March 15, 2014 / Vol. 39, No. 6 / OPTICS LETTERS
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Tunable orbital angular momentum mode filter based on optical geometric transformation Hao Huang,1,* Yongxiong Ren,1 Guodong Xie,1 Yan Yan,1 Yang Yue,1 Nisar Ahmed,1 Martin P. J. Lavery,2 Miles J. Padgett,2 Sam Dolinar,3 Moshe Tur,4 and Alan E. Willner1 1
Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089, USA 2
School of Physics and Astronomy, SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom 3 Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, California 91109, USA 4
School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel *Corresponding author: [email protected]
Received January 13, 2014; revised February 14, 2014; accepted February 15, 2014; posted February 18, 2014 (Doc. ID 204653); published March 14, 2014 We present a tunable mode filter for spatially multiplexed laser beams carrying orbital angular momentum (OAM). The filter comprises an optical geometric transformation-based OAM mode sorter and a spatial light modulator (SLM). The programmable SLM can selectively control the passing/blocking of each input OAM beam. We experimentally demonstrate tunable filtering of one or multiple OAM modes from four multiplexed input OAM modes with vortex charge of l −9, −4, 4, and 9. The measured output power suppression ratio of the propagated modes to the blocked modes exceeds 14.5 dB. © 2014 Optical Society of America OCIS codes: (120.2440) Filters; (330.6110) Spatial filtering; (090.2890) Holographic optical elements. http://dx.doi.org/10.1364/OL.39.001689
Laser beams carrying orbital angular momentum (OAM) have recently gained interest due to their potential beneficial use in a number of applications [1]. For example, one application would be to multiplex many different data channels, each of which is carried on a different OAM beam, thereby increasing system capacity and spectral efficiency [2–4]. In terms of capacity, this might represent another orthogonal dimension of system growth, which is similar to how wavelength-division multiplexing (WDM) has been utilized to increase capacity by assigning different data channels to different wavelengths. In general, the electromagnetic field of an OAM beam is identified by a phase term expressed as expilφ, where φ is the azimuthal angle in the transverse plane of the beam, and the integer l is the azimuthal index indicating the topologic charge of the beam [5]. Spatially co-located OAM beams with different charges are principally orthogonal to one another, allowing the efficient multiplexing and demultiplexing of multiple data channels. Recent reports have demonstrated a multiplexing/ demultiplexing of 32 OAM channels in free space and two OAM channels in a 1.1 km vortex fiber [6,7]. Similar to the field of WDM, where tunable filters are frequently used to select a certain wavelength channel or to remove out-of-band noise, a tunable mode filter could be useful in an OAM-multiplexed system, e.g., to extract the desired information carried on certain modes. Note that an OAM beam may be distorted after transmission, and the power of a single mode is then distributed to multiple OAM modes. In this case, a mode filter would be helpful to remove power of unwanted modes. Presently, various techniques have been proposed to manipulate wavelength channels in WDM or specific modes for non-OAM mode division multiplexing [8], while very few basic reconfigurable elements exist to enable OAM-based systems. Recently, an OAM mode sorter (i.e., demultiplexer) was demonstrated to convert the incoming beam’s OAM value into different output spatial positions [9,10], and a spatial add/drop multiplexer is 0146-9592/14/061689-04$15.00/0
described to extract a spatial mode from a light beam, leaving the other orthogonal modes undisturbed [11]. It might be interesting to further demonstrate a tunable optical filter that can selectively output one OAM mode from multiple-input multiplexed OAM modes and keep its OAM characteristics [12]. In this Letter, we present a tunable mode-passing/ blocking filter for spatially multiplexed OAM beams using a log-polar transformation-based mode sorter and a spatial light modulator (SLM). Multiple multiplexed OAM beams are mapped to different positions on the screen of the SLM and are reflected back to the sorter. The programmable SLM can selectively control the passing or blocking of each OAM beam. Mode filtering or blocking of one or multiple OAM modes from four OAM modes (l −9, −4, 4, and 9) is demonstrated. The performance of the filter including the beam quality, loss, and resolution are discussed. The concept of an OAM mode filter is shown in Fig. 1. Similar to the tunable filter in the wavelength domain, as shown in Fig. 1, an OAM mode filter is expected to be able to transmit (i.e., a “bandpass filter”) or block (i.e., a band-block filter) the selected OAM modes from
