Rapid complex mode decomposition of vector beams by common path interferometry

Fredrik K. Fatemi; Guy Beadie

doi:10.1364/OE.21.032291

1. Introduction

Cylindrical vector beams (CVBs) and higher order modes of optical fibers have found use in a wide range of applications in recent years [1–11]. For instance, CVBs can be used for real-time polarimetry [2], dark-field imaging [3], optical trapping [4], atomic physics [5], and entanglement of spatial and polarization degrees of freedom [6]. Selective excitation of higher order modes in fibers has also been of interest for atom guiding [7, 8] and trapping along nanofibers [9, 10], femtosecond laser applications [11, 12], and mode-division multiplexing [13]. Mode decomposition techniques that can both analyze the beams and be operated in real time to shape the beam should be valuable.

Among the more recent imaging techniques for decomposing the complex mode content of fibers are those based on direct intensity measurements [14–18] and on correlation measurements from specially designed digital holograms, be they fixed holograms [19] or active ones defined by, e.g. spatial light modulators [20, 21]. Although intensity measurements alone do not provide phase information, the beam patterns are assumed to arise from the sum of a small basis set compared with the number of pixels in a typical CCD image so that algorithms may extract reasonable estimates of the complex mode weights [14–18]. Holographic techniques, with direct access to phase information, can provide rapid mode decomposition of linearly polarized modes [19, 20], but require either spatial light modulators or custom diffractive optical elements. An instructive paper comparing the relative merits of intensity-only and holographic methods is presented in [22]. Decomposition techniques have been successful at studying high-speed nonlinear dynamics [16] and optimizing fiber coupling [23]. We note that there are other powerful techniques for rapid mode decomposition requiring either low coherence or broadly tunable light sources, such as spatially and spectrally resolved imaging [25–27] or cross-correlation methods [28], but these are not suitable for fixed-frequency sources.

In this paper, we have used interferometry for decomposition into optical modes (IDIOM) for stable, sensitive, and unambiguous measurement of the electric field and complex mode coefficients of vector beams. We first verify the technique with free-space beams of known modal content, and then use it to analyze vector beams from fibers supporting up to 10 modes. Although interferometry has been used for mode assessment [13, 24], quantitative decomposition of vector beams using interferometry has been unexplored to the best of our knowledge. Interferometry for scalar mode decomposition has recently been investigated [29]. High stability is achieved in a common path arrangement so that relative intermodal phase shifts are resilient against fluctuations. After obtaining the field through interferometry, it is decomposed by least-squares-fitting into a desired basis using the Moore-Penrose pseudoinverse. We first test IDIOM with known free-space modes, and then use it to measure the mode content of fibers containing up to 10 modes as the fiber input mode superposition is varied. Furthermore, we construct a fiber scattering matrix that transforms the input complex coefficients to the output coefficients. Finally, we highlight the sensitivity advantage of IDIOM for weak signals by measuring the complex modal content of beams with signal-to-noise ratio (SNR) << 1. The optical setup is inexpensive and compact, using only one polarization component. By directly imaging the vector field, exotic polarization states of light such as Poincare beams [30] may be analyzed even if mode decomposition is not required.

2. Background

In this work, after testing the technique with free-space beams of known modal content, we illustrate the technique by decomposing beams exiting few-mode optical fibers. In general, the transverse field at a given plane and time may be expanded as

E (r, θ) = \sum_{i = 1}^{n} c_{i} f_{i} (r, θ)

where the time-dependence has been removed, the c_i are complex amplitudes, and the f_i(r, θ) are guided-mode basis functions for the fiber. The f_i(r, θ) may be described by TM, TE, HE, and EH vector modes, which have a spatially-varying polarization profile. These vector modes are required for propagation in fibers exhibiting strong guiding, such as in tapered nanofibers [9, 10]. For a weakly-guided fiber, such as standard single-mode telecommunications fiber, the vector modes can be decomposed approximately using the linearly-polarized (LP) basis. The mode profiles of both bases are shown in Fig. 1.

Fig. 1 (top) LP and (bottom) vector modes (HE, EH, TM, TE) used in this paper. Different angular orientations of degenerate modes are labeled with ‘e’ and ‘o.’ For the LP-modes, only the vertical polarization is shown; the horizontal polarizations have the same intensity profiles.

