effect of polarization on symmetry of focal spot of a plasmonic lens

Effect of polarization on symmetry of focal spot of a plasmonic lens

Jun Wang, Wei Zhou and Anand K. Asundi

Precision Engineering and Nanotechnology Centre School of Mechanical and Aerospace Engineering

Nanyang Technological University, Singapore 639798 [email protected]; [email protected]

Abstract: When one uses a metallic nanostructure to excite surface plasmon polaritons (SPPs) for subwavelength focusing, there is the distinct shape of the light emerging from the plasmonic lens in the transverse directions due to 2D confinement of SPPs. To study the tuning of symmetry of a focal spot, we consider an annular plasmonic lens incident with a polarized plane wave having various polarization states, including circular polarization (CP), elliptical polarization (EP), and radial polarization (RP), compared to linear polarization (TM). We find that plasmonic modes are independent of the polarization approach and the different polarization states enable to tune transverse electric field. More specifically, for CP case where the phase

function ( )[ ] 0expRe =θi , the total-electric-field intensity is distributed

uniformly in the transverse plane, while for RP case where the phase

function ( )[ ] 1expRe =θi , a significant intensity contrast is observed in the

two diagonal directions. We show an agreement between the analytical description and numerical simulation.

2009 Optical Society of America

OCIS codes: (050.6624) Subwavelength structures; (240.6680) Surface plasmons; (260.5430) Polarization

References and links

1. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 1999).

2. S. Kawata, Near-field Optics and Surface Plasmon Polaritons (Springer, Berlin, 2001). 3. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin,

1988). 4. W. Knoll, “Interfaces and thin films as seen by bound electromagnetic waves,” Ann. Rev. Phys. Chem. 49,

569-638 (1998). 5. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189-193

(2006). 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, 1726-1729 (2005). 7. N. Fang, Z. Liu, T. J. Yen, and X. Zhang, “Regenerating evanescent waves from a silver superlens,” Opt.

Express 11, 682-687 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-11-7-682. 8. Z. Sun and H. K. Kim, “Refractive transmission of light and beam shaping with metallic nano-optic lenses,”

Appl. Phys. Lett. 85, 642-644 (2004). 9. L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, H. Hua, U. Welp, E. Brown, and C. W. Kimball,

“Subwavelength focusing and guiding of surface plasmons,” Nano. Lett. 5, 1399-1402 (2005). 10. Y. Fu, W. Zhou, L. E. N. Lennie, C. Du, and X. Luo, “Plasmonic microzone plate: superfocusing at visible

regime,” Appl. Phys. Lett. 87, 061124 (2007). 11. H. Ko, H. C. Kim, and M. Cheng, “Light transmission through a metallic/dielectric nano-optic lens,” J. Vac.

Sci. Technol. B 26, 62188-2191 (2008). 12. H. C. Kim, H. Ko, and M. Cheng, “Optical focusing of plasmonic Fresnel zone plate-based metallic structure

covered with a dielectric layer,” J. Vac. Sci. Technol. B 26, 2197-2203 (2008). 13. J. Wang and W. Zhou, “Subwavelength beaming using depth-tuned annular nanostructures,” J. Mod. Opt.

(DOI: 10.1080/09500340902812094, in press) (2009). 14. S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.

179, 1–7 (2000). 15. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

#109361 - $15.00 USD Received 30 Mar 2009; revised 23 Apr 2009; accepted 26 Apr 2009; published 29 Apr 2009

(C) 2009 OSA 11 May 2009 / Vol. 17, No. 10 / OPTICS EXPRESS 8137

mailto:[email protected]

mailto:[email protected]

http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-11-7-682

16. D. F. V. James and E. Wolf, “Determination of the degree of coherence of light from spectroscopic measurements,” Opt. Commun. 145, 1-4 (1998).

