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Orbital angular momentum filter of photon based on spin-orbital angular momentum coupling
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Orbital angular momentum filter of photon based on spin-orbital angular momentum coupling Dong-Xu Chen, Pei Zhang ∗ , Rui-Feng Liu, Hong-Rong Li, Hong Gao, Fu-Li Li MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, Department of Applied Physics, Xi’an Jiaotong University, Xi’an 710049, China
a r t i c l e
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Article history: Received 18 May 2015 Received in revised form 8 June 2015 Accepted 9 June 2015 Available online xxxx Communicated by V.A. Markel Keywords: Orbital angular momentum Filter Interferometer
a b s t r a c t Determination of the orbital angular momentum (OAM) of vortex beams has been hotly discussed. We propose a new type of method to determine the orbital angular momentum of photons, filtering. We present an OAM filter scheme which consists of a cavity with a polarization-based Mach–Zehnder interferometer inside. Our scheme can purify the specific OAM with unitary efficiency theoretically without the pre-knowledge of the OAM spectrum of the input light. We also implemented a proof-ofprinciple experiment to demonstrate the feasibility of our scheme by cascading three interferometers. Our method offers a new way to determine the OAM spectrum of a light and this method can also be exploited to prepare the eigenstate of vortex beams. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The orbital angular momentum (OAM) of photons has been drawning more and more attention since it was discovered by L. Allen et al. in 1992 [1]. The infinite dimension of this degree of freedom of photons has critical applications in many fields [2,3], for example, high dimensional entanglement [4–7], optical tweezers [8–10], high capacity communication [11–13], quantum algorithm [14–16]. The OAM of photons is characterized by the spiral phase exp(i φ) in the transverse plane, where specifies the topological number of the vortex beams, ranging from −∞ to ∞, φ is the azimuthal phase. A common example of such beams is the Laguerre–Gaussian (LG) beam. There are many ways to generate LG beams. A mode converter converts Hermite–Gaussian (HG) beams to LG beams by use of two cylindrical lens [17–20]. A vortex phase plate (VPP) adds an exp(i φ) phase to the fundamental TEM00 ( = 0) mode from laser [21–23]. A computer generated hologram can also generate specific OAM [24,25]. However, as the imperfection of the devices, the beams generated are usually not perfect as expected. Given a beam, without the knowledge of the OAM spectrum, how does one extract the specific OAM one needs or purify the spatial mode? Here we propose an OAM filtering scheme which will solve this problem.
*
Corresponding author. E-mail address: [email protected] (P. Zhang).
http://dx.doi.org/10.1016/j.physleta.2015.06.022 0375-9601/© 2015 Elsevier B.V. All rights reserved.
Fig. 1. Schematic of the OAM filter. When input an LG beam in mixed state or pure state, a black box with an external control variable xvar will filter out the desired mode with OAM of x . Changing the variable xvar , the output LG beam changes.
Considering a device as depicted in Fig. 1, an OAM filter can be a black box which has an external control variable xvar . If we input LG beam in whatever pure state or mixed state, this box will output LG beam with = x which is related to xvar . In other words, an OAM filter can output only one specific OAM light as desired, which is analogous to polarizer for polarization. To realize such an OAM filter, we need to consider the determination of OAM. So far, the principle of the determination of OAMs can be generally classified into three categories. The first one is projection-based schemes. Photons to be measured incident on a device that performs a translation to the state from nonzero to = 0. Following by a spatial filter (a single mode fiber, for example), only photons with = 0 will arrive at the detector. The second one is diffraction-based schemes [26–34] in which the diffraction pattern of the photons through an obstacle is observed. The magnitude of can be decided according to the patterns. The third one is the interferometer-based schemes [35–38]. Photons enter an interferometer in which different modes undergo different
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Fig. 2. The cavity scheme of our experiment. Two switchable mirrors (SM) form a cavity where photon cycles inside. The solid line frame is the polarization-based interferometer which rotates the polarization of the photon. A Faraday rotation material (FRM) compensate the polarization of the desired OAM back to horizontal. The detailed description is shown in article. M: mirror, PBS: polarization beam splitter, QWP: quarter wave plate, Dove: Dove lens.