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The first 10 LP basis functions, f₁(r, θ) to f₁₀(r, θ), respectively, are:

\begin{array}{l} L P_{01}^{x} (r, θ) = F_{01} (r) \hat{x} \\ L P_{01}^{y} (r, θ) = F_{01} (r) \hat{y} \\ L P_{11}^{x e} (r, θ) = F_{11} (r) \cos θ \hat{x} \\ L P_{11}^{x o} (r, θ) = F_{11} (r) \sin θ \hat{x} \\ L P_{11}^{y e} (r, θ) = F_{11} (r) \cos θ \hat{y} \\ L P_{11}^{y o} (r, θ) = F_{11} (r) \sin θ \hat{y} \\ L P_{21}^{x e} (r, θ) = F_{21} (r) \cos 2 θ \hat{x} \\ L P_{21}^{x o} (r, θ) = F_{21} (r) \sin 2 θ \hat{x} \\ L P_{21}^{y e} (r, θ) = F_{21} (r) \cos 2 θ \hat{y} \\ L P_{21}^{y o} (r, θ) = F_{21} (r) \sin 2 θ \hat{y} \end{array}

In the paper, we say that modes of different polarizations but having the same subscripts belong the same family. The vector basis uses the same radial profiles, and can be formed from superpositions of these expressions. For example, the radially-polarized

T E_{01} = L P_{11}^{x e} + L P_{11}^{y o}

. In general, the radial functions, F₀₁(r), F₁₁(r), and F₂₁(r) are Bessel functions inside the fiber medium; however, after propagating through free-space to the CCD camera, the radial profiles are altered, and different radial functions may be more appropriate. Without loss of generality for IDIOM as a technique, we choose the following Gaussian functions, with waist parameter ω, as convenient approximations to the radial profiles detected from few-mode fibers in the manner described in our setup discussion in Section 3:

\begin{array}{l} F_{01} (r) = \exp (- \frac{r^{2}}{ω^{2}}) \\ F_{11} (r) = r exp (- \frac{r^{2}}{ω^{2}}) \\ F_{21} (r) = r^{2} exp (- \frac{r^{2}}{ω^{2}}) \end{array}

We have dropped the normalization factors for clarity. For few-mode fibers, exact far-field expressions of the basis modes could be determined empirically by exciting individual modes [5, 31] and measuring the beam profiles. Alternatively, one could derive analytic expressions that predict mode profiles in the image plane – tailored for whatever imaging arrangement is chosen in the setup. However, we note that while these simple expressions in Eq. (3) could be improved, especially for higher order mode fibers, we demonstrate that for the family of modes considered in this work, the symmetries of the angular terms are sufficient to distinguish among the different modes.

3. Algorithm

3.1 Background

An important distinction between this work and some other techniques is that we decompose based on the field pattern, not intensity patterns. Since the field phase and symmetry are principal discriminators among different modes, the mode field decomposition can be computed unambiguously. To determine the complex vector electric field, we employ interferometry with a uniform, Gaussian reference field. Because we consider vector fields, the combined reference and signal beams are split into both horizontal and vertical polarization components; the reference beam starts out linearly polarized at 45° to have equal contribution on each. For each polarization component, the field reconstruction method employed below is similar to interferometric techniques for ultrashort pulse measurement [32] and holography, so we only briefly outline the procedure here. In the following, the field polarization direction (x or y) is suppressed and the fields indicate the complex amplitude appropriate for a single polarization component.

A reference field, E_ref, interferes with the unknown signal field, E_sig, which we wish to write in terms of the basis functions in Eqs. (2), to form a total field E_tot, as shown in Fig. 2(a). E_ref is tilted by a small wavevector k_tilt with respect to the propagation direction of E_sig to impart phase-sensitive interference fringes A charge-coupled device (CCD) camera detects the corresponding intensities, I_ref, I_sig, and I_tot:

I_{t o t} \propto {| E_{s i g} (r) + E_{r e f} (r) exp (i k_{t i l t} \cdot r) |}^{2}

= I_{s i g} + I_{r e f} + 2 R e [E_{s i g}^{*} (r) E_{r e f} (r) exp (i k_{t i l t} \cdot r)]

where

E_{s i g}^{*}

is the complex conjugate of_{$E_{s i g}$}. Thus:

I_{t o t} - I_{s i g} - I_{r e f} = 2 R e {| E_{s i g}^{*} (r) E_{r e f} (r) | exp [i φ (r, θ) + i k_{t i l t} \cdot r]}

where φ(r,θ) is the relative phase between E_ref and E_sig, and where the proportionality constants connecting field to intensity have been dropped for convenience. As mentioned above, much of the symmetry information is encapsulated by the phase function, φ(r,θ). This information can be calculated unambiguously by Fourier transform. If we take the Fourier transform of Eq. (6), we find