Fig. 1. Concept of a tunable OAM mode filter. In analogy with WDM, an OAM filter transmits or blocks OAM beam(s) of choice from multiple spatially multiplexed OAM beams. © 2014 Optical Society of America
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multiple spatially multiplexed OAM modes. After filtering, the output data channels are required to maintain their input OAM property. The principle of the OAM mode filter is based on double-passing through an OAM mode sorter. The mode sorter, as recently reported in [9,10], essentially comprises a log-polar transformer [13,14] and a convex lens (CL), the combination of which separates different OAM states into spatial positions. The log-polar transformer maps from the polar coordinates to the perpendicular coordinates. Specifically, it maps a position x; y in the input planepto a position u; v in the output plane, where u −a ln x2 y2 ∕b and v a tan−1 y∕x, a and b are scaling constants [9]. As a result, an OAM “ring” is unfolded and converted into a rectangular-shaped plane wave in the forward propagation. Accordingly, a rectangular-shaped beam with a tilted phase profile can be folded and reconverted to a ring-shaped OAM beam in the backward propagation, as indicated in Fig. 2(a). As a result of the transformation in the forward pass, the multiplexed OAM beams at the input plane, as described in Fig. 2(b) by a set of concentric rings in terms of the beam intensity, are mapped into a set of spatially overlapped rectangular-shaped plane waves. Note that each plane wave after the transformation has a tilt, whose value depends on the vortex charge of the corresponding input OAM beam. A lens can then focus these differently tilted plane waves into spatially distinct elongated spots at its focal plane, with an inter spot separation of λf ∕2πa, where λ is the wavelength, and f is the focal length of the lens. On the other hand, the beams can be reflected by a mirror, which is placed perpendicular to the propagating direction at the focal plane of the lens, and pass through the mode sorter again in the opposite direction, as shown in Fig. 2(c). Since the mirror is at the focal plane, the reflected beams with the elongated shape are collimated by the CL and are converted to rectangular plane waves with different tilts. Following that, the geometric transformer performs inverse transformation (from Cartesian to log-polar coordinates) and converts the tilt plane waves back into vortex beams with a ring-shaped intensity. Furthermore, if the mirror at the focal plane of the lens is replaced with a programmable mirror array, we can selectively control the passing or blocking of each OAM mode, and the tunable OAM mode-filtering function can be achieved. Figure 2(c) shows an example of a “band blocking” filter (blocking OAM beam with l l3 ). The schematic overview of the OAM filter setup is shown in Fig. 3. The light from a laser source is collimated and then launched onto a liquid crystal on a silicon-based diffractive SLM. The SLM is loaded with a designed phase hologram to generate multiple superimposed OAM modes. The generated OAM beams are sent to the OAM mode filter as the inputs. The mode transformer in the filter is achieved by using two reflective optical elements [10], each of which has an aperture size of 8 mm. A CL with a focal length of 1 m is placed right after the mode transformer to focus the beams. At the focus plane, we use a reflective SLM instead of the mirror array to reflect the beams. The SLM surface is divided into different regions, each of which encompasses the
Fig. 2. Principle of the OAM mode filter. (a) A log-polar geometrical transformation transforms an OAM beam to a rectangular-shaped plane wave and vice versa. (b) Multiplexed OAM beams are mapped to different positions at the focal plane of the convex lens (CL) after passing through the mode sorter. (c) The reflected beams (after filtering) are converted back to ring-shapes while backpropagating through the mode sorter.
spot corresponding to a specific OAM mode. Each region of the SLM can be programmed to either reflect the beam back to the lens or to diffract the beam away from the other beams, effectively blocking it. All reflected beams backpropagate through the optical system and are converted back into the desired selection of superimposed OAM beams. We use a beam splitter to separate the backward-propagating beam from the input beams. We note that the charge of the OAM beams after double passing through the mode sorter are inversed. Due to the additional reflection of the beam splitter, the final output beams are expected to have the same vortex charges as the input OAM beams. We first tested the OAM mode filter by sending a single OAM beam as the input. The filter is set to pass all the input beams. Therefore an OAM beam with the same vortex charge is expected to be obtained at the output port of the filter. Simulation results in Fig. 4 illustrate the
Fig. 3. Schematic overview of the OAM filter setup. SLM, spatial light modulator. BS, nonpolarization beam splitter.
