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PHYSICAL REVIEW LETTERS
PRL 112, 153601 (2014)
Mimicking Faraday Rotation to Sort the Orbital Angular Momentum of Light Wuhong Zhang, Qianqian Qi, Jie Zhou, and Lixiang Chen* Department of Physics and Laboratory of Nanoscale Condensed Matter Physics, Xiamen University, Xiamen 361005, People’s Republic of China (Received 12 February 2014; published 14 April 2014) The efficient separation of the orbital angular momentum (OAM) is essential to both the classical and quantum applications with twisted photons. Here we devise and demonstrate experimentally an efficient method of mimicking the Faraday rotation to sort the OAM based on the OAM-to-polarization coupling effect induced by a modified Mach-Zehnder interferometer. Our device is capable of sorting the OAM of positive and negative numbers, as well as their mixtures. Furthermore, we report the first experimental demonstration to sort optical vortices of noninteger charges. The possibility of working at the photon-count level is also shown using an electron-multiplying CCD camera. Our scheme holds promise for quantum information applications with single-photon entanglement and for high-capacity communication systems with polarization and OAM multiplexing. DOI: 10.1103/PhysRevLett.112.153601
PACS numbers: 42.50.Tx, 07.60.Ly, 42.25.Ja, 42.79.Sz
The study of the angular momentum of light has a long history [1]. Early in 1909, Poynting first associated spin angular momentum with circular polarization [2]. However, it was only in 1992 that Allen et al. [3] recognized that a light beam with a helical phase front of expðilϕÞ carried a well-defined orbital angular momentum (OAM) of lℏ per photon, where ϕ is the azimuthal angle and l is the OAM quantum number. For a single photon, the spin and OAM are almost separable, at least within the paraxial approximation, so it is permissible to write down a state vector in the form jσi ⊗ jli as eigenstates of the angular momentum operator: Jˆz jσi ⊗ jli ¼ ðσ þ lÞjσi ⊗ jli, where the spin (σℏ) and orbital (lℏ) contributions to the total angular momentum are readily identified [4]. As the eigenstates of photon spin, the left- and right-handed circular polarizations are indicated as jσ ¼ þ1i ¼ jLi and jσ ¼ −1i ¼ jRi, respectively. While for the OAM eigenstate jli, theoretically, l can take any integers so that twisted photons offer the opportunity to realize a high-dimensional Hilbert space [5]. The past two decades have witnessed a rapidly growing interest in OAM ranging from optical manipulation and free-space communications, to quantum optics and quantum information [6,7]. Recently, considerable attention was also paid to the OAM separation, which plays a fundamental role in a variety of applications with twisted photons. Leach et al. [8] devised an interferometric method that could distinguish individual photons in different OAM states and route single photons according to the modulus of the individual OAM. They further extended this method to sort photons according to their spin, orbital, or total angular momentum [9]. Based on a multipoint interferometer, Berkhout and Beijersbergen [10] presented an efficient method for probing the OAM from astronomical objects. Additionally, Hickmann et al. [11] succeeded in revealing the magnitude 0031-9007=14=112(15)=153601(5)
and sign of the topological charge by observing a triangular lattice of spots diffracted by a triangular aperture. As many photons are required to form the full interferograms, these schemes in Refs. [10,11] could not be effectively applied to sort the OAM states at the single-photon level. By a clever geometric transformation, Berkhout et al. [12] achieved the efficient sorting of the OAM states. Mirhosseini et al. [13] recently reported the enhancement of the mode separation efficiency for these transformed OAM. It is noted that the OAM is inevitably changed during the mode transformation. In contrast, in the interferometric method outlined by Leach et al. [8,9], the helical phase structures are well preserved such that they can be used further for information processing with the OAM. Based on the OAMto-polarization coupling effect induced by a modified Mach-Zehnder interferometer, here we demonstrate a new experimental method to sort the OAM efficiently, which mimics the Faraday effect to rotate the polarization of a light beam by an angle proportional to its own OAM number. We distinguish our sorting scheme from those also employing the interferometric technique, where they took advantage of the phase symmetry of the OAM beams but did not involve the OAM-to-polarization coupling effect [8,9]. Our scheme may have potential in quantum information protocols with single-photon entanglement and in optical communications with both polarization and OAM multiplexing. Discovered by Michael Faraday in 1845, the Faraday effect describes a rotation of polarization caused by the interaction of light with a magnetic field in a chiral material [14]. The rotation angle θ is proportional to the magnetic field B in the direction of propagation, namely, θ ¼ vBd, where v is the Verdet constant for the material, and d is the length of interaction, see Fig. 1(a). This effect can be well understood from the theory of circular birefringence (see
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FIG. 1 (color online). The function of our device. (a) Polarization rotation due to the Faraday effect. The rotation angle θ is proportional to the magnetic field B and the interaction length d. (b) Polarization rotation produced by our device. The rotation angle θ is proportional to the OAM number l and the relative orientation α of Dove prisms. (c) Experimental setup for mimicking the Faraday effect to sort the OAM (see the text for more details), where the inset shows the hologram addressed by the SLM to generate a fractional vortex of M ¼ 9=2.
