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Insight of limitations of effective media theory for metal–dielectric multilayer metamaterials
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Insight of limitations of effective media theory for metal–dielectric multilayer metamaterials Q1
P. Zhu a,n, P. Jin a, L. Jay Guo b a b
Research Center of Ultra-precision Optoelectronic Instrumentation, Harbin Institute of Technology, Harbin 150080, China Department of Electrical Engineering and Computer Science, C-PHOM, University of Michigan, 1301 Beal Avenue, Ann Arbor, MI 48109, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 23 November 2012 Received in revised form 3 May 2013 Accepted 5 May 2013
Effective media theory (EMT) is usually used in the simulations and descriptions of one type of metamaterials which are composed of metal–dielectric multilayer films because of its simplicity and intuitiveness. It is consistent with the rigorous electromagnetic field analysis when the approximation conditions are strictly satisfied. However, there are several noticeable, and sometimes significant failures for the predicted properties that are caused by the approximation dealing with the finite layer of metal/ dielectric multilayer structures. These inaccuracies and errors including surface plasmon polaritons (SPPs) spreading angles, reflection properties and interference patterns for SPPs lithography, originate from the differences of isofrequency contours between ideal hyperbolic multilayer structures and the actually fabricated multilayer structures with finite layer of films. Simulation results of EMT with metal/ dielectric multilayer structures are analyzed and compared with the rigorous field analysis approach. Being aware of these issues can help to reduce the errors and improve the designs of the plasmonic devices for many applications. & 2013 Published by Elsevier B.V.
Keywords: Effective media theory Surface plasmon polaritions Metal–dielectric multilayer films Interference patterns
1. Introduction Effective media theory is widely used to describe the macroscopic properties of complicated structures. In the area of plasmonics, it has been successfully applied to explain the novel and unique properties of hyperbolic metamaterials with metal–dielectric multilayer films, such as high photon density of states [1–3] and deep subwavelength SPPs interference [4,5]. Recently, Harish et al. demonstrated an optical topological transition in anisotropic multilayer metamaterials that results in a dramatic increase in the photon density of states with the analyses of EMT [1]. In 2009 Ting et al. proposed a SPPs interference method based on hyperbolic metal–dielectric multilayer films which also employ the effective permittivity to describe the optical property of multilayer metamaterials [4]. On the other hand, multi-film structures can also be analyzed by the rigorous field analysis approach [6–8], like the coupled mode theory (CMT), finite difference time domain (FDTD) or finite element method (FEM), which can describe the detailed distributions of electro-magnetic fields in each layer of the metamaterials. These two approaches will definitely match well with each other for most cases when the conditions for EMT are strictly satisfied, which require extreme thin films (smaller than 1/10 of incident wavelength) and infinite stacks of metal/dielectric layers
Q4
n
Corresponding author. Tel.: +86 18704639265. E-mail address: [email protected] (P. Zhu).
[9]. However, the actual devices always have limited stacks of films and prefer fewer layer numbers from the practical fabrication point of view. Besides, EMT only describes the macroscopic property of the structures as a whole without considering the influence of fabrication defects, such as fluctuations of film thickness and film roughness. Therefore there are some spectacular optical phenomena which are not correctly predicted or analyzed by EMT when it is applied to finite layer metal–dielectric multilayer films. In this paper, we take an insight onto the failures of predictions and limitations of the EMT approach in metal/ dielectric multilayer metamaterials by comparing the differences of SPPs spreading angles and dispersion relations. Inaccurate reflection property and unexpected interference patterns of SPPs lithography by the analyses of EMT are pointed out. Influence of fabrication imperfections which cannot be analyzed by EMT are also presented by FEM simulations. These results may attract more attentions to the limitations of EMT for the analyses of hyperbolic metal–dielectric multilayer structures and give some guidelines for the plasmonic structure design. 2. Analyses of beam spreading angle in multilayer metamaterials Incident light passing through the metal/dielectric hyperbolic metamaterials with nano-slits will spread into two lobes [10] and the interference patterns for plasmonic lithography are
0030-4018/$ - see front matter & 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.optcom.2013.05.005
Please cite this article as: P. Zhu, et al., Optics Communications (2013), http://dx.doi.org/10.1016/j.optcom.2013.05.005i
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Fig. 1. Schematic of the MgF2/Al multilayer structure with film thickness of 20 nm/ 15 nm and the width of the slit is 80 nm and the incident wavelength is 365 nm.