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Fig. 4. Simulated beam profiles at each position of the setup. (a), (b), (c), and (d) correspond to ⓐ, ⓑ, ⓒ, and ⓓ in Fig. 3, respectively. (a) The intensity (left) and phase front (right) of the input beam. (b) “ring” is unfolded to a rectangular shape after the mode transformation. (c) Rectangular-shaped beam is focused. (d) Intensity (left) and phase (right) of the beam at the filter output.
beam profile evolution when we send in an OAM with l 4. Figures 4(a), 4(b), 4(c), and 4(d) (corresponding to the positions ⓐ, ⓑ, ⓒ, and ⓓ, respectively, of the setup in Fig. 3) shows the simulated beam profile of the input beam, the beam after transformation, the beam at the focal plane, and the output beam, respectively. We compare the input beam and the output beam by calculating their intensity correlation and wavefront (including both intensity and phase front) correlation coefficient, which is ∼0.98 and ∼0.97, respectively. These two numbers indicate that the ring-shaped intensity and the helical phase of the beam can be maintained fairly well after filtering. Figure 5 shows the observed interferograms of both the input and output beams. The interferograms are obtained by interfering the target beam with a plane wave (approximated by an expanded Gaussian beam) from the same laser source. We experimentally explored the
Fig. 5. Observed intensities and interferograms of both the input and the output beams of the OAM filter in the experiment. Only one OAM beam is sent to the filter each time. The filter is set to pass all the modes.
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Fig. 6. (a) Simulated wavefront correlation coefficient between the input and output beams with l value varying from −20 to 20. (b) Simulated wavefront correlation between the input (l 4 and 9, respectively) and output beam as a function of the reflector (mirror array or SLM) position on the beam propagating axis (0 is the focal plane position).
tuning range of the filter by varying the l value of the input OAM beam from 0 to 9 (including −4 and −9). A comparison between the input and output column in Fig. 5 implies that the output beams have the same vortex charge as the input OAM beams. A wider range exploration by simulation is shown in Fig. 6(a). We simulated the wavefront correlation coefficient between the input and output beam with a specific beam waist but with different l values. The curve indicates that an OAM beam with a larger l value experiences a larger distortion after the filtering. The major reason is that the geometric transformation is designed to work perfectly for normally incident light, while a higher l value OAM beam has a larger angular deviation. The theoretical upper limit of the filter tuning range is given by l ≪ 2πr 2 ∕λL, where r is the beam radius, λ is the wavelength, and L is the separation between the two elements in the sorter [15]. Since the reflector (mirror array or SLM) is required to be placed right at the focal plane of the lens, a position shift away from the focal plane may cause distortions on the output beams of the filter. The simulation result in Fig. 6(b) indicates that the wavefront correlation coefficient between the input and output beam varies almost symmetrically with the reflector shift along the beam propagating axis. We then send multiple multiplexed OAM modes as the filter input and analyze the output beams by a second aligned mode sorter. Figure 7(a) shows the superimposed intensity profile of the input beams, including OAM with l 4 and 9. After the first passage through the mode sorter, four OAM beams are mapped to four elongated spots at the focal plane, as shown in Fig. 7(b). Figure 7(c) shows the image of the beams at the output of the filter without blocking any beam. A further log-polar transformation maps the output beams into four spots again, as shown in Fig. 7(d).
Fig. 7. (a) Input of the OAM filter, including OAM beams with l 4 and 9. (b) Intensity distribution after the first mode sorter. (c) The output of the OAM filter without blocking any mode. (d) The intensity distribution after the second mode sorter.
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At the focal plane of mode filter
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Fig. 8. Experimental results of the tunable OAM mode filtering. The grid areas in the left column correspond to the regions where we applied the phase grating to block the light. The middle column shows the filter output analyzed by the second mode sorter. The right column shows the normalized power spectrum of the filter output. (a1)–(a3) All modes pass. (b1)–(b3) Block l −9. (c1)–(c3) Block l −4. (d1)–(d3) Block l −9 and l 9. (e1)–(e3) l −4 pass. (f1)–(f3) l 4 pass.
