## Coupling of spin and angular momentum of light in plasmonic vortex |

Optics Express, Vol. 20, Issue 9, pp. 10083-10094 (2012)

http://dx.doi.org/10.1364/OE.20.010083

Acrobat PDF (1526 KB)

### Abstract

We present that two distinct optical properties of light, the spin angular momentum (SAM) and the orbital angular momentum (OAM), can be coupled in the plasmonic vortex. If a plasmonic vortex lens (PVL) is illuminated by the helical vector beam (HVB) with the SAM and OAM, then those distinct angular momenta contribute to the generation of the plasmonic vortex together. The analytical model reveals that the total topological charge of the generated plasmonic vortex is given by a linear summation of those of the SAM and OAM, as well as the geometric charge of the PVL. The generation of the plasmonic vortex and the manipulation of the fractional topological charge are also presented.

© 2012 OSA

## 1. Introduction

3. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. **9**(3), 193–204 (2010). [CrossRef] [PubMed]

4. L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. **90**(16), 167401 (2003). [CrossRef] [PubMed]

5. F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. **3**(5), 324–328 (2007). [CrossRef]

6. Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. **5**(9), 1726–1729 (2005). [CrossRef] [PubMed]

7. H. L. Offerhaus, B. van den Bergen, M. Escalante, F. B. Segerink, J. P. Korterik, and N. F. van Hulst, “Creating focused plasmons by noncollinear phasematching on functional gratings,” Nano Lett. **5**(11), 2144–2148 (2005). [CrossRef] [PubMed]

8. S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. **86**(14), 3008–3011 (2001). [CrossRef] [PubMed]

9. A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett. **91**(18), 183901 (2003). [CrossRef] [PubMed]

6. Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. **5**(9), 1726–1729 (2005). [CrossRef] [PubMed]

7. H. L. Offerhaus, B. van den Bergen, M. Escalante, F. B. Segerink, J. P. Korterik, and N. F. van Hulst, “Creating focused plasmons by noncollinear phasematching on functional gratings,” Nano Lett. **5**(11), 2144–2148 (2005). [CrossRef] [PubMed]

10. H. Kim and B. Lee, “Diffractive slit patterns for focusing surface plasmon polaritons,” Opt. Express **16**(12), 8969–8980 (2008). [CrossRef] [PubMed]

13. B. Lee, I.-M. Lee, S. Kim, D.-H. Oh, and L. Hesselink, “Review on subwavelength confinement of light with plasmonics,” J. Mod. Opt. **57**(16), 1479–1497 (2010). [CrossRef]

14. K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. **101**(3), 030404 (2008). [CrossRef] [PubMed]

17. H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. **10**(2), 529–536 (2010). [CrossRef] [PubMed]

18. E. Lombard, A. Drezet, C. Genet, and T. W. Ebbesen, “Polarization control of non-diffractive helical optical beams through subwavelength metallic apertures,” New J. Phys. **12**(2), 023027 (2010). [CrossRef]

23. J. Trevino, H. Cao, and L. Dal Negro, “Circularly symmetric light scattering from nanoplasmonic spirals,” Nano Lett. **11**(5), 2008–2016 (2011). [CrossRef] [PubMed]

17. H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. **10**(2), 529–536 (2010). [CrossRef] [PubMed]

## 2. Helical vector beam

*l*is the helix winding number of 2π cycles in phase around the circumference. Light with the helix winding number

*l*is regarded as a flux of photons with the OAM

*l*correspond to the direction of counter-clockwise and clockwise rotating electromagnetic fields, respectively. There are various kinds of optical beam with helicity such as the Laguerre-Gaussian beams and the Bessel beam [25].