Fig. 2. (a) Distributions of the intensity of electrical field in the equivalent structure with effective permittivities calculated from the Al/MgF2 multilayer structure of Fig. 1. (b) Intensity of electrical field in the actual Al/MgF2 multilayer structure with the same parameters of Fig. 1.
significantly influenced by the spreading angle of the two lobes [11]. In most cases EMT is used to analyze this effect without considering the potential issues for the actual finite layer metal/ dielectric multi-film structures. Detailed comparisons of EMT and rigorous electromagnetic field analyses are presented. We use the metal–dielectric multilayer structure with nanopatterned chromium mask as a basic unit for the following demonstrations and discussions, as shown in Fig. 1. Al and MgF2 are used in the metal/dielectric stacks and their thicknesses are 15 nm and 20 nm, respectively. A Cr mask with nano-slit is placed on top of the multilayer structure. Metal–dielectric multilayer structure can be considered as a single anisotropic medium with effective dielectric permittivity [10]: εx ¼ εy ¼
εm þ ηεd ; 1þη
1 1 1 η ; ¼ þ εz 1 þ η εm εd
ð1Þ
ð2Þ
where η is the ratio of the two layer thickness η¼hd/hm, εm and εd are the permittivity of the metal and dielectric, respectively. The effective parameters for the structure in Fig. 1 are calculated as εx ¼−7.1+1.5i and εz ¼3.7+0.05i according to Eqs. (1) and (2). In an anisotropic system, the incident waves from one nano-slit excite high-kx modes which propagate almost parallel to each other, with the angle between propagation direction and optical axis given by the ratio of Poynting vector component [12] as shown in Fig. 2(a). We can note that each sidewall of the slit excites two spreading beams and the two beams in the center are much weaker. There will be only two main lobes when the slit is narrow enough, like 50 nm. The spreading angle can be express as sffiffiffiffiffiffiffiffiffi εx k x ε′ tan θ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3Þ ≃ − x′ 2 εz 2 k εz εx ωc2 − εxz 0
with ε ¼ Re½ε. The calculated angle using effective permittivity is 621. Meanwhile, the spreading angle of the incident light can be understood by the coupling of SPPs between the multilayer waveguide, which is estimated by tan θ ¼ L=D, where L equals to the coupling length and D is the distance that SPPs propagate along the z direction [12]. In the above designed structure, L is 436 nm and D equals to 175 nm according to the method described in Ref. [12]. The calculated spreading angle in the same structure is 68.11 which is a little larger than the value calculated by EMT. It is because of the inaccurate approximation of Eqs. (1) and (2) when dealing with multilayer structures with finite layers and will be discussed in detail later.
Fig. 3. Beam spreading angles in the multilayer structures. The thicknesses of dielectric/metal are 12 nm/9 nm, 16 nm/12 nm and 20 nm/15 nm.
There is one more phenomenon that is beyond the anticipation of EMT when the film thicknesses of the multilayer structure vary with the same ratio η. According to Eqs. (1)–(3), the spreading angle of waves in hyperbolic material should be the same if the ratio η remains a constant. However, it is not quite the case when considering the actual multilayer structures with finite layers. We give one example for demonstration: Al/MgF2 multilayer structures with thicknesses of 15 nm/20 nm, 12 nm/16 nm and 9 nm/ 12 nm (the ratio is a constant of 0.75). We find that the beam spreading angle changes slightly instead of remaining a constant, as shown in Fig. 3. To further demonstrate the difference between EMT approximated and actual multilayer structures, we plot the schematic of isofrequency contours (or dispersion relation) of different multilayer structures in Fig. 4 [10]. The circle in the center represents the dispersion of conventional multilayer structures with both positive permittivities εx and εz and all the propagation wave vectors kx and kz are confined in a small range. The dispersion curve of ideal metamaterial shows a hyperbola with infinite large wave vector kx while the actual metal/dielectric multilayer structure with finite periodic stacks only support limited propagation wave-vectors in a certain range as indicated by Lk. They match very well at small kx range and diverge with each other when the kx becomes larger. As we know, the optical properties are directly dependent on structure's dispersion relation. This is the reason that causes the inaccuracy and errors predicted by the EMT method for actual multilayer structures.