Ref. [15] and references therein). An incident horizontal polarization H can be decomposed into an equally weighted superposition of left- and right-handed p circularly ffiffiffi polarized components, e.g., jHi ¼ ðjLi þ jRiÞ= 2 (note here the polarization change is induced by the magnetooptic effect such that the OAM ket have been trivially discarded). When propagating, the two components experience different refractive indices, nL and nR , respectively, so that they suffer a relative phase shift, △φ ¼ ð2π=λÞðnR − nL Þd, which equivalently induces polarization rotation of the output light by an angle of θ ¼ △φ=2. The effect of circular birefringence can be described as follows: 1 jHi → pffiffiffi ðexp ðiθÞjLi þ exp ð−iθÞjRiÞ ¼ cos θjHi 2 þ sin θjVi;
(1)
where θ ¼ △φ=2 ¼ vBd. As illustrated in Fig. 1(b), our interferometric device can realize the analogous polarization rotation as the Faraday effect. The rotation angle θ can be expressed as θ ¼ lα, where l is the OAM number of the incoming light, and α is the relative orientation of the two Dove prisms in Fig. 1(c). Our approach benefits from the OAM-to-polarization coupling effect, which was first outlined theoretically by us to encode the OAM onto multiple spin states [16]. We assume the incoming photons have a definite horizontal polarization and a definite OAM number l, whose pffiffiffi state can then be denoted as jHi ⊗ jli ¼ ð1= 2ÞðjLi þ jRiÞ ⊗ jli. We first need to separate the left- and right-handed circular components, L and R, and then impart them with a different phase delay, which is OAM dependent. For convenience, the circular polarization beam splitter is substituted by a traditional polarizing beam splitter (PBS1) with a preposed quarter-wave plate
(QWP1). The QWP1 converts L and R to the horizontal and vertical polarizations, H and V, respectively, while PBS1 transmits H photons and reflects V photons. After entering the interferometer, the H and V components traverse the embedded Dove prisms, DP1 and DP2, respectively. The total internal reflection from the base of each prism flips the transverse cross section of the transmitted image. If DP1 and DP2 are set to have a relative orientation angle α, then a phase difference of exp ði2lαÞ is imparted to the H and V components. The two components recombine in PBS2, and the following quarter-wave plate (QWP2) converts H and V back to L and R, respectively. Finally, the state of the emerging photons effectively becomes 1 jHi ⊗ jli → pffiffiffi ðexp ðilαÞjLi þ exp ð−ilαÞjRiÞ ⊗ jli 2 ¼ ðcosðlαÞjHi þ sinðlαÞjViÞ ⊗ jli: (2) By comparing Eq. (2) with Eq. (1), we conclude that our device mimics the Faraday effect to rotate the polarization, and, more interestingly, the rotation angle θ ¼ lα is proportional to the OAM number of the input photons. Such OAM-to-polarization coupling effect, therefore, enables us to sort the OAM efficiently. For example, if α is preset at 90°, then an even OAM beam attains the horizontal polarization, while the odd OAM beam attains the vertical polarization. Thus, the even and odd OAM beams can be directly routed by the following PBS3. More generally, given two arbitrary OAM, l1 and l2 , no matter if they are positive or negative, integer or noninteger, if α ¼ 90 °=ðl1 − l2 Þ, then their polarizations will pose perpendicularly to each other and can be easily separated by PBS3. As is shown below, our experiment confirms the good performance of our device to serve as an even-odd
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OAM sorter, and, for the first time, to sort optical vortices of noninteger topological charges. Before showing our experimental results, we describe briefly the generation of noninteger optical vortices and multiple OAM superpositions using a computer-controlled spatial light modulator (SLM, Hamamatsu). A linearly polarized fundamental Gaussian light from a 5 mW, 633 nm HeNe laser is expanded and incident on the SLM. A frequently used design is to add a blazed grating modulo 2π to a spiral phase of expðilϕÞ, then we obtain a forked hologram, whose first-order diffracted beam carries lℏ OAM per photon [17]. This design is readily adapted to noninteger vortices of expðiMϕÞ, giving an additional radial discontinuity to the pattern [18]. Although the SLM we use is a phase-only modulator, it can be utilized to shape the intensity also. This is achieved by multiplying the phase hologram with the desired intensity distribution, and the hologram addressed by the SLM is given by [18,19] Φðr; ϕÞSLM ¼ ½Φðr; ϕÞDesired þ Φðr; ϕÞLinear mod2π × sin c2 ½1 − πIðr; ϕÞDesired ;
(3)
