## Abstract

A general design method of Fresnel zone plate-like surface plasmon polariton launching lenses (SPPLLs) with an arbitrary phase correction step is proposed, which can implement imaging between the far-field light and surface plasmon polaritons (SPPs). The imaging properties of half-wave SPPLLs, including the resolution power, object-image relationships, and aberrations, are investigated by using a simulation method based on Huygens-Fresnel principle with SPP point source model. The results show that SPPLLs are suitable for the connection between plasmonic and conventional diffraction-limited photonic devices.

© 2010 OSA

## 1. Introduction

Surface plasmons polaritons (SPPs), which are electromagnetic excitations propagating on the metal/dielectric interfaces, are well known as a promising approach to achieve nanoscale photonic integration, and open the possibilities for breaking the classical diffraction limit [1,2]. Recently, plasmon-based planar optical devices have drawn the attention of many researchers. Optical components such as Bragg mirrors [3,4], demultiplexer [5], SPP based far-field optical hyperlens [6], and superlens [7] have been demonstrated. Previous studies showed that nanoparticle ensembles on metal films can be used to focus SPPs [8]. Yin et al. [9] used nanoholes arranged on a quarter-circle on metal films to focus SPPs. Liu et al. [10] demonstrated the possibility of focusing SPPs by circular and elliptical structures milled into metallic films. For use in future applications, the SPP devices must be connected to conventional diffraction-limited photonic devices [2]. A SPP lens with good imaging quality can satisfy this demand to implement far/near-field conversion for multiple channels in parallel. Although a surface plasmon polariton launching lens (SPPLL) has been demonstrated experimentally [11], its imaging properties are still not very clear. In this paper, we present a general design method of SPPLLs with an arbitrary phase correction step. The imaging properties of half-wave SPPLLs, including the resolutions, object-image relationships, and aberrations, are investigated by using numerical simulation. The results show that SPPLLs have distinctive imaging properties between three dimensional (3D) far-field light and two dimensional (2D) SPPs and potentials to apply in plasmonic devices.

## 2. Design method of SPPLLs and the simulation method

A SPPLL is composed of Fresnel zones, which are grooves engraved on a metal film and can implement conversion between light and SPPs by providing the wavevector conservation. In practice, the groove width is usually set to be *λ*
_{SPP}/2 to obtain a high coupling efficiency, where *λ*
_{SPP} is the SPP wavelength [12]. The schematic of a SPPLL is shown in Fig. 1(a)
, which is symmetric with respect to the *y* axis. The distance between the outer edge of the *k*th zone and the origin is given by

*f*is the focal length and

*q*(

*q ≥*2) is the number of the phase correction step (e.g.,

*q*= 2 is half-wave,

*q*= 4 is quarter-wave). The

*y*coordinate of the

*k*th zone is

*a*

_{k}and

*a*

_{1}is assumed to be 0. The phase correction step

*q*can be described as the division of two integers

*q = m/n*(

*m*and

*n*are the possible smallest integers). For the focal spot on the upper side of the SPPLL, one method to determine the

*k*th zone’s position

*a*

_{k}is: For any

*k*, we have$p=\mathrm{mod}(nk,m)$, where the function mod(

*x*,

*y*) gives the remainder on division of

*x*by

*y*, and

*a*

_{k}= (

*p*–

*n*)

*λ*

_{SPP}/

*m*. As a result, the difference of

*a*

_{k + 1}and

*a*

_{k}is

*nλ*

_{SPP}

*/m*or (

*n*-

*m*)

*λ*

_{SPP}

*/m*. Such a design guarantees constructive interference of the SPPs exited from all zones at the focal spot when the SPPLL is normally illuminated by a plane wave.

