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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 21 — Oct. 10, 2011
  • pp: 20634–20641
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Modeling and designing metallic superlens with metallic objects

Guillaume Tremblay and Yunlong Sheng  »View Author Affiliations


Optics Express, Vol. 19, Issue 21, pp. 20634-20641 (2011)
http://dx.doi.org/10.1364/OE.19.020634


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Abstract

When the metallic near-field superlens is to image a planar object, which is itself metallic, such as that in the near-field lithography applications, the object nanometer features will act as the Hertzian dipole sources and launch homogeneous and evanescent waves. The imaging system can be modeled as a dielectric Fabry-Perot cavity with the two surface plasmon resonant mirrors. We show the expressions of the transfer function and optimize the imaging system configuration using the genetic algorithm. The effectiveness of the design is confirmed by the image intensity profile computed with the numerical finite difference in time domain method.

© 2011 OSA

1. Introduction

The superlens is a slab of negative index material (NIM) which can restore the phase of propagating waves and the amplitude of the evanescent waves [1

1. V. G. Veselago, “The electromagnetics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]

] to achieve near-field imaging with sub-wavelength resolution beyond the diffraction limit. The metallic superlens is an alternative to the NIM lens made with noble metals existing in the nature like silver, gold and aluminum, which have a sole negative real part of dielectric function Re{εm}<0 at operating optical frequency, but a normal permeability μm~1 [2

2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

]. The metallic superlens is easier to implement, experimentally demonstrated [3

3. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005). [CrossRef] [PubMed]

,5

5. H. Qin, X. Li, and S. Shen, “Novel optical lithography using silver superlens,” Chin. Opt. Lett. 6(2), 149–151 (2008). [CrossRef]

] and has potential applications to near-field lithography of nanometre resolution [4

4. D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express 13(6), 2127–2134 (2005). [CrossRef] [PubMed]

7

7. C. P. Moore, R. J. Blaikie, and M. D. Arnold, “An improved transfer-matrix model for optical superlenses,” Opt. Express 17(16), 14260–14269 (2009). [CrossRef] [PubMed]

]. The principle of the metallic near field superlens is based on the amplification of the surface plasmon polaritons (SPPs) by the resonance of the multiply reflected SPP modes between the two interfaces of the superlens. The SP resonant amplification is necessary to compensate for the exponential decay of the evanescent waves away from the object plane [2

2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

,8

8. H. Raether, Surface Plasmons (Springer, Berlin, 1988).

], but enhances disproportionally the evanescent part of spatial spectrum, resulting in narrow peaks in the transfer function and high sidelobes in the images [6

6. Z. Shi, V. Kochergin, and F. Wang, “193nm Superlens imaging structure for 20nm lithography node,” Opt. Express 17(14), 11309–11314 (2009). [CrossRef] [PubMed]

]. Research efforts have been made for improving the imaging performance of the near field superlens in terms of its transfer function by optimizing the parameters of the superlens imaging system configuration [9

9. V. A. Podolskiy, N. A. Kuhta, and G. W. Milton, “Optimizing the superlens: manipulating geometry to enhance the resolution,” Appl. Phys. Lett. 87(23), 231113 (2005). [CrossRef]

,10

10. G. Tremblay and Y. Sheng, “Designing the metallic superlens close to the cutoff of the long-range mode,” Opt. Express 18(2), 740–745 (2010). [CrossRef] [PubMed]

]. It has been found that the typical two dominant feature peaks in the transfer function of the metallic near field superlens correspond to the long-range and short-range SPP waveguide modes, respectively, propagating along the interfaces of the superlens [11

11. G. Tremblay and Y. Sheng, “Improving imaging performance of a metallic superlens using the long-range surface plasmon polariton mode cutoff technique,” Appl. Opt. 49(7), A36–A41 (2010). [CrossRef] [PubMed]

]. Therefore, the metallic superlens can be designed by approaching the condition of the cut-off of the long-range SPP mode [12

12. J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B Condens. Matter 33(8), 5186–5201 (1986). [CrossRef] [PubMed]

] in order to flatten the transfer function and suppress the sidelobes in the image [10

10. G. Tremblay and Y. Sheng, “Designing the metallic superlens close to the cutoff of the long-range mode,” Opt. Express 18(2), 740–745 (2010). [CrossRef] [PubMed]

].