where Φðr; ϕÞDesired and Iðr; ϕÞDesired are the desired phase and intensity distributions, respectively, Φðρ; ϕÞLinear is the phase of the linear grating, and sin c2 ð·Þ accounts for the mapping of the phase depth to the diffraction efficiency of the spatially dependent blazing function. To generate the modified Laguerre-Gaussian (LG) beams carrying noninteger vortices, we choose Φðr; ϕÞDesired ¼ ðl − ð1=2ÞÞϕ and Iðr; ϕÞDesired ¼ jLGlp¼0 ðr; ϕÞj2 to obtain the desired hologram. Then we know that the firstorder diffracted beam carries the desired vortex of M ¼ l − ð1=2Þ. A typical hologram is shown in Fig. 1(c), which is used to generate a M ¼ 9=2 vortex. We can also prepare a variety of multiple OAM superpositions with selectivity. For example, by P setting Φðr; ϕÞDesired ¼ arg½ l¼6;7 LGlp¼0 ðr; ϕÞ and P Iðr; ϕÞDesired ¼ j l¼6;7 LGlp¼0 ðr; ϕÞj2 , we readily prepare a quadruple superposition of l ¼ 6, 7. Sorting the OAM of even and odd numbers.—With two Dove prisms simply preset at 90°, our device is able to serve as an even-odd OAM sorter. We first guide a LG beam bearing a single OAM into our device, where the polarization undergoes the following transformation: jHi ⊗ jl ¼ eveni → jHi ⊗ jl ¼ eveni and jHi ⊗ jl ¼ oddi → jVi ⊗ jl ¼ oddi. Thus, the even and odd OAM emerge from the output ports A and B of PBS3, respectively. Two color CCD cameras are placed there to monitor the intensity of the light. We present the experimental observations in Fig. 2 from which one can see that our device can sort both positive and negative OAM. The reliability of our results can also be verified pffiffiffiby the bright intensity rings, whose radius scales with l for the LG beams [20]. We further investigate a more general case when the input light is a mixture of multiple OAM. It is expected that
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FIG. 2 (color online). Experimental results of the even-odd OAM sorter with a single OAM input. The even OAM, e.g., l ¼ −2, 12, 20, appear at port A while the odd OAM, e.g., l ¼ 5, −17, appear at port B.
all the even OAM modes will assemble at port A while the odd ones will assemble at port B. Without loss of generality, we consider the input light comprising two pairs of opposite OAM. The intensity pattern of the superposition of opposite l OAM essentially results from the interference of an expðilϕÞ beam with its mirror image, where the complete constructive and destructive interference occurs at the angle ϕ determined by cosð2lϕÞ ¼ 1, respectively, and, as a consequence, a pattern like a 2l-petal flower is formed [21]. This also suggests a way to acquire the information about the OAM numbers straightforwardly by counting the petal numbers. The flower patterns in the upper and bottom panels in Fig. 3 are recorded at ports A and B, respectively, each of which are exactly the superposition of opposite l OAM, and the petal numbers reveal the l values. For example, for l ¼ 6 OAM, a flower of 12 petals is observed at port A. While for l ¼ 5, a ten-petal flower is at port B. Sorting the vortices of noninteger charges.—The terms of the optical vortices and OAM are often, yet not always correctly, used interchangeably. They are both associated with the light beams with an expðiMϕÞ phase structure [22]. For integer M, the number of the OAM and the topological charge of the vortex, indeed, take the same value M. However, for noninteger M, the two quantities are
FIG. 3 (color online). Experimental results of the even-odd OAM sorter with multiple OAM input. The top and bottom panels are the flower patterns with different petal numbers recorded at ports A and B, respectively.
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generally unequal [18]. At this point, it is more appropriate to refer to the beams with noninteger charges as noninteger vortices rather than noninteger OAM [22]. Here we report a proof-of-principle experiment to sort the optical vortices, e.g., M 1 and M2 . According to Eq. (2), the resultant polarization of the M 1 vortex beam points at angle M 1 α while that of the M2 vortex beam points at M2 α. Similarly, by presetting α ¼ 90 °=ðM 1 − M 2 Þ, we are able to make their polarizations pose orthogonally to each other. We employ an additional half-wave plate (HWP) in its fast axis aligning at angle M 1 α=2 in front of PBS3 to perform a unitary polarization rotation: M 1 α → H and M2 α → V. Subsequently, they can be easily separated by PBS3. We take the half-integer vortices, i.e., M ¼ l − ð1=2Þ, for example, to demonstrate the application. In order to distinguish these half-integer vortices from their intensity patterns, we modify the standard LG beams to bear an additional phase discontinuity, whose light field is readily tailored as uM ðr; ϕÞ ¼
1 −i ϕ : 2
LGlp¼0 ðr; ϕÞ exp
(4)
Because of the phase discontinuity, the propagation of the vortex beam will give rise to a radial opening to the ring, in our case, a line of low intensity along the ϕ ¼ 0 direction. By setting α ¼ 90 ° and HWP at 22.5 °, we observe that the vortices M ¼ 1=2, 5=2, 9=2 emerge from port A, while M ¼ 3=2, 7=2 escape from port B, as shown in Fig. 4. Besides, one can see that the ring gaps of the vortices from port A always face right, whereas those from port B always face left. This is because an additional reflection is introduced by PBS3 when the beams are recorded at port B. As far as we know, this is the first experiment to sort the optical vortices of noninteger charges, which holds promise for quantum applications,
FIG. 4 (color online). Experimental results for sorting the halfinteger optical vortices. The top row shows the intensity profiles of the half-integer vortices carried by the modified LG beams. The middle and bottom rows record the intensity profiles of the vortices output from ports A and B, respectively.