Please cite this article as: P. Zhu, et al., Optics Communications (2013), http://dx.doi.org/10.1016/j.optcom.2013.05.005i
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Fig. 4. Schematic of the isofrequency contours of different multilayer structures. The green dotted line is the dispersion curve of ideal hyperbolic materials and the red circle in the center is the dispersion of conversional structure. The curve shows the dispersion of M/D multilayer structure with finite layers. (For interpretation of the references to color in this figure legend, the reader is referred to the web Q3 version of this article.)
The hyperbolic isofrequency contour also demonstrate why the incident light spreads into two beams in the metal/dielectric multilayer structure, which is because the direction of propagating wave is always perpendicular to the isofrequency contour [13], as shown by the arrows in Fig. 4. Different positions in the x coordinate correspond to different propagation wave vectors kx and the beam spreading angles vary slightly because of the curvature in the isofrequency contour.
3. Reflection property of metal/dielectric multilayer structures One main application of metal/dielectric multilayer metamaterial is SPP interference lithography [14,15]. However, it suffers from low field intensity and penetration depth because SPPs become very weak after propagating through the multilayer structures [16]. One direct method to increase the intensity of SPP is to reduce the reflection of top Cr mask and make more light to get into the multilayer structures. Therefore we study the reflection property of the metal/dielectric multilayer structures and compare the reflection spectrum calculated by the EMT approximation method and rigorous electromagnetic field analysis with FEM simulations in Fig. 5. We found that the reflection of the multilayer structure for the SPP lithography is lower than 0.2 within a broad range from 400 nm to 500 nm and no larger than 0.4 in the whole visible range, as shown in Fig. 5(b). However, the reflection becomes much higher when we use equivalent effective media parameters calculated from Eqs. (1) and (2) to replace the Al/MgF2 multilayer structure. Result from the EMT approximation does show the resonance and low reflection as the rigorous electromagnetic field analysis. It is because the low reflection is caused by the resonance of metal/dielectric/metal (MIM) cavities [17] and obvious resonance dips can be observed in the spectra simulated M/D multilayer structures. However, the EMT method omits the details and treats the structure as a whole. No resonant cavity is remained in the EMT approximation and much more light is reflected according to the simulation result, which means the EMT approximation method is not accurate for analyzing the reflection property of M/D multilayer structure in this case.
4. Limitations of EMT on SPP interference Plasmonic lithography with SPPs interference is a potential technology for the lithography of next generation. More analyses
Fig. 5. (a) Schematic of the MgF2/Al multilayer structure with thickness of 30 nm/ 20 nm. The period of top Cr grating is 220 nm and the structure using the equivalent hyperbolic effective media parameters to replace the multilayer films. (b) Reflection spectra of the above structures.
