203-2001, (2001).
  6. S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett. 34(16), 2525–2527 (2009).
    [Crossref] [PubMed]
  7. S. Golowich and S. Ramachandran, “Impact of fiber design on polarization dependence in microbend gratings,” Opt. Express 13(18), 6870–6877 (2005).
    [Crossref] [PubMed]
  8. 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] [PubMed]
  9. A. Volyar, “Algebra and geometry of optical vortices: measuring of a mode spectrum in optical fibres,” in First International Conference on Advanced Optoelectronics and Lasers, 2003. Proceedings of CAOL 2003,  1, 30–35 (2003).
  10. M. Skorobogatiy, C. Anastassiou, S. G. Johnson, O. Weisberg, T. D. Engeness, S. A. Jacobs, R. U. Ahmad, and Y. Fink, “Quantitative characterization of higher-order mode converters in weakly multimoded fibers,” Opt. Express 11(22), 2838–2847 (2003).
    [Crossref] [PubMed]
  11. B. A. Alvi, M. Asif, and A. Israr, “Measurement of complex mode amplitudes in multimode fiber,” in SPIE OPTO, 82840K–82840K (2012).
    [Crossref]
  12. T. Kaiser, D. Flamm, S. Schrter, and M. Duparr, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express 17(11), 9347–9356 (2009).
    [Crossref] [PubMed]
  13. J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16(10), 7233–7243 (2008).
    [Crossref] [PubMed]
  14. J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 61–70 (2009).
    [Crossref]
  15. D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (C2) imaging of fiber and waveguide modes,” Opt. Express 19(14), 13,008–13,019 (2011).
    [Crossref]
  16. V. Inavalli and N. Viswanathan, “Switchable vector vortex beam generation using an optical fiber,” Opt. Commun. 283(6), 861–864 (2010).
    [Crossref]
  17. A. Snyder and J. Love, Optical waveguide theory, vol. 190 (Springer, 1983).
  18. J. J. Kapany and N. S. Burke, Optical Waveguides (Academic Press, 1972).
  19. A. V. Volyar and T. A. Fadeeva, “Vortex nature of optical fiber modes: I. structure of the natural modes,” Tech. Phys. Lett. 22, 330–332 (1996).
  20. R. J. Black and L. Gagnon, Optical Waveguide Modes: Polarization, Coupling and Symmetry, 1st ed. (McGraw-Hill Professional, 2010).

2012 (3)

2011 (1)

D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (C2) imaging of fiber and waveguide modes,” Opt. Express 19(14), 13,008–13,019 (2011).
[Crossref]

2010 (1)

V. Inavalli and N. Viswanathan, “Switchable vector vortex beam generation using an optical fiber,” Opt. Commun. 283(6), 861–864 (2010).
[Crossref]

2009 (4)

2008 (1)

2005 (2)

2004 (1)

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Vortex-preserving weakly guiding anisotropic twisted fibres,” J. Opt. A-Pure Appl. Op. 6(5), S162–S165 (2004).
[Crossref]

2003 (2)

A. Volyar, “Algebra and geometry of optical vortices: measuring of a mode spectrum in optical fibres,” in First International Conference on Advanced Optoelectronics and Lasers, 2003. Proceedings of CAOL 2003,  1, 30–35 (2003).

M. Skorobogatiy, C. Anastassiou, S. G. Johnson, O. Weisberg, T. D. Engeness, S. A. Jacobs, R. U. Ahmad, and Y. Fink, “Quantitative characterization of higher-order mode converters in weakly multimoded fibers,” Opt. Express 11(22), 2838–2847 (2003).
[Crossref] [PubMed]

1996 (1)

A. V. Volyar and T. A. Fadeeva, “Vortex nature of optical fiber modes: I. structure of the natural modes,” Tech. Phys. Lett. 22, 330–332 (1996).

Ahmad, R. U.

Alexeyev, C. N.

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Vortex-preserving weakly guiding anisotropic twisted fibres,” J. Opt. A-Pure Appl. Op. 6(5), S162–S165 (2004).
[Crossref]

Alvi, B. A.

B. A. Alvi, M. Asif, and A. Israr, “Measurement of complex mode amplitudes in multimode fiber,” in SPIE OPTO, 82840K–82840K (2012).
[Crossref]

Anastassiou, C.

Asif, M.

B. A. Alvi, M. Asif, and A. Israr, “Measurement of complex mode amplitudes in multimode fiber,” in SPIE OPTO, 82840K–82840K (2012).
[Crossref]

Barankov, R. A.