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as they correspond to the vector states defined in the highdimensional OAM Hilbert space [23–26]. Of interest is that our device can be effectively used for the generation of single-photon spin-orbit entanglement. It is noted that the generalized concept of entanglement is not limited to different particles but is applicable to different degrees of freedom in single particles [27,28]. The ability of our device to induce the OAM-to-polarization coupling provides a way to create and manipulate the single-photon entanglement. As was shown above, the even and odd OAM will lead to the horizontal and vertical polarizations, respectively. In other words, a single-photon hybrid entangled state is created, 1 jψi ¼ pffiffiffi ðjHi ⊗ jl ¼ eveni þ jVi ⊗ jl ¼ oddiÞ; (5) 2 which should be described in terms of a tensor product Hilbert space, H ¼ Hs ⊗ Ho , where Hs and Ho are the disjoint Hilbert spaces corresponding to the degrees of freedom of the spin and OAM, respectively. We carry out another experiment by using a low-noise electron multiplier CCD (EMCCD) camera (E2 V) instead of the color CCD camera, to show the ability of our device working at the photon-count level. This is achieved by inserting a series of neutral-density filters to attenuate the laser power down to a very faint level, with the average photon flux about 1.0 per pixel. As shown in Fig. 5, the observed single rings and petal-like flowers are still of clear edge and good contrast. These results imply that our scheme may be applied to engineer hybrid qudit states encoded in both polarization and OAM and to realize a hybrid space of much higher dimensions [29]. It has been recognized that using several degrees of freedom of single photons has potential in building a deterministic quantum information processor [30,31]. A natural extension of our scheme is to sort arbitrarily many OAM states by cascading multiple units of our devices. Based on such an extension, it is also
FIG. 5 (color online). Experimental results when working at the photon-count level. The left panel shows the intensity simulations of the input OAM mixtures. The middle and right panels display the patterns recorded by EMCCD at ports A and B, respectively.
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possible to employ both polarization and OAM in the mode division multiplexing and demultiplexing for high-capacity optical communications [32,33]. Besides, because of the spin-orbit coupling effect, it may have potential in analyzing the hyperentanglement for photon pairs entangled simultaneously in the polarization and OAM and in realizing a quantum interface between the OAM and path entanglement for high-dimensional quantum information transfer [34]. In summary, we have devised an interferometric device to mimic the Faraday rotation to sort the OAM of light, and the sorting algorithm is insensitive to the positive and negative OAM values. We have sorted both a single OAM and a mixture of multiple OAM into individual even and odd subsets and observed the characteristic single bright rings and the petal-like flowers, which verifies the reliability of our scheme. Furthermore, we succeeded in sorting the optical vortices of noninteger topological charges for the first time. The OAM-to-polarization coupling effect also enables our device to create the single-photon hybrid entanglement when working at the photon-count level. The potential of our work in highdimensional quantum information science was briefly discussed. We are grateful to Professor Miles Padgett and his optics group at the University of Glasgow for their kind help, and we thank Dr. Xiancong Lu for a careful reading of the manuscript. This work is supported by the National Natural Science Foundation of China (NSFC) (Grant No. 11104233), the Fundamental Research Funds for the Central Universities (Grants No. 2011121043, No. 2012121015), the Doctoral Fund of the Ministry of Education of China (Grant No. 2011012112003), and the program for New Century Excellent Talents in University of China (Grant No. NCET-13-0495).