2. Transfer function

Consider a near-field imaging system with a metallic superlens of dielectric function εs = ε4 with Re{εs}<0 The object is also a metal layer, such as a metal coated lithographic mask, of dielectric function εo = ε2 with Re{εo}<0, as shown in Fig. 1
Fig. 1 Metallic near-field superlens with a metallic object layer
. In practice there can be multiple dielectric layers between the object and the superlens, such as index matching layers [6

6. Z. Shi, V. Kochergin, and F. Wang, “193nm Superlens imaging structure for 20nm lithography node,” Opt. Express 17(14), 11309–11314 (2009). [CrossRef] [PubMed]

]. We consider a single dielectric layer of dielectric function εd = ε3 and assume that a photosensitive layer is in contact with the superlens, for the sake of simplicity. Although in practice there can be a spacer of thickness d5 between the superlens and the image plane, the impact of such a spacer is well known as introducing an exponential decay of the evanescent waves over an additional distance d5 away from the superlens interface εm5. Consider also the dielectric media 1 and 5 as semi-infinite. For the purpose of the image analysis, we consider a sub-wavelength feature slit perforated through the metal object layer, and that the illuminating light is a plane wave of TM polarization incident normally to the object layer from the dielectric medium ε1.

The transmission coefficient of the imaging system shown in Fig. 1 can be written by considering the multiple reflections at the interfaces of each of the 3 layers of the media of εo, εd and εs, respectively, as

τtot=τoτdτs=[eot01tod1eo2rodro1][ed1ed2ρs+ρo][estdsts51es2rsdrs5]
(1)

This is the transmission coefficient of a Fabry-Perot resonator, which consists of the dielectric medium εd sandwiched by two SP resonant mirrors: the metal object layer of εo and the superlens of εs. The propagation factor ei = exp(ikzidi) with the sub-index i = o, d, s describing the phase shift of the homogeneous waves propagating over the distance di and the exponent decay in amplitude of the evanescent waves over di, respectively. Equation (1) can describe the impact of both homogeneous and inhomogeneous waves on the transmission coefficient, where the Fresnel reflection and transmission coefficients from medium i to medium j are
rij=εjkziεikzjεjkzi+εikzj,tij=2εjkziεjkzi+εikzj
(2)
with kzi=εik02kx2 as the wave vector component in z, and rij=rji, tij=tjiεjkzi/εikzj. The reflection coefficient of the superlensρs+and that of the metal object ρo expressed respectively as:

ρs+=rds+es2rs51es2rsdrs5andρo=rdo+eo2ro11eo2ro1rod
(3)

In fact, ρs+ is the reflected wave from the superlens to the layer εd with the Fresnel reflection coefficient rds, summed with the waves, which are multiply reflected between two interfaces of the superlens and transmitted through the interface εs-εd to the layer εd, and ρo is the reflected wave from the object to the layer εd with the Fresnel reflection coefficient rdo summed with the waves, which are multiply reflected between two interfaces of the object layer and transmitted through the interface εo-εd to the layer εd. The superscripts + and – of the reflection coefficients in Eqs. (3) indicate the propagation direction of the incident wave relative to the z axis. Moreover, application of the Maxwell’s equations and the boundary conditions at the interfaces to the electromagnetic fields in each layer in the system would result in the same transfer function as expressed in Eq. (1) [11

11. G. Tremblay and Y. Sheng, “Improving imaging performance of a metallic superlens using the long-range surface plasmon polariton mode cutoff technique,” Appl. Opt. 49(7), A36–A41 (2010). [CrossRef] [PubMed]

]. Equation (1) can also be obtained as a transfer function using the transfer matrix approach [14

14. C. C. Katsidis and D. I. Siapkas, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference,” Appl. Opt. 41(19), 3978–3987 (2002). [CrossRef] [PubMed]

].

The metallic slab superlens does not focus the propagating waves, so that the image is formed by the projection of the plane wave passing through the slit. If we remove the constant Fresnel transmission coefficient of the interface ε1o for the incident plane wave, t1o, in Eq. (1), then we obtain the transfer function of the system due to the source B located on the first interface of metal object layer:

τB=[eotod1eo2rodro1][ed1ed2ρs+ρo][estdsts51es2rsdrs5]
(4)

In the same way, the transfer function of the system due to the source A is written as

τA=τdτs=[ed1ed2ρs+ρo][estdsts51es2rsdrs5]
(5)

3. Optimal design with genetic algorithm

In most GA implementations, the chromosomes are encoded with strings of binary numbers. However, the parameters to optimize in the structure of the metallic near field superlens with metallic object are the thickness and dielectric functions of the layers, in which do, dd, ds and εd are real valued, while the real part of the dielectric function, εo, and εs, depends on the wavelength. The real and imaginary parts of the complex-valued εo, and εs may not be optimized separately. Thus, we use the GA, in which the chromosomes are encoded in vectors with the real valued parameters to be optimized as the vector elements (genes).