phase differences and interfere at different outputs of the interferometer. In the first two categories, the spatial mode of the photons is destroyed due to the projection process or the diffraction. Although the OAM is successfully measured the photons cannot be used in future, which makes these two types of measurement fail to be candidates for OAM filter. In the third category, the OAM of light is unchanged, so the photons can be used after then. We will briefly introduce the interferometer-based scheme and put forward our method to construct an OAM filter which is based on an interferometer. In Refs. [35,36], the OAMs of photons were sorted by Mach– Zehnder interferometer with Dove prisms inserted. However, to filter out one specific OAM, one needs to know the OAM spectrum previously to avoid degeneracy. In Ref. [37] W. Zhang et al. proposed an experiment sorting the OAM in a polarization-based Mach–Zehnder interferometer. The device mimics the Faraday effect in the spatial mode degree of freedom of photons, rotating the polarization of photons by α , α is the relative angle of the Dove prisms in the interferometer. The net result is that vortex beams with different OAMs in the same polarization turn into different polarizations with OAMs unchanged. By choosing appropriate α , the polarizations in the output port of the interferometer can be discriminated by a polarization beam splitter (PBS). The device transforms the problem of discriminating different OAMs into the issue of discriminating different polarizations. Because of the binary dimension of polarization, the polarization-based interferometer can only distinguish two different OAMs one time. In the following, we shall propose a scheme of OAM filter based on the method of Ref. [37], and show how it works. 2. Scheme description
Fig. 3. Theoretical results of the loops of the photon passing through the interferometer vs. the purity of the requested modes for initial state in (a) ±1, ±2 and ±3 equally superposed initial state and (b) ±1, ±2, ±3, ±4 and ±5 equally superposed initial state with different α and requested modes, with α = 5◦ , 0 = 1 (dotted red line), α = 5◦ , 0 = 3 (solid black line), α = 10◦ , 0 = 1 (solid blue line), and α = 10◦ , 0 = 3 (dotted black line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
without loss, while other modes will be projected onto horizontal polarization by cos[( − 0 )α ] attenuation in amplitude. What should be noted is that unlike the PBS3 in Ref. [37], we don’t aim to rotate the polarizations of different modes to orthogonal states, the PBS4 in our scheme is to rotate the specified mode to horizontal polarization. To this stage, the spin-orbital angular momentum state of photons becomes | H , again. Then SM2 reflects photon back and reverse OAM to −. When passing through the FRM the second time, the polarization becomes −θ = 0 α shifted from horizontal, so the spin-orbital angular momentum state is:
|ψ2 = [cos(0 α )| H + sin(0 α )| V ]| − .
The interferometer rotates polarization by −α when photon with − is injected inversely. So when the photon gets out of the interferometer the second time, the spin-orbital angular momentum state is:
|ψ2 = [cos(0 − )α | H + sin(0 − )α | V ]| − .
|ψ0 =
|ψ1 = [cos(α )| H + sin(α )| V ]|.
|ψ f =
Then the Faraday rotation material (FRM) rotates the polarization by θ = −0 α , where 0 is the OAM number of light we want. The polarization of the requested OAM photon will be compensated by FRM, that is, in horizontal polarization, while for other photons with different OAMs, the polarization will be ( −0 )α shifted from horizontal. After PBS4, the requested spatial mode will undergo
(3)
Accompanied by PBS1, it decays by cos[(0 − )α ] once again. After being reflected by SM1, the spin-orbital angular momentum of photon becomes | H , , the same as the initial state except for different attenuation coefficients of different OAMs. Then the photon experiences next cycle. After passing through the interferometer N times, the photons will be attenuated by cos N [( − 0 )α ]. Then SM2 is OFF, the photon gets out of the cavity. Suppose the input is a superposition state, different modes will undergo different polarization rotation angles:
Fig. 2 shows the cavity scheme of our OAM filter. In the beginning, the first switchable mirror (SM1) is OFF and the second switchable mirror (SM2) is ON. When photon enters the cavity, SM1 is ON. The solid line frame is the polarization based interferometer in Ref. [37]. Let the initial state be horizontal polarized and the magnitude of the OAM equal to , so the input state can be written as |ψ0 = | H , . The relative angle of the two Dove prisms is set to be α . After passing through the interferometer, the polarization of photon is rotated by α , and the spin-orbital angular momentum state is:
(1)
(2)
c | H |
→ |ψ0 =
c [cos(α )| H + sin(α )| V ]|,
(4)
where c = |c | exp(i θ ) is the relative complex coefficients of different modes, |c | is the relative amplitude, θ is the relative phase. After being attenuated by the post-selection of polarization, the amplitude is attenuated by factor cos[( − 0 )α ]. Repeating this process N times, the final state is:
c cos N [( − 0 )α ]| H |
unnormalized.
(5)
With n → ∞, c cos N [( − 0 )α ] → 0 if = 0 . Thus |ψ f → c 0 | H |0 = | H |0 . The derivation is also adaptable when the input is in mix state. Fig. 3 plots the loops N of the photon passing through the interferometer vs. the purity η of the specific mode with different α ,
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Fig. 4. Experimental setup. Three cascaded interferometers demonstrate the feasibility of our scheme. Each interferometer is presented by a black box the detail of which is shown in the inset. BS: beam splitter. SLM: spatial light modulator, BD: beam deviator.
