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OPTICS LETTERS / Vol. 39, No. 19 / October 1, 2014
Temporal soliton excitation in an ε-near-zero plasmonic metamaterial C. Argyropoulos,1 P.-Y. Chen,1 G. D’Aguanno,2 and A. Alù1,* 1
Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, Texas, 78712, USA 2 AEgis Technologies, 410 Jan Davis Dr., Huntsville, Alabama 35806, USA *Corresponding author: [email protected] Received May 19, 2014; revised August 22, 2014; accepted August 25, 2014; posted August 26, 2014 (Doc. ID 212312); published September 22, 2014 The excitation of temporal solitons in a metamaterial formed by an array of ε-near-zero (ENZ) plasmonic channels loaded with a material possessing a cubic (χ 3 ) nonlinearity are theoretically explored. The unique interplay between the peculiar dispersion properties of ENZ channels and their enhanced effective nonlinearity conspires to yield low threshold intensities for the formation of slow group velocity solitons. © 2014 Optical Society of America OCIS codes: (160.3918) Metamaterials; (250.5403) Plasmonics; (190.0190) Nonlinear optics. http://dx.doi.org/10.1364/OL.39.005566
Since their theoretical predictions over 40 years ago [1], temporal solitons in optical fibers have played a key role in optical communications [2]. The relatively recent advent of metamaterials has triggered a great deal of new research activity aimed at the study of the properties of temporal and/or spatial soliton propagation in such exotic materials [3–9]. For example, negative-index materials have been utilized to tailor the propagation of bright and dark gap solitons [10], and new nonlinear Schrödinger equations (NLSEs) were derived [11–13] adapted to material possessing Drude-like dispersion for both the electric permittivity and the magnetic permeability. More specifically, in Ref. [12], a NLSE for temporal soliton propagation through a generic metamaterial with Drude-like electric and magnetic permeability dispersion and simultaneous electric and magnetic cubic nonlinearities was derived, and soliton propagation in ideal ε-near-zero (ENZ) and in the μ-near-zero (MNZ) regimes was thoroughly investigated. In this Letter, we discuss temporal soliton propagation in a realistic ENZ metamaterial geometry, as depicted in Fig. 1. The structure is composed of an array of narrow periodic rectangular slits of width w, height t ≪ w, length l, and periods a; b ≫ t, carved in a silver screen with relative Drude permittivity dispersion εAg ε∞ − f 2p ∕ f f iγ, f p 2175 THz, γ 0 THz, ε∞ 5 [14]. For simplicity of analysis, the losses of silver are not included in the first part of our analysis; their effects will be considered later in this Letter. The slits are loaded with a generic nonlinear Kerr material, with relative permittivity εch εL χ 3 jE ch j2 , εL 2.2, χ 3 4.4 × 10−20 m2 ∕V2 (typical glass nonlinearity) [15], and jE ch j is the magnitude of the local electric field in the channel. The slit width is designed to operate at the cut-off of its dominant quasi-TE10 mode, i.e., at the frequency for which its guided wave number β has near-zero real part [16]. This geometry was originally introduced in [17], where it was shown that it ensures two relevant conditions: first, the guided wavelength in each slit is effectively infinite, since λg 2π∕Reβ; second, the corresponding modal impedance in each channel is very large, since it is inversely proportional to Reβ for TE modes. The large modal impedance can be successfully used to compensate 0146-9592/14/195566-04$15.00/0
the geometrical mismatch at the transition between each narrow slit and free-space. These properties result in an anomalous impedance-matching phenomenon, able to produce large coupling of the incident light inside each channel, together with a large field enhancement and uniform phase distribution, a combination of properties extremely well suited to boost optical nonlinearities [17]. We also notice that this anomalous matching phenomenon depends only on the interface properties, i.e., on the aperture to period ratio of the array, and it is therefore independent of the channels’ length l or shape [17]. First, we consider the linear operation of the rectangular plasmonic waveguide (εrect εL ). In this case, a dominant quasi-transverse electric (quasi-TE) mode propagates along the z axis with a wavenumber βrect , which can be computed by the dispersion equation [16]: q q β2rect − k2Ag w q ; tan β2pp − β2rect 2 β2 − β2 pp
(1)
rect
where βpp is the wavenumber of an equivalent parallel plate channel of width t calculated as
Fig. 1. Rectangular plasmonic channels loaded with a nonlinear optical material. The nonlinear channels are carved in a metal screen. © 2014 Optical Society of America
October 1, 2014 / Vol. 39, No. 19 / OPTICS LETTERS
Fig. 2. Dispersion of the normalized guided wavenumber β (normalized to the free-space wavevector k0 ) versus frequency in a lossless rectangular plasmonic channel with a cut-off frequency at approximately 278 THz.