*
[email protected] [1] M. Padgett, J. Courtial, and L. Allen, Phys. Today 57, 35 (2004). [2] J. H. Poynting, Proc. R. Soc. A 82, 560 (1909). [3] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992). [4] S. M. Barnett and R. Zambrini, in Orbital Angular Momentum of Light in Quantum Imaging, edited by M. I. Kolobov (Springer-Verlag, New York, 2006), p. 290. [5] G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2001). [6] G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
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[7] S. Franke-Arnold, L. Allen, and M. Padgett, Laser Photonics Rev. 2, 299 (2008). [8] J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002). [9] J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. FrankeArnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004). [10] G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett. 101, 100801 (2008). [11] J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, Phys. Rev. Lett. 105, 053904 (2010). [12] G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010). [13] M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, Nat. Commun. 4, 2781 (2013). [14] M. Faraday, in Faradays Diary, Nov 12, 1839 June 26, 1947, edited by T. Martin (Bell, London, 1933), Vol. 4. [15] M. Mansuripur, Opt. Photonics News 10, 32 (1999). [16] L. Chen and W. She, New J. Phys. 11, 103002 (2009). [17] S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, New J. Phys. 6, 103 (2004). [18] J. Leach, E. Yao, and M. J. Padgett, New J. Phys. 6, 71 (2004). [19] L. Chen, W. Zhang, Q. Lu, and X. Lin, Phys. Rev. A 88, 053831 (2013). [20] M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995). [21] M. Padgett, J. Arlt, N. Simpson, and L. Allen, Am. J. Phys. 64, 77 (1996). [22] M. R. Dennis, K. O’Holleran, and M. J. Padgett, Prog. Opt. 53, 293 (2009). [23] S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, Phys. Rev. Lett. 92, 217901 (2004). [24] S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, Phys. Rev. Lett. 95, 240501 (2005). [25] J. B. Gotte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, J. Mod. Opt. 54, 1723 (2007). [26] L. Chen, J. Lei, and J. Romero, Light: Sci. Appl. 3, e153 (2014). [27] S. J. van Enk, Phys. Rev. A 72, 064306 (2005). [28] L. Chen and W. She, J. Opt. Soc. Am. B 27, A7 (2010). [29] S. Straupe and S. Kulik, Nat. Photonics 4, 585 (2010). [30] B. G. Englert, C. Kurtsiefer, and H. Weinfurter, Phys. Rev. A 63, 032303 (2001). [31] M. Fiorentino and F. N. C. Wong, Phys. Rev. Lett. 93, 070502 (2004). [32] G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004). [33] J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Nat. Photonics 6, 488 (2012). [34] R. Fickler, R. Lapkiewicz, M. Huber, M. Lavery, M. Padgett, and A. Zeilinger, arXiv:1402.2423.
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Mimicking Faraday Rotation to Sort the Orbital Angular Momentum of Light Wuhong Zhang, Qianqian Qi, Jie Zhou, and Lixiang Chen* Department of Physics and Laboratory of Nanoscale Condensed Matter Physics, Xiamen University, Xiamen 361005, People’s Republic of China (Received 12 February 2014; published 14 April 2014) The efficient separation of the orbital angular momentum (OAM) is essential to both the classical and quantum applications with twisted photons. Here we devise and demonstrate experimentally an efficient method of mimicking the Faraday rotation to sort the OAM based on the OAM-to-polarization coupling effect induced by a modified Mach-Zehnder interferometer. Our device is capable of sorting the OAM of positive and negative numbers, as well as their mixtures. Furthermore, we report the first experimental demonstration to sort optical vortices of noninteger charges. The possibility of working at the photon-count level is also shown using an electron-multiplying CCD camera. Our scheme holds promise for quantum information applications with single-photon entanglement and for high-capacity communication systems with polarization and OAM multiplexing. DOI: 10.1103/PhysRevLett.112.153601
PACS numbers: 42.50.Tx, 07.60.Ly, 42.25.Ja, 42.79.Sz
The study of the angular momentum of light has a long history [1]. Early in 1909, Poynting first associated spin angular momentum with circular polarization [2]. However, it was only in 1992 that Allen et al. [3] recognized that a light beam with a helical phase front of expðilϕÞ carried a well-defined orbital angular momentum (OAM) of lℏ per photon, where ϕ is the azimuthal angle and l is the OAM quantum number. For a single photon, the spin and OAM are almost separable, at least within the paraxial approximation, so it is permissible to write down a state vector in the form jσi ⊗ jli as eigenstates of the angular momentum operator: Jˆz jσi ⊗ jli ¼ ðσ þ lÞjσi ⊗ jli, where the spin (σℏ) and orbital (lℏ) contributions to the total angular momentum are readily identified [4]. As the eigenstates of photon spin, the left- and right-handed circular polarizations are indicated as jσ ¼ þ1i ¼ jLi and jσ ¼ −1i ¼ jRi, respectively. While for the OAM eigenstate jli, theoretically, l can take any integers so that twisted photons offer the opportunity to realize a high-dimensional Hilbert space [5]. The past two decades have witnessed a rapidly growing interest in OAM ranging from optical manipulation and free-space communications, to quantum optics and quantum information [6,7]. Recently, considerable attention was also paid to the OAM separation, which plays a fundamental role in a variety of applications with twisted photons. Leach et al. [8] devised an interferometric method that could distinguish individual photons in different OAM states and route single photons according to the modulus of the individual OAM. They further extended this method to sort photons according to their spin, orbital, or total angular momentum [9]. Based on a multipoint interferometer, Berkhout and Beijersbergen [10] presented an efficient method for probing the OAM from astronomical objects. Additionally, Hickmann et al. [11] succeeded in revealing the magnitude 0031-9007=14=112(15)=153601(5)
and sign of the topological charge by observing a triangular lattice of spots diffracted by a triangular aperture. As many photons are required to form the full interferograms, these schemes in Refs. [10,11] could not be effectively applied to sort the OAM states at the single-photon level. By a clever geometric transformation, Berkhout et al. [12] achieved the efficient sorting of the OAM states. Mirhosseini et al. [13] recently reported the enhancement of the mode separation efficiency for these transformed OAM. It is noted that the OAM is inevitably changed during the mode transformation. In contrast, in the interferometric method outlined by Leach et al. [8,9], the helical phase structures are well preserved such that they can be used further for information processing with the OAM. Based on the OAMto-polarization coupling effect induced by a modified Mach-Zehnder interferometer, here we demonstrate a new experimental method to sort the OAM efficiently, which mimics the Faraday effect to rotate the polarization of a light beam by an angle proportional to its own OAM number. We distinguish our sorting scheme from those also employing the interferometric technique, where they took advantage of the phase symmetry of the OAM beams but did not involve the OAM-to-polarization coupling effect [8,9]. Our scheme may have potential in quantum information protocols with single-photon entanglement and in optical communications with both polarization and OAM multiplexing. Discovered by Michael Faraday in 1845, the Faraday effect describes a rotation of polarization caused by the interaction of light with a magnetic field in a chiral material [14]. The rotation angle θ is proportional to the magnetic field B in the direction of propagation, namely, θ ¼ vBd, where v is the Verdet constant for the material, and d is the length of interaction, see Fig. 1(a). This effect can be well understood from the theory of circular birefringence (see
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FIG. 1 (color online). The function of our device. (a) Polarization rotation due to the Faraday effect. The rotation angle θ is proportional to the magnetic field B and the interaction length d. (b) Polarization rotation produced by our device. The rotation angle θ is proportional to the OAM number l and the relative orientation α of Dove prisms. (c) Experimental setup for mimicking the Faraday effect to sort the OAM (see the text for more details), where the inset shows the hologram addressed by the SLM to generate a fractional vortex of M ¼ 9=2.
