Parallel Polarization State Generation

Abstract

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.

Figure 1. Concept.

(a) An illustration showing the general, modular implementation of the described method for a parallel polarization state generator (PSG). An input beam is (i) split into four beams of different polarizations, which are then (ii) intensity modulated either in reflection or transmission, (iii) and finally combined to form a single output beam, the polarization and phase of which can be tuned with a precision and speed limited by the modulator. (b) A schematic of PSG architecture is shown, in which modulators are placed after light sources A_i with well-defined states of polarization (SOP) and relative phase, and their weighted linear superposition produces the desired output signal. (c) Generation of horizontally polarized light using this method is illustrated. The electric fields of four propagating electromagnetic waves (red, green, blue, and yellow) with elliptical polarizations are superimposed and plotted as function of wave propagation position. They are intensity modulated and beam combined to generate the desired horizontal polarization (black).

Figure 2. Simulations.

Two systems of Stokes basis vectors (SBVs) were simulated– one with degenerate SOPs and another with SOPs mapped to a regular tetrahedron on the Poincaré sphere. (a) SBVs of four degenerate SOPs: linear horizontal (C1), vertical (C2), +45° with a 180° phase shift (C3), and right circular polarization (C4), are shown. A Monte Carlo simulation (blue points) by randomly varying the intensity modulation parameters showed complete, yet non-uniform coverage of SOPs over the Poincaré sphere. Polarization trajectories between C3 and C4 are shown for coherent (blue line) and incoherent combination (red line). Incoherent trajectories are geodesics. (b) SBVs optimized for uniformity of SOP coverage are shown, corresponding to vertices of a regular tetrahedron inscribed in the Poincaré sphere. In Jones notation, the SBVs here were [0.7071, 0.7071i], [−9.856, 0.1691i], [0.5141, 0.7941 −0.3242i], and [0.5141, −0.7941 −0.3242i], labeled C1-4, respectively. (c) The degenerate system of (a) is mapped using a Mercator projection of the Poincaré sphere, where θ is the polar angle and ϕ is the azimuthal angle. All coherent and incoherent trajectories are shown in black dotted and red solid lines, respectively. The coherent trajectories connected to C1 are warped by increasing the relative phase difference between C1 and other SBVs by 6° (blue dotted lines). The colored regions show the regions of SOPs enabled by combining sets of three SBVs: combining C1, C2, and C4 with varying intensities generates SOPs in the blue region; similarly, (C1, C3, C4) and (C2, C3, C4) generate the red and green regions, respectively. However, (C1, C2, C3) generate a region of no area because these SBVs are not linearly independent in this system. (d) The Mercator projection of the regular tetrahedron system of (b), where coherent and incoherent trajectories between SBVs are shown with black and red dotted lines, respectively. In this case, SOP regions generated have similar size and great overlap, yielding better overall uniformity. Due to overlap between regions, they are color-coded and labeled as the following: C1, C2, C3 combine to cover regions (a–c); similarly: C1, C2, C4 (a,d,e); C1, C3, C4 (c,e,f); and C2, C3, C4 (b,d,f).

Figure 3. Experimental setup.

Light from a HeNe laser is prepared in the linear +45° polarization using a wire-grid polarizer. The beam is then split into two beams by a non-polarizing beam splitter (BS). Each of these beams is split again using variable circular polarizers (VCPs) into two elliptical polarization states. The resultant SOP of the four beams is tuned by rotating the quarter wave plate embedded in the VCPs. Variable neutral-density filters (VNDFs) are placed directly after the VCPs to balance the four beam intensities. The four beams are then directed onto four quadrants of the surface of a computer controlled Texas Instruments DLP3000 digital micromirror device (DMD). The DMD is composed of an array of polarization-insensitive mirrors that can be switched in one of two positions. Mirrors that point in the direction of the output beam contribute to the total intensity and all other light is directed into a beam dump. The DMD behaves as a 2-D diffraction grating for the incident laser light. An iris is used to select the strongest diffraction order. The path length differences of the four intensity-modulated beams passing through the iris are adjusted to be less than the coherence length of the laser (<20 cm) with a series of mirrors. They are combined using three non-polarizing beam splitters to form a single beam. Finally, this beam is passed through a 100-μm pinhole, in order to select a small uniform portion of the wavefront of the combined beam to maximize the degree of polarization, to form the PSG output.

Figure 4. Experimental results.

(a) Data from the experimental setup of Fig. 3. The Stokes basis vectors (SBVs) are set to SOPs approximating (within the error of tuning the variable circular polarizers) a regular tetrahedron on the Poincaré sphere. The SBVs C1, C2, C3, and C4 were measured and the resulting tetrahedron is drawn. Coherent polarization trajectories from each SBV to every other SBV are generated by modulating SBV intensities in 20 discrete increments spanning 20 seconds, and the raw data as measured by the polarimeter are shown. (b) The results of a Monte Carlo experiment, in which 200 random intensity modulation parameters α were used, are shown on the Poincaré sphere, indicating good uniformity of coverage of SOPs. (c) Time series data of a coherent polarization trajectory between two SBVs (C2 to C4) in (a) are compared to theoretical calculation (dotted line) and show good agreement. S₁, S₂, and S₃ are elements of the Stokes vector. (d) An eye pattern is generated for a polarization signal that switches between linear horizontal and vertical polarizations using the DLP3000. The data are shown for a pseudorandom bitstream modulated at 1 kHz. The inset is a larger view of the red rectangle and shows the measured settling time (eye rise and fall time) to be 3.5 μs following an exponential.