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Fig. 5. Theoretical (on the left of the pictures) and experimental results (on the right of the pictures) of the change of the intensity pattern after the first interferometer as HWP4 rotates. The titles are the orientation of the optics axis of HWP4.
where Fig. 3(a) for ±1, ±2 and ±3 equally superposed initial state √ |ψ0 = (|1 + |−1 + |2 + |−2 + |3 + |−3)/ 6 and Fig. 3(b) for ±1, ±2, ±3, ±4 and ±5 equally superposed initial state |ψ√ 0 = (|1 + |−1 + |2 + |−2 + |3 + |−3 + |4 + |−4 + |5 + |−5)/ 10. From the chart, we can see that as N increases, η tends to unity, and with larger α , η increases faster. 3. Experiment and discussion To demonstrate the feasibility of our scheme, we conduct an experiment that consists of three polarization-based Mach–Zehnder interferometers as is shown in Fig. 4. The black box stands for the interferometer, and the inset shows the details of the interferometer. In our setup, light in 632.8 nm wavelength is emitted from a He–Ne laser and then passes through a half-wave plate (HWP1) and a PBS to adjust the intensity of the light incidents perpendicularly on the spatial light modulator (SLM, HAMAMATSU, X10468) which is loaded with computer-generated hologram. An aperture on the Fourier plane of lens (L1) selects the first order of the diffracted beam after SLM. Then PBS2 projects the photon onto horizontal polarization to prepare the initial polarization state. Light undergoes three interferometers to filter out the specific OAM state then it is recorded by a charge-coupled device (CCD) camera. In each interferometer, HWP2 transforms the polarization of the photon to diagonal polarization state which can be equally separated by the calcite beam deviators (BD). As the difference between diagonal polarization and circular polarization is that the relative phase of the horizontal component and vertical component is π and π /2, respectively, which can be compensated in the two arms of the polarization-based interferometer. This allows us replace the QWP1 in Ref. [37] by HWP2. The two BDs make up the polarization-based Mach–Zehnder interferometer. Inside the interferometer, we put two Dove lenses to add different phases to different modes and an HWP3 to rotate the polarization by 90◦ . When light gets out of the interferometer, its polarization is converted back to linear polarization by a quarter wave plate (QWP). Then HWP4 rotates the desired 0 mode to H polarization and PBS3 attenuate the intensity of other modes by cos[( − 0 )α ]. As the size of the dove lens is 5 mm × 5 mm, to avoid the diameter of the beam being greater than the size of the dove lens, we zoom out the beam using two lenses (L3 and L4) in each level. First, we observe the evolution of the intensity pattern of the beam after the first interferometer as HWP4 rotates. Fig. 5 shows the intensity change√of the output beam with initial state |ψ0 = (|1 + |−2 + |3)/ 3 and α = 30◦ . The optical axis of HWP4 is calibrated by sending = 0 into the light path and rotating HWP4 when the output of the interferometer reaches the maximum. Rotating HWP4, the polarization of the beam changes. When light passes through PBS3, the relative amplitudes of different OAMs also change, as a consequence, the intensity pat-
Fig. 6. Experimental results (on the right of the pictures) and theoretical patterns √ (on the left of the pictures) of the mode with 0 = 1 and |ψ0 = (|1 + |−2)/ 2 and |ψ0 = (|1 + |−2 + |3 + |−4)/2. The relative angle of the two Dove lenses is 30◦ for (a) and 40◦ for (b), with the orientation of HWP4 equals to 15◦ and 20◦ , respectively, to compensate the polarization of the specific mode.Pictures are taken after the first, second and third interferometer, respectively.