about the limitations of EMT for SPP interference lithography are presented. Fig. 6 shows the intensity of electrical field distributions of interference patterns in a MgF2/Al multilayer structure with different thicknesses. The ratio of hMgF2 : hAl remains a constant 4:3. The exposure wavelength of 365 nm, which is commonly used in commercial lithography products. A layer of Al film is used under the photoresist layer in order to increase the pattern contrast and penetration depth of the pattern [12]. The structure in Fig. 6(a) with MgF2/Al film thickness 20 nm/ 15 nm is our optimized parameters for SPP interference lithography and is used as a reference here. We increase and reduce the film thicknesses with the same ratio 4:3 to 24 nm/18 nm, 16 nm/ 12 nm and 12 nm/9 nm. We found an interesting phenomenon that interference patterns keep uniform for the thicker films of 24 nm/18 nm, yet non-uniform for the thinner films of 16 nm/ 12 nm and 12 nm/9 nm, as shown in Fig. 6(b), (c) and (d), respectively. These results correspond to the difference of isofrequency contours discussed in Fig. 4 well. Multilayer films can be considered as multiple waveguides. The thicker waveguide support comparatively smaller propagation wave vector [18] and vice versa. Because of the inaccuracy of EMT for finite metal/dielectric multilayer films, wave vector kx in small range matches the ideal hyperbolic curve well while diverge with each other when kx becomes large. Therefore, interference patterns for thinner films than the reference will not be uniform even though they keep the same film thickness ratio of metal/dielectric. Other limitations of EMT is that it cannot be use to analyze the influence of surface roughness of the multilayer structure, which are very important for the SPPs lithography application. The energy loss caused by the roughness is fatal to the propagation of SPPs. Strict analysis of the electromagnetic field can give better
Please cite this article as: P. Zhu, et al., Optics Communications (2013), http://dx.doi.org/10.1016/j.optcom.2013.05.005i
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Fig. 6. Intensity of electrical fields of the SPP interference patterns with different thicknesses of MgF2/Al films: (a) 24 nm/18 nm, (b) 20 nm/15 nm, (c) 16 nm/12 nm and (d) 12 nm/9 nm.
Fig. 7. (a) Electric fields of the multilayer structure with roughness of 2 nm and (b) 5 nm, the films in the dotted line are without any roughness and the pattern is used for comparison.
description of the formation of the patterns. We add a random roughness of average fluctuation of 2 nm and 5 nm to the film surface in the multilayer structure, leaving one column of pattern without roughness for comparison. The intensities of electrical filed distributions are shown in Fig. 7(a) and (b). Intensity of electrical filed distribution in the vertical direction changes significantly when the roughness increases from 2 nm to 5 nm because of the interference of scattered light from the rough film surface. The final interference patterns in photoresist do not change significantly, which means this kind of structure has good robustness for fabrication.
dielectric multilayer structures. An MgF2/Al multilayer structure for SPPs lithography is used as an example to demonstrate their difference. Beam spreading angles of SPPs and reflection properties of the multilayer structure are analyzed and compared using the both methods. Influences of film thickness and roughness for interference patterns are also presented. Above analyses and results show that the EMT method is not always accurate for the design and analysis of metal–dielectric multilayer structures with finite stacks. The rigorous electromagnetic approach is essential and more accurate in some cases. These conclusions can provide some useful guidelines for the design of plasmonic devices.
5. Conclusion Acknowledgments In summary, we analyze the consistency and exceptions between the effective media theory and the rigorous electromagnetic field approach when dealing with the hyperbolic metal–
This work was supported by the National Science Foundation Materials Research Science and Engineering Center (Program DMR
Please cite this article as: P. Zhu, et al., Optics Communications (2013), http://dx.doi.org/10.1016/j.optcom.2013.05.005i
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1120923). P. Zhu also acknowledges the support from China Scholarship Council (CSC) and the University of Michigan. References [1] H.N.S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, V.M. Menon, Science 336 (6078) (2012) 205. [2] G. D'Aguanno, N. Mattiucci, A. Alu, C. Argyropoulos, J.V. Foreman, M.J. Bloemer, Optics Express 20 (9) (2012). [3] Z. Jacob, J.Y. Kim, G.V. Naik, A. Boltasseva, E.E. Narimanov, V.M. Shalaev, Applied Physics B: Lasers and Optics 100 (1) (2010) 215. [4] T. Xu, Y.H. Zhao, J.X. Ma, C.T. Wang, J.H. Cui, C.L. Du, X.G. Luo, Optics Express 16 (18) (2008) 13579. [5] Y. Xiong, Z.W. Liu, X. Zhang, Applied Physics Letters 93 (11) (2008). [6] X. Fan, G.P. Wang, J.C.W. Lee, C.T. Chan, Physical Review Letters 97 (7) (2006) 073901.