The experimental results of the tunable filtering functions are shown in Fig. 8. Figures 8(a1)–8(a3) show the scenario when all four modes pass through the filter. Through the programming of the SLM at the focal plane of the first mode sorter where all the input modes are spatially separated, we selectively block the channels carried on corresponding OAM modes, as shown in the left column of Fig. 8. We demonstrated a “band-block” filter, which blocks the OAM beam with l −9 and l −4, as shown in Figs. 8(b) and 8(c), respectively. We analyzed the power of each mode at the focal plane of the second mode sorter using a camera, and the power of OAM beam with l −9 and l −4 are suppressed by ∼17 and ∼15.5 dB, respectively, as shown in Figs. 8(b3) and 8(c3). The filter we proposed is also “bandwidth” tunable. We demonstrated the blocking–filtering of two OAM modes with l 4 instead of one, as shown in Figs. 8(d1) to 8(d3). At the filter output, the power of the OAM beams with l 9 and −9 are suppressed by ∼14.5 and ∼16 dB, respectively. We also present the “bandpass” filtering of OAM − 4 and OAM 4, respectively, by blocking the three other modes using the SLM, as shown in Figs. 8(e1) and 8(f1), respectively. Figures 8(e3) and 8(f3) indicate that the output of the filter has majority of the
power from a single OAM mode, such as OAM with l −4 and l 4, respectively. The suppression ratio of the other modes are >15 dB. In this proof-of-concept experiment, we demonstrated the filtering of four OAM modes (l 4 and 9) with a minimum vortex charge spacing of five. However, the minimum OAM mode charge spacing that can be distinguished by the filter (i.e., filtering resolution) is fundamentally limited by the crosstalk errors of the mode sorter. In the present form of the mode sorter, the transformed light spots of the neighboring modes slightly overlap, and ∼20% of the energy of an OAM mode is leaked to the other modes after the mode sorter [10]. We note that this crosstalk of the mode sorter can be reduced to ∼5% by modifying the transformation to give multiple transverse cycles [16], and the filtering resolution can be also potentially improved accordingly. We note that the current filter scheme only works for a single linearly polarized beam due to the polarization dependence of SLM. If the SLM is replaced with a mirror array, then the filter would be polarizationindependent. In addition, current design of the filter has 6 dB power loss (3 dB for the forward beam and 3 dB for the backward beam) due to the use of the beam splitter, which could potentially be mitigated by using an additional mode sorter to convert the elongated beams back to the OAM beams, instead of propagating the beams through the same mode sorter back and force. This work is supported by DARPA under the Inpho program. References 1. A. Yao and M. Padgett, Adv. Opt. Photon. 3, 161 (2011). 2. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004). 3. J. H. Shapiro, S. Guha, and B. I. Erkmen, J. Opt. Netw. 4, 501 (2005). 4. P. Boffi, P. Martelli, A. Gatto, and M. Martinelli, Proc. SPIE 8647, 864705 (2013). 5. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992). 6. J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, Nat. Photonics 6, 488 (2012). 7. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, A. Willner, and S. Ramachandran, Science 340, 1545 (2013). 8. J. D. Love and N. Riesen, Opt. Lett. 37, 3990 (2012). 9. G. Berkhout, M. Lavery, J. Courtial, M. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010). 10. M. Lavery, D. J. Robertson, G. C. G. Berkhout, G. n. D. Love, M. J. Padgett, and J. Courtial, Opt. Express 20, 2110 (2012). 11. D. A. B. Miller, Opt. Express 21, 20220 (2013). 12. H. Huang, Y. Ren, G. Xie, Y. Yan, Y. Yue, N. Ahmed, M. Lavery, M. Padgett, S. Dolinar, and A. E. Willner, in CLEO: Applications and Technology (Optical Society of America, 2013), paper JTu4A.89. 13. O. Bryngdahl, J. Opt. Soc. Am. 64, 1092 (1974). 14. Y. Saito, S. Komatsu, and H. Ohzu, Opt. Commun. 47, 8 (1983). 15. M. Lavery, D. Robertson, A. Sponselli, J. Courtial, N. Steinhoff, G. Tyler, A. Willner, and M. Padgett, New J. Phys. 15, 013024 (2013). 16. M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, Nat. Commun. 4, 2781 (2013).