tern after PBS3 changes. In Fig. 5, the titles are the rotating angles of HWP4. The experimental results (on the right of the pictures) are accompanied by the corresponding theoretical results (on the left of the pictures). We can see from Fig. 5 that the pattern changes as predicted. In our experiment, we set the specific mode with 0 = 1, so the angle of HWP4 in each interferometer is to be 0 α /2 = α /2. Fig. 6 shows the results of each √ level when we input different initial states, |ψ0 = (|1 + |−2)/ 2 and |ψ0 = (|1 + |−2 + |3 + |−4)/2. In Fig. 6(a) α is set to be 30◦ in Fig. 6(b) α is set to be 40◦ . We can see the consistency between experimental results (on the right of the pictures) and the corresponding theoretical results (on the left of the pictures). In Fig. 6, when |ψ0 = (|1 + |−2 + |3 + |−4)/2, the proportion of 0 = 1 is increased from level 1 to level 3, the reason why the intensity pattern of the output is different from that of the mode with = 1 is that the rest of other modes has great influence on the output pattern. When enough steps have been taken, the pattern of = 1 mode will emerge. The fan shape in our experimental results is due to the difference of Gouy phase shifts for different modes. The different beam diameters are due to different positions of the CCD camera and different ratios of L3 and L4 in each level. Note that our method does not need the knowledge of the OAM spectrum of the input state only if the input state contains the requested mode (if the input doesn’t contain the request mode, the output will be null, One special case is when the specific mode
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is rarely in the input, we may need hundreds of steps of filtering otherwise the specific mode won’t be apparent in the output and we will not know whether the mode exists in the input. So we need to decide an error boundary according to the practical in this case.). What we need to do is to adjust the relate angle of the Dove lenses and the orientation of HWP4, then we will get the specific OAM we need. This is the characteristic of a filter and is different to the previous works on interferometer-based OAM sorting. It should be emphasized that when (1 − 2 )α = nπ (n is an integer), we will not be able to separate 1 and 2 because of the periodicity of polarization. If (1 − 2 )α is closed to nπ , there will be close-talk for a certain cycle time. By choosing appropriate α , we can filter one specific OAM from as many OAMs as possible. Theoretically, when α is an irrational number, we can filter out all the different OAMs. 4. Conclusion In conclusion, we propose a scheme that can filter large number of OAMs with unity purity theoretically. Our method can output the specific mode without the knowledge of the initial state, just by adjusting the relative angle of the Dove lenses α and the angle of HWP4 (γ ), setting γ = 0 α /2, we will get the spatial mode with = 0 . Our method offers a way to filter the OAM of photons and it provides a useful tool to manipulate OAM state and may have potential applications in OAM communications and quantum information science. Acknowledgements This work is supported by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China (Grant Nos. 11374008, 11174233, 11374238 and 11374239). References [1] L. Allen, M.W. Beijersbergen, R. Spreeuw, J. Woerdman, Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes, Phys. Rev. A 45 (1992) 8185, http://dx.doi.org/10.1103/PhysRevA.45.8185. [2] S. Franke-Arnold, L. Allen, M. Padgett, Advances in optical angular momentum, Laser Photonics Rev. 2 (2008) 299, http://dx.doi.org/10.1002/lpor.200810007. [3] A.M. Yao, M.J. Padgett, Orbital angular momentum: origins, behavior and applications, Adv. Opt. Photonics 3 (2011) 161, http://dx.doi.org/10.1364/ AOP.3.000161. [4] A. Mair, A. Vaziri, G. Weihs, A. Zeilinger, Entanglement of the orbital angular momentum states of photons, Nature 412 (2001) 313, http://dx.doi.org/ 10.1038/35085529. [5] G. Molina-Terriza, J.P. Torres, L. Torner, Twisted photons, Nat. Phys. 3 (2007) 305, http://dx.doi.org/10.1038/nphys607. [6] J. Leach, B. Jack, J. Romero, A.K. Jha, A.M. Yao, S. Franke-Arnold, D.G. Ireland, R.W. Boyd, S.M. Barnett, M.J. Padgett, Quantum correlations in optical angle– orbital angular momentum variables, Science 329 (2010) 662, http://dx.doi.org/ 10.1126/science.1190523. [7] A.C. Dada, J. Leach, G.S. Buller, M.J. Padgett, E. Andersson, Experimental highdimensional two-photon entanglement and violations of generalized Bell inequalities, Nat. Phys. 7 (2011) 677, http://dx.doi.org/10.1038/nphys1996. [8] H. He, M. Friese, N. Heckenberg, H. Rubinsztein-Dunlop, Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity, Phys. Rev. Lett. 75 (1995) 826, http://dx.doi.org/10.1103/ PhysRevLett.75.826. [9] N. Simpson, K. Dholakia, L. Allen, M. Padgett, Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner, Opt. Lett. 22 (1997) 52, http://dx.doi.org/10.1364/OL.22.000052. [10] M. Padgett, R. Bowman, Tweezers with a twist, Nat. Photonics 5 (2011) 343, http://dx.doi.org/10.1038/nphoton.2011.81. [11] G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, S. FrankeArnold, Free-space information transfer using light beams carrying orbital angular momentum, Opt. Express 12 (2004) 5448, http://dx.doi.org/10.1364/ OPEX.12.005448. [12] J. Wang, J.-Y. Yang, I.M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, et al., Terabit free-space data transmission employing orbital angular momentum multiplexing, Nat. Photonics 6 (2012) 488, http:// dx.doi.org/10.1038/nphoton.2012.138.