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[7] H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, J.S. Aitchison, Physical Review Letters 81 (16) (1998) 3383. [8] X.B. Fan, G.P. Wang, Optics Letters 31 (9) (2006) 1322. [9] D.R. Smith, P. Kolinko, D. Schurig, Journal of the Optical Society of America B 21 (2004) 1032. [10] S. Thongrattanasiri, V.A. Podolskiy, Optics Letters 34 (7) (2009) 890. [11] P. Zhu, H.F. Shi, L.J. Guo, Optics Express 20 (11) (2012) 12521. [12] B. Wood, J.B. Pendry, D.P. Tsai, Physical Review B 74 (11) (2006) 115116. [13] D.R. Smith, D. Schurig, Physical Review Letters 90 (2003) 077405. [14] K.V. Sreekanth, V.M. Murukeshan, J.K. Chua, Applied Physics Letters 93 (9) (2008). [15] B.B. Zeng, X.D. Yang, C.T. Wang, X.G. Luo, Optics Express 17 (19) (2009) 16783. [16] X. Wan, Q.K. Wang, H.H. Tao, Journal of the Optical Society of America A: Optics Image Science and Vision 27 (5) (2010) 973. [17] J.M. Hao, J. Wang, X.L. Liu, W.J. Padilla, L. Zhou, M. Qiu, Applied Physics Letters 96 (25) (2010). [18] J.A. Dionne, L.A. Sweatlock, H.A. Atwater, A. Polman, Physical Review B 73 (3) (2006).
Please cite this article as: P. Zhu, et al., Optics Communications (2013), http://dx.doi.org/10.1016/j.optcom.2013.05.005i
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Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Insight of limitations of effective media theory for metal–dielectric multilayer metamaterials Q1
P. Zhu a,n, P. Jin a, L. Jay Guo b a b
Research Center of Ultra-precision Optoelectronic Instrumentation, Harbin Institute of Technology, Harbin 150080, China Department of Electrical Engineering and Computer Science, C-PHOM, University of Michigan, 1301 Beal Avenue, Ann Arbor, MI 48109, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 23 November 2012 Received in revised form 3 May 2013 Accepted 5 May 2013
Effective media theory (EMT) is usually used in the simulations and descriptions of one type of metamaterials which are composed of metal–dielectric multilayer films because of its simplicity and intuitiveness. It is consistent with the rigorous electromagnetic field analysis when the approximation conditions are strictly satisfied. However, there are several noticeable, and sometimes significant failures for the predicted properties that are caused by the approximation dealing with the finite layer of metal/ dielectric multilayer structures. These inaccuracies and errors including surface plasmon polaritons (SPPs) spreading angles, reflection properties and interference patterns for SPPs lithography, originate from the differences of isofrequency contours between ideal hyperbolic multilayer structures and the actually fabricated multilayer structures with finite layer of films. Simulation results of EMT with metal/ dielectric multilayer structures are analyzed and compared with the rigorous field analysis approach. Being aware of these issues can help to reduce the errors and improve the designs of the plasmonic devices for many applications. & 2013 Published by Elsevier B.V.
Keywords: Effective media theory Surface plasmon polaritions Metal–dielectric multilayer films Interference patterns
1. Introduction Effective media theory is widely used to describe the macroscopic properties of complicated structures. In the area of plasmonics, it has been successfully applied to explain the novel and unique properties of hyperbolic metamaterials with metal–dielectric multilayer films, such as high photon density of states [1–3] and deep subwavelength SPPs interference [4,5]. Recently, Harish et al. demonstrated an optical topological transition in anisotropic multilayer metamaterials that results in a dramatic increase in the photon density of states with the analyses of EMT [1]. In 2009 Ting et al. proposed a SPPs interference method based on hyperbolic metal–dielectric multilayer films which also employ the effective permittivity to describe the optical property of multilayer metamaterials [4]. On the other hand, multi-film structures can also be analyzed by the rigorous field analysis approach [6–8], like the coupled mode theory (CMT), finite difference time domain (FDTD) or finite element method (FEM), which can describe the detailed distributions of electro-magnetic fields in each layer of the metamaterials. These two approaches will definitely match well with each other for most cases when the conditions for EMT are strictly satisfied, which require extreme thin films (smaller than 1/10 of incident wavelength) and infinite stacks of metal/dielectric layers
Q4
n
Corresponding author. Tel.: +86 18704639265. E-mail address: [email protected] (P. Zhu).