Once the parent chromosomes were selected, our GA with real parameter genes performed the reproduction with two distinct crossover mechanisms with equal probability [15

15. G. Tremblay, J. N. Gillet, Y. Sheng, M. Bernier, and G. Paul-Hus, “Optimizing fiber Bragg gratings using the genetic algorithm with fabrication-constraint encoding,” J. Lightwave Technol. 23(12), 4382–4386 (2005). [CrossRef]

]. The first mechanism generated an offspring by randomly selecting genes from its two parents. The offspring c is reproduced by randomly selecting genes from the parents a and b as:
a=[a1,a2,a3,...,an]b=[b1,b2,b3,...,bn]}c=[x1,x2,x3,...,xn],
(7)
where every “xi” with 1≤ i ≤ n in c has an equal probability of being an “ai” or a “bi”. The second mechanism generating an offspring c is a linear crossover of the real-valued chromosomes by performing a linear sum of the two parents, a and b, using
c=(0,5+g)a+(1,5g)b,
(8)
where g is a random parameter within the interval 0 ≤ g ≤ 2. The probability of the reproduction was set to 0.9. In the case where reproduction does not occur, the offspring is cloned from its second parent.

Once the reproduction process was done, new genes were introduced into the population. The mutations were performed on the newly generated chromosomes with a probability of 0.5. This value is higher than typical values of 0.15-0.3 for real-valued GA [15

15. G. Tremblay, J. N. Gillet, Y. Sheng, M. Bernier, and G. Paul-Hus, “Optimizing fiber Bragg gratings using the genetic algorithm with fabrication-constraint encoding,” J. Lightwave Technol. 23(12), 4382–4386 (2005). [CrossRef]

] in order to increase genetic diversity. The mutations consist in randomly choosing a single gene inside a chromosome and changing the gene randomly. A real valued chromosome element m is mutated to generate mm by
mm=0.25(2h1)+0.75m,
(9)
where h’ is a random value, 0< h’ <1, such that the mutated gene is kept close to the original value of the gene within an interval controlled by h’. Note that two elites were cloned for each new generation and were never mutated to avoid losing precious genetic material.

For design example we have chosen deliberately the superlens studied in Ref [6

6. Z. Shi, V. Kochergin, and F. Wang, “193nm Superlens imaging structure for 20nm lithography node,” Opt. Express 17(14), 11309–11314 (2009). [CrossRef] [PubMed]

], which is a five-layer structure with an Al superlens of thickness ds = 13 nm. The object mask is also in Al and of thickness do = 20 nm. Thus, εo = εs = −4.42 + 0.43i at λ = 193 nm. In between the object mask and superlens, there are two layers of SiO2 with εd1 = 2.4 and MgO with Re(εd2) = 4.08, which serves as an index matching layer. Both layers are of thickness dd1 = dd2 = 10 nm. There is a spacer layer of thickness d5 = 8 nm and ε5 = 2.89 between the superlens and photoresist layer of ε6 = 2.89. The system showed a nice transfer function, as shown in Fig. (1) in Ref [6

6. Z. Shi, V. Kochergin, and F. Wang, “193nm Superlens imaging structure for 20nm lithography node,” Opt. Express 17(14), 11309–11314 (2009). [CrossRef] [PubMed]

], with a high SPP resonance peak in the region of frequency 1.5 < k/k0 < 2.0. The superlens showed a resolution of 20 nm for a periodic object. However, for an object of two-slit structure, which had a larger spatial spectrum bandwidth, there were two high side lobes lying outside of the image, which can be suppressed only by placing assistant features on both sides of the slits in the object plane, as shown in Fig. (5) in Ref [6

6. Z. Shi, V. Kochergin, and F. Wang, “193nm Superlens imaging structure for 20nm lithography node,” Opt. Express 17(14), 11309–11314 (2009). [CrossRef] [PubMed]

]. The results in Ref [6

6. Z. Shi, V. Kochergin, and F. Wang, “193nm Superlens imaging structure for 20nm lithography node,” Opt. Express 17(14), 11309–11314 (2009). [CrossRef] [PubMed]

]. have been obtained by numerical FDTD simulation.