D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (C2) imaging of fiber and waveguide modes,” Opt. Express 19(14), 13,008–13,019 (2011).
[Crossref]

Black, R. J.

R. J. Black and L. Gagnon, Optical Waveguide Modes: Polarization, Coupling and Symmetry, 1st ed. (McGraw-Hill Professional, 2010).

Bolle, C.

Bozinovic, N.

Burke, N. S.

J. J. Kapany and N. S. Burke, Optical Waveguides (Academic Press, 1972).

Burrows, E. C.

Dimarcello, F. V.

Duparr, M.

Engeness, T. D.

Esmaeelpour, M.

Essiambre, R.-J.

Fadeeva, T. A.

A. V. Volyar and T. A. Fadeeva, “Vortex nature of optical fiber modes: I. structure of the natural modes,” Tech. Phys. Lett. 22, 330–332 (1996).

Fini, J. M.

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 61–70 (2009).
[Crossref]

Fink, Y.

Flamm, D.

Fleming, J.

Gagnon, L.

R. J. Black and L. Gagnon, Optical Waveguide Modes: Polarization, Coupling and Symmetry, 1st ed. (McGraw-Hill Professional, 2010).

Ghalmi, S.

Gnauck, A. H.

Golowich, S.

Inavalli, V.

V. Inavalli and N. Viswanathan, “Switchable vector vortex beam generation using an optical fiber,” Opt. Commun. 283(6), 861–864 (2010).
[Crossref]

Inavalli, V. V. G.

Israr, A.

B. A. Alvi, M. Asif, and A. Israr, “Measurement of complex mode amplitudes in multimode fiber,” in SPIE OPTO, 82840K–82840K (2012).
[Crossref]

Jacobs, S. A.

Johnson, S. G.

Kaiser, T.

Kapany, J. J.

J. J. Kapany and N. S. Burke, Optical Waveguides (Academic Press, 1972).

Kristensen, P.

Lingle, R.

Love, J.

A. Snyder and J. Love, Optical waveguide theory, vol. 190 (Springer, 1983).

McCurdy, A. H.

Mermelstein, M. D.

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 61–70 (2009).
[Crossref]

Monberg, E.

Mumtaz, S.

Nicholson, J. W.

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 61–70 (2009).
[Crossref]

J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16(10), 7233–7243 (2008).
[Crossref] [PubMed]

Peckham, D. W.

Ramachandran, S.

Randel, S.

Ryf, R.

Schimpf, D. N.

D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (C2) imaging of fiber and waveguide modes,” Opt. Express 19(14), 13,008–13,019 (2011).
[Crossref]

Schrter, S.

Sierra, A.

Skorobogatiy, M.

Snyder, A.

A. Snyder and J. Love, Optical waveguide theory, vol. 190 (Springer, 1983).

Viswanathan, N.

V. Inavalli and N. Viswanathan, “Switchable vector vortex beam generation using an optical fiber,” Opt. Commun. 283(6), 861–864 (2010).
[Crossref]

Viswanathan, N. K.

Volyar, A.

A. Volyar, “Algebra and geometry of optical vortices: measuring of a mode spectrum in optical fibres,” in First International Conference on Advanced Optoelectronics and Lasers, 2003. Proceedings of CAOL 2003,  1, 30–35 (2003).

Volyar, A. V.

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Vortex-preserving weakly guiding anisotropic twisted fibres,” J. Opt. A-Pure Appl. Op. 6(5), S162–S165 (2004).
[Crossref]

A. V. Volyar and T. A. Fadeeva, “Vortex nature of optical fiber modes: I. structure of the natural modes,” Tech. Phys. Lett. 22, 330–332 (1996).

Weisberg, O.

Winzer, P. J.

Wisk, P.

Yablon, A. D.

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 61–70 (2009).
[Crossref]

J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16(10), 7233–7243 (2008).
[Crossref] [PubMed]

Yan, M. F.

Yavorsky, M. A.

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Vortex-preserving weakly guiding anisotropic twisted fibres,” J. Opt. A-Pure Appl. Op. 6(5), S162–S165 (2004).
[Crossref]

First International Conference on Advanced Optoelectronics and Lasers, 2003. Proceedings of CAOL 2003 (1)

A. Volyar, “Algebra and geometry of optical vortices: measuring of a mode spectrum in optical fibres,” in First International Conference on Advanced Optoelectronics and Lasers, 2003. Proceedings of CAOL 2003,  1, 30–35 (2003).