[9]. However, the actual devices always have limited stacks of films and prefer fewer layer numbers from the practical fabrication point of view. Besides, EMT only describes the macroscopic property of the structures as a whole without considering the influence of fabrication defects, such as fluctuations of film thickness and film roughness. Therefore there are some spectacular optical phenomena which are not correctly predicted or analyzed by EMT when it is applied to finite layer metal–dielectric multilayer films. In this paper, we take an insight onto the failures of predictions and limitations of the EMT approach in metal/ dielectric multilayer metamaterials by comparing the differences of SPPs spreading angles and dispersion relations. Inaccurate reflection property and unexpected interference patterns of SPPs lithography by the analyses of EMT are pointed out. Influence of fabrication imperfections which cannot be analyzed by EMT are also presented by FEM simulations. These results may attract more attentions to the limitations of EMT for the analyses of hyperbolic metal–dielectric multilayer structures and give some guidelines for the plasmonic structure design. 2. Analyses of beam spreading angle in multilayer metamaterials Incident light passing through the metal/dielectric hyperbolic metamaterials with nano-slits will spread into two lobes [10] and the interference patterns for plasmonic lithography are
0030-4018/$ - see front matter & 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.optcom.2013.05.005
Please cite this article as: P. Zhu, et al., Optics Communications (2013), http://dx.doi.org/10.1016/j.optcom.2013.05.005i
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Fig. 1. Schematic of the MgF2/Al multilayer structure with film thickness of 20 nm/ 15 nm and the width of the slit is 80 nm and the incident wavelength is 365 nm.
Fig. 2. (a) Distributions of the intensity of electrical field in the equivalent structure with effective permittivities calculated from the Al/MgF2 multilayer structure of Fig. 1. (b) Intensity of electrical field in the actual Al/MgF2 multilayer structure with the same parameters of Fig. 1.
significantly influenced by the spreading angle of the two lobes [11]. In most cases EMT is used to analyze this effect without considering the potential issues for the actual finite layer metal/ dielectric multi-film structures. Detailed comparisons of EMT and rigorous electromagnetic field analyses are presented. We use the metal–dielectric multilayer structure with nanopatterned chromium mask as a basic unit for the following demonstrations and discussions, as shown in Fig. 1. Al and MgF2 are used in the metal/dielectric stacks and their thicknesses are 15 nm and 20 nm, respectively. A Cr mask with nano-slit is placed on top of the multilayer structure. Metal–dielectric multilayer structure can be considered as a single anisotropic medium with effective dielectric permittivity [10]: εx ¼ εy ¼
εm þ ηεd ; 1þη
1 1 1 η ; ¼ þ εz 1 þ η εm εd
ð1Þ
ð2Þ
where η is the ratio of the two layer thickness η¼hd/hm, εm and εd are the permittivity of the metal and dielectric, respectively. The effective parameters for the structure in Fig. 1 are calculated as εx ¼−7.1+1.5i and εz ¼3.7+0.05i according to Eqs. (1) and (2). In an anisotropic system, the incident waves from one nano-slit excite high-kx modes which propagate almost parallel to each other, with the angle between propagation direction and optical axis given by the ratio of Poynting vector component [12] as shown in Fig. 2(a). We can note that each sidewall of the slit excites two spreading beams and the two beams in the center are much weaker. There will be only two main lobes when the slit is narrow enough, like 50 nm. The spreading angle can be express as sffiffiffiffiffiffiffiffiffi εx k x ε′ tan θ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3Þ ≃ − x′ 2 εz 2 k εz εx ωc2 − εxz 0
with ε ¼ Re½ε. The calculated angle using effective permittivity is 621. Meanwhile, the spreading angle of the incident light can be understood by the coupling of SPPs between the multilayer waveguide, which is estimated by tan θ ¼ L=D, where L equals to the coupling length and D is the distance that SPPs propagate along the z direction [12]. In the above designed structure, L is 436 nm and D equals to 175 nm according to the method described in Ref. [12]. The calculated spreading angle in the same structure is 68.11 which is a little larger than the value calculated by EMT. It is because of the inaccurate approximation of Eqs. (1) and (2) when dealing with multilayer structures with finite layers and will be discussed in detail later.