Ref. [15] and references therein). An incident horizontal polarization H can be decomposed into an equally weighted superposition of left- and right-handed p circularly ffiffiffi polarized components, e.g., jHi ¼ ðjLi þ jRiÞ= 2 (note here the polarization change is induced by the magnetooptic effect such that the OAM ket have been trivially discarded). When propagating, the two components experience different refractive indices, nL and nR , respectively, so that they suffer a relative phase shift, △φ ¼ ð2π=λÞðnR − nL Þd, which equivalently induces polarization rotation of the output light by an angle of θ ¼ △φ=2. The effect of circular birefringence can be described as follows: 1 jHi → pffiffiffi ðexp ðiθÞjLi þ exp ð−iθÞjRiÞ ¼ cos θjHi 2 þ sin θjVi;
(1)
where θ ¼ △φ=2 ¼ vBd. As illustrated in Fig. 1(b), our interferometric device can realize the analogous polarization rotation as the Faraday effect. The rotation angle θ can be expressed as θ ¼ lα, where l is the OAM number of the incoming light, and α is the relative orientation of the two Dove prisms in Fig. 1(c). Our approach benefits from the OAM-to-polarization coupling effect, which was first outlined theoretically by us to encode the OAM onto multiple spin states [16]. We assume the incoming photons have a definite horizontal polarization and a definite OAM number l, whose pffiffiffi state can then be denoted as jHi ⊗ jli ¼ ð1= 2ÞðjLi þ jRiÞ ⊗ jli. We first need to separate the left- and right-handed circular components, L and R, and then impart them with a different phase delay, which is OAM dependent. For convenience, the circular polarization beam splitter is substituted by a traditional polarizing beam splitter (PBS1) with a preposed quarter-wave plate
(QWP1). The QWP1 converts L and R to the horizontal and vertical polarizations, H and V, respectively, while PBS1 transmits H photons and reflects V photons. After entering the interferometer, the H and V components traverse the embedded Dove prisms, DP1 and DP2, respectively. The total internal reflection from the base of each prism flips the transverse cross section of the transmitted image. If DP1 and DP2 are set to have a relative orientation angle α, then a phase difference of exp ði2lαÞ is imparted to the H and V components. The two components recombine in PBS2, and the following quarter-wave plate (QWP2) converts H and V back to L and R, respectively. Finally, the state of the emerging photons effectively becomes 1 jHi ⊗ jli → pffiffiffi ðexp ðilαÞjLi þ exp ð−ilαÞjRiÞ ⊗ jli 2 ¼ ðcosðlαÞjHi þ sinðlαÞjViÞ ⊗ jli: (2) By comparing Eq. (2) with Eq. (1), we conclude that our device mimics the Faraday effect to rotate the polarization, and, more interestingly, the rotation angle θ ¼ lα is proportional to the OAM number of the input photons. Such OAM-to-polarization coupling effect, therefore, enables us to sort the OAM efficiently. For example, if α is preset at 90°, then an even OAM beam attains the horizontal polarization, while the odd OAM beam attains the vertical polarization. Thus, the even and odd OAM beams can be directly routed by the following PBS3. More generally, given two arbitrary OAM, l1 and l2 , no matter if they are positive or negative, integer or noninteger, if α ¼ 90 °=ðl1 − l2 Þ, then their polarizations will pose perpendicularly to each other and can be easily separated by PBS3. As is shown below, our experiment confirms the good performance of our device to serve as an even-odd
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OAM sorter, and, for the first time, to sort optical vortices of noninteger topological charges. Before showing our experimental results, we describe briefly the generation of noninteger optical vortices and multiple OAM superpositions using a computer-controlled spatial light modulator (SLM, Hamamatsu). A linearly polarized fundamental Gaussian light from a 5 mW, 633 nm HeNe laser is expanded and incident on the SLM. A frequently used design is to add a blazed grating modulo 2π to a spiral phase of expðilϕÞ, then we obtain a forked hologram, whose first-order diffracted beam carries lℏ OAM per photon [17]. This design is readily adapted to noninteger vortices of expðiMϕÞ, giving an additional radial discontinuity to the pattern [18]. Although the SLM we use is a phase-only modulator, it can be utilized to shape the intensity also. This is achieved by multiplying the phase hologram with the desired intensity distribution, and the hologram addressed by the SLM is given by [18,19] Φðr; ϕÞSLM ¼ ½Φðr; ϕÞDesired þ Φðr; ϕÞLinear mod2π × sin c2 ½1 − πIðr; ϕÞDesired ;