The near-field superlens could be designed with the numerical simulation, but by scanning the configuration parameters or by trials with randomly chosen parameters. The close-to-cutoff of the long-range SPP waveguide mode technique has been proposed as a rule of thumb for the design [10

10. G. Tremblay and Y. Sheng, “Designing the metallic superlens close to the cutoff of the long-range mode,” Opt. Express 18(2), 740–745 (2010). [CrossRef] [PubMed]

,11

11. G. Tremblay and Y. Sheng, “Improving imaging performance of a metallic superlens using the long-range surface plasmon polariton mode cutoff technique,” Appl. Opt. 49(7), A36–A41 (2010). [CrossRef] [PubMed]

]. The technique was applied to the superlens of Shi [6

6. Z. Shi, V. Kochergin, and F. Wang, “193nm Superlens imaging structure for 20nm lithography node,” Opt. Express 17(14), 11309–11314 (2009). [CrossRef] [PubMed]

] with the modification in the configuration that first put εd1 = εd2 = 4.08, for simplicity, and then decreased the permittivity of the spacer layer to ε5 = 1.93 in order to approach the cutoff condition [10

10. G. Tremblay and Y. Sheng, “Designing the metallic superlens close to the cutoff of the long-range mode,” Opt. Express 18(2), 740–745 (2010). [CrossRef] [PubMed]

]. Figure 2
Fig. 2 (a) Transfer functions of the structure designed using the close-to-cutoff technique computed with Eqs. (4) and (5) considering the presence of metal object mask; (b) Image calculated by FDTD method.
shows the transfer function of this superlens structure computed with Eqs. (4) and (5) where the presence of the metal object mask was considered, and the thickness of the metal object layer was set to do = 20 nm, and the image intensity profile computed with the FDTD simulation. We see that the high SPP resonance peak shown in Ref [6

6. Z. Shi, V. Kochergin, and F. Wang, “193nm Superlens imaging structure for 20nm lithography node,” Opt. Express 17(14), 11309–11314 (2009). [CrossRef] [PubMed]

] was effectively suppressed, as shown in Fig. 2a, that led to a significant reduction of the high side lobes lying outside the image of the two-slit, as shown in Fig. 2b, although the DC component and the central lobe in the image plane were still high. Note that the close-to-cutoff technique is based on the three-layer IMI structure without considering the presence of metal object mask, so that several trials were still needed to make the design useful for the real structure with the presence of metal object.

The intensity profile of the image in Fig. 3b shows a ratio of the maximal image intensity of the two slits over the background “DC” noise of about 10, which is a great improvement compared to the results shown in Fig. 3b, for which this ratio never exceeds 2. Also, the sidelobes are much weaker. The reduction of the background noise is due to the 31.5 nm thickness of the Al object layer that attenuates most of the incident signal. The improvement of the image quality is due to the optimization of the transfer functions, shown in Fig. 3a, where, for source A the blue line shows better field amplification for the frequency range kx0 > 2 and a significant reduction of the peak at the frequency of the peak kx0~1 as compared with Fig. 2a. For source B, the transfer function is free of any feature peaks with a profile broader than that illustrated in Fig. 2a.

4. Conclusion

We have proposed a general theoretical model of the metallic near-field superlens with a metallic object layer as a Fabry-Perot dielectric cavity with two SPP resonant mirrors, and introduced expressions of the transfer function, which are valid for both homogeneous and evanescent waves. We have proposed to design the superlens system configuration by optimizing the transfer function of the system using the GA, which is advantageous over the numerical simulations based on random trials and the close-to-cutoff of long-range SPP mode technique

References and links

1.

V. G. Veselago, “The electromagnetics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]

2.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

3.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005). [CrossRef] [PubMed]

4.

D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express 13(6), 2127–2134 (2005). [CrossRef] [PubMed]

5.

H. Qin, X. Li, and S. Shen, “Novel optical lithography using silver superlens,” Chin. Opt. Lett. 6(2), 149–151 (2008). [CrossRef]

6.

Z. Shi, V. Kochergin, and F. Wang, “193nm Superlens imaging structure for 20nm lithography node,” Opt. Express 17(14), 11309–11314 (2009). [CrossRef] [PubMed]

7.

C. P. Moore, R. J. Blaikie, and M. D. Arnold, “An improved transfer-matrix model for optical superlenses,” Opt. Express 17(16), 14260–14269 (2009). [CrossRef] [PubMed]

8.

H. Raether, Surface Plasmons (Springer, Berlin, 1988).

9.