ℱ {2 R e [E_{s i g}^{*} (r) E_{r e f} (r) exp (i k_{t i l t} \cdot r)]} = g^{*} (k - k_{t i l t}) + g (k + k_{t i l t})

where g(k) is the Fourier transform of

E_{s i g}^{*} (r) E_{r e f} (r)

[32]. However, we know that the tilt only comes from one direction, so as long as these two terms are well separated in k-space, we may apply a digital filter to keep only one of them [27]. Choosing the wrong sign of k_tilt leads to complex conjugation of E_sig, but the correct sign is determined by calibration with a known circular polarization state. Figure 2(a) shows an example of this Fourier transform with well-separated peaks. Achieving adequate separation in k-space only requires that the tilt be large enough that the spatial frequency of the interference oscillations is faster than all other spatial frequencies in E_ref or E_sig.

Fig. 2 (a) Interferometer setup for scalar beams. For Gaussian input, the FFT of the interferogram has Gaussian features that are fit to extract k_tilt (b) Modified setup for vector modes. The signal beam is split: One arm (black) is interfered with a reference beam (blue), and the other arm (red) is displaced vertically so that four images fit on one CCD image. Actual reflection angles are shallow to avoid differential phase and amplitude shifts between horizontal and vertical components. A sample image is shown, along with the FFT of the vertical component. One of the FFT features is digitally removed as described in the text. Apart from aspheric lenses to collimate the fiber outputs of the signal and reference arms, no other imaging lenses are used.

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With one Fourier component rejected, the inverse Fourier transform immediately gives $E_{s i g}^{*} (r) E_{r e f} (r) exp (i k_{t i l t} \cdot r)$ . k_tilt determines $E_{s i g}^{*} (r) E_{r e f} (r)$ unambiguously. In our setup, we find k_tilt by the center of the features on the Fourier Transform [e.g. Figure 2(a)] for symmetric signal beams, but since it is simply a linear phase ramp it could also be removed by more sophisticated image processing or other direct measurements of the tilt angle. We also know E_ref through the intensity measurement I_ref and the assumption that it has uniform phase over the signal beam extent; this assumption is readily met by using a large reference beam and can be verified if needed by commercially available shearing interferometers, though these were not used in this work. Relative phase curvature (i.e. collimation differences) can also be removed by additional image processing if needed; we typically measure phase variations of less than 0.1 rad over the beam extent. Therefore, $E_{s i g}^{*} (r)$ is recovered. Any uncertainty in the choice of sign of k_tilt is removed by calibrating the system with a known circular polarization state.

The setup in Fig. 2(a) must be modified to accommodate vector beams, as we now need to determine the two polarization components of E_sig, as well as the relative phase between them. Figure 2(b) shows our modification to the scalar setup for vector modes. To determine E_sig, we measure $I_{r e f}^{x, y}$ , $I_{s i g}^{x, y}$ , and $I_{t o t}^{x, y}$ for each polarization, where the superscripts indicate polarization direction. The reference intensities $I_{r e f}^{x, y}$ can be measured independently so that only two images are needed for each polarization. We record these four images simultaneously. A signal beam is split and recombined with a vertical offset (red and black beams); the splitting ratio is measured for calibration. $E_{r e f}^{x, y}$ (blue) is combined with the lower of these two beams so that both $I_{s i g}^{x, y}$ and $I_{t o t}^{x, y}$ are recorded simultaneously. The two beams then pass through the calcite polarizing beam displacement prism (PBDP) creating four total beams, to a CCD camera. As mentioned previously, the reference beam is linearly polarized at 45° with respect to the PBDP axes. Crucial to stable function of this analyzer is that, although the reference and signal beams are separated, they travel common paths in the PBDP so that the relative phase between the horizontal and vertical components is preserved. Although air currents and vibrations, etc., may cause fluctuations in the relative phase between the reference and signal fields prior to the PBDP, the relative shift between φ^x (phase between $E_{r e f}^{x}$ and $E_{s i g}^{x}$ ) and φ^y (phase between $E_{r e f}^{y}$ and $E_{s i g}^{y}$ ) remains fixed. This immunity from turbulence is exhibited clearly in the later sections. We use short exposure times (~1 ms) to eliminate the effect of turbulence and preserve good interference contrast. The setup is inexpensive and compact, as it only requires one polarization optic (PBDP).

When analyzing the few-mode fibers, the signal fiber is simply collimated by an asphere. For fibers with higher order mode content, 4f imaging could be employed to observe the waveguide directly and one would use exact waveguide basis functions; for few mode fibers this is unnecessary and would only increase the size and complexity of the setup, in part because their core sizes can be very small and would require very high quality microscope objectives.