IEEE J. Sel. Top. Quantum Electron. (1)

J. W. Nicholson, A. D. Yablon, J. M. Fini, and M. D. Mermelstein, “Measuring the modal content of large-mode-area fibers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 61–70 (2009).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. A-Pure Appl. Op. (1)

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Vortex-preserving weakly guiding anisotropic twisted fibres,” J. Opt. A-Pure Appl. Op. 6(5), S162–S165 (2004).
[Crossref]

Opt. Commun. (1)

V. Inavalli and N. Viswanathan, “Switchable vector vortex beam generation using an optical fiber,” Opt. Commun. 283(6), 861–864 (2010).
[Crossref]

Opt. Express (5)

Opt. Lett. (4)

SPIE OPTO (1)

B. A. Alvi, M. Asif, and A. Israr, “Measurement of complex mode amplitudes in multimode fiber,” in SPIE OPTO, 82840K–82840K (2012).
[Crossref]

Tech. Phys. Lett. (1)

A. V. Volyar and T. A. Fadeeva, “Vortex nature of optical fiber modes: I. structure of the natural modes,” Tech. Phys. Lett. 22, 330–332 (1996).

Other (4)

R. J. Black and L. Gagnon, Optical Waveguide Modes: Polarization, Coupling and Symmetry, 1st ed. (McGraw-Hill Professional, 2010).

A. Snyder and J. Love, Optical waveguide theory, vol. 190 (Springer, 1983).

J. J. Kapany and N. S. Burke, Optical Waveguides (Academic Press, 1972).

“Launched Power Distribution Measurement Procedure for Graded-Index Multimode Fiber Transmitters (FOTP-203),” ANSI/TIA/EIA-455-203-2001, (2001).

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Figures (10)

Fig. 1
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.

Fig. 2
Fig. 2

Spatial polarization patterns of the Vector basis set.

Fig. 3
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).

Fig. 4
Fig. 4

Simulated polarization filtered NFI images for the example with dominant V 21 ( H E + ) state.

Fig. 5
Fig. 5

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

Fig. 6
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.

Fig. 7
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.

Fig. 8
Fig. 8

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

Fig. 9
Fig. 9

(a) Power fraction in the fundamental, transverse, and HE21 pairs; (b) Power estimated in the HE1,1 modes by regression analysis (red crosses) and by coupling into a SMF (blue curve).

Fig. 10
Fig. 10

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

Equations (20)