q q 2 2 t εslab βpp − kAg 2 2 q ; − tanh βpp − kslab εAg β2 − k2 2 pp
(2)
slab
p p where kslab ω εslab ε0 μ0 k0 εslab is the wavenumber p in the dielectric slab, and kAg k0 εAg is the wavenumber in silver, which is almost purely imaginary at optical frequencies. The width and height dimensions of the rectangular channels are chosen to be w 200 nm and t 40 nm, respectively. The rectangular waveguide wavenumber βrect , calculated according to Eq. (1), is shown in Fig. 2. The ENZ operation is achieved at the cut-off frequency f 0 278 THz (λ0 1.079 μm), for which βrect 0. When the rectangular channels are loaded with Kerr nonlinear χ 3 materials, the effective permittivity is given by [17] εeff εL −
π2 b2 a2 3 χ jE in j2 ; k20 w2 w2 t2
(3)
where E in is the electric field impinging on the rectangular channels, and a and b are the periods along y and x, respectively. From Eq. (3), it is evident that the effective nonlinearity inside each channel of the array is effectively enhanced by a factor that depends on the inverse 2 3 of the open area of the array: χ 3 eff ba∕wt χ . For the particular design proposed here, and assuming a b 800 nm, a 1600 enhancement factor in the effective nonlinearity of the material filling the channels is achieved, together with full impedance matching to free-space. In Fig. 3, we show the group velocity vg ∂ω∕∂β of a pulse propagating along one of the channels. As expected [12], ultraslow pulse propagation can be achieved close to the ENZ operation. In this case a group velocity of c∕26 is achieved at 279 THz, where c is the velocity of light in free-space. The group velocity dispersion (GVD) is given by β2 d1∕vg ∕dω, and it measures the strength of dispersion of the metamaterial, determining the threshold intensity needed to launch temporal solitons. The GVD parameter, normalized to 1∕cω0 , ω0 2πf 0 , is plotted in Fig. 4. The normalized GVD parameter at our design frequency (279 THz) is β2 −1820.
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Fig. 3. Normalized group velocity vg versus frequency for a lossless rectangular plasmonic channel. In this case we operate at 279 THz close to the cut-off frequency (278 THz) where vg c∕26.
The fundamental soliton solution can be found by solving the NLSE as discussed in [12]. Two types of solitons can be supported: (i) bright solitons, obtained when β2 < 0 combined with a focusing nonlinearity χ 3 > 0; and (ii) dark solitons, formed when β2 > 0 and a defocusing nonlinearity χ 3 < 0. In our case, bright solitons are supported at the ENZ operation β2 < 0; χ 3 > 0. The threshold intensity to launch solitons is given by [12] S thr
jβ2 jε ; 2 μT 0 ω0 jχ 3 eff j
(4)
where ε, μ are the relative permittivity and permeability of the rectangular plasmonic waveguide, and T 0 is the input pulse duration. In our case, μ 1 and ε βrect ∕k0 2 , which is equal to ε 0.027 at the fre2 3 quency of interest, while χ 3 1600χ 3 . eff ba∕wt χ The GVD parameter of a standard optical fiber at telecommunication wavelengths λ 1.55 μm expressed in units of 1∕cω0 (similar to Fig. 4) is approximately ≃ −0.01∕cω0 and the threshold intensity βfiber 2 3 2 2 S fiber thr ≃ 0.03∕T 0 ω0 cjχ fiber j, where the fiber’s refractive index is n ≃ 1.4 [12]. The ratio between the power required to launch solitons in the plasmonic ENZ structure S ENZ thr and the optical is computed at the frequency of interest fiber S fiber thr
Fig. 4. Normalized GVD parameter β2 of a lossless rectangular plasmonic channel. In the figure, the value of β2 at the design frequency (279 THz) is reported.