With the intensity images acquired, the fields for each polarization component are determined by the steps above. $I_{s i g}^{x, y}$ and $I_{r e f}^{x, y}$ need to be subtracted from the combined $I_{t o t}^{x, y}$ as in Eq. (6). To accurately do this, the vertically-displaced $I_{s i g}^{x, y}$ and $I_{t o t}^{x, y}$ beams need proper registration. This is done in advance by blocking the reference beam so that the upper and lower traces in Fig. 2(b) have identical profiles and their displacements can be determined with subpixel accuracy by proper fitting. The relative power of the signal beam in the interference arm is also determined from this image. Also shown in Fig. 2(b) is the Fourier transform for a typical beam with higher order mode content showing well-separated g*(k-k_tilt) and g(k + k_tilt).

The signal and reference beams are derived from the same laser source by a fiber-coupled 50:50 beam splitter. The two output fibers of the splitter are polarization maintaining, but the output polarizations of the splitter are purified by linear polarizers with extinction ratios of 1000:1. The reference beam is collimated to a 1/e² diameter of 8.0 mm at the entrance of the setup in Fig. 2. To test the few-mode fibers, the signal beam is passed through a mode-shaping element (discussed in later sections) and the fiber under test prior to entering the analyzer.

The laser used in this work is a New Focus Vortex diode laser operating at 795 nm. It has a sub-MHz linewidth, for a coherence length >300 meters. Because this is an interferometric technique, IDIOM is ideally suited to lasers with linewidths <1 GHz that have coherence lengths longer than ~30 cm. Shorter coherence lengths are still possible to analyze with this apparatus, as long as care is taken to keep the path lengths to the PBDP between the reference and signal beams within the coherence length.

3.2 Decomposition

In this work, in addition to free-space beam decomposition, we have analyzed optical fibers supporting up to 10 waveguide modes. The beam profiles typically span at least a 128 x 128 pixel area. Since the field is determined over several thousand pixels, we can perform a least-squares fit of these 10 complex amplitudes using the Moore-Penrose pseudoinverse analysis in MATLAB® [33]. This pseudoinverse technique is ideally suited for overdetermined problems such as our case in which the image is completely determined by a small number of coefficients. It conducts a least-squares fit of the complex amplitudes given the data over thousands of pixels. This matrix manipulation is fast, and the pseudoinverse gives the mode amplitudes of the chosen basis. The inversion can be run at 15 Hz on a PC with Intel i7 CPU at 2.9 GHz using 128x128 pixels. The choice of vector or LP basis depends on the application, but does not affect the speed or accuracy of the inversion; converting between bases is straightforward.

With the complete field determined, a more direct alternative to using the pseudoinverse is to simply calculate the overlap integral with the basis functions in Eqs. (2). With aberration-free or low-noise data, this approach would be preferred. However, we have found that when the beams have low SNR (as in Section 6), the results are unreliable; by conducting the least squares fit, the decomposition is robust. Furthermore, the pseudoinverse can be calculated so quickly that it is far from the rate-limiting step in the data processing. Our rate limiting step is the data acquisition and other manipulation of the image files.

For few mode fibers, the waist parameter, ω, of F₀₁, F₁₁, and F₂₁ is determined empirically by adjusting the launch into the fiber under test so that the signal beam is strongly Gaussian or hollow. High purity is not needed to extract the waist parameter by a fit to the appropriate functional form in Eq. (3). Similarly, we could choose radial functions which more closely match theoretically diffracted fiber solutions, but we find that this simpler family with identical symmetry properties is sufficient to identify the underlying fiber modes. We note that high purity signal beams for few mode fibers are relatively easy to produce: Gaussian output beams can be generated by mandrel wrapping to attenuate higher orders, and hollow beams are readily produced with high purity using a phase plate [5].

Since the modes have a well-defined common axis, it is important that the fit origins are chosen carefully. For calibration, estimates of beam centers are determined from signal beams with symmetric profiles. Then, the decomposition algorithm optimizes this location by minimizing error as a function of beam center. This step only needs to be done once. Because only the relative phases between the modes are important and we cannot measure absolute phase, we choose the phase of one of the modes to be 0. One choice is a fundamental mode, but in many studies where higher order modes are of interest, the LP₀₁ modes may be strongly suppressed so other modes are more appropriate for referencing.