*σ*=

*s*= + 1 for the right-handed circular polarization (RCP) and

*s*= −1 for the left-handed circular polarization (LCP), respectively [15

15. Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. **101**(4), 043903 (2008). [CrossRef] [PubMed]

15. Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. **101**(4), 043903 (2008). [CrossRef] [PubMed]

26. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. **27**(5), 285–287 (2002). [CrossRef] [PubMed]

*l*varies from −1 to 2 with increment of 1. The abscissa and the ordinate denote the

*x*and

*y*coordinates with arbitrary unit, respectively. In addition to the polarization factor, the azimuthal phase term of

*J*(

_{j}*ρ*), where

*J*and

_{j}*ρ*are the

*j*th-order Bessel function with the first kind and the radial coordinate (

*ρ*= (

*x*

^{2}+

*y*

^{2})

^{1/2}). Even though the polarization is different for each panel, they look quite similar in some cases. For example, the LCP with

*l*= 1 (Fig. 1(c)), the radial polarization with

*l*= 0 (Fig. 1(f)), and the RCP with

*l*= −1 (Fig. 1(i)) look similar at a fixed time. However, the field distributions with time development show definitely unique patterns. We thus present a movie clip for Fig. 1 that clearly shows the electric field distribution with varying time. In the movie clip, the handedness symmetry is observed for two combinations: the LCP with

*l*= 1 (Fig. 1(c)) and the RCP with

*l*= −1 (Fig. 1(i)), and the LCP with

*l*= −1 (Fig. 1(a)) and the RCP with

*l*= 1 (Fig. 1(k)). In addition, it will be shown below that there are such combinations that result in the same topological charges due to coupling of the spin and angular momenta in the PVL. From these observations, an important remark can be made that the polarization states and helix winding number of phase can be separately manipulated so that we obtain any arbitrary combination of SAM and OAM.

## 3. SP excitation at metal slit

20. P. S. Tan, X.-C. Yuan, J. Lin, Q. Wang, T. Mei, R. E. Burge, and G. G. Mu, “Surface plasmon polaritons generated by optical vortex beams,” Appl. Phys. Lett. **92**(11), 111108 (2008). [CrossRef]

*r*from the center of the PVL to the slit with the azimuthal angle

*a*is the inner radius of the PVL,

*m*is the geometrical charge of the PVL, and

*λ*is the effective wavelength of SPs. According to this equation, the distance from slit to the center is proportional to the value of azimuthal angle

_{SP}*x*,

*y*) represents the remainder of the division of

*x*by

*y*. The PVL slit pattern based on the modified equation has

*m*partial slit segments and the distance between the center of geometry and slits is in a range from

*a*to (

*a*+

*λ*), providing a moderate uniformity of SP energy.

_{SP}_{||}of incident light toward the center of the PVL reaches its maximum (

*l*= 0. For the initial state with

*ωt*= 0, the in-plane electric field heads toward the center of the PVL. The resultant phase is

*ωt*= π) exhibits the in-plane electric field directed outward of the PVL and the resultant phase is

*ωt*= 3π/2 gives rise to

*l*= 1, Fig. 3(b)), then the SPs excited at the slit also carry the phase retardation proportional to

*m*= 1 illuminated by the full-symmetric HVB with the radial polarization and no helical phase. Note that the initial phases at the slit are the same. In this case, however, the distance from the center of the PVL to the slit is linearly proportional to the azimuthal angle of the slit. Therefore, another type of phase retardation occurs: the propagation phase retardation. This relation is represented by the arrows between the PVL slit and the hypothetical circle (denoted by dashed line). Note that the number of arrows is dependent on the azimuthal angle. The resultant phase distribution along the dashed circle exhibits the same phase distribution as those of the case in Figs. 3(b) and 3(c). From discussion above based on the phase diagram, it can be inferred that the phase distribution of the SPs inside the PVL is affected by three factors: the helical phase of the HVB, the polarization state, and the geometrical charge of the PVL.