17. FDTD Solutions, from Lumerical Solutions Inc., http://www.lumerical.com. 18. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370-4379 (1972). 19. Z. W. Liu, J. M. Steele, H. Lee, and X. Zhang, “Tuning the focus of a plasmonic lens by the incident angle,”

Appl. Phys. Lett. 88, 171108 (2006).

1. Introduction

In optical imaging system, a lens’s spot size (∆r) and depth of field (DOF) are related to

numerical aperture (NA) and wavelength (λ) [1]: NAr /61.0 λ=∆ and NADOF /5.0 λ±= .

Either decreasing wavelength or increasing aperture can reduce the spot size, but the improvement in spatial resolution is achieved at the expense of reducing the depth of focus.

In the field of nano-optics [2], a surface bound traveling wave along metal-dielectric interface is referred to as surface plasmon polaritons (SPPs). The evanescent wave has shorter wavelength than excitation radiation according to the dispersion condition [3,4]. A plasmonic lens uses metallic nanostructures to excite SPPs and manipulates the coupling of SPPs into propagation waves for subwavelength focusing [5-9].

When a transverse polarization component (TM component) is used to excite SPPs in an annular plasmonic lens, beam profiles may be asymmetric in the transverse plane. For

example, at the wavelength λ = 632.8 nm, the focal spot of a plasmonic micro-zone plate has 250 nm in full-width half-maximum (FWHM) along the x direction and 210 nm along the y direction [10]. It is assumed that the x direction is the polarization direction and the values are calculated based on electric field intensity, |Ex|

2. It is noted that different polarization states

instead of a TM approach enable to tune the focal spot. In the work reported in Refs. 11-12, a circularly polarized plane wave is used as illumination source. But the roles of polarization in beaming effect are not clear.

Herein we address the issue by evaluating how the focal spot is tuned by different polarization states, including circular polarization (CP), elliptical polarization (EP), and radial polarization (RP), compared to TM polarization. An annular plasmonic lens having a depth-tuned structure [13] is adopted to provide a collimated focal beam for analysis and discussion. It is found that plasmonic modes are independent of the polarization state, that is, the resonant wavelengths remain the same as those of TM polarization, and the different polarization states enables to tune transverse electric field along the two diagonal directions.

2. Electric field modeling

Figure 1(a) is schematic of an annular plasmonic lens with a depth-tuned structure. With incidence of a TM polarized plane wave, asymmetric beam profiles emerging from the lens is observed in the transverse plane, as shown in Fig. 1(b) and 1(c). At the side lobe, the peak's amplitude change and shift are observed. It is reasoned that the shift in side lobes reflects the change in width of the main lobe, that is, the width of focal spot, as the energy can be more concentrated in the main lobe [14].

To make it clear, we first model the case using the electromagnetic approach [15]. The

electric vector for the TM polarization can be jEiEEyx

��

,, ⊥⊥⊥+= and that for the TE is

jEiEEyx

��

||,||,||+= , where i

�

and j�

are unit vectors in the x and y directions. x

E,⊥

and y

E,⊥

(x

E||,

and y

E||,

) are the electric-field components of the TM- (TE-) polarized light in the

corresponding directions in the free space. With a complex phase function ( )θiexp , 1−=i ,

the superposition of electric fields along an angular direction ϕ having unit vector ϕi�

is given

by ( )( ) ( ) ( )( )ϕϕθϕϕθϕ

sincosexpsincosexp||,||,,,|| yxyx

EEiEEiEiEE +++=⋅+=⊥⊥⊥

��

. 0=z

E

is assumed. The total electric field intensity along the angular direction is given as,

( )( )[ ].expRe2sin||sin||cos ||,

*

,||,

*

,

2

||

222222

yyxxyx EEEEiEEEEE ⊥⊥⊥ +++=+= θϕϕϕ (1)



http://www.lumerical.com/

In the equation, the asterisk indicates complex conjugation, and Ex and Ey are the electric-field components in the directions. We then have the following description and discussion.