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Tunable orbital angular momentum mode filter based on optical geometric transformation Hao Huang,1,* Yongxiong Ren,1 Guodong Xie,1 Yan Yan,1 Yang Yue,1 Nisar Ahmed,1 Martin P. J. Lavery,2 Miles J. Padgett,2 Sam Dolinar,3 Moshe Tur,4 and Alan E. Willner1 1
Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089, USA 2
School of Physics and Astronomy, SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom 3 Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, California 91109, USA 4
School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel *Corresponding author: [email protected]
Received January 13, 2014; revised February 14, 2014; accepted February 15, 2014; posted February 18, 2014 (Doc. ID 204653); published March 14, 2014 We present a tunable mode filter for spatially multiplexed laser beams carrying orbital angular momentum (OAM). The filter comprises an optical geometric transformation-based OAM mode sorter and a spatial light modulator (SLM). The programmable SLM can selectively control the passing/blocking of each input OAM beam. We experimentally demonstrate tunable filtering of one or multiple OAM modes from four multiplexed input OAM modes with vortex charge of l −9, −4, 4, and 9. The measured output power suppression ratio of the propagated modes to the blocked modes exceeds 14.5 dB. © 2014 Optical Society of America OCIS codes: (120.2440) Filters; (330.6110) Spatial filtering; (090.2890) Holographic optical elements. http://dx.doi.org/10.1364/OL.39.001689
Laser beams carrying orbital angular momentum (OAM) have recently gained interest due to their potential beneficial use in a number of applications [1]. For example, one application would be to multiplex many different data channels, each of which is carried on a different OAM beam, thereby increasing system capacity and spectral efficiency [2–4]. In terms of capacity, this might represent another orthogonal dimension of system growth, which is similar to how wavelength-division multiplexing (WDM) has been utilized to increase capacity by assigning different data channels to different wavelengths. In general, the electromagnetic field of an OAM beam is identified by a phase term expressed as expilφ, where φ is the azimuthal angle in the transverse plane of the beam, and the integer l is the azimuthal index indicating the topologic charge of the beam [5]. Spatially co-located OAM beams with different charges are principally orthogonal to one another, allowing the efficient multiplexing and demultiplexing of multiple data channels. Recent reports have demonstrated a multiplexing/ demultiplexing of 32 OAM channels in free space and two OAM channels in a 1.1 km vortex fiber [6,7]. Similar to the field of WDM, where tunable filters are frequently used to select a certain wavelength channel or to remove out-of-band noise, a tunable mode filter could be useful in an OAM-multiplexed system, e.g., to extract the desired information carried on certain modes. Note that an OAM beam may be distorted after transmission, and the power of a single mode is then distributed to multiple OAM modes. In this case, a mode filter would be helpful to remove power of unwanted modes. Presently, various techniques have been proposed to manipulate wavelength channels in WDM or specific modes for non-OAM mode division multiplexing [8], while very few basic reconfigurable elements exist to enable OAM-based systems. Recently, an OAM mode sorter (i.e., demultiplexer) was demonstrated to convert the incoming beam’s OAM value into different output spatial positions [9,10], and a spatial add/drop multiplexer is 0146-9592/14/061689-04$15.00/0
described to extract a spatial mode from a light beam, leaving the other orthogonal modes undisturbed [11]. It might be interesting to further demonstrate a tunable optical filter that can selectively output one OAM mode from multiple-input multiplexed OAM modes and keep its OAM characteristics [12]. In this Letter, we present a tunable mode-passing/ blocking filter for spatially multiplexed OAM beams using a log-polar transformation-based mode sorter and a spatial light modulator (SLM). Multiple multiplexed OAM beams are mapped to different positions on the screen of the SLM and are reflected back to the sorter. The programmable SLM can selectively control the passing or blocking of each OAM beam. Mode filtering or blocking of one or multiple OAM modes from four OAM modes (l −9, −4, 4, and 9) is demonstrated. The performance of the filter including the beam quality, loss, and resolution are discussed. The concept of an OAM mode filter is shown in Fig. 1. Similar to the tunable filter in the wavelength domain, as shown in Fig. 1, an OAM mode filter is expected to be able to transmit (i.e., a “bandpass filter”) or block (i.e., a band-block filter) the selected OAM modes from
