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We demonstrate a technique for generating azimuthally and radially polarized beams using a nematic liquid crystal spatial light modulator and a π phase step. The technique is similar in concept to prior techniques that interfere TEM01 and TEM10 laser modes, but the presented technique removes the requirement of interferometric stability. We calculate an overlap integral of >0.96 with >70% efficiency from an input Gaussian mode. The technique can easily switch between beams with azimuthal and radial polarization.
©2010 Optical Society of America
1. Introduction
In recent years considerable attention has been devoted to light beams with radially and azimuthally polarized fields (RPFs and APFs) due to their unusual properties and potential applications. For example, focusing RPFs with a high numerical aperture lens produces a strong longitudinal field component in the focus that is smaller than the diffraction limit for the linearly polarized light [1–6]. Some of the applications of these beams include optical guiding and trapping [7,8], metal cutting [9,10], determination of the orientation of single molecules [11,12], optimizing photon collection of molecular-based single-photon sources [13,14], charged-particle acceleration [15], and scanning optical microscopy [16,17].
Numerous techniques have been employed to generate these beams externally to the laser cavity, having different degrees of stability, complexity, efficiency, and cost. Techniques with segmented waveplates [18], spatially-varying liquid crystal plates [19], and custom optical fibers [20,21] can efficiently generate these fields, but are not commercially available or are not versatile. Standard multimode optical fibers can produce vector beams, but lack intrinsic stability and are inefficient unless the input beam is already preshaped. Sub-wavelength gratings [22–24] work well in the infrared, but have not been demonstrated for visible light. Ferroelectric liquid crystal spatial light modulators (FLC SLMs) can produce arbitrary polarization states only with low efficiency [25]. Efficient and arbitrary polarization modification can be done with a pair of nematic liquid crystal SLMs [26,27], but have the highest cost.
One of the earliest and simplest techniques combines orthogonal TEM01 and TEM10 laser modes to create a beam with azimuthal symmetry [28,29], commonly called the TEM01* doughnut mode. In those works, the TEM10 and TEM01 modes are generated approximately in separate arms of a Michelson interferometer by inserting π - phase steps into the paths of the Gaussian beams in each path. While this technique is efficient and uses relatively inexpensive, commercially available components, a significant drawback is the requirement of interferometric stability.
In this work, we present a stabilized version of the interference of modes technique using polarization multiplexing on a nematic SLM and a π-phase step. Our approach eliminates the requirement on the phase stability by effectively combining two interferometer arms into a single path. While this technique cannot produce truly arbitrary 2D polarization encoding, it allows easy reconfiguration between radially, azimuthally, or hybrid vector beams. Our analysis shows that the mode purity and mode purification through spatial filtering can achieve beams with excellent cylindrical symmetry and over 98% TEM01* mode purity.
2. Experimental setup
The schematic of our technique is depicted in the Fig. 1 . To assist with the visualization, we show in Fig. 2 the relative phases of the light for x- and y-polarizations in four quadrants corresponding to points A, B, and D of the Fig. 1, and the phase function of the SLM (Boulder Nonlinear Systems Model P512) at point C. Initially our technique requires that horizontally and vertically polarized light components are equal, so a half-wave plate is used to orient the linear polarization of the 780 nm, TEM00 laser light to 45 degrees as shown in Fig. 1. Figure 2(a) shows the identical phases, which we chose as 0 in all four quadrants for both polarizations. The black arrows in Fig. 2 depict the phase of each polarization component in each quadrant. Also shown is the resultant polarization (third column). Figure 2(b) shows the phases after passing through the π - phase step operating on the left (x < 0) portion of the beam (see Fig. 1). The phase step affects both polarizations equally. The π - phase step was made by depositing a 390nm-thick layer of SiN on half of a quartz wafer. By adjusting the angle of the phase step plate we could use it with different optical wavelengths. Next we change the phases for the y-polarization only using the SLM as shown in the Fig. 2(c). The combined operation results in the quadrant approximation to the radial polarization field shown in the third column of Fig. 2(d). Mathematically, the generated field is
where C 0 = is a normalization constant, and the waist, ω 0, is the 1/e2 radius of the Gaussian (TEM00) input beam. This is an approximation to the desired RPF formed by the superposition of TEM01 and TEM10 modes:where C 1 = is the normalization constant, θ is the azimuthal coordinate, and ω 1 is the waist parameter. The lenses L1 and L2 in Fig. 1 are used to relay and register the image of the phase step onto the SLM; in principle a π/2 phase step could be used directly in front of the reflective SLM, and these lenses could be removed. The lens L3 and the CCD camera can both be moved relative to L2 either to image the SLM or the focal plane of L2. With our choice of the phase step orientation and the SLM phases shown in Fig. 2(c) we create the condition needed for RPF generation similar to that of the interference of modes approach. The main advantage of our technique is realized by keeping both polarizations in the same optical beam, which eliminates interferometric stability requirements. Overall measured efficiency here is only 50% due to uncoated lenses and imperfect SLM reflectivity.
Fig. 2 Parameters required for producing radially polarized beams. Row letters (A - D) correspond to the positions in Fig. 1. Columns depict x-, y-, and combined polarizations at those points.