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[37] W. Zhang, Q. Qi, J. Zhou, L. Chen, Mimicking Faraday rotation to sort the orbital angular momentum of light, Phys. Rev. Lett. 112 (2014) 153601, http:// dx.doi.org/10.1103/PhysRevLett.112.153601.
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Orbital angular momentum filter of photon based on spin-orbital angular momentum coupling Dong-Xu Chen, Pei Zhang ∗ , Rui-Feng Liu, Hong-Rong Li, Hong Gao, Fu-Li Li MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, Department of Applied Physics, Xi’an Jiaotong University, Xi’an 710049, China
a r t i c l e
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Article history: Received 18 May 2015 Received in revised form 8 June 2015 Accepted 9 June 2015 Available online xxxx Communicated by V.A. Markel Keywords: Orbital angular momentum Filter Interferometer
a b s t r a c t Determination of the orbital angular momentum (OAM) of vortex beams has been hotly discussed. We propose a new type of method to determine the orbital angular momentum of photons, filtering. We present an OAM filter scheme which consists of a cavity with a polarization-based Mach–Zehnder interferometer inside. Our scheme can purify the specific OAM with unitary efficiency theoretically without the pre-knowledge of the OAM spectrum of the input light. We also implemented a proof-ofprinciple experiment to demonstrate the feasibility of our scheme by cascading three interferometers. Our method offers a new way to determine the OAM spectrum of a light and this method can also be exploited to prepare the eigenstate of vortex beams. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The orbital angular momentum (OAM) of photons has been drawning more and more attention since it was discovered by L. Allen et al. in 1992 [1]. The infinite dimension of this degree of freedom of photons has critical applications in many fields [2,3], for example, high dimensional entanglement [4–7], optical tweezers [8–10], high capacity communication [11–13], quantum algorithm [14–16]. The OAM of photons is characterized by the spiral phase exp(i φ) in the transverse plane, where specifies the topological number of the vortex beams, ranging from −∞ to ∞, φ is the azimuthal phase. A common example of such beams is the Laguerre–Gaussian (LG) beam. There are many ways to generate LG beams. A mode converter converts Hermite–Gaussian (HG) beams to LG beams by use of two cylindrical lens [17–20]. A vortex phase plate (VPP) adds an exp(i φ) phase to the fundamental TEM00 ( = 0) mode from laser [21–23]. A computer generated hologram can also generate specific OAM [24,25]. However, as the imperfection of the devices, the beams generated are usually not perfect as expected. Given a beam, without the knowledge of the OAM spectrum, how does one extract the specific OAM one needs or purify the spatial mode? Here we propose an OAM filtering scheme which will solve this problem.
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Corresponding author. E-mail address: [email protected] (P. Zhang).
http://dx.doi.org/10.1016/j.physleta.2015.06.022 0375-9601/© 2015 Elsevier B.V. All rights reserved.
Fig. 1. Schematic of the OAM filter. When input an LG beam in mixed state or pure state, a black box with an external control variable xvar will filter out the desired mode with OAM of x . Changing the variable xvar , the output LG beam changes.
Considering a device as depicted in Fig. 1, an OAM filter can be a black box which has an external control variable xvar . If we input LG beam in whatever pure state or mixed state, this box will output LG beam with = x which is related to xvar . In other words, an OAM filter can output only one specific OAM light as desired, which is analogous to polarizer for polarization. To realize such an OAM filter, we need to consider the determination of OAM. So far, the principle of the determination of OAMs can be generally classified into three categories. The first one is projection-based schemes. Photons to be measured incident on a device that performs a translation to the state from nonzero to = 0. Following by a spatial filter (a single mode fiber, for example), only photons with = 0 will arrive at the detector. The second one is diffraction-based schemes [26–34] in which the diffraction pattern of the photons through an obstacle is observed. The magnitude of can be decided according to the patterns. The third one is the interferometer-based schemes [35–38]. Photons enter an interferometer in which different modes undergo different
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Fig. 2. The cavity scheme of our experiment. Two switchable mirrors (SM) form a cavity where photon cycles inside. The solid line frame is the polarization-based interferometer which rotates the polarization of the photon. A Faraday rotation material (FRM) compensate the polarization of the desired OAM back to horizontal. The detailed description is shown in article. M: mirror, PBS: polarization beam splitter, QWP: quarter wave plate, Dove: Dove lens.