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Fig. 5. Ratio of threshold intensities required to launch solitons in the plasmonic waveguide and the optical fiber ENZ S fiber thr versus the frequency S thr of operation. In the figure, it fiber at the operational is highlighted the value of S ENZ thr ∕S thr frequency (279 THz). fiber f 279 THz and is equal to S ENZ thr ∕S thr 1.57. Interestingly, the threshold intensity to launch these slow-group velocity solitons in the proposed plasmonic metamaterial is comparable to the intensity required to launch standard solitons in a conventional fiber. The high dispersion of the plasmonic waveguide is translated to high values of β2 at the ENZ frequency. However, this large dispersion is compensated by the low effective permittivity and by the enhanced effective nonlinearity of the plasmonic ENZ waveguide. Finally, we plot in Fig. 5 the ratio fiber of threshold intensities S ENZ thr ∕S thr for different frequencies, assuming no frequency dispersion for the optical fiber and constant enhancement of nonlinearity for the plasmonic waveguide. It is interesting that even lower intensities are needed to launch solitons in the plasmonic metamaterial compared to a fiber over a broad frequency range, in addition to the remarkable advantages of large power coupling into the slits, due to impedance matching, and largely enhanced soliton-matter interaction due to the strong field enhancement in the slits (here solitons are launched in slits that may be orders of magnitude smaller than the wavelength). Note that in the aforementioned comparison we have made the fair assumption of considering the nonlinearity of the material filling the plasmonic channel to be the same as the one of a glass fiber. Clearly, the inherent losses of silver, or of other metals, at optical frequencies may play a detrimental role to the creation of temporal and spatial solitons inside the channels. Nevertheless, material scientists are currently in the pursuit of realizing lower loss plasmonic materials [18]. Losses can also be reduced, or even totally disappear, when gain media are appropriately introduced in plasmonic devices [19]. Furthermore, alternative designs based on superconducting metamaterials have been proposed to alleviate losses from DC to the terahertz range [20]. Even considering realistic losses of plasmonic metals, the proposed structure can realistically support soliton propagation in the low-frequency regime, e.g., in the 100 GHz frequency range, considering a damping term in the Drude model γ 4.35 [14]. The design dimensions in this case need to be retuned to: w 1 mm, t 0.4 μm, a 4 μm, b 2 mm, and l 160 mm. The wavenumber
Fig. 6. Real part of dispersion of the normalized guided wavenumber β (normalized to the free-space wavevector k0 ) versus frequency in a lossy rectangular waveguide with a cut-off frequency at approximately 96 GHz.
is computed using Eq. (1), and the corresponding real part of β is now plotted in Fig. 6. For this new geometry, the cut-off frequency has a lower value, approximately equal to f 0 96 GHz. Again, ENZ operation is possible and, when nonlinearities are included, the effective permittivity is given by Eq. (3). As a result, temporal solitons arise despite the presence of losses. The group velocity, normalized to the speed of light, is plotted in Fig. 7 (blue line) for the lossy rectangular channels. Again, we choose to operate slightly off the ENZ frequency at f > f 0 98 GHz, in order to avoid high dispersion. The obtained group velocity is vg c∕7, larger than the previous example at optical frequencies, but still quite small. The GVD parameter β2 is also shown in Fig. 7 (red line). At the frequency of operation the normalized β2 −104.6, which leads to bright soliton formation for χ 3 > 0. The calculated propagation length at the operation frequency is La 1∕2 × Imβ 16.04 cm, and the second order dispersion length Ld T 20 ∕jβ2 j 15.6 cm, where the pulse duration is chosen to be T 0 300 ps. We emphasize that the soliton dynamics within the metamaterial channels are similar to the ones in conventional guided structures, and are therefore not explicitly reported in this Letter. We instead focused here on showing the large nonlinear response and key parameters to
Fig. 7. Normalized group velocity vg (blue line) versus frequency and normalized GVD parameter β2 (red line) of the lossy rectangular waveguide. In this case, we operate at 98 GHz close to the cut-off frequency (96 GHz) where vg c∕7 and β2 −104.6.