Fig. 1. Concept of a tunable OAM mode filter. In analogy with WDM, an OAM filter transmits or blocks OAM beam(s) of choice from multiple spatially multiplexed OAM beams. © 2014 Optical Society of America
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multiple spatially multiplexed OAM modes. After filtering, the output data channels are required to maintain their input OAM property. The principle of the OAM mode filter is based on double-passing through an OAM mode sorter. The mode sorter, as recently reported in [9,10], essentially comprises a log-polar transformer [13,14] and a convex lens (CL), the combination of which separates different OAM states into spatial positions. The log-polar transformer maps from the polar coordinates to the perpendicular coordinates. Specifically, it maps a position x; y in the input planepto a position u; v in the output plane, where u −a ln x2 y2 ∕b and v a tan−1 y∕x, a and b are scaling constants [9]. As a result, an OAM “ring” is unfolded and converted into a rectangular-shaped plane wave in the forward propagation. Accordingly, a rectangular-shaped beam with a tilted phase profile can be folded and reconverted to a ring-shaped OAM beam in the backward propagation, as indicated in Fig. 2(a). As a result of the transformation in the forward pass, the multiplexed OAM beams at the input plane, as described in Fig. 2(b) by a set of concentric rings in terms of the beam intensity, are mapped into a set of spatially overlapped rectangular-shaped plane waves. Note that each plane wave after the transformation has a tilt, whose value depends on the vortex charge of the corresponding input OAM beam. A lens can then focus these differently tilted plane waves into spatially distinct elongated spots at its focal plane, with an inter spot separation of λf ∕2πa, where λ is the wavelength, and f is the focal length of the lens. On the other hand, the beams can be reflected by a mirror, which is placed perpendicular to the propagating direction at the focal plane of the lens, and pass through the mode sorter again in the opposite direction, as shown in Fig. 2(c). Since the mirror is at the focal plane, the reflected beams with the elongated shape are collimated by the CL and are converted to rectangular plane waves with different tilts. Following that, the geometric transformer performs inverse transformation (from Cartesian to log-polar coordinates) and converts the tilt plane waves back into vortex beams with a ring-shaped intensity. Furthermore, if the mirror at the focal plane of the lens is replaced with a programmable mirror array, we can selectively control the passing or blocking of each OAM mode, and the tunable OAM mode-filtering function can be achieved. Figure 2(c) shows an example of a “band blocking” filter (blocking OAM beam with l l3 ). The schematic overview of the OAM filter setup is shown in Fig. 3. The light from a laser source is collimated and then launched onto a liquid crystal on a silicon-based diffractive SLM. The SLM is loaded with a designed phase hologram to generate multiple superimposed OAM modes. The generated OAM beams are sent to the OAM mode filter as the inputs. The mode transformer in the filter is achieved by using two reflective optical elements [10], each of which has an aperture size of 8 mm. A CL with a focal length of 1 m is placed right after the mode transformer to focus the beams. At the focus plane, we use a reflective SLM instead of the mirror array to reflect the beams. The SLM surface is divided into different regions, each of which encompasses the
Fig. 2. Principle of the OAM mode filter. (a) A log-polar geometrical transformation transforms an OAM beam to a rectangular-shaped plane wave and vice versa. (b) Multiplexed OAM beams are mapped to different positions at the focal plane of the convex lens (CL) after passing through the mode sorter. (c) The reflected beams (after filtering) are converted back to ring-shapes while backpropagating through the mode sorter.
spot corresponding to a specific OAM mode. Each region of the SLM can be programmed to either reflect the beam back to the lens or to diffract the beam away from the other beams, effectively blocking it. All reflected beams backpropagate through the optical system and are converted back into the desired selection of superimposed OAM beams. We use a beam splitter to separate the backward-propagating beam from the input beams. We note that the charge of the OAM beams after double passing through the mode sorter are inversed. Due to the additional reflection of the beam splitter, the final output beams are expected to have the same vortex charges as the input OAM beams. We first tested the OAM mode filter by sending a single OAM beam as the input. The filter is set to pass all the input beams. Therefore an OAM beam with the same vortex charge is expected to be obtained at the output port of the filter. Simulation results in Fig. 4 illustrate the
Fig. 3. Schematic overview of the OAM filter setup. SLM, spatial light modulator. BS, nonpolarization beam splitter.