phase differences and interfere at different outputs of the interferometer. In the first two categories, the spatial mode of the photons is destroyed due to the projection process or the diffraction. Although the OAM is successfully measured the photons cannot be used in future, which makes these two types of measurement fail to be candidates for OAM filter. In the third category, the OAM of light is unchanged, so the photons can be used after then. We will briefly introduce the interferometer-based scheme and put forward our method to construct an OAM filter which is based on an interferometer. In Refs. [35,36], the OAMs of photons were sorted by Mach– Zehnder interferometer with Dove prisms inserted. However, to filter out one specific OAM, one needs to know the OAM spectrum previously to avoid degeneracy. In Ref. [37] W. Zhang et al. proposed an experiment sorting the OAM in a polarization-based Mach–Zehnder interferometer. The device mimics the Faraday effect in the spatial mode degree of freedom of photons, rotating the polarization of photons by α , α is the relative angle of the Dove prisms in the interferometer. The net result is that vortex beams with different OAMs in the same polarization turn into different polarizations with OAMs unchanged. By choosing appropriate α , the polarizations in the output port of the interferometer can be discriminated by a polarization beam splitter (PBS). The device transforms the problem of discriminating different OAMs into the issue of discriminating different polarizations. Because of the binary dimension of polarization, the polarization-based interferometer can only distinguish two different OAMs one time. In the following, we shall propose a scheme of OAM filter based on the method of Ref. [37], and show how it works. 2. Scheme description
Fig. 3. Theoretical results of the loops of the photon passing through the interferometer vs. the purity of the requested modes for initial state in (a) ±1, ±2 and ±3 equally superposed initial state and (b) ±1, ±2, ±3, ±4 and ±5 equally superposed initial state with different α and requested modes, with α = 5◦ , 0 = 1 (dotted red line), α = 5◦ , 0 = 3 (solid black line), α = 10◦ , 0 = 1 (solid blue line), and α = 10◦ , 0 = 3 (dotted black line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
without loss, while other modes will be projected onto horizontal polarization by cos[( − 0 )α ] attenuation in amplitude. What should be noted is that unlike the PBS3 in Ref. [37], we don’t aim to rotate the polarizations of different modes to orthogonal states, the PBS4 in our scheme is to rotate the specified mode to horizontal polarization. To this stage, the spin-orbital angular momentum state of photons becomes | H , again. Then SM2 reflects photon back and reverse OAM to −. When passing through the FRM the second time, the polarization becomes −θ = 0 α shifted from horizontal, so the spin-orbital angular momentum state is:
|ψ2 = [cos(0 α )| H + sin(0 α )| V ]| − .
The interferometer rotates polarization by −α when photon with − is injected inversely. So when the photon gets out of the interferometer the second time, the spin-orbital angular momentum state is:
|ψ2 = [cos(0 − )α | H + sin(0 − )α | V ]| − .
|ψ0 =
|ψ1 = [cos(α )| H + sin(α )| V ]|.
|ψ f =
Then the Faraday rotation material (FRM) rotates the polarization by θ = −0 α , where 0 is the OAM number of light we want. The polarization of the requested OAM photon will be compensated by FRM, that is, in horizontal polarization, while for other photons with different OAMs, the polarization will be ( −0 )α shifted from horizontal. After PBS4, the requested spatial mode will undergo
(3)
Accompanied by PBS1, it decays by cos[(0 − )α ] once again. After being reflected by SM1, the spin-orbital angular momentum of photon becomes | H , , the same as the initial state except for different attenuation coefficients of different OAMs. Then the photon experiences next cycle. After passing through the interferometer N times, the photons will be attenuated by cos N [( − 0 )α ]. Then SM2 is OFF, the photon gets out of the cavity. Suppose the input is a superposition state, different modes will undergo different polarization rotation angles:
Fig. 2 shows the cavity scheme of our OAM filter. In the beginning, the first switchable mirror (SM1) is OFF and the second switchable mirror (SM2) is ON. When photon enters the cavity, SM1 is ON. The solid line frame is the polarization based interferometer in Ref. [37]. Let the initial state be horizontal polarized and the magnitude of the OAM equal to , so the input state can be written as |ψ0 = | H , . The relative angle of the two Dove prisms is set to be α . After passing through the interferometer, the polarization of photon is rotated by α , and the spin-orbital angular momentum state is:
(1)
(2)
c | H |
→ |ψ0 =
c [cos(α )| H + sin(α )| V ]|,
(4)
where c = |c | exp(i θ ) is the relative complex coefficients of different modes, |c | is the relative amplitude, θ is the relative phase. After being attenuated by the post-selection of polarization, the amplitude is attenuated by factor cos[( − 0 )α ]. Repeating this process N times, the final state is:
c cos N [( − 0 )α ]| H |
unnormalized.
(5)
With n → ∞, c cos N [( − 0 )α ] → 0 if = 0 . Thus |ψ f → c 0 | H |0 = | H |0 . The derivation is also adaptable when the input is in mix state. Fig. 3 plots the loops N of the photon passing through the interferometer vs. the purity η of the specific mode with different α ,
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Fig. 4. Experimental setup. Three cascaded interferometers demonstrate the feasibility of our scheme. Each interferometer is presented by a black box the detail of which is shown in the inset. BS: beam splitter. SLM: spatial light modulator, BD: beam deviator.