(3)
where Φðr; ϕÞDesired and Iðr; ϕÞDesired are the desired phase and intensity distributions, respectively, Φðρ; ϕÞLinear is the phase of the linear grating, and sin c2 ð·Þ accounts for the mapping of the phase depth to the diffraction efficiency of the spatially dependent blazing function. To generate the modified Laguerre-Gaussian (LG) beams carrying noninteger vortices, we choose Φðr; ϕÞDesired ¼ ðl − ð1=2ÞÞϕ and Iðr; ϕÞDesired ¼ jLGlp¼0 ðr; ϕÞj2 to obtain the desired hologram. Then we know that the firstorder diffracted beam carries the desired vortex of M ¼ l − ð1=2Þ. A typical hologram is shown in Fig. 1(c), which is used to generate a M ¼ 9=2 vortex. We can also prepare a variety of multiple OAM superpositions with selectivity. For example, by P setting Φðr; ϕÞDesired ¼ arg½ l¼6;7 LGlp¼0 ðr; ϕÞ and P Iðr; ϕÞDesired ¼ j l¼6;7 LGlp¼0 ðr; ϕÞj2 , we readily prepare a quadruple superposition of l ¼ 6, 7. Sorting the OAM of even and odd numbers.—With two Dove prisms simply preset at 90°, our device is able to serve as an even-odd OAM sorter. We first guide a LG beam bearing a single OAM into our device, where the polarization undergoes the following transformation: jHi ⊗ jl ¼ eveni → jHi ⊗ jl ¼ eveni and jHi ⊗ jl ¼ oddi → jVi ⊗ jl ¼ oddi. Thus, the even and odd OAM emerge from the output ports A and B of PBS3, respectively. Two color CCD cameras are placed there to monitor the intensity of the light. We present the experimental observations in Fig. 2 from which one can see that our device can sort both positive and negative OAM. The reliability of our results can also be verified pffiffiffiby the bright intensity rings, whose radius scales with l for the LG beams [20]. We further investigate a more general case when the input light is a mixture of multiple OAM. It is expected that
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FIG. 2 (color online). Experimental results of the even-odd OAM sorter with a single OAM input. The even OAM, e.g., l ¼ −2, 12, 20, appear at port A while the odd OAM, e.g., l ¼ 5, −17, appear at port B.
all the even OAM modes will assemble at port A while the odd ones will assemble at port B. Without loss of generality, we consider the input light comprising two pairs of opposite OAM. The intensity pattern of the superposition of opposite l OAM essentially results from the interference of an expðilϕÞ beam with its mirror image, where the complete constructive and destructive interference occurs at the angle ϕ determined by cosð2lϕÞ ¼ 1, respectively, and, as a consequence, a pattern like a 2l-petal flower is formed [21]. This also suggests a way to acquire the information about the OAM numbers straightforwardly by counting the petal numbers. The flower patterns in the upper and bottom panels in Fig. 3 are recorded at ports A and B, respectively, each of which are exactly the superposition of opposite l OAM, and the petal numbers reveal the l values. For example, for l ¼ 6 OAM, a flower of 12 petals is observed at port A. While for l ¼ 5, a ten-petal flower is at port B. Sorting the vortices of noninteger charges.—The terms of the optical vortices and OAM are often, yet not always correctly, used interchangeably. They are both associated with the light beams with an expðiMϕÞ phase structure [22]. For integer M, the number of the OAM and the topological charge of the vortex, indeed, take the same value M. However, for noninteger M, the two quantities are
FIG. 3 (color online). Experimental results of the even-odd OAM sorter with multiple OAM input. The top and bottom panels are the flower patterns with different petal numbers recorded at ports A and B, respectively.