October 1, 2014 / Vol. 39, No. 19 / OPTICS LETTERS
support temporal soliton formation in ENZ metamaterials. To summarize, we have studied temporal slow-group velocity solitons sustained by an array of plasmonic channels at the ENZ regime. We believe that our results may inspire further experimental and theoretical efforts in the emerging field of nonlinear plasmonics combined with slow light applications, such as all-optical buffering and switching. This work was partially supported by the National Science Foundation with grant no. ECCS-1348049, by DARPA SBIR project Nonlinear Plasmonic Devices, by the Army Research Office with grant no. W911NF-111-0447, and by the Air Force Office of Scientific Research with grant no. FA9550-13-1-0204. References 1. A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973). 2. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1995). 3. N. A. Zharova, I. Shadrivov, A. A. Zharov, and Y. S. Kivshar, Opt. Express 13, 1291 (2005). 4. Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, Phys. Rev. Lett. 99, 153901 (2007). 5. R. Driben and J. Herrmann, Opt. Lett. 35, 2529 (2010).
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6. N. N. Rosanov, N. V. Vysotina, A. N. Shatsev, A. S. Desyatnikov, and Y. S. Kivshar, Phys. Rev. Lett. 108, 133902 (2012). 7. P. Ginzburg, A. V. Krasavin, and A. V. Zayats, New J. Phys. 15, 013031 (2013). 8. M. G. Silveirinha, Phys. Rev. B 87, 235115 (2013). 9. C. Rizza, A. Ciattoni, and E. Palange, Phys. Rev. A 83, 053805 (2011). 10. G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, Phys. Rev. Lett. 93, 213902 (2004). 11. M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Polyakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005). 12. G. D’Aguanno, N. Mattiucci, and M. J. Bloemer, J. Opt. Soc. Am. B 25, 1236 (2008). 13. S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, Phys. Rev. A 75, 033815 (2007). 14. P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972). 15. R. W. Boyd, Nonlinear Optics (Academic, London, 1992). 16. A. Alù and N. Engheta, Phys. Rev. B 78, 035440 (2008). 17. C. Argyropoulos, P. Y. Chen, G. D’Aguanno, N. Engheta, and A. Alù, Phys. Rev. B 85, 045129 (2012). 18. A. Boltasseva and H. A. Atwater, Science 331, 290 (2011). 19. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, Nature 466, 735 (2010). 20. R. Singh, Z. Tian, J. G. Han, C. Rockstuhl, J. Q. Gu, and W. L. Zhang, Appl. Phys. Lett. 96, 071114 (2010).
OPTICS LETTERS / Vol. 39, No. 19 / October 1, 2014
Temporal soliton excitation in an ε-near-zero plasmonic metamaterial C. Argyropoulos,1 P.-Y. Chen,1 G. D’Aguanno,2 and A. Alù1,* 1
Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, Texas, 78712, USA 2 AEgis Technologies, 410 Jan Davis Dr., Huntsville, Alabama 35806, USA *Corresponding author: [email protected] Received May 19, 2014; revised August 22, 2014; accepted August 25, 2014; posted August 26, 2014 (Doc. ID 212312); published September 22, 2014 The excitation of temporal solitons in a metamaterial formed by an array of ε-near-zero (ENZ) plasmonic channels loaded with a material possessing a cubic (χ 3 ) nonlinearity are theoretically explored. The unique interplay between the peculiar dispersion properties of ENZ channels and their enhanced effective nonlinearity conspires to yield low threshold intensities for the formation of slow group velocity solitons. © 2014 Optical Society of America OCIS codes: (160.3918) Metamaterials; (250.5403) Plasmonics; (190.0190) Nonlinear optics. http://dx.doi.org/10.1364/OL.39.005566
Since their theoretical predictions over 40 years ago [1], temporal solitons in optical fibers have played a key role in optical communications [2]. The relatively recent advent of metamaterials has triggered a great deal of new research activity aimed at the study of the properties of temporal and/or spatial soliton propagation in such exotic materials [3–9]. For example, negative-index materials have been utilized to tailor the propagation of bright and dark gap solitons [10], and new nonlinear Schrödinger equations (NLSEs) were derived [11–13] adapted to material possessing Drude-like dispersion for both the electric permittivity and the magnetic permeability. More specifically, in Ref. [12], a NLSE for temporal soliton propagation through a generic metamaterial with Drude-like electric and