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Fig. 4. Simulated beam profiles at each position of the setup. (a), (b), (c), and (d) correspond to ⓐ, ⓑ, ⓒ, and ⓓ in Fig. 3, respectively. (a) The intensity (left) and phase front (right) of the input beam. (b) “ring” is unfolded to a rectangular shape after the mode transformation. (c) Rectangular-shaped beam is focused. (d) Intensity (left) and phase (right) of the beam at the filter output.
beam profile evolution when we send in an OAM with l 4. Figures 4(a), 4(b), 4(c), and 4(d) (corresponding to the positions ⓐ, ⓑ, ⓒ, and ⓓ, respectively, of the setup in Fig. 3) shows the simulated beam profile of the input beam, the beam after transformation, the beam at the focal plane, and the output beam, respectively. We compare the input beam and the output beam by calculating their intensity correlation and wavefront (including both intensity and phase front) correlation coefficient, which is ∼0.98 and ∼0.97, respectively. These two numbers indicate that the ring-shaped intensity and the helical phase of the beam can be maintained fairly well after filtering. Figure 5 shows the observed interferograms of both the input and output beams. The interferograms are obtained by interfering the target beam with a plane wave (approximated by an expanded Gaussian beam) from the same laser source. We experimentally explored the
Fig. 5. Observed intensities and interferograms of both the input and the output beams of the OAM filter in the experiment. Only one OAM beam is sent to the filter each time. The filter is set to pass all the modes.
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Fig. 6. (a) Simulated wavefront correlation coefficient between the input and output beams with l value varying from −20 to 20. (b) Simulated wavefront correlation between the input (l 4 and 9, respectively) and output beam as a function of the reflector (mirror array or SLM) position on the beam propagating axis (0 is the focal plane position).
tuning range of the filter by varying the l value of the input OAM beam from 0 to 9 (including −4 and −9). A comparison between the input and output column in Fig. 5 implies that the output beams have the same vortex charge as the input OAM beams. A wider range exploration by simulation is shown in Fig. 6(a). We simulated the wavefront correlation coefficient between the input and output beam with a specific beam waist but with different l values. The curve indicates that an OAM beam with a larger l value experiences a larger distortion after the filtering. The major reason is that the geometric transformation is designed to work perfectly for normally incident light, while a higher l value OAM beam has a larger angular deviation. The theoretical upper limit of the filter tuning range is given by l ≪ 2πr 2 ∕λL, where r is the beam radius, λ is the wavelength, and L is the separation between the two elements in the sorter [15]. Since the reflector (mirror array or SLM) is required to be placed right at the focal plane of the lens, a position shift away from the focal plane may cause distortions on the output beams of the filter. The simulation result in Fig. 6(b) indicates that the wavefront correlation coefficient between the input and output beam varies almost symmetrically with the reflector shift along the beam propagating axis. We then send multiple multiplexed OAM modes as the filter input and analyze the output beams by a second aligned mode sorter. Figure 7(a) shows the superimposed intensity profile of the input beams, including OAM with l 4 and 9. After the first passage through the mode sorter, four OAM beams are mapped to four elongated spots at the focal plane, as shown in Fig. 7(b). Figure 7(c) shows the image of the beams at the output of the filter without blocking any beam. A further log-polar transformation maps the output beams into four spots again, as shown in Fig. 7(d).
Fig. 7. (a) Input of the OAM filter, including OAM beams with l 4 and 9. (b) Intensity distribution after the first mode sorter. (c) The output of the OAM filter without blocking any mode. (d) The intensity distribution after the second mode sorter.
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At the focal plane of mode filter
Power spectrum
Output of mode analyzer
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Fig. 8. Experimental results of the tunable OAM mode filtering. The grid areas in the left column correspond to the regions where we applied the phase grating to block the light. The middle column shows the filter output analyzed by the second mode sorter. The right column shows the normalized power spectrum of the filter output. (a1)–(a3) All modes pass. (b1)–(b3) Block l −9. (c1)–(c3) Block l −4. (d1)–(d3) Block l −9 and l 9. (e1)–(e3) l −4 pass. (f1)–(f3) l 4 pass.