V. A. Podolskiy, N. A. Kuhta, and G. W. Milton, “Optimizing the superlens: manipulating geometry to enhance the resolution,” Appl. Phys. Lett. 87(23), 231113 (2005). [CrossRef]

10.

G. Tremblay and Y. Sheng, “Designing the metallic superlens close to the cutoff of the long-range mode,” Opt. Express 18(2), 740–745 (2010). [CrossRef] [PubMed]

11.

G. Tremblay and Y. Sheng, “Improving imaging performance of a metallic superlens using the long-range surface plasmon polariton mode cutoff technique,” Appl. Opt. 49(7), A36–A41 (2010). [CrossRef] [PubMed]

12.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B Condens. Matter 33(8), 5186–5201 (1986). [CrossRef] [PubMed]

13.

J. Chen, G. A. Smolyakov, S. R. J. Brueck, and K. J. Malloy, “Surface plasmon modes of finite, planar, metal-insulator-metal plasmonic waveguides,” Opt. Express 16(19), 14902–14909 (2008). [CrossRef] [PubMed]

14.

C. C. Katsidis and D. I. Siapkas, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference,” Appl. Opt. 41(19), 3978–3987 (2002). [CrossRef] [PubMed]

15.

G. Tremblay, J. N. Gillet, Y. Sheng, M. Bernier, and G. Paul-Hus, “Optimizing fiber Bragg gratings using the genetic algorithm with fabrication-constraint encoding,” J. Lightwave Technol. 23(12), 4382–4386 (2005). [CrossRef]

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(260.3910) Physical optics : Metal optics
(310.2790) Thin films : Guided waves

ToC Category:
Optics at Surfaces

History
Original Manuscript: August 18, 2011
Revised Manuscript: September 16, 2011
Manuscript Accepted: September 19, 2011
Published: October 3, 2011

Citation
Guillaume Tremblay and Yunlong Sheng, "Modeling and designing metallic superlens with metallic objects," Opt. Express 19, 20634-20641 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20634


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References

  1. V. G. Veselago, “The electromagnetics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp.10, 509–514 (1968). [CrossRef]
  2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000). [CrossRef] [PubMed]
  3. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science308(5721), 534–537 (2005). [CrossRef] [PubMed]
  4. D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express13(6), 2127–2134 (2005). [CrossRef] [PubMed]
  5. H. Qin, X. Li, and S. Shen, “Novel optical lithography using silver superlens,” Chin. Opt. Lett.6(2), 149–151 (2008). [CrossRef]
  6. Z. Shi, V. Kochergin, and F. Wang, “193nm Superlens imaging structure for 20nm lithography node,” Opt. Express17(14), 11309–11314 (2009). [CrossRef] [PubMed]
  7. C. P. Moore, R. J. Blaikie, and M. D. Arnold, “An improved transfer-matrix model for optical superlenses,” Opt. Express17(16), 14260–14269 (2009). [CrossRef] [PubMed]
  8. H. Raether, Surface Plasmons (Springer, Berlin, 1988).
  9. V. A. Podolskiy, N. A. Kuhta, and G. W. Milton, “Optimizing the superlens: manipulating geometry to enhance the resolution,” Appl. Phys. Lett.87(23), 231113 (2005). [CrossRef]
  10. G. Tremblay and Y. Sheng, “Designing the metallic superlens close to the cutoff of the long-range mode,” Opt. Express18(2), 740–745 (2010). [CrossRef] [PubMed]
  11. G. Tremblay and Y. Sheng, “Improving imaging performance of a metallic superlens using the long-range surface plasmon polariton mode cutoff technique,” Appl. Opt.49(7), A36–A41 (2010). [CrossRef] [PubMed]
  12. J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B Condens. Matter33(8), 5186–5201 (1986). [CrossRef] [PubMed]
  13. J. Chen, G. A. Smolyakov, S. R. J. Brueck, and K. J. Malloy, “Surface plasmon modes of finite, planar, metal-insulator-metal plasmonic waveguides,” Opt. Express16(19), 14902–14909 (2008). [CrossRef] [PubMed]
  14. C. C. Katsidis and D. I. Siapkas, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference,” Appl. Opt.41(19), 3978–3987 (2002). [CrossRef] [PubMed]
  15. G. Tremblay, J. N. Gillet, Y. Sheng, M. Bernier, and G. Paul-Hus, “Optimizing fiber Bragg gratings using the genetic algorithm with fabrication-constraint encoding,” J. Lightwave Technol.23(12), 4382–4386 (2005). [CrossRef]

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