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Fig. 5. Theoretical (on the left of the pictures) and experimental results (on the right of the pictures) of the change of the intensity pattern after the first interferometer as HWP4 rotates. The titles are the orientation of the optics axis of HWP4.
where Fig. 3(a) for ±1, ±2 and ±3 equally superposed initial state √ |ψ0 = (|1 + |−1 + |2 + |−2 + |3 + |−3)/ 6 and Fig. 3(b) for ±1, ±2, ±3, ±4 and ±5 equally superposed initial state |ψ√ 0 = (|1 + |−1 + |2 + |−2 + |3 + |−3 + |4 + |−4 + |5 + |−5)/ 10. From the chart, we can see that as N increases, η tends to unity, and with larger α , η increases faster. 3. Experiment and discussion To demonstrate the feasibility of our scheme, we conduct an experiment that consists of three polarization-based Mach–Zehnder interferometers as is shown in Fig. 4. The black box stands for the interferometer, and the inset shows the details of the interferometer. In our setup, light in 632.8 nm wavelength is emitted from a He–Ne laser and then passes through a half-wave plate (HWP1) and a PBS to adjust the intensity of the light incidents perpendicularly on the spatial light modulator (SLM, HAMAMATSU, X10468) which is loaded with computer-generated hologram. An aperture on the Fourier plane of lens (L1) selects the first order of the diffracted beam after SLM. Then PBS2 projects the photon onto horizontal polarization to prepare the initial polarization state. Light undergoes three interferometers to filter out the specific OAM state then it is recorded by a charge-coupled device (CCD) camera. In each interferometer, HWP2 transforms the polarization of the photon to diagonal polarization state which can be equally separated by the calcite beam deviators (BD). As the difference between diagonal polarization and circular polarization is that the relative phase of the horizontal component and vertical component is π and π /2, respectively, which can be compensated in the two arms of the polarization-based interferometer. This allows us replace the QWP1 in Ref. [37] by HWP2. The two BDs make up the polarization-based Mach–Zehnder interferometer. Inside the interferometer, we put two Dove lenses to add different phases to different modes and an HWP3 to rotate the polarization by 90◦ . When light gets out of the interferometer, its polarization is converted back to linear polarization by a quarter wave plate (QWP). Then HWP4 rotates the desired 0 mode to H polarization and PBS3 attenuate the intensity of other modes by cos[( − 0 )α ]. As the size of the dove lens is 5 mm × 5 mm, to avoid the diameter of the beam being greater than the size of the dove lens, we zoom out the beam using two lenses (L3 and L4) in each level. First, we observe the evolution of the intensity pattern of the beam after the first interferometer as HWP4 rotates. Fig. 5 shows the intensity change√of the output beam with initial state |ψ0 = (|1 + |−2 + |3)/ 3 and α = 30◦ . The optical axis of HWP4 is calibrated by sending = 0 into the light path and rotating HWP4 when the output of the interferometer reaches the maximum. Rotating HWP4, the polarization of the beam changes. When light passes through PBS3, the relative amplitudes of different OAMs also change, as a consequence, the intensity pat-
Fig. 6. Experimental results (on the right of the pictures) and theoretical patterns √ (on the left of the pictures) of the mode with 0 = 1 and |ψ0 = (|1 + |−2)/ 2 and |ψ0 = (|1 + |−2 + |3 + |−4)/2. The relative angle of the two Dove lenses is 30◦ for (a) and 40◦ for (b), with the orientation of HWP4 equals to 15◦ and 20◦ , respectively, to compensate the polarization of the specific mode.Pictures are taken after the first, second and third interferometer, respectively.