To form an APF, both the SLM phase pattern and the π - phase step are rotated by 90 degrees. Figure 3 shows the phases needed to produce an APF. In analogy with the RPF:
where C 0 = . In addition to the APF beam described by Eq. (3), other modes can also be produced using this technique by the appropriate phase choices in the four quadrants of the SLM. For example, to form the HE21 cylindrical waveguide mode, which alternates between radial and azimuthal polarization, either the SLM phase pattern or the π - phase step is rotated by 90 degrees. We note that one can also convert between radial, azimuthal, and hybrid modes simply by adding two standard half waveplates.Fig. 3 Parameters required for producing azimuthally polarized beams. Row letters (A - D) correspond to the positions in Fig. 1. Columns depict x-, y-, and combined polarizations at those points.
We note that a ferroelectric liquid crystal SLM (FLC SLM) may be used in place of the nematic SLM to generate these approximate polarization profiles with one caveat. In FLC SLMs, each pixel acts as an orientable half-wave-plate. For most materials used in FLC SLMs, the maximum polarization rotation is ~90 degrees, so that the π - phase step used here is still required in order to have 180 degree polarization rotations. However, to DC balance the FLC SLM, the waveplate orientation of each pixel must oscillate on frequency scales of ~1 Hz or faster, such that the desired polarization field is only present with 50% duty cycle. For some applications this duty cycle may still be acceptable.
3. Results and theory
The resulting images in the focus of L3 with and without a polarizer present are shown in Fig. 4(a) , 4(b), along with calculated images. The arrows in the images show polarizer transmission axis. The images lack cylindrical symmetry because, like some of the techniques presented in the introduction, we are not combining pure TEM10 and TEM01 modes. Mode purity is calculated through the overlap integral of these target modes with our approximate fields ERPF or EAPF. By symmetry, we consider only the overlap integral of x-polarization components of Eq. (2) and Eq. (1) (the overlap integral for the APF will be identical). Specifically
where we have used the symmetry of this integral in x (x sgn(x) = |x|) to integrate over θ from –π/2 to π/2. The purity of the TEM10 mode is determined by maximizing Eq. (4) with respect to ω 1. A straightforward calculation shows that this is achieved when , at which value the overlap integral is = 0.87, or a mode purity of = 0.75.Fig. 4 Experimental (top row) and calculated (bottom row) RPF images at the focus of lens L3. When the polarizer is present, its direction is shown by the arrows.
To produce purer TEM01 modes efficiently we have to remove higher order modes selectively. To achieve this, we spatially filter the beam using a pinhole [30]. Because the π-phase step strictly eliminates the TEM00 mode contribution, and the TEM01 mode is the first higher order mode, this traditional spatial filtering is very effective at isolating the TEM01 modes. Figure 5 shows the setup used to spatially filter the beam shown in the Fig. 4(a). After the pinhole we also used an aperture to reject the diffraction rings and re-imaged the pinhole with a triplet lens L4. Experimentally, we also adjusted the aperture size in Fig. 1 to control the size of the focused beam on the pinhole to improve the final beam quality. The resultant image is shown together with the calculated image in Fig. 5. Comparing Figs. 4 and 5 it is clear that the quality of the mode in the experiment is also significantly improved. In Fig. 5, we also show an image of the beam interfered with a horizontally polarized TEM00 beam. In this case, the generated beam was an APF, so interference contrast is maximized at the top and bottom of the beam, where the polarization is purely horizontal, and is zero on the left and right sides, where polarization is vertical. Furthermore, the interferogram explicitly shows the π phase shift between the top and bottom halves of the beam as a lateral shift in the interference pattern by half a fringe period.
Fig. 5 Schematic of the setup used to spatially filter the beam shown in the Fig. 4(a). The experimental and calculated final beams are shown. Also shown is the interferogram of the APF beam with the horizontally polarized TEM00 beam, and horizontally polarized APF beam before and after the pinhole.
The TEM01 mode purity is increased by spatial filtering at the expense of power throughput. In Fig. 6 we show overlap integral and the pinhole throughput as a function of the pinhole diameter. The curves in this figure were calculated with parameters approximating experimental conditions. The solid red curve shows the overlap integral, which achieves a maximum value near 300 microns. The measured mode purities before and after the pinhole, given by the square of the overlap integral of the measured APF beam with the perfect calculated mode, are 0.85 and 0.97, respectively.
Fig. 6 Calculated overlap integral and the pinhole throughput as a function of the pinhole size for optimizing the TEM01 mode. The unmodified TEM00 waist diameter was 130 microns.
Although making the pinhole smaller generates a beam with better cylindrical symmetry, the overlap integral is reduced due to the extra diffraction rings that are generated. The aperture in Fig. 5 eliminates these and, therefore, the overlap integral with the “useful” portion of the beam continues to increase with smaller pinhole size (dashed red line). For scale we point out that the beam waist diameter for an unmodified TEM00 beam propagating through the system (i.e. no phase step or SLM present) is ~130 microns, so that the pinhole size is approximately 2.3 times larger than the beam at the peak of the solid red curve. Also shown in Fig. 6 is the relative power transmitted through the pinhole (dashed blue line). In practice, overall power throughput is also reduced by the reflection efficiency of SLM, which in our case was 60% at 780 nm.
4. Conclusion
In summary we demonstrated an improved interference of modes technique that does not require interferometric stability, is reconfigurable and could be used at different wavelengths. We also demonstrated a technique that can be used to improve the mode quality by stripping away higher order modes. Experimental results agree well with the calculations.
Acknowledgments
This work was supported by the Office of Naval Research.
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