We have used common path interferometry for rapid determination of the electric field and complex modal content of vector beams, which have spatially-varying polarization. We combine a reference beam with a signal beam prior to a polarization beam splitter for stable interferograms that preserve intermodal phase shifts even in noisy environments. Interferometric decomposition into optical modes (IDIOM) provides a direct, sensitive measure of the complete electric field, enabling rapid modal decomposition and is ideally suited to single-frequency laser sources. We apply the technique to beams exiting optical fibers that support up to 10 modes. We also use the technique to characterize the fibers by determining a scattering matrix that transforms an input superposition of modes into an output superposition. Furthermore, because interferograms are linear in the field, this technique is very sensitive and can accurately reconstruct beams with signal-to-noise << 1.
© 2013 Optical Society of America
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 , dark-field imaging , optical trapping , atomic physics , and entanglement of spatial and polarization degrees of freedom . 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 . 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  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 . Decomposition techniques have been successful at studying high-speed nonlinear dynamics  and optimizing fiber coupling . 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 , 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 . 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  may be analyzed even if mode decomposition is not required.
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 as9, 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.
The first 10 LP basis functions, f1(r, θ) to f10(r, θ), respectively, are: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.
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  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, Eref, interferes with the unknown signal field, Esig, which we wish to write in terms of the basis functions in Eqs. (2), to form a total field Etot, as shown in Fig. 2(a). Eref is tilted by a small wavevector ktilt with respect to the propagation direction of Esig to impart phase-sensitive interference fringes A charge-coupled device (CCD) camera detects the corresponding intensities, Iref, Isig, and Itot:Eq. (6), we find32]. 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 . Choosing the wrong sign of ktilt leads to complex conjugation of Esig, 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 Eref or Esig.
With one Fourier component rejected, the inverse Fourier transform immediately gives . ktilt determines unambiguously. In our setup, we find ktilt 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 Eref through the intensity measurement Iref 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, is recovered. Any uncertainty in the choice of sign of ktilt 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 Esig, as well as the relative phase between them. Figure 2(b) shows our modification to the scalar setup for vector modes. To determine Esig, we measure,, and for each polarization, where the superscripts indicate polarization direction. The reference intensities 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. (blue) is combined with the lower of these two beams so that both and 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 φx (phase between and) and φy (phase between and ) 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. and need to be subtracted from the combined as in Eq. (6). To accurately do this, the vertically-displaced and 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-ktilt) and g(k + ktilt).
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.
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® . 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 .
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.
4.1. 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.
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.
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  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 , 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.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: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 Sj4 and Sj6 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 Sj3 and Sj5 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 CM1 = (, 0, , 0, 0, 0); CM2 = (0, , 0, 0, , 0); and CM3 = (,,, 0, , 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, CM1, readily determines columns 1 and 3 (6 unknowns), as we may choose ϕM1 = 0:
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.
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 . 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.
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.
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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