4. Results

4.1. Initial test

To verify the technique, we first apply it to a known, free-space input beam that approximates an LP₁₁ mode. We begin with a collimated Gaussian beam from a single mode optical fiber. This beam passes through a π-phase-plate, as shown in Fig. 3(a), and is then spatially filtered, resulting in a profile that agrees well with an LP₁₁ mode [5, 31] with very little contribution from the two fundamental LP₀₁ modes. This beam passes through a rotatable quarter-wave-plate and half-wave-plate before entering the mode analyzer. For this initial test, we experimentally determine the complex coefficients of the first 6 modes of the LP-basis in Eq. (2) as a function of the orientations of the waveplates, and compare with the expected coefficients determined by using simulated input to the decomposition routine. Each waveplate is rotated through 180 degrees in 15 degree increments, giving a 13 x 13 matrix of amplitudes for each of the 6 coefficients. So that the powers of all 13 x 13 x 6 values can be viewed simultaneously, we unfold each 13 x 13 matrix into a single column to form a 169 x 6 matrix. As expected, only 2 of the LP₁₁ modes in Fig. 3 are strong, corresponding to c₃ and c₅ (following the order of functions in Eqs. (2). For this reason, only the relative phase between these two coefficients is important, which we show as the 13 x 13 matrix for c₃; the phase of c₅ is set to 0. The agreement between the expected amplitudes and phases and the measured values is excellent. The standard deviation of the difference between the expected and measured amplitudes is only 2.1%. There is a slight phase offset between the simulated and measured phases, likely due to either phase changes upon reflections or small offset of the zero position of the waveplates. Phase changes due to reflections can be calibrated with known polarization inputs.

Fig. 3 (a) Creation of a known, free-space LP11 mode. The waveplates are rotated to create different superpositions of $L P_{11}^{y e}$ and $L P_{11}^{x e}$ modes. Not shown is an aperture after the phase plate for mild spatial filtering. (b) (left) Measured and simulated coefficient amplitudes. The fundamental modes are strongly suppressed by the phase plate. At right are the measured and simulated phases of c₃ for each waveplate orientation.

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4.2 Fiber modes

In this subsection, we use IDIOM to analyze the output of a fiber supporting 6 vector modes, and a fiber supporting 10 vector modes. We use fiber lengths of 30-50 cm, and they are kept straight during the measurement. First, we characterize the performance of a fiber-based vector mode generator [5, 31]. This uses the same free-space LP₁₁ mode of Sec. 4.1, which is then coupled into Corning HI1060 fiber. This fiber has a numerical aperture of 0.14 and a core diameter of 5.3 μm so that it supports the LP₀₁ and LP₁₁ mode families from our 795 nm diode laser. Using strain-induced birefringence, one can generate pure CVBs (e.g. HE₂₁, TM₀₁, and TE₀₁) [5, 31]. Under ideal coupling with a centered phase plate, there is again zero contribution from the LP₀₁ family because the field on the axis is cancelled, but the relative weights of these families can be adjusted by decentering the plate. Interfamily mode mixing (i.e. between LP₀₁ and LP₁₁, etc.) in few mode fibers is extremely weak unless the fiber is highly perturbed because the propagation constants are very different. We do not notice any mixing even when the fiber sags by 3-4 cm over a 30 cm length. Strain and torsion, however, can cause intrafamily mixing among the LP₁₁ states.

For this demonstration, we scan the phase plate in 25 μm increments across a Gaussian beam with 1/e² radius of 490 μm [Fig. 4(a)], and use our technique to determine the modal powers as a function of phase plate position. The π phase step is initially set to −1.0 mm, so that the initial beam is almost purely Gaussian with no higher order mode content.

Fig. 4 (a) Results for a fiber supporting only the LP₀₁ and LP₁₁ mode families. The input beam profile for the centered phase plate is shown at left, along with typical output interferograms for the centered phase step for horizontal and vertical polarization components. Modal powers in the LP₀₁ family (black) and LP₁₁ family (red) are shown, with input (solid lines) and output (points) mode powers. (b) Measured (left) and reconstructed (right) beam profiles for phase step position at −150, 0, and 150 μm (Media 1, 0.7 MB). The media file shows the horizontal and vertical interferograms. (c) Results for SMF-28 fiber, which supports up to the LP₂₁ mode family. As in (a), mode powers for the LP₀₁, LP₁₁, and LP₂₁ families are shown. (d) Measured and reconstructed images for phase step position at −150, 0, and 150 μm (Media 2, 0.7 MB). The media file shows the horizontal and vertical interferograms.

