Parallel Polarization State Generation

The control of polarization, an essential property of light, is of wide scientific and technological interest. The general problem of generating arbitrary time-varying states of polarization (SOP) has always been mathematically formulated by a series of linear transformations, i.e. a product of matrices, imposing a serial architecture. Here we show a parallel architecture described by a sum of matrices. The theory is experimentally demonstrated by modulating spatially-separated polarization components of a laser using a digital micromirror device that are subsequently beam combined. This method greatly expands the parameter space for engineering devices that control polarization. Consequently, performance characteristics, such as speed, stability, and spectral range, are entirely dictated by the technologies of optical intensity modulation, including absorption, reflection, emission, and scattering. This opens up important prospects for polarization state generation (PSG) with unique performance characteristics with applications in spectroscopic ellipsometry, spectropolarimetry, communications, imaging, and security.

( ) (S1) The complex amplitude coefficients of equation (S1) are known as the Jones vector: E 0 x e iδ x , E 0 y e iδ y ( ) . The Stokes parameters, which were defined by G. Stokes in 1852 to mathematically describe polarized light, including partial polarization 1 , are extremely useful today but were historically hampered by the inability to quantify optical intensity measurements. To derive the Stokes parameters, the equation of the polarization ellipse 2 , can be rearranged such that the grouped terms of (E 0 x 2 + E 0 y 2 ) 2 − (E 0 x 2 − E 0 y 2 ) 2 − (2E 0 x E 0 y cosδ) 2 = (2E 0 x E 0 y sinδ) 2 can be written as the following Stokes parameters: S 0 is the total intensity, and S 1 , S 2 , S 3 ( ) is the Stokes vector describing the SOP. The Stokes parameters simplify measurement of SOP enormously by requiring only 4 intensity measurements. A triangle inequality exists, in which the total intensity S 0 2 ≥ S 1 2 + S 2 2 + S 3 2 . The ratio of the length of the Stokes vector to the total intensity is the degree of . The Stokes vector that is normalized to a unit vector traces all possible SOPs on a mathematical object called the Poincaré sphere.

Regions of coverage of Poincaré Sphere using degenerate SBVs
The degenerate SBVs take on any four of the six degenerate polarizations, which are the linear horizontal, linear vertical, linear +45°, linear -45°, right circular, and left circulation polarizations. We explored a system of SBVs in the four following polarizations: linear horizontal, linear vertical, linear +45°, and right circular. Coverage of the Poincaré sphere by possible SOP states using the above system is shown in Fig. 2a,c. Fig. S1 plots the Mercator projection of Fig. 2c on the Poincaré sphere. Each system is uniquely defined by the SOPs and the global phases ϕ n of the SBVs that comprise it. In this particular system, all SBVs have ϕ=0°, except the SBV with linear +45° SOP, for which ϕ=180°. Figure S1: Degenerate SBV coverage. The Poincaré sphere is shown, covered by possible SOPs, as generated by linear combinations of four degenerate SBVs, in the following polarizations: linear horizontal, linear vertical, linear +45°, and right circular. All SBVs have global phases ϕ=0°, except that of the linear +45° polarization with ϕ=180°. This system is identical to that of Fig. 2a,c.

Discussion of metric tensor
The uniformity of the system can be described by the metric on the Poincaré sphere, where angular separation between SOPs in Stokes space is given by By describing nearby states as E i = C ij α j and E i = C ij α j + δα j ( ) , it is possible to construct a metric tensor, where For the case of coherent combination, this is the Fubini-Study metric, and for incoherent combination this is the Bures metric.

S4 Coherent versus incoherent combination
As described in the main text, coherent combination ( Fig. S2) produces polarization trajectories that are sensitive to the difference in global phases ϕ n between SBVs (Fig. S3), whereas incoherent combination necessitates a geodesic trajectory that is insensitive to ϕ n (Fig. S4). It is possible to switch between these two combination methods to generate trajectories with degrees of coherence intermediate between the two limits by changing the mutual coherence between SBVs. Mutual coherence, or the degree of cross-correlation, can be tuned by varying the optical path length between SBVs. By having the optical length exceed the coherence length of the light source, the SBVs no longer have a fixed phase relation and incoherently combine. Fig. S4 further illustrates the difference in polarization trajectories as generated by the two methods of combination.
The SOP measured following incoherent combination can be described as a linear sum of Stokes vectors: ( ) , in which S C 1 ( ) is the Stokes vector corresponding to the SOP of C 1 , and so on. This can be seen as the simultaneous detection of the intensities of non-interfering beams with their unaltered SOPs; hence the intensities of Stokes vectors add linearly on the detector side. The incoherent trajectory is insensitive to the relative phase difference between SBVs. Finally, the degree of polarization of the generated SOP as measured by a polarimeter has two contributors: (a) any unpolarized background originating from the unpolarized parts of the sources' signals and (b) any less than unity value of the degree of cross-correlation (mutual coherence) between combined SBVs.   This example system is implemented using the experimental setup described in the main text. A polarization trajectory is generated by coherent combination by keeping the optical path length between the two SBVs C2 and C3 well below the coherence length of the laser (~20 cm). The trajectory is measured by the polarimeter and the data are shown. c) An incoherent polarization trajectory following the geodesic path between SBVs C2 and C4 is generated by making the optical path length between SBVs much longer than the coherence length of the laser (hence reducing the mutual coherence).

Comparison of the performance of our experiment with a commercial PSG
Performance characteristics of our implementation are promising, with an SOP settling time (representing speed and stability) of 3.5 µs compared to a state-of-the-art device (Thorlabs DPC5500) with 150 µs for < 10° deviation and 1 ms for < 1° deviation. However the SOP accuracy of our embodiment (5.9° error) is limited by the unstable relative phase between the four SBVs, whereas the DPC5500 can be as accurate as 0.25°. However, there is room for major improvements, in terms of both speed and accuracy, such as by using faster modulators and miniaturization; the latter would greatly increase the phase stability between SBVs and reduce the error. Realistically, we expect PSGs in the visible and telecom wavelengths, for example, to achieve the speeds of the fastest modulators available, e.g. greater than 40 GHz (lithium niobate), pushing PSG technology from the kiloradians/second regime into the gigaradians/second.

S6 Insertion loss calculation
The fundamental limitation to the efficiency of the parallel architecture stems from inefficient beam combining. The theoretical coherent beam combining efficiency for a system with N ports has been derived as the following 3 : where P m is the power and φ m the phase of the mth beam. This assumes perfect alignment and coherence, so, in practice, there will be additional losses. Theoretical insertion loss (IL) was calculated using equation (S3), where IL = −10 log(η) , for a large number of SOPs which represented uniform coverage of the Poincaré sphere. Descriptive statistics are shown in Table S1. A PSG constructed from a set of 4 degenerate SBVs has a greater range of insertion loss compared to that of regular tetrahedral SBVs ( Fig. S5 and see standard deviation values in Table S1). This can be explained by the increased uniformity of the regular tetrahedral SBVs, which are well-separated SOPs in state space, in contrast to the degenerate SBVs, implying that on average each generable SOP will be much closer to one single SBV than the other three, leading to a large yet frequent power imbalance between the beams when combining. With the set of 4 degenerate SBVs, there is greater variability in the state space distance from each generable SOP to the SBVs.