## 4. Numerical simulation

27. H. Kim, I.-M. Lee, and B. Lee, “Extended scattering-matrix method for efficient full parallel implementation of rigorous coupled-wave analysis,” J. Opt. Soc. Am. A **24**(8), 2313–2327 (2007). [CrossRef] [PubMed]

*m*is 1. The wavelength of laser is 660 nm and the corresponding wavelength of SPs is 629 nm. The slit width is 250 nm, thickness of gold layer is 300 nm, and inner radius,

*a*, is 4

*µ*m. The laser beam is assumed to be modulated by the phase SLM. When the modulated laser beam passing through a quarter wave plate or a radial polarizer illuminates the PVL from the backside, SPs are excited at the slit and propagate toward the center of the geometry. The simulation is carried out with altering three variables: the variable for geometrical effect of the PVL

*m*, the polarization effect

*s*, and the helix winding number of azimuthal phase term

*l*. The key factor governing the properties of the plasmonic vortex is the topological charge. We can extract the topological charge by measuring the primary ring size of the generated plasmonic vortex [17

17. H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. **10**(2), 529–536 (2010). [CrossRef] [PubMed]

*m*= 1) which is shown in Fig. 4(b). In this figure, the top panel corresponds to the LCP (

*s*= −1), the middle to the radial polarization (

*s*= 0), and the bottom to the RCP (

*s*= + 1). From the left panel, helix winding number

*l*varies from −1 to 2 with increment of one. For the sake of clear comparison, the arrangement of panels in Fig. 5 is the same as that in Fig. 1. As shown in Fig. 5, for the PVLs with the topological charge of 2, such as (

*l*,

*m*,

*s*) = (1, 1, 0), (0, 1, 1), and (2, 1, −1), the primary ring sizes of the plasmonic vortices are the same. Other PVL group with the same topological charge exhibited the same vortex radius as expected.

*s*and the helix winding number

*l*are fixed, the geometric effect

*m*of the PVL is varied. It turns out that the change of the primary ring sizes of the plasmonic vortex also depends on the variation in geometric charge of PVL as shown in Fig. 6(a) with (

*l*,

*m*,

*s*) = (0, 1, 0) and Fig. 6(b) with (

*l*,

*m*,

*s*) = (0, 3, 0). In this case the total vortex charges are different from each other, resulting in the different sizes of the plasmonic vortices. However, even if the sum of the polarization effect and the helix winding number is different, vortex radii can be the same when PVLs are appropriately used as shown in Fig. 6(c) with (

*l*,

*m*,

*s*) = (−1, 3, −1). We note that the primary ring sizes of the plasmonic vortices in Figs. 6(a) and 6(c) are the same. This is because the total vortex charges of those configurations are the same (

*j*= 1).

*Q*are aligned along the circle with the radius

_{SRC}*b*and the angle

*k*. When an observation point is represented by

_{SP}*P*(

*r*,

*φ*) inside the PVL, the wave function

*f*(

*r*,

*φ*) at

*P*(

*r*,

*φ*) is given byBy using Taylor’s expansion and the integral form of the Bessel function of the first kind, we obtain approximation equation,where

*j*=

*l*+

*m*+

*s*. The quantitative plasmonic vortex pattern is given with intensity of electrical field, |

*f*(

*r*,

*φ*)|. Therefore, the primary ring size is given with solution of the Bessel function of the first kind.

*l*. Each figure displays different geometric effect, such as

*m*= −1 in Fig. 7(a) and

*m*= 0 in Fig. 7(b). The blue dashed, green dotted, and red solid lines with diamond, cross, and circle markers denote results of the RCP, the radial polarization, and the LCP, respectively. The left vertical axis of ordinates displays radius of vortex and the right vertical axis, solution of Bessel function of the first kind. As expected, the result shows the radii of vortices are identical when the PVLs have same topological charge

*j*, which is the sum of geometrical charge

*m*, polarization effect

*s*, and azimuthal phase term

*l*. Moreover, this figure shows that the primary ring sizes of simulation result are almost identical to solution of the Bessel function of the first kind.

*l*,

*m*and

*s*are given, we can estimate the primary ring size. On the contrary, even though if we know the primary ring size of vortex, it is difficult to individually estimate the parameter

*l*,

*m*and

*s*. The geometrical charge

*m*can be easily obtained by using microscope. When

*l*is 0, the method to find polarization of illuminated laser using PVL whose

*m*is 1 is reported [28

28. S. Yang, W. Chen, R. L. Nelson, and Q. Zhan, “Miniature circular polarization analyzer with spiral plasmonic lens,” Opt. Lett. **34**(20), 3047–3049 (2009). [CrossRef] [PubMed]

*l*is non-zero, however, it is difficult to distinguish

*l*and

*s*from

*l*+

*s*because

*l*and

*s*equally contribute to deciding the primary ring size.