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A typical output interferogram for the horizontal and vertical components is shown at left and in Fig. 4 (Media 1). Instead of showing the individual mode powers, Fig. 4(a) shows the total power in the two LP₀₁ and four LP₁₁ mode families. Additionally, we show the expected total mode powers determined by decomposition of the input beam to the fiber (solid lines). Fiber coupling variations among the different mode families can change the total amplitude of the transmitted modes, but not the shape of the curve. We typically observe a root-mean-squared difference between the reconstructed and experimental intensity images of ~5%. As expected, the fundamental mode contribution is minimized when the phase step is centered. When the phase plate is centered on the input beam, we find that only 0.04% of the power exits the fiber in the fundamental mode. Although we do not know what the exact fundamental mode contribution is without optimizing the basis functions, previous studies with tapered nanofibers showed high purity mode propagation with less than 1% fundamental mode contamination [9] using a similar launching technique.

Furthermore, to show that the decomposition of amplitudes and phases within the LP₁₁ mode families is accurate, we also show total reconstructed images (i.e the sum of horizontal and vertical polarization components) for three different phase plate positions based on the retrieved complex coefficients of the 6 LP modes, and compare with the recorded output modes in Fig. 4(b) and Fig. 4 (Media 1). Section 4 discusses the complex mode amplitudes from the fibers in more detail. As is seen in Fig. 4 (Media 1), air currents and vibrations cause the fringes to be unstable; however, there is no relative movement of the fringes between the horizontal and vertical components because the reference and signal beams travel common paths after polarization analysis, allowing operation even in a noisy environment.

We have also applied the decomposition technique to SMF-28 fiber, which has a core diameter of 8.2 μm and a numerical aperture of 0.14, supporting the LP₂₁ family of modes (which includes the vector basis functions HE₃₁ and EH₁₁). In this case, we use a phase plate as shown in Fig. 4(c) that has π phase steps in each quadrant. This phase profile matches that of the LP₂₁ mode family for strong overlap. We use SMF-28 fiber, which supports all 10 modes in Eq. (2) for 795 nm light. Again, by scanning the phase plate across the beam in 25 μm increments, we find the powers in the three mode families shown in Fig. 4(c). Along with these data are the mode decompositions of the input beam (solid lines). We observe little coupling between the different families because the fiber is held fixed and straight. As in the prior case, the images of the reconstructed beam are in excellent agreement with the CCD images, showing that the decomposition within each mode family is also accurate [Fig. 4(d) and Fig. 4 (Media 2)]. In principle, fibers with higher order mode content could be measured using this technique provided suitable basis functions are used, as discussed in Sec. 3.

5. Fiber characterization

Typically, a single mode optical fiber will convert an input linear polarization to an elliptical polarization due to intrinsic and strain-induced birefringence. For single mode fiber, the birefringence can be compensated for with inline polarization controllers or, for free-space coupling, waveplates at the entrance or output. For fibers supporting higher order modes, mode conversion also occurs, and has been used to generate pure CVBs from an input superposition of modes [5, 31]. Characterization of the scattering matrix of highly multimode fibers has also been of interest for multimode fiber imaging applications [34], but those applications have only considered scalar fields.

Here, we show how IDIOM can be used to determine a fiber scattering matrix, S, that transforms an input superposition of coefficients, C, to an output one, D.

D = S C

In a fiber supporting two fundamental modes and four higher order modes, S is a 6 x 6 matrix. Although it is possible for a Gaussian input beam to couple to the LP₁₁ modes by launching the light off-axis [23], the mode spacing between these families is large enough that interfamily mixing in the fiber, and matrix terms that connect different families, are small. Because of this, and because for many applications the desired output is in a specific mode family, we will treat S as a 6 x 6 matrix that only has intrafamily mixing:

(\begin{matrix} S_{11} & S_{12} & 0 & 0 & 0 & 0 \\ S_{21} & S_{22} & 0 & 0 & 0 & 0 \\ 0 & 0 & S_{33} & S_{34} & S_{35} & S_{36} \\ 0 & 0 & S_{43} & S_{44} & S_{45} & S_{46} \\ 0 & 0 & S_{53} & S_{54} & S_{55} & S_{56} \\ 0 & 0 & S_{63} & S_{64} & S_{65} & S_{66} \end{matrix}) (\begin{matrix} c_{1} \\ c_{2} \\ c_{3} \\ c_{4} \\ c_{5} \\ c_{6} \end{matrix}) = (\begin{matrix} D_{1} \\ D_{2} \\ D_{3} \\ D_{4} \\ D_{5} \\ D_{6} \end{matrix}) e^{i ϕ}