To analyze the performance of SPPLLs, a simulation method based on Huygens-Fresnel principle with SPP point source model is developed. When a SPPLL is illuminated by a point source of light or a plane wave, SPPs are launched on all the Fresnel zones. According to the Huygens-Fresnel principle, each point on the grooves can be considered as a linear point source of the secondary wavelets of SPPs. And an approximate complex phasor form of a SPP point source was given in Ref [13]. The resulting field can be obtained by adding up all the wavelets taking into account the phase and amplitude. Figure 1(b) shows the configuration of imaging between two points of light and SPPs (*S*
_{1} and *S*
_{2}). In this paper, SPPLLs are assumed to be fabricated on the air/gold interface (*z* = 0), and the gold film is semi-infinite thick. The origin of the coordinates *O* is located at the midpoint of SPPLLs and the *x* axis is oriented along the SPPLL. In general, the object-image relationships of SPPLLs can be assumed to be the same as a thick lens whose first and second principal planes are located at distances of *OP*
_{1} = *r*
_{1} and *OP*
_{2} = *r*
_{2}, respectively. *F*
_{1} and *F*
_{2} are the first and second focal points with *f*
_{1} and *f*
_{2} as the corresponding first and second focal length, which are reckoned with respect to the principal planes, respectively. When a *y*-polarized point source *S*
_{1} is located at a height of *h* (object distance) above the center of the SPPLL, the light waves radiated from the source impinge on the SPPLL and excite SPPs with a phase distribution 2π/*λ*
_{0}·(*x*
^{2}+*h*
^{2})^{1/2} along the *x*-axis. As a result, the SPPs launched on the grooves converge towards *S*
_{2} with an image distance *d*. The case of *h* → ∞ corresponds to the source of a plane wave. Furthermore, the case of an oblique incident plane wave at a tilt angle *α* with respect to normal (in the *x*-*z* plane) can be simulated by inducing a phase delay *x*sin*α*·2π/*λ*
_{0} along the *x* axis. In the calculation, the SPP propagation loss is considered.

In order to verify our simulation method, the numerical SPP intensity distribution of a SPPLL, whose design parameters are described in detail in Ref [11], is compared with the experimental result when the SPPLL is normally illuminated by a plane wave. The experimental and simulated SPP intensity distributions around the focus are shown in Fig. 2(a) and 2(b), respectively. It should be noted that all fine features in Fig. 2(a) are precisely reproduced in Fig. 2(b), which shows the good agreement between the experiment and simulation.

## 3. Imaging properties of SPPLLs

We have demonstrated the bidirectional hybrid imaging between the far-field light and SPPs by using the proposed SPPLL experimentally [11]. In this section, the light to SPPs imaging properties of half-wave SPPLLs, including the resolution power, object-image relationships, and aberrations, are numerically investigated. The simulation conditions are summarized and listed in Table 1
. The geometry of a SPPLL is determined by the design wavelength *λ*
_{SPP}, focal length *f*, and numerical aperture NA, which is defined as *n*
_{SPP}sin*θ*, where *θ* is the semi angle formed by the focal spot and the most outer edges of the SPPLL [Fig. 1 (a)], *n*
_{SPP} = [(ε_{0}ε_{M})/(ε_{0}+ε_{M})]^{1/2} is the effective refractive index of SPPs with ε_{0} and ε_{M} being the dielectric function of the vacuum and gold, respectively. *λ*
_{0} is the wavelength in vacuum.

#### 3.1 Resolution

In analogy with a microscope image system, whose full width at half maximum (FWHM) resolution is 0.51*λ*
_{0}/NA, a similar formula for SPPLLs can be expected. The FWHMs with respect to 1/NA, *f*, and *λ*
_{0} are calculated and shown in Fig. 3
. In Fig. 3(a), the FWHM is proportionate to 1/NA and saturates at 344 nm when 1/NA ≤ 1, corresponding to a semi angle *θ* ≥80°. The saturation originates from that the intensity of the SPP point source has cos^{2}(*φ*) angular dependence, where *φ* is the azimuth with respect to the direction of the polarization of the incident light [9,13–15]. The FWHM with respect to *f* is also shown in Fig. 3(a) for NA = 0.7. In this case, the FWHM keeps constant at 500 nm for *f* ≥ 15 μm and slightly increases when *f* < 15 μm. For a fixed NA of 0.7, the FWHM increases linearly with the increase of the incident wavelength *λ*
_{0} [Fig. 3 (b)]. Furthermore, a relationship FWHM = 0.42*λ*
_{0}/NA is obtained by linearly fitting the data in Fig. 3. The smaller coefficient 0.42 indicates that SPPLLs enjoy a higher resolve power than a traditional imaging system when they work at the same *λ*
_{0} and NA. The reason is that, in the air at the gold/air interface, the intensity ratio of the out-of-plane to in-plane electric field components (*E*
_{z} and *E*
_{//}) ${\left|{E}_{\text{z}}\right|}^{2}/{\left|{E}_{\text{//}}\right|}^{2}={\left|{\epsilon}_{\text{M}}\right|}^{2}/{\left|{\epsilon}_{\text{0}}\right|}^{2}~12$ [16], hence the intensity of SPPs is dominated by that of the out-of-plane component, whose direction of the electric field vector is always along the *z* axis. Therefore, the out-of-plane electric field components from all Fresnel zones constructively interfere at the focus and bring to a sharply focused spot [17].