*j*. However, the propagation loss of the SP wave becomes dominant for a PVL with a large diameter [6

6. Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. **5**(9), 1726–1729 (2005). [CrossRef] [PubMed]

## 5. Fractional plasmonic vortex

*j*,

*l*,

*m*, and

*s*are the integers. A question for the non-integer (or fractional) cases naturally arises. This section covers the case when those parameters have non-integer values and what would happen in that case. We also examine if the superposition rule (

*j*=

*l*+

*m*+

*s*) is still valid in the non-integer cases.

*l*and

*m*arise when the wavelength of the laser beam does not match that for the designed optical system. A spiral phase plate or an SLM induces a certain amount of phase delay and the amount of the modulated phase is intrinsically dependent on the wavelength of light. The parameters

*l*and

*m*are associated with the winding number of phase around the center axis, and thus they are integers when the modulated phase is an integer multiple of 2π. If the wavelength of incident light is different from the wavelength designed for the optical device, then the resultant amount of the modulated phase is not 2π anymore; it could go beyond or under 2π. For example, if the laser beam with the 600 nm wavelength illuminates the spiral phase plate or an SLM that induces 2π phase delay (

*l*= 1) for the 660 nm, then the phase modulation would exceed 2π and the resultant helix winding number would be greater than one (

*λ*

_{660}) for free space wavelength of 660 nm (

*m*= 1) is illuminated by a laser beam with a different wavelength of 600 nm, then the SP wave

*feels*the distance

*λ*

_{660}more than 2π, which leads to a non-integer

*m*(1 <

*m*< 2).

*l*or

*m*in the plasmonic vortex, we carried out numerical simulations. Let us first consider cases where either

*l*or

*m*is non-integer. If the helix winding number

*l*is not an integer, the phase discontinuities are generated by the HVB along the axis and they cause a singular line linking the discontinuities in free space [29

29. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. **6**, 71 (2004). [CrossRef]

30. S. H. Tao, W. M. Lee, and X.-C. Yuan, “Dynamic optical manipulation with a higher-order fractional bessel beam generated from a spatial light modulator,” Opt. Lett. **28**(20), 1867–1869 (2003). [CrossRef] [PubMed]

*l*, the PVL with fractional

*m*generates a singular line. Recall that the PVL is designed by using equation

*m*thus makes discontinuity along

*x*-axis. This discontinuity makes a singular line and breaks the plasmonic vortex pattern as shown in Fig. 8(a) with (

*l*,

*m*,

*s*) = (0, 1.7, 0). In order to explain the effect of fractional

*l*and

*m*, phase profiles inside of vortex pattern are displayed in Figs. 8(c)-8(f). The phase profile is obtained along the circle whose center is coincident with center of vortex pattern as shown in Fig. 8(b), which represents the phase of surface plasmon at the center of PVL. Figures 8(c) and 8(d) show the phase profiles in vortex patterns with (

*l*,

*m*,

*s*) = (0, 1.7, 0) and (

*l*,

*m*,

*s*) = (1.3, 0, 0), respectively. They show that the phase varies nonlinearly when

*m*and

*l*have fractional values.

*j*is determined by sum of

*l*,

*m*and

*s*when each value is integer (

*j*=

*l*+

*m*+

*s*) and this property originates from the superposition of the phase profile. It is not trivial whether the aforementioned superposition rule is still valid in the non-integer case. To examine this issue, we carried out simulations in which

*l*and

*m*are both non-integers whereas their sum

*l*+

*m*is an integer. Figure 8(e) shows the phase profile in vortex pattern with (

*l*,

*m*,

*s*) = (1.3, 1.7, 0). Note that this can be regarded as a superposition of (

*l*,

*m*,

*s*) = (0, 1.7, 0) (Fig. 8(c)) and (

*l*,

*m*,

*s*) = (1.3, 0, 0) (Fig. 8(d)). The topological charge

*j*of Fig. 8(e) is obtained as 3. It is noteworthy that its phase profile is the same as that of (

*l*,

*m*,

*s*) = (2, 0, 1) (Fig. 8(f)), whose topological charge

*j*is 3.