Fig. 3. Beam spreading angles in the multilayer structures. The thicknesses of dielectric/metal are 12 nm/9 nm, 16 nm/12 nm and 20 nm/15 nm.
There is one more phenomenon that is beyond the anticipation of EMT when the film thicknesses of the multilayer structure vary with the same ratio η. According to Eqs. (1)–(3), the spreading angle of waves in hyperbolic material should be the same if the ratio η remains a constant. However, it is not quite the case when considering the actual multilayer structures with finite layers. We give one example for demonstration: Al/MgF2 multilayer structures with thicknesses of 15 nm/20 nm, 12 nm/16 nm and 9 nm/ 12 nm (the ratio is a constant of 0.75). We find that the beam spreading angle changes slightly instead of remaining a constant, as shown in Fig. 3. To further demonstrate the difference between EMT approximated and actual multilayer structures, we plot the schematic of isofrequency contours (or dispersion relation) of different multilayer structures in Fig. 4 [10]. The circle in the center represents the dispersion of conventional multilayer structures with both positive permittivities εx and εz and all the propagation wave vectors kx and kz are confined in a small range. The dispersion curve of ideal metamaterial shows a hyperbola with infinite large wave vector kx while the actual metal/dielectric multilayer structure with finite periodic stacks only support limited propagation wave-vectors in a certain range as indicated by Lk. They match very well at small kx range and diverge with each other when the kx becomes larger. As we know, the optical properties are directly dependent on structure's dispersion relation. This is the reason that causes the inaccuracy and errors predicted by the EMT method for actual multilayer structures.
Please cite this article as: P. Zhu, et al., Optics Communications (2013), http://dx.doi.org/10.1016/j.optcom.2013.05.005i
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Fig. 4. Schematic of the isofrequency contours of different multilayer structures. The green dotted line is the dispersion curve of ideal hyperbolic materials and the red circle in the center is the dispersion of conversional structure. The curve shows the dispersion of M/D multilayer structure with finite layers. (For interpretation of the references to color in this figure legend, the reader is referred to the web Q3 version of this article.)
The hyperbolic isofrequency contour also demonstrate why the incident light spreads into two beams in the metal/dielectric multilayer structure, which is because the direction of propagating wave is always perpendicular to the isofrequency contour [13], as shown by the arrows in Fig. 4. Different positions in the x coordinate correspond to different propagation wave vectors kx and the beam spreading angles vary slightly because of the curvature in the isofrequency contour.
3. Reflection property of metal/dielectric multilayer structures One main application of metal/dielectric multilayer metamaterial is SPP interference lithography [14,15]. However, it suffers from low field intensity and penetration depth because SPPs become very weak after propagating through the multilayer structures [16]. One direct method to increase the intensity of SPP is to reduce the reflection of top Cr mask and make more light to get into the multilayer structures. Therefore we study the reflection property of the metal/dielectric multilayer structures and compare the reflection spectrum calculated by the EMT approximation method and rigorous electromagnetic field analysis with FEM simulations in Fig. 5. We found that the reflection of the multilayer structure for the SPP lithography is lower than 0.2 within a broad range from 400 nm to 500 nm and no larger than 0.4 in the whole visible range, as shown in Fig. 5(b). However, the reflection becomes much higher when we use equivalent effective media parameters calculated from Eqs. (1) and (2) to replace the Al/MgF2 multilayer structure. Result from the EMT approximation does show the resonance and low reflection as the rigorous electromagnetic field analysis. It is because the low reflection is caused by the resonance of metal/dielectric/metal (MIM) cavities [17] and obvious resonance dips can be observed in the spectra simulated M/D multilayer structures. However, the EMT method omits the details and treats the structure as a whole. No resonant cavity is remained in the EMT approximation and much more light is reflected according to the simulation result, which means the EMT approximation method is not accurate for analyzing the reflection property of M/D multilayer structure in this case.