The experimental results of the tunable filtering functions are shown in Fig. 8. Figures 8(a1)–8(a3) show the scenario when all four modes pass through the filter. Through the programming of the SLM at the focal plane of the first mode sorter where all the input modes are spatially separated, we selectively block the channels carried on corresponding OAM modes, as shown in the left column of Fig. 8. We demonstrated a “band-block” filter, which blocks the OAM beam with l −9 and l −4, as shown in Figs. 8(b) and 8(c), respectively. We analyzed the power of each mode at the focal plane of the second mode sorter using a camera, and the power of OAM beam with l −9 and l −4 are suppressed by ∼17 and ∼15.5 dB, respectively, as shown in Figs. 8(b3) and 8(c3). The filter we proposed is also “bandwidth” tunable. We demonstrated the blocking–filtering of two OAM modes with l 4 instead of one, as shown in Figs. 8(d1) to 8(d3). At the filter output, the power of the OAM beams with l 9 and −9 are suppressed by ∼14.5 and ∼16 dB, respectively. We also present the “bandpass” filtering of OAM − 4 and OAM 4, respectively, by blocking the three other modes using the SLM, as shown in Figs. 8(e1) and 8(f1), respectively. Figures 8(e3) and 8(f3) indicate that the output of the filter has majority of the
power from a single OAM mode, such as OAM with l −4 and l 4, respectively. The suppression ratio of the other modes are >15 dB. In this proof-of-concept experiment, we demonstrated the filtering of four OAM modes (l 4 and 9) with a minimum vortex charge spacing of five. However, the minimum OAM mode charge spacing that can be distinguished by the filter (i.e., filtering resolution) is fundamentally limited by the crosstalk errors of the mode sorter. In the present form of the mode sorter, the transformed light spots of the neighboring modes slightly overlap, and ∼20% of the energy of an OAM mode is leaked to the other modes after the mode sorter [10]. We note that this crosstalk of the mode sorter can be reduced to ∼5% by modifying the transformation to give multiple transverse cycles [16], and the filtering resolution can be also potentially improved accordingly. We note that the current filter scheme only works for a single linearly polarized beam due to the polarization dependence of SLM. If the SLM is replaced with a mirror array, then the filter would be polarizationindependent. In addition, current design of the filter has 6 dB power loss (3 dB for the forward beam and 3 dB for the backward beam) due to the use of the beam splitter, which could potentially be mitigated by using an additional mode sorter to convert the elongated beams back to the OAM beams, instead of propagating the beams through the same mode sorter back and force. This work is supported by DARPA under the Inpho program. References 1. A. Yao and M. Padgett, Adv. Opt. Photon. 3, 161 (2011). 2. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004). 3. J. H. Shapiro, S. Guha, and B. I. Erkmen, J. Opt. Netw. 4, 501 (2005). 4. P. Boffi, P. Martelli, A. Gatto, and M. Martinelli, Proc. SPIE 8647, 864705 (2013). 5. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992). 6. J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, Nat. Photonics 6, 488 (2012). 7. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, A. Willner, and S. Ramachandran, Science 340, 1545 (2013). 8. J. D. Love and N. Riesen, Opt. Lett. 37, 3990 (2012). 9. G. Berkhout, M. Lavery, J. Courtial, M. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010). 10. M. Lavery, D. J. Robertson, G. C. G. Berkhout, G. n. D. Love, M. J. Padgett, and J. Courtial, Opt. Express 20, 2110 (2012). 11. D. A. B. Miller, Opt. Express 21, 20220 (2013). 12. H. Huang, Y. Ren, G. Xie, Y. Yan, Y. Yue, N. Ahmed, M. Lavery, M. Padgett, S. Dolinar, and A. E. Willner, in CLEO: Applications and Technology (Optical Society of America, 2013), paper JTu4A.89. 13. O. Bryngdahl, J. Opt. Soc. Am. 64, 1092 (1974). 14. Y. Saito, S. Komatsu, and H. Ohzu, Opt. Commun. 47, 8 (1983). 15. M. Lavery, D. Robertson, A. Sponselli, J. Courtial, N. Steinhoff, G. Tyler, A. Willner, and M. Padgett, New J. Phys. 15, 013024 (2013). 16. M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, Nat. Commun. 4, 2781 (2013).