*l*and

*m*. One may be inclined to ask about the effect of the fractional spin parameter

*s*. By using the spin operator, the spin of the polarization is represented as

*s*= sin(2

*θ*)sin(

*φ*

_{y}_{-}

*), where the angle*

_{x}*θ*describes the angle between the amplitudes of the electric field components in the

*x*and

*y*directions, and

*φ*

_{y}_{-}

*is the relative phase difference of the electric field components in the*

_{x}*x*and

*y*directions. In the case of the RCP, which is represented by

*θ*is π/4,

*φ*

_{y-x}is

*π*/2, and

*s*= 1. For the

*x*-directional linear polarization (

*θ*is 0 and

*s*= 0. In this equation,

*s*is a real number between −1 and 1 when light has an elliptical polarization. Although the HVB has a fractional spin parameter

*s*, light with a fractional

*s*is unsuitable for PVL system. Because an elliptically polarized beam has different amplitudes between

*x*and

*y*directions, the intensity of excited SP waves is different with the position. Like the linearly polarized beam, the elliptically polarized HVB breaks uniformity of the SPs and it makes the asymmetric plasmonic vortex pattern. The radially polarized beam is represented by

**10**(2), 529–536 (2010). [CrossRef] [PubMed]

31. G. M. Lerman, L. Stern, and U. Levy, “Generation and tight focusing of hybridly polarized vector beams,” Opt. Express **18**(26), 27650–27657 (2010). [CrossRef] [PubMed]

*s*of radial polarization is obtained as 0. Because the radially polarized beam excites the SP waves uniformly, it can be used in PVL system. Consequently, spin parameter

*s*is one of only 1, 0 and −1 in the PVL system, and superposition property of PVL topology

*j*=

*l*+

*m*+

*s*is satisfied at real number

*l*,

*m*and

*s*= 1, 0, −1.

## 6. Conclusion

*s*, helix winding number

*l*and the geometric charge of PVL

*m*are explained.

*s*and

*l*are associated with the spin angular momentum and the orbital angular momentum, respectively. The topological charge

*j*in the plasmonic vortex is given by the superposition rule (

*j*=

*l*+

*m*+

*s*), where

*l*and

*m*are given by real numbers and

*s*is given by one of 1, 0, −1. It is shown that the quantitative primary ring size of vortex pattern is one of the solutions of the Bessel function of the first kind and the primary ring sizes of simulation result coincide well with the solutions. We believe that our finding can pave a novel way to the generation and manipulation of plasmonic hot spots and vortices.

## Acknowledgment

## References and links

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26. | Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. |

27. | H. Kim, I.-M. Lee, and B. Lee, “Extended scattering-matrix method for efficient full parallel implementation of rigorous coupled-wave analysis,” J. Opt. Soc. Am. A |

28. | S. Yang, W. Chen, R. L. Nelson, and Q. Zhan, “Miniature circular polarization analyzer with spiral plasmonic lens,” Opt. Lett. |

29. | J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. |

30. | S. H. Tao, W. M. Lee, and X.-C. Yuan, “Dynamic optical manipulation with a higher-order fractional bessel beam generated from a spatial light modulator,” Opt. Lett. |

31. | G. M. Lerman, L. Stern, and U. Levy, “Generation and tight focusing of hybridly polarized vector beams,” Opt. Express |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(050.4865) Diffraction and gratings : Optical vortices

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: March 12, 2012

Revised Manuscript: April 7, 2012

Manuscript Accepted: April 10, 2012

Published: April 18, 2012

**Citation**

Seong-Woo Cho, Junghyun Park, Seung-Yeol Lee, Hwi Kim, and Byoungho Lee, "Coupling of spin and angular momentum of light in plasmonic vortex," Opt. Express **20**, 10083-10094 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-9-10083

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