tern after PBS3 changes. In Fig. 5, the titles are the rotating angles of HWP4. The experimental results (on the right of the pictures) are accompanied by the corresponding theoretical results (on the left of the pictures). We can see from Fig. 5 that the pattern changes as predicted. In our experiment, we set the specific mode with 0 = 1, so the angle of HWP4 in each interferometer is to be 0 α /2 = α /2. Fig. 6 shows the results of each √ level when we input different initial states, |ψ0 = (|1 + |−2)/ 2 and |ψ0 = (|1 + |−2 + |3 + |−4)/2. In Fig. 6(a) α is set to be 30◦ in Fig. 6(b) α is set to be 40◦ . We can see the consistency between experimental results (on the right of the pictures) and the corresponding theoretical results (on the left of the pictures). In Fig. 6, when |ψ0 = (|1 + |−2 + |3 + |−4)/2, the proportion of 0 = 1 is increased from level 1 to level 3, the reason why the intensity pattern of the output is different from that of the mode with = 1 is that the rest of other modes has great influence on the output pattern. When enough steps have been taken, the pattern of = 1 mode will emerge. The fan shape in our experimental results is due to the difference of Gouy phase shifts for different modes. The different beam diameters are due to different positions of the CCD camera and different ratios of L3 and L4 in each level. Note that our method does not need the knowledge of the OAM spectrum of the input state only if the input state contains the requested mode (if the input doesn’t contain the request mode, the output will be null, One special case is when the specific mode
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is rarely in the input, we may need hundreds of steps of filtering otherwise the specific mode won’t be apparent in the output and we will not know whether the mode exists in the input. So we need to decide an error boundary according to the practical in this case.). What we need to do is to adjust the relate angle of the Dove lenses and the orientation of HWP4, then we will get the specific OAM we need. This is the characteristic of a filter and is different to the previous works on interferometer-based OAM sorting. It should be emphasized that when (1 − 2 )α = nπ (n is an integer), we will not be able to separate 1 and 2 because of the periodicity of polarization. If (1 − 2 )α is closed to nπ , there will be close-talk for a certain cycle time. By choosing appropriate α , we can filter one specific OAM from as many OAMs as possible. Theoretically, when α is an irrational number, we can filter out all the different OAMs. 4. Conclusion In conclusion, we propose a scheme that can filter large number of OAMs with unity purity theoretically. Our method can output the specific mode without the knowledge of the initial state, just by adjusting the relative angle of the Dove lenses α and the angle of HWP4 (γ ), setting γ = 0 α /2, we will get the spatial mode with = 0 . Our method offers a way to filter the OAM of photons and it provides a useful tool to manipulate OAM state and may have potential applications in OAM communications and quantum information science. Acknowledgements This work is supported by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China (Grant Nos. 11374008, 11174233, 11374238 and 11374239). References [1] L. Allen, M.W. Beijersbergen, R. Spreeuw, J. Woerdman, Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes, Phys. Rev. A 45 (1992) 8185, http://dx.doi.org/10.1103/PhysRevA.45.8185. [2] S. Franke-Arnold, L. Allen, M. Padgett, Advances in optical angular momentum, Laser Photonics Rev. 2 (2008) 299, http://dx.doi.org/10.1002/lpor.200810007. [3] A.M. Yao, M.J. Padgett, Orbital angular momentum: origins, behavior and applications, Adv. Opt. Photonics 3 (2011) 161, http://dx.doi.org/10.1364/ AOP.3.000161. [4] A. Mair, A. Vaziri, G. Weihs, A. Zeilinger, Entanglement of the orbital angular momentum states of photons, Nature 412 (2001) 313, http://dx.doi.org/ 10.1038/35085529. [5] G. Molina-Terriza, J.P. Torres, L. Torner, Twisted photons, Nat. Phys. 3 (2007) 305, http://dx.doi.org/10.1038/nphys607. [6] J. Leach, B. Jack, J. Romero, A.K. Jha, A.M. Yao, S. Franke-Arnold, D.G. Ireland, R.W. Boyd, S.M. Barnett, M.J. Padgett, Quantum correlations in optical angle– orbital angular momentum variables, Science 329 (2010) 662, http://dx.doi.org/ 10.1126/science.1190523. [7] A.C. Dada, J. Leach, G.S. Buller, M.J. Padgett, E. Andersson, Experimental highdimensional two-photon entanglement and violations of generalized Bell inequalities, Nat. Phys. 7 (2011) 677, http://dx.doi.org/10.1038/nphys1996. [8] H. He, M. Friese, N. Heckenberg, H. Rubinsztein-Dunlop, Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity, Phys. Rev. Lett. 75 (1995) 826, http://dx.doi.org/10.1103/ PhysRevLett.75.826. [9] N. Simpson, K. Dholakia, L. Allen, M. Padgett, Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner, Opt. Lett. 22 (1997) 52, http://dx.doi.org/10.1364/OL.22.000052. [10] M. Padgett, R. Bowman, Tweezers with a twist, Nat. Photonics 5 (2011) 343, http://dx.doi.org/10.1038/nphoton.2011.81. [11] G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, S. FrankeArnold, Free-space information transfer using light beams carrying orbital angular momentum, Opt. Express 12 (2004) 5448, http://dx.doi.org/10.1364/ OPEX.12.005448. [12] J. Wang, J.-Y. Yang, I.M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, et al., Terabit free-space data transmission employing orbital angular momentum multiplexing, Nat. Photonics 6 (2012) 488, http:// dx.doi.org/10.1038/nphoton.2012.138.
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