JTu5A.14
CLEO:2015 © OSA 2015 JTu5A.14.pdf
CLEO:2015 © OSA 2015
Measuring Orbital Angular Momentum of Light With a Single, Stationary Lens Samuel N. Alperin,1 Mark E. Siemens,1 Robert D. Niederriter,2 and Juliet T. Gopinath3 1
Department of Physics and Astronomy, University of Denver, 2112 East Wesley Avenue, Denver, Colorado 80208, USA 2 Department of Physics, University of Colorado, Boulder, Colorado 80309, USA 3 Department of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder, Colorado 80309, USA Corresponding author:
[email protected]
We demonstrate that average orbital angular momentum (OAM) can be measured with a simplified twist parameter measurement technique. This technique uses a stationary apparatus composed of only a cylindrical lens and a CCD. OCIS codes: (080.4865) Optical vortices, (140.3295) Laser beam characterization
1. Introduction Light with orbital angular momentum (OAM) is characterized by a helical phasefront, and has proven itself to be highly promising in multiple research areas including astronomical measurements of angular frequency [1], the design of optical tweezers [2], and ultra high bandwidth telecommunications [3]. Accurate, simple, and efficient measurements of OAM are critical for any future utilization of the technology outside of the lab; a measurement technique involving only a few stationary optics is therefore desirable. Current techniques to measure OAM do so by measuring the strengths of the individual component modes. This is executed with either several spatial light modulators [4], which are expensive and bulky, or with complicated arrays of dove prisms [5]. These techniques can reveal more detailed information about the beam, but the size, complexity, and fragility of the required setups makes them undesirable in some situations. In this paper, we show that the average OAM can be measured simply and accurately with a simplified twist measurement, which represents the amount of rotation of the beam axis in the focal plane. In the past, this parameter has been found in the context of a full M2 beam quality measurement, which involves recording a series of images with a moving CCD, and a cylindrical lens that rotates by 90° midprocedure. In contrast, the demonstrated technique uses a single, stationary cylindrical lens and a CCD. We connect our measurements with an analytical model for Laguerre Gaussian modes carrying OAM, clearly relating the twist parameter to OAM.
2. Experimental Methods We generated beams with orbital angular momentum by passing a collimated zero order Gaussian beam from a HeNe laser through a forked diffraction grating displayed on a programmable LCD panel, controlled as part of a disassembled projector. The images of Figure 1 show the results of interfering a Gaussian reference beam with beams carrying each of the first five Laguerre Gaussian modes, respectively. These images show that for each of the programmed LG modes, the number of phase rotations matches the expected value.
Figure 1: Experimental interference of Laguerre Gaussian Beam with controllable OAM with a Gaussian reference at 633 nm.
A beam carrying orbital angular momentum then passed through a 1 meter focal length cylindrical lens. A CCD was placed at the focus, as is shown in Fig. 2B. With this apparatus, a series of images was taken for OAM values l=0 to l=5. The choice of a cylindrical lens with a relatively long focal length leads to a large beam focus, allowing us to both increase the signal to noise ratio by having more pixels within the beam width and show that the appearance of the twisted modes imaged by the CCD match those in our models. The images (Fig. 2A) were then processed as 2D intensity arrays to find their xy covariances to enter into the equation for the twist parameter T:
JTu5A.14
CLEO:2015 © OSA 2015 JTu5A.14.pdf
T=
2fV xy d2
CLEO:2015 © OSA 2015
.
Using the relationship between the beam twist and OAM in units of ℏ per photon,
OAM = 2πT λ ,
we experimentally determined the OAM of each programmed OAM mode. The average value of the measured OAM is plotted as a function of the expected value (Fig. 2C). Figure 2A shows how the measured images compare to an analytical model for calculating the mode of an OAM beam at the focus of a cylindrical lens, via a 1dimensional Fourier transform. The measured images have integer numbers of dark slits equal to the OAM values, in accordance with the model.
Figure 2: Measuring “twist” to measure OAM. A) Comparison of measured and modeled laser modes at the focus of the cylindrical lens. B) Schematic optical setup. C) Experimental OAM measurement; error bars are the standard deviation of five measurements.
As is evident from the plot of the measured OAM as a function of the expected OAM (Fig. 2C), our measurement alone is highly linear for low order modes, but has a significant systematic error. This error is caused by a slight rotation of the cylindrical lens with respect to the yaxis of the CCD. This misalignment can be corrected by introducing a rotation correction factor based on a rotation of the analytical model and calculation of the resulting error in twist and OAM measurement, the result of which is indicated by the red data points in Fig. 2C. These corrected measurements are accurate for low order modes, which suggests that our experimental method is off only by a small rotation of the cylindrical lens, but also that accurate measurements can be taken without a good rotational calibration.
3. Conclusions The similarity between the modeled 1D Fourier transformed OAM beams and the images of the beams after the 1D optical transform of the cylindrical lens demonstrates the accuracy of this model. This means that OAM can be measured with a stationary lens and camera, and calibration of the OAM measurement can be performed without mechanical movement.
4. References [1] M. Lavery, F. Speirits, S. Barnett, M. Padgett, “Detection Of A Spinning Object Using Light's Orbital Angular Momentum,” Science 341, 537540 (2013). [2] H. He, M. Friese, N. Heckenberg, H. RubinszteinDunlop, “Direct Observation Of Transfer Of Angular Momentum To Absorptive Particles From A Laser Beam With A Phase Singularity,” Physical Review Letters 75, 826831 (1995). [3] Nenad Bozinovic, Yang Yue, Yongxiong Ren, Moshe Tur, Poul Kristensen, Hao Huang, Alan E. Willner, and Siddharth Ramachandran, “TerabitScale Orbital Angular Momentum Mode Division Multiplexing in Fibers,” Science 340, 15451548 (2013). [4] G.C.G. Berkhout, M.P.J. Lavery, M.J. Padgett, and M.W. Beijersbergen, “Measuring orbital angular momentum superpositions of light by mode transformation,” Optics Letters 36, 18631865 (2011). [5] J. Leach, M. Padgett, S. Barnett, S. FrankeArnold, J. Courtial, “Measuring The Orbital Angular Momentum Of A Single Photon,” Physical Review Letters 88, 25790114 (2002).