Here, c_1-6 are the complex coefficients for

L P_{01}^{x}

,

L P_{01}^{y}

,

L P_{11}^{e x}

,

L P_{11}^{o x}

,

L P_{11}^{e y}

, and

L P_{11}^{o y}

, respectively (the order of Eqs. (2). In our experimental setup, the input beam to the fiber is as shown in Fig. 3 so that c₄ and c₆ are zero, and the columns S_j4 and S_j6 are not interrogated. If the phase plate is centered, c₁ and c₂ are also 0, but for generality we will consider the case of a decentered phase plate so that c₁ and c₂ contribute. Thus, the 2x2 submatrix block in S, and the columns S_j3 and S_j5 comprise 12 matrix unknowns. The global phase, ϕ, does not affect the observed intensity profiles and is usually absorbed into D, but each waveplate orientation leads to a different phase imparted by the fiber that is lost if it is absorbed, so we keep it as an unknown parameter. In general, c₄ and c₆ may not be zero, but the approach should be readily extendable to those situations.

To determine the 12 matrix elements, we make 3 measurements, M1, M2, and M3, with the following input vectors C^M1 = ( $c_{1}^{M 1}$ , 0, $c_{3}^{M 1}$ , 0, 0, 0); C^M2 = (0, $c_{2}^{M 2}$ , 0, 0, $c_{5}^{M 2}$ , 0); and C^M3 = ( $c_{1}^{M 3}$ , $c_{2}^{M 3}$ , $c_{3}^{M 3}$ , 0, $c_{5}^{M 3}$ , 0). We generate and determine these three input states as in Sec. 4.1 with half- and quarter-wave plates, giving input horizontal, vertical, and circular polarizations. The first input state, C^M1, readily determines columns 1 and 3 (6 unknowns), as we may choose ϕ^M1 = 0:

S_{11} = \frac{D_{1}^{M 1}}{C_{1}^{M 1}}; S_{21} = \frac{D_{2}^{M 1}}{C_{1}^{M 1}}; S_{j 3} = \frac{D_{j}^{M 1}}{C_{3}^{M 1}}

for j = 3-6. While it might appear that an additional pure state, C^M2, can determine the remaining 6 unknowns in columns 2 and 5, each measurement generates a different, unknown ϕ, requiring the third measurement. For C^M2, we have:

S_{12} = \frac{D_{1}^{M 2}}{C_{2}^{M 2}} exp (i ϕ_{M 2}); S_{22} = \frac{D_{2}^{M 2}}{C_{2}^{M 2}} exp (i ϕ_{M 2}); S_{j 5} = \frac{D_{j}^{M 2}}{C_{5}^{M 2}} exp (i ϕ_{M 2})

Any two of the six similar relationships given by the third measurement, C^M3, can be used to eliminate exp(iϕ_M3) and solve for exp(iϕ_M2), after some algebra:

e^{i ϕ_{M 2}} = \frac{C_{3}^{M 2} C_{1}^{M 3} [S_{11} D_{2}^{M 2} - S_{21} D_{1}^{M 3}]}{C_{3}^{M 3} [D_{1}^{M 3} D_{2}^{M 2} - D_{2}^{M 3} D_{1}^{M 2}]}

All relevant elements of S are thus determined for our experimental setup.

Since S has been fully determined based on the three measurements, it should be able to predict the outputs for any waveplate combinations. To test this, in Fig. 5, we show the results of this procedure applied to Corning HI1060 fiber for the two-lobed input shown in Fig. 3. To assess the accuracy of the above procedure, we first measure the output for 169 different quarter- and half-wave plate orientations as in Sec. 4.1 and compare with the predicted outputs using S. As is shown in Fig. 5(a), the agreement between the measured and predicted coefficients is good. The images in Fig. 5(b), and Fig. 5 (Media 3), comparing the measured images and those determined by S also highlight the good agreement.

Fig. 5 (a) Measured (left) and predicted (right) mode powers and phases of the LP₁₁ output modes of the fiber over all orientations of the input half- and quarter-wave-plates. CCD images (left) and predicted (right) profiles calculated by S (Media 3, 0.8 MB).

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We cannot generate all possible combinations of output modes solely with bulk optics because we are incident on the fiber with a two-lobed profile that is approximated by superpositions of c₃ and c₅ (with no contribution from c₄ and c₆), and because bulk waveplates affect the entire beam uniformly. This is apparent from Fig. 5(a) – there is no orientation of the waveplates for which only one of c₃ – c₆ is generated with high power. Because we have not generated the entire space of coefficients on the input, we cannot generate the entire space on the output either with bulk optics alone.