Fig. 1. (a). An annular plasmonic lens having a depth-tuned structure (groove depths, t1 = 130 nm, t2 = 80 nm and t3 = 30 nm) milled in the output side of a Ag thin film (thickness, h = 200 nm). Other structure parameters are: central hole diameter = 200 nm, groove width = 200 nm, and groove period = 420 nm. The structure is incident with TM-polarized light having electric vector along the x direction. (b). Total electric-field intensity |E|2 of the collimated beam spot at

z = 1.35 µm. (c). Plot of |E|2 as a function of distance (normalized with respect to wavelength) measured along 4 different directions. It shows asymmetry of focal spot.

The spectral density in the output space is expressed by [15, 16],

( ) ( ) ( ) ( ) ( ) ( ) ( )ωωωωωωω||

*

||

**EEEEEES +==

⊥⊥, (2)

as function of angular frequency ω, where the asterisk has the same expression in Eq. (1), and the angular brackets denotes ensemble averaging. The spectrum in the free space is

independent of the polarization phase angle θ.

In addition, it is noted that the intensity profile in the x or y direction is equal to |E⊥|2 or

|E|||2, that is,

2

0

2

0 =⊥==

ϕϕEE and

2

2/||

2

2/ πϕπϕ === EE , if one substitutes ϕ = 0 or ϕ = π/2 into

Eq. (1). For an annular structure, 2

2/||

2

0 πϕϕ ==⊥= EE as the similar 2D confinement of SPPs

using the TM or TE polarization component appears along the polarization direction.



Furthermore, the beam profile along the diagonal directions in 45° and 135° with respect

to the x shows an intensity contrast, ( ) ( )2

135

2

45

2

135

2

45 °°°° ====+−=

ϕϕϕϕEEEEC . By substituting

Eq. (1) into the contrast expression, the contrast becomes,

[ ] ( )2

||

2

||,

*

,||,

*

,Recos2 EEEEEEC

yyxx++=

⊥⊥⊥θ , (3)

showing the normalized correlation (yyxx

EEEE||,

*

,||,

*

, ⊥⊥+ ) of the electric field between TM and

TE polarization states using the total electric field 2

||

2

EE +⊥

. In the case θ = π/3, the

contrast is the half of that in the case θ = 0.

400 450 500 550 600 650 700 7500

0.01

0.02

0.03

0.04

0.05

0.06

Wavelength (nm)

T (

arb

un

it)

λ1

λ2

TM: linear polarizationCP: circular polarizationEP: elliptical polarizationRP: radial polarization

Fig. 2. Plasmonic modes supported by the structure incident with a polarized plane wave, showing that the plasmonic modes have no shift with different polarization states including CP,

EP and RP, compared with those of TM-polarization. Herein corresponding phase difference θ

= π/2, π/3, and 0 in the CP, EP and RP cases.

3. Numerical results and discussion

In our simulation, the incident plane wave is formed by the superposition of two electric

components (TE and TM) with difference phase angles, θ = π/2, π/3, or 0 corresponding to the CP, EP or RP cases. The TM (TE) polarization component along the x (y) direction is assumed. Through the annular structure, the transmitted electromagnetic fields are analyzed by 3-D finite-difference time-domain (FDTD) simulations using absorption boundary condition. A commercial FDTD software package, FDTD Solutions, provided by Lumerical Solutions Inc. [17] is used. The dispersive data are based on the experimental data of Johnson and Christy on optical constants of noble metals [18]. For example, at the wavelength of

incidence, λ = 632.8 nm (frequency f = c/λ = 473.76 THz), the dielectric constant of the silver

material used in the FDTD is εm = -18.295 + 0.498i. Although the current polarization case considered here is right-hand, the discussion and analysis are also suitable for a left-hand polarization approach.