4. Limitations of EMT on SPP interference Plasmonic lithography with SPPs interference is a potential technology for the lithography of next generation. More analyses
Fig. 5. (a) Schematic of the MgF2/Al multilayer structure with thickness of 30 nm/ 20 nm. The period of top Cr grating is 220 nm and the structure using the equivalent hyperbolic effective media parameters to replace the multilayer films. (b) Reflection spectra of the above structures.
about the limitations of EMT for SPP interference lithography are presented. Fig. 6 shows the intensity of electrical field distributions of interference patterns in a MgF2/Al multilayer structure with different thicknesses. The ratio of hMgF2 : hAl remains a constant 4:3. The exposure wavelength of 365 nm, which is commonly used in commercial lithography products. A layer of Al film is used under the photoresist layer in order to increase the pattern contrast and penetration depth of the pattern [12]. The structure in Fig. 6(a) with MgF2/Al film thickness 20 nm/ 15 nm is our optimized parameters for SPP interference lithography and is used as a reference here. We increase and reduce the film thicknesses with the same ratio 4:3 to 24 nm/18 nm, 16 nm/ 12 nm and 12 nm/9 nm. We found an interesting phenomenon that interference patterns keep uniform for the thicker films of 24 nm/18 nm, yet non-uniform for the thinner films of 16 nm/ 12 nm and 12 nm/9 nm, as shown in Fig. 6(b), (c) and (d), respectively. These results correspond to the difference of isofrequency contours discussed in Fig. 4 well. Multilayer films can be considered as multiple waveguides. The thicker waveguide support comparatively smaller propagation wave vector [18] and vice versa. Because of the inaccuracy of EMT for finite metal/dielectric multilayer films, wave vector kx in small range matches the ideal hyperbolic curve well while diverge with each other when kx becomes large. Therefore, interference patterns for thinner films than the reference will not be uniform even though they keep the same film thickness ratio of metal/dielectric. Other limitations of EMT is that it cannot be use to analyze the influence of surface roughness of the multilayer structure, which are very important for the SPPs lithography application. The energy loss caused by the roughness is fatal to the propagation of SPPs. Strict analysis of the electromagnetic field can give better
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Fig. 6. Intensity of electrical fields of the SPP interference patterns with different thicknesses of MgF2/Al films: (a) 24 nm/18 nm, (b) 20 nm/15 nm, (c) 16 nm/12 nm and (d) 12 nm/9 nm.
Fig. 7. (a) Electric fields of the multilayer structure with roughness of 2 nm and (b) 5 nm, the films in the dotted line are without any roughness and the pattern is used for comparison.
description of the formation of the patterns. We add a random roughness of average fluctuation of 2 nm and 5 nm to the film surface in the multilayer structure, leaving one column of pattern without roughness for comparison. The intensities of electrical filed distributions are shown in Fig. 7(a) and (b). Intensity of electrical filed distribution in the vertical direction changes significantly when the roughness increases from 2 nm to 5 nm because of the interference of scattered light from the rough film surface. The final interference patterns in photoresist do not change significantly, which means this kind of structure has good robustness for fabrication.
dielectric multilayer structures. An MgF2/Al multilayer structure for SPPs lithography is used as an example to demonstrate their difference. Beam spreading angles of SPPs and reflection properties of the multilayer structure are analyzed and compared using the both methods. Influences of film thickness and roughness for interference patterns are also presented. Above analyses and results show that the EMT method is not always accurate for the design and analysis of metal–dielectric multilayer structures with finite stacks. The rigorous electromagnetic approach is essential and more accurate in some cases. These conclusions can provide some useful guidelines for the design of plasmonic devices.
5. Conclusion Acknowledgments In summary, we analyze the consistency and exceptions between the effective media theory and the rigorous electromagnetic field approach when dealing with the hyperbolic metal–
This work was supported by the National Science Foundation Materials Research Science and Engineering Center (Program DMR
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