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E t ( r , θ , t ) = e i ω t i = 1 6 γ i e i ( r , θ ) ,
H E 11 ( e ) ( r , θ ) = F 01 ( r ) x ^ H E 11 ( o ) ( r , θ ) = F 01 ( r ) y ^ H E 21 ( e ) ( r , θ ) = ( x ^ cos θ y ^ sin θ ) F 11 ( r ) H E 21 ( o ) ( r , θ ) = ( x ^ sin θ + y ^ cos θ ) F 11 ( r ) T M 01 ( r , θ ) = ( x ^ cos θ + y ^ sin θ ) F 11 ( r ) T E 01 ( r , θ ) = ( x ^ sin θ y ^ cos θ ) F 11 ( r ) .
V 11 ( H E + ) ( r , θ ) = ( H E 11 ( e ) + i H E 11 ( o ) ) / 2 = ( x ^ + i y ^ ) F 01 / 2 V 11 ( H E ) ( r , θ ) = ( H E 11 ( e ) i H E 11 ( o ) ) / 2 = ( x ^ i y ^ ) F 01 / 2 V 21 ( H E + ) ( r , θ ) = ( H E 21 ( e ) + i H E 21 ( o ) ) / 2 = e i θ ( x ^ + i y ^ ) F 11 / 2 V 21 ( H E ) ( r , θ ) = ( H E 21 ( e ) i H E 21 ( o ) ) / 2 = e i θ ( x ^ i y ^ ) F 11 / 2 V 01 ( T + ) ( r , θ ) = ( T M 01 i T E 01 ) / 2 = e i θ ( x ^ + i y ^ ) F 11 / 2 V 01 ( T ) ( r , θ ) = ( T M 01 + i T E 01 ) / 2 = e i θ ( x ^ i y ^ ) F 11 / 2 .
L P 01 ( x ) ( r , θ ) = x ^ F 01 ( r ) L P 01 ( y ) ( r , θ ) = y ^ F 01 ( r ) L P 11 ( x + ) ( r , θ ) = x ^ e i θ F 11 ( r ) L P 11 ( x ) ( r , θ ) = x ^ e i θ F 11 ( r ) L P 11 ( y + ) ( r , θ ) = y ^ e i θ F 11 ( r ) L P 11 ( y ) ( r , θ ) = y ^ e i θ F 11 ( r ) .
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 ) .
γ i = | γ i | e i ϕ i .
| 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 ( γ 3 γ 5 * ) cos ( 2 θ ) F 11 ( r ) 2 2 ( γ 3 γ 5 * ) sin ( 2 θ ) F 11 ( r ) 2 + 2 ( γ 1 ( γ 3 + γ 5 ) * ) ) cos ( θ ) F 01 ( r ) F 11 ( r ) + 2 ( γ 1 ( γ 3 γ 5 ) * ) ) sin ( θ ) F 01 ( r ) F 11 ( r ) .
| P E t ( r , θ ) | 2 = | γ 2 | 2 F 01 ( r ) 2 + ( | γ 4 | 2 + | γ 6 | 2 ) F 11 ( r ) 2 + 2 ( γ 4 γ 6 * ) cos ( 2 θ ) F 11 ( r ) 2 + 2 ( γ 4 γ 6 * ) sin ( 2 θ ) F 11 ( r ) 2 + 2 ( γ 2 ( γ 6 + γ 4 ) * ) ) cos ( θ ) F 01 ( r ) F 11 ( r ) + 2 ( γ 2 ( γ 6 γ 4 ) * ) ) sin ( θ ) F 01 ( r ) F 11 ( r ) .
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 = γ 1 ( γ 3 + γ 5 ) * ; Ξ 26 + 4 = γ 2 ( γ 6 + γ 4 ) * ; Ξ 13 5 = γ 1 ( γ 3 γ 5 ) * ; Ξ 26 4 = γ 2 ( γ 6 γ 4 ) * .
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 | ) .
δ a b = ϕ a ϕ b .
2 | P 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 ) ( γ 3 + γ 6 ) ( γ 5 + γ 4 ) * F 11 ( r ) 2 2 sin ( 2 θ ) ( γ 3 + γ 6 ) ( γ 5 + γ 4 ) * F 11 ( r ) 2 + 2 cos ( θ ) ( γ 1 + γ 2 ) ( γ 3 + γ 6 + γ 5 + γ 4 ) * F 01 ( r ) F 11 ( r ) + 2 sin ( θ ) ( γ 1 + γ 2 ) ( γ 3 + γ 6 γ 5 γ 4 ) * F 01 ( r ) F 11 ( r )
2 | P 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 ) ( γ 3 γ 6 ) ( γ 5 γ 4 ) * F 11 ( r ) 2 2 sin ( 2 θ ) ( γ 3 γ 6 ) ( γ 5 γ 4 ) * F 11 ( r ) 2 + 2 cos ( θ ) ( γ 1 γ 2 ) ( γ 3 γ 6 + γ 5 γ 4 ) * F 01 ( r ) F 11 ( r ) + 2 sin ( θ ) ( γ 1 γ 2 ) ( γ 3 γ 6 γ 5 + γ 4 ) * F 01 ( r ) F 11 ( r ) .
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 ) = ( γ 1 + γ 2 ) ( γ 3 + γ 6 + γ 5 + γ 4 ) * ; Ξ 1236 + 54 ( y ) = ( γ 1 γ 2 ) ( γ 3 γ 6 + γ 5 γ 4 ) * ; Ξ 1236 54 ( x ) = ( γ 1 + γ 2 ) ( γ 3 + γ 6 γ 5 γ 4 ) * ; Ξ 1236 54 ( y ) = ( γ 1 γ 2 ) ( γ 3 γ 6 γ 5 + γ 4 ) * .
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 .
( z z ) = ( | 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 | ) ( cos δ 43 sin δ 43 ) .
( Ξ 13 + 5 / | γ 1 | Ξ 13 5 / | γ 1 | ) = ( | γ 5 | cos δ 53 + | γ 3 | | γ 5 | sin δ 53 | γ 5 | sin δ 53 | γ 5 | cos δ 53 | γ 3 | ) ( cos ϕ 3 sin ϕ 3 ) ,
( Ξ 26 + 4 / | γ 2 | Ξ 26 4 / | γ 2 | ) = ( | γ 6 | cos δ 64 + | γ 4 | | γ 6 | sin δ 64 | γ 6 | sin δ 64 | γ 6 | cos δ 64 | γ 4 | ) ( cos δ 24 sin δ 24 ) ,
γ ˜ 2 = γ 2 * ; γ ˜ 3 = γ 5 * ; γ ˜ 5 = γ 3 * ; γ ˜ 4 = γ 6 * ; γ ˜ 6 = γ 4 * .
γ ^ = argmin γ p , x | I p ( x ) P p E t ( x , γ ) | 2 ,

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