Figure 2 shows the transmission through the structure incident with a polarized plane wave

having various polarization states. The transmission peaks show the plasmonic modes at λ1

and λ2 (we reported elsewhere). They are independent of the polarization states, as described in Eq. (2). As reported [13], each plasmonic mode includes multiple wavelengths and, at the



Fig. 3. Total electric field (left) transmitted through the structure under illumination using different polarization states, including (a) CP, (b) EP, and (c) RP, showing the phase modulation effect on the beam profile (right) along the transverse direction in x and y and

diagonal directions along 45° and 135° with respect to the x. Refer to Fig. 1 for the directions.



wavelength λ1 (420-440 nm), the plasmonic lens is suitable for collimation. Hence we fix the wavelengths and calculate the focal spot under different polarization states.

At λ = 420 nm, Fig. 3 shows the total electric-field intensity |E|2 = |Ex|

2 + |Ey|

2 + |Ez|

2 in

the x-y plane along the longitudinal direction at z = 1.35 µm for the (a) EP, (b) CP, and (c) RP cases. The intensity of transverse electric field, |Ex|

2 + |Ey|

2, is significantly tuned. In the

figure, the intensity along the horizontal (x) is equal to that along the vertical (y) as described

in Section 2, while the intensity along the diagonal directions (45° and 135° with respect to

the x) is tuned. In Fig. 3(a), in the direction of 45°, the peak shift is observed at the side lobe,

which is <0.1λ for the EP case. And the beam in 45° is narrower than that in the x or y directions.

In addition, the phase function Re(exp(iθ)) in Eq. (3) indicates tuning capability.

Re(exp(iθ)) = 0.5, where θ = π/3, for the EP case, and the phase function becomes 0 for the CP and 1 for the RP case. For example, in Fig. 3(b), the uniformly distributed total-electric-

field intensity is observed in the x-y plane, while, in Fig. 3(c), the peak shifts 0.2λ in the 45°, larger than that for the EP case, and much narrow beam is observed in the same direction.

Table 1 shows the measured FWHM normalized with respect to the incident wavelength λ

= 440 nm. In CP case, an approximately symmetric focal spot forms, e.g. the beam profile in

the 4 directions is within the range of 1.18λ-1.20λ. The beam tuning in the diagonal directions

is most obvious when the phase function ( )[ ] 1expRe =θi and there is a much smaller spot

having 1.04λ along the 45°. In EP case, the degree of tuning is between that of CP and RP

cases and FWHM in the 45° is equal to 1.12λ, larger than 1.04λ of RP case but smaller than

1.20λ of CP case. This is similar with the change in intensity contrast vs. polarization states, as in Eq. (3).

Throughout this study, it is assumed that the input beam is normally incident and that the beam is not misaligned. It should be noted, however, that misalignment of the input beam may affect symmetry of the output beam [19]. It would be interesting to carry out further researches to understand how different modes of polarization affect the output beam profiles under the obliquely incident beams.

Table 1. Measured FWHM (×λ) in the x-y plane at z = 1.35 µm.

TM EP CP RP

X 1.06 1.18 1.18 1.14

Y 1.24 same as x same as x same as x

45° 1.18 1.12 1.20 1.04

135° same as 45° 1.24 1.20 1.24

4. Conclusion

We analytically described the transverse electric field transmitted through an annular plasmonic lens incident with a polarized plane wave and numerically measured the focal spot using a polarization approach. There is an agreement between the analytical description and simulation. The same plasmonic modes are observed for CP, EP, or RP polarization cases as for TM case. Using a polarized plane wave the transverse electric field is tuned; the tuning

effect on focus spot is observed along the diagonal directions in 45° and 135° with respect to

the x direction. Of the cases, RP approach forms the smallest focus spot along the 45° using

the phase function Re[exp(iθ )] = 1, showing maximum tuning capability, while CP approach

using Re[exp(iθ )] = 0 forms a symmetrically electric field distribution in the focal plane. Phase function indicates the tuning capability.



Acknowledgments

This work is financially supported by A*STAR (Agency for Science, Technology and Research), Singapore, under SERC Grant No. 072 101 0023.



effect of polarization on symmetry of focal spot of a plasmonic lens

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