#### 3.2 Object-image relationships

As shown in Fig. 1 (b), the object-image distance formula of SPPLLs is assumed to be the same as a thick optical lens [18]:

For the light to SPPs imaging, the object distance*h*is changed to obtain the corresponding image distance

*d*, and the result is fitted by Eq. (2). A rather small residual sum of squares (~2.5 × 10

^{−21}) of the fitting is obtained, which indicates that the object-image relationship of SPPLLs can be fully described by Eq. (2). The corresponding parameters are

*f*

_{1}= 45.6 μm,

*r*

_{1}= −21.7 μm,

*f*

_{2}= 31.5 μm, and

*r*

_{2}= −1.53 μm.

For the SPPs to light imaging, the object-image relationship is also investigated. Due to the conversion from two dimensions (SPPs) to three dimensions (light), a SPP point source will be imaged to a light line [11]. The parameters of the same SPPLL as above for the SPPs to light imaging are *f*
_{1} = 42.9 μm, *r*
_{1} = −13.2 μm, *f*
_{2} = 30.9 μm, and *r*
_{2} = −1.53 μm. It should be noted that these values differ from that of the light to SPPs imaging, which means that SPPLLs are not conjugate imaging systems. This is attributed to the difference wavelength of the light and SPPs, and the geometry of SPPLLs depends on the design wavelength according to Eq. (1). As a result, the two cases have difference object-image distance formula parameters for the same SPPLL.

#### 3.3 Aberrations

For an optical lens, the monochromatic aberrations are the result of the paraxial treatment of the rays. The five primary aberrations include spherical aberration, coma, astigmatism, field curvature and distortion [18]. The spherical aberration corresponds to the dependence of focal length on aperture (NA) for nonparaxial rays. For SPPLLs, the spherical aberration is effectively suppressed at the designed wavelength due to the design principle.

On the other hand, coma, astigmatism, field curvature and distortion are the subsequence of large inclination angle illumination. For the imaging of a light point source, the consequence is the position deviation of the SPP focus from the aberration-free imaging and the degradation of the resolution. Here, the intensity of SPPs for plane wave light source with *α* = 0, 5, 11, 20, and 45 degree are calculated and shown in Fig. 4 (a), (b), (c), (d), and (e)
, respectively. The focal spot deviates gradually from the focal line *y* = 30 μm with the increase of *α*. At the same time, the transverse deviation from the aberration-free imaging also exists. Moreover, a large *α* provides an extended depth of focus accompanied by the decrease of the intensity of the focal spot. For a tilt angle *α* ≤ 11 degree, the focal spot orients almost along the *y* axis. When *α* increases to 20 and 45 degree, the angle between the longer axes of the focal spot and the y axis is 9.5 and 34.3 degree, respectively. The rotation of the focal spot can be explained by considering the excitation conditions. For the normal incidence (*α* = 0 degree), the phase of the incident light along the SPPLL is the same, therefore the focal spot is symmetric with respect to the *y* axis. However, at inclined incidence, the induced phase difference along the SPPLL results in the deflection of the SPP energy flow at the focal spot and the rotation of the focal spot. Figure 4 (f) displays the transverse position and longitudinal deviation from *y* = 30 μm for *α* = 0 to 11 degree. The red solid line is the aberration-free imaging position. In this range, the transverse aberration is fairly small, and the longitudinal aberration increases for *α* > 4°. The FWHM, which is shown in the inset of Fig. 4(f), increases with the increase of *α*. When SPPLLs are used to connect far field devices and SPP waveguides, the aberrations become insignificant because the position of the input end of a SPP waveguide can be adjusted to obtain the best coupling.

One of the main shortcomings of the traditional Fresnel zone plates is their high chromatic aberration. Here, we investigate the chromatic aberration of SPPLLs by calculating the monochromatic intensity along the optical axis. The results of a SPPLL with design parameters *λ*
_{SPP} = 813.5 nm (*λ*
_{0} = 830 nm) and *f* = 30 μm are shown in Fig. 5
for seven different incident vacuum wavelengths. As can be seen, the focal length of a SPPLL is inversely proportional to the wavelength, which is similar to the tradition Fresnel zone plates. When the incident wavelength is longer than the design wavelength, the intensity curve has a long tail. For the incident wavelength of 950 nm, the intensity at the design focal spot (*y* = 30 μm) is ~20% of the maximum intensity. The tail of the intensity curve provides an extended depth of focus, which means that the SPPLL is less sensitive to the chromatic aberration in a large range.