The retrieved matrix only applies to a particular configuration of the fiber. If additional output modes are desired, strain-induced birefringence [5, 31] or other orientations of the phase plate may be used to generate a new matrix for additional input superpositions. Spatial light modulators could also be used to create different input superpositions than are available with bulk optics, in which case the S-matrix does not need to be recalculated [20]. In Fig. 6, we show two different output coefficient matrices for two different amounts of strain. While it is not clear if all combinations of output mode superpositions are possible, we show in the bottom panel of Fig. 6 that we can generate any basis mode with >90% purity with suitable strain; each image in the bottom of Fig. 6 used a different amount of strain. Additionally, we have demonstrated this purity whether in the LP basis or in the vector basis. Once the beam exits the fiber, intrafamily conversion between some modes is easily performed with bulk optics. For example, since two half-wave-plates with relative orientation of θ will rotate an arbitrary linear polarization by 2θ, one can convert between a purely radially-polarized beam to a purely azimuthally-polarized beam with standard waveplates. However, in situations where a particular mode is sought inside the fiber [9, 10], this option is not available.

Fig. 6 (top) Two different outputs of the fiber using strain-induced birefringence. Bottom: Pure modes in both the LP and vector bases can be generated with strain-induced birefringence. The purity of each of the modes is above 90%. Profiles can be compared with Fig. 1.

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6. Sensitivity

Because the signal beam is homodyned with a reference field to form the interference, IDIOM is linear in the field and has excellent sensitivity; as seen in Eq. (6), the signal field is multiplied by the reference field, which can be much stronger than the signal field. A detailed analysis of the effect of noise is outside the scope of this paper, but we have observed excellent results with noisy data. To show this, we reduce the power of the signal beam by 4 orders of magnitude so that the peak SNR on the signal arm is << 1, where we define the peak SNR as the ratio of the peak intensity of the beam to the root-mean-square fluctuations. During these measurements, we keep the exposure duration, signal beam profile, and reference beam power constant. Images of the recorded interferograms and signals are shown in Fig. 7(a), along with the recovered images. At the lowest power in Fig. 7, the SNR on the signal arm is ~0.1, as estimated by knowing the SNR at full power. The coefficients are shown in Fig. 7(b) and Fig. 7(c). The relative power in each mode stays the same indicating good agreement. Even for the fundamental modes, which have ~1% of the total power, the recovered coefficients are consistent; the effective SNR for those modes is ~10⁻³ for the lowest power. The relative phases of the 4 higher order modes are consistent within 0.2 rad. Such high sensitivity could be especially useful in systems with high loss or technical noise.

Fig. 7 (a) Interferograms, signal images, and reconstructed images for four different beam powers over 4 orders of magnitude. Color contrast is adjusted so that the interferograms are observed. Slight fringes on some signal images are due to other optical components but are filtered out in processing. The signal image on the bottom row has a peak SNR of ~0.1. At right are the recovered mode powers (top) and phases (bottom) for the 6 modes at 6 different powers spanning 4 orders of magnitude. The normalization is chosen so that c3 is 1.0 at full power so that the coefficients can be easily compared. In decreasing order, the normalized beam powers are 1.0 (black), 0.138 (red), 0.041 (green), 0.010 (blue), 0.0017 (dashed), and 0.00021 (magenta). The lowest power in (b) has SNR ~0.1 on the signal arm. We chose c₅ as the reference mode, so its phase is always 0.

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7. Conclusion

We have demonstrated a powerful technique for rapid, unambiguous determination of the electric field and complex modal content in vector beams. Interferometric decomposition into optical modes (IDIOM) is demonstrated with free-space beams and beams exiting optical fibers that support up to 10 vector modes, though it may be applied generally to any beam with suitable basis. By making measurements of the fiber output with three different input superpositions, we form the fiber scattering matrix that transforms an input superposition into the output superposition. IDIOM is very sensitive, capable of determining modal content even when the signal beam has a SNR << 1. We acknowledge helpful discussions with Jonathan Hoffman and technical assistance during data collection from Eunkeu Oh. This work was funded by the Office of Naval Research and by the Defense Advanced Research Projects Agency.

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Rapid complex mode decomposition of vector beams by common path interferometry

Abstract

1. Introduction

2. Background

3. Algorithm

3.1 Background

3.2 Decomposition

4. Results

4.1. Initial test

4.2 Fiber modes

5. Fiber characterization

6. Sensitivity

7. Conclusion

References and links

Supplementary Material (3)

Cited By

Figures (7)

Equations (12)

Optics Express