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generally unequal [18]. At this point, it is more appropriate to refer to the beams with noninteger charges as noninteger vortices rather than noninteger OAM [22]. Here we report a proof-of-principle experiment to sort the optical vortices, e.g., M 1 and M2 . According to Eq. (2), the resultant polarization of the M 1 vortex beam points at angle M 1 α while that of the M2 vortex beam points at M2 α. Similarly, by presetting α ¼ 90 °=ðM 1 − M 2 Þ, we are able to make their polarizations pose orthogonally to each other. We employ an additional half-wave plate (HWP) in its fast axis aligning at angle M 1 α=2 in front of PBS3 to perform a unitary polarization rotation: M 1 α → H and M2 α → V. Subsequently, they can be easily separated by PBS3. We take the half-integer vortices, i.e., M ¼ l − ð1=2Þ, for example, to demonstrate the application. In order to distinguish these half-integer vortices from their intensity patterns, we modify the standard LG beams to bear an additional phase discontinuity, whose light field is readily tailored as uM ðr; ϕÞ ¼
1 −i ϕ : 2
LGlp¼0 ðr; ϕÞ exp
(4)
Because of the phase discontinuity, the propagation of the vortex beam will give rise to a radial opening to the ring, in our case, a line of low intensity along the ϕ ¼ 0 direction. By setting α ¼ 90 ° and HWP at 22.5 °, we observe that the vortices M ¼ 1=2, 5=2, 9=2 emerge from port A, while M ¼ 3=2, 7=2 escape from port B, as shown in Fig. 4. Besides, one can see that the ring gaps of the vortices from port A always face right, whereas those from port B always face left. This is because an additional reflection is introduced by PBS3 when the beams are recorded at port B. As far as we know, this is the first experiment to sort the optical vortices of noninteger charges, which holds promise for quantum applications,
FIG. 4 (color online). Experimental results for sorting the halfinteger optical vortices. The top row shows the intensity profiles of the half-integer vortices carried by the modified LG beams. The middle and bottom rows record the intensity profiles of the vortices output from ports A and B, respectively.
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as they correspond to the vector states defined in the highdimensional OAM Hilbert space [23–26]. Of interest is that our device can be effectively used for the generation of single-photon spin-orbit entanglement. It is noted that the generalized concept of entanglement is not limited to different particles but is applicable to different degrees of freedom in single particles [27,28]. The ability of our device to induce the OAM-to-polarization coupling provides a way to create and manipulate the single-photon entanglement. As was shown above, the even and odd OAM will lead to the horizontal and vertical polarizations, respectively. In other words, a single-photon hybrid entangled state is created, 1 jψi ¼ pffiffiffi ðjHi ⊗ jl ¼ eveni þ jVi ⊗ jl ¼ oddiÞ; (5) 2 which should be described in terms of a tensor product Hilbert space, H ¼ Hs ⊗ Ho , where Hs and Ho are the disjoint Hilbert spaces corresponding to the degrees of freedom of the spin and OAM, respectively. We carry out another experiment by using a low-noise electron multiplier CCD (EMCCD) camera (E2 V) instead of the color CCD camera, to show the ability of our device working at the photon-count level. This is achieved by inserting a series of neutral-density filters to attenuate the laser power down to a very faint level, with the average photon flux about 1.0 per pixel. As shown in Fig. 5, the observed single rings and petal-like flowers are still of clear edge and good contrast. These results imply that our scheme may be applied to engineer hybrid qudit states encoded in both polarization and OAM and to realize a hybrid space of much higher dimensions [29]. It has been recognized that using several degrees of freedom of single photons has potential in building a deterministic quantum information processor [30,31]. A natural extension of our scheme is to sort arbitrarily many OAM states by cascading multiple units of our devices. Based on such an extension, it is also
FIG. 5 (color online). Experimental results when working at the photon-count level. The left panel shows the intensity simulations of the input OAM mixtures. The middle and right panels display the patterns recorded by EMCCD at ports A and B, respectively.
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possible to employ both polarization and OAM in the mode division multiplexing and demultiplexing for high-capacity optical communications [32,33]. Besides, because of the spin-orbit coupling effect, it may have potential in analyzing the hyperentanglement for photon pairs entangled simultaneously in the polarization and OAM and in realizing a quantum interface between the OAM and path entanglement for high-dimensional quantum information transfer [34]. In summary, we have devised an interferometric device to mimic the Faraday rotation to sort the OAM of light, and the sorting algorithm is insensitive to the positive and negative OAM values. We have sorted both a single OAM and a mixture of multiple OAM into individual even and odd subsets and observed the characteristic single bright rings and the petal-like flowers, which verifies the reliability of our scheme. Furthermore, we succeeded in sorting the optical vortices of noninteger topological charges for the first time. The OAM-to-polarization coupling effect also enables our device to create the single-photon hybrid entanglement when working at the photon-count level. The potential of our work in highdimensional quantum information science was briefly discussed. We are grateful to Professor Miles Padgett and his optics group at the University of Glasgow for their kind help, and we thank Dr. Xiancong Lu for a careful reading of the manuscript. This work is supported by the National Natural Science Foundation of China (NSFC) (Grant No. 11104233), the Fundamental Research Funds for the Central Universities (Grants No. 2011121043, No. 2012121015), the Doctoral Fund of the Ministry of Education of China (Grant No. 2011012112003), and the program for New Century Excellent Talents in University of China (Grant No. NCET-13-0495).
*
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