## 4. Conclusion

In conclusion, we have presented the general design method of SPPLLs with an arbitrary phase correction step and numerically investigated the imaging properties of half-wave SPPLLs. The object-image relationships of SPPLLs can be described by a lens equation of a thick optical lens. However, SPPLLs are not conjugate imaging system due to the difference of refractive indexes of SPPs and light. For light to SPPs imaging, SPPLLs can have a higher resolution than that of a traditional optical lens with the same NA and wavelength. Furthermore, the investigation of aberrations of SPPLLs provides important information for applications. As microlenses to connect 3D far-field and 2D SPPs, SPPLLs can be expected to implement multiple-channel connection in parallel between SPPs and diffraction-limited photonic devices, such as a fiber array.

## Acknowledgments

This work was supported by the Research Fund for the Doctoral Program of Higher Education under Grant No. 20090001110010 and the National Basic Research Program of China under Grant Nos. 2009CB623703 and 2007CB307001.

## References and links

**1. **W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**(6950), 824–830 (2003). [CrossRef] [PubMed]

**2. **E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science **311**(5758), 189–193 (2006). [CrossRef] [PubMed]

**3. **J. C. Weeber, Y. Lacroute, A. Dereux, E. Devaux, T. Ebbesen, C. Girard, M. U. González, and A. L. Baudrion, “Near-field characterization of Bragg mirrors engraved in surface plasmon waveguides,” Phys. Rev. B **70**(23), 235406 (2004). [CrossRef]

**4. **J. C. Weeber, M. U. González, A. L. Baudrion, and A. Dereux, “Surface plasmon routing along right angle bent metal strips,” Appl. Phys. Lett. **87**(22), 221101 (2005). [CrossRef]

**5. **A. Drezet, D. Koller, A. Hohenau, A. Leitner, F. R. Aussenegg, and J. R. Krenn, “Plasmonic crystal demultiplexer and multiports,” Nano Lett. **7**(6), 1697–1700 (2007). [CrossRef] [PubMed]

**6. **Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-Field Optical Hyperlens Magnifying Sub-Diffraction-Limited Objects,” Science **315**(5819), 1686 (2007). [CrossRef] [PubMed]

**7. **Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-Field Optical Superlens,” Nano Lett. **7**(2), 403–408 (2007). [CrossRef] [PubMed]

**8. **W. Nomura, M. Ohtsu, and T. Yatsui, “Nanodot coupler with a surface plasmon polariton condenser for optical far/near-field conversion,” Appl. Phys. Lett. **86**(18), 181108 (2005). [CrossRef]

**9. **L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. **5**(7), 1399–1402 (2005). [CrossRef] [PubMed]

**10. **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]

**11. **J. Wang, J. Zhang, X. Wu, H. Luo, and Q. Gong, “Subwavelength-resolved bidirectional imaging between two and three dimensions using a surface plasmon launching lens,” Appl. Phys. Lett. **94**(8), 081116 (2009). [CrossRef]

**12. **A. Giannattasio, I. R. Hooper, and W. L. Barnes, “Dependence on surface profile in grating-assisted coupling of light to surface plasmon-polaritons,” Opt. Commun. **261**(2), 291–295 (2006). [CrossRef]

**13. **S. H. Chang, S. K. Gray, and G. C. Schatz, “Surface plasmon generation and light transmission by isolated nanoholes and arrays of nanoholes in thin metal films,” Opt. Express **13**(8), 3150–3165 (2005). [CrossRef] [PubMed]

**14. **B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, Scattering, and interference of surface plasmons,” Phys. Rev. Lett. **77**(9), 1889–1892 (1996). [CrossRef] [PubMed]

**15. **A. Bouhelier, Th. Huser, H.-J. Güntherodt, and D. W. Pohl, “Plasmon optics of structured silver films,” Phys. Rev. B **63**(15), 155404 (2001). [CrossRef]

**16. **H. Raether, *Surface Plasmons on Smooth and Rough Surfaces and on Gratings* (Springer, 1988).

**17. **G. M. Lerman, A. Yanai, and U. Levy, “Demonstration of nanofocusing by the use of plasmonic lens illuminated with radially polarized light,” Nano Lett. **9**(5), 2139–2143 (2009). [CrossRef] [PubMed]

**18. **E. Hecht, *Optics* 4th (Addison-Wesley, New York, 2002). [PubMed]