magnetic permeability dispersion and simultaneous electric and magnetic cubic nonlinearities was derived, and soliton propagation in ideal ε-near-zero (ENZ) and in the μ-near-zero (MNZ) regimes was thoroughly investigated. In this Letter, we discuss temporal soliton propagation in a realistic ENZ metamaterial geometry, as depicted in Fig. 1. The structure is composed of an array of narrow periodic rectangular slits of width w, height t ≪ w, length l, and periods a; b ≫ t, carved in a silver screen with relative Drude permittivity dispersion εAg ε∞ − f 2p ∕ f f iγ, f p 2175 THz, γ 0 THz, ε∞ 5 [14]. For simplicity of analysis, the losses of silver are not included in the first part of our analysis; their effects will be considered later in this Letter. The slits are loaded with a generic nonlinear Kerr material, with relative permittivity εch εL χ 3 jE ch j2 , εL 2.2, χ 3 4.4 × 10−20 m2 ∕V2 (typical glass nonlinearity) [15], and jE ch j is the magnitude of the local electric field in the channel. The slit width is designed to operate at the cut-off of its dominant quasi-TE10 mode, i.e., at the frequency for which its guided wave number β has near-zero real part [16]. This geometry was originally introduced in [17], where it was shown that it ensures two relevant conditions: first, the guided wavelength in each slit is effectively infinite, since λg 2π∕Reβ; second, the corresponding modal impedance in each channel is very large, since it is inversely proportional to Reβ for TE modes. The large modal impedance can be successfully used to compensate 0146-9592/14/195566-04$15.00/0
the geometrical mismatch at the transition between each narrow slit and free-space. These properties result in an anomalous impedance-matching phenomenon, able to produce large coupling of the incident light inside each channel, together with a large field enhancement and uniform phase distribution, a combination of properties extremely well suited to boost optical nonlinearities [17]. We also notice that this anomalous matching phenomenon depends only on the interface properties, i.e., on the aperture to period ratio of the array, and it is therefore independent of the channels’ length l or shape [17]. First, we consider the linear operation of the rectangular plasmonic waveguide (εrect εL ). In this case, a dominant quasi-transverse electric (quasi-TE) mode propagates along the z axis with a wavenumber βrect , which can be computed by the dispersion equation [16]: q q β2rect − k2Ag w q ; tan β2pp − β2rect 2 β2 − β2 pp
(1)
rect
where βpp is the wavenumber of an equivalent parallel plate channel of width t calculated as
Fig. 1. Rectangular plasmonic channels loaded with a nonlinear optical material. The nonlinear channels are carved in a metal screen. © 2014 Optical Society of America
October 1, 2014 / Vol. 39, No. 19 / OPTICS LETTERS
Fig. 2. Dispersion of the normalized guided wavenumber β (normalized to the free-space wavevector k0 ) versus frequency in a lossless rectangular plasmonic channel with a cut-off frequency at approximately 278 THz.
q q 2 2 t εslab βpp − kAg 2 2 q ; − tanh βpp − kslab εAg β2 − k2 2 pp
(2)
slab
p p where kslab ω εslab ε0 μ0 k0 εslab is the wavenumber p in the dielectric slab, and kAg k0 εAg is the wavenumber in silver, which is almost purely imaginary at optical frequencies. The width and height dimensions of the rectangular channels are chosen to be w 200 nm and t 40 nm, respectively. The rectangular waveguide wavenumber βrect , calculated according to Eq. (1), is shown in Fig. 2. The ENZ operation is achieved at the cut-off frequency f 0 278 THz (λ0 1.079 μm), for which βrect 0. When the rectangular channels are loaded with Kerr nonlinear χ 3 materials, the effective permittivity is given by [17] εeff εL −
π2 b2 a2 3 χ jE in j2 ; k20 w2 w2 t2
(3)
where E in is the electric field impinging on the rectangular channels, and a and b are the periods along y and x, respectively. From Eq. (3), it is evident that the effective nonlinearity inside each channel of the array is effectively enhanced by a factor that depends on the inverse 2 3 of the open area of the array: χ 3 eff ba∕wt χ . For the particular design proposed here, and assuming a b 800 nm, a 1600 enhancement factor in the effective nonlinearity of the material filling the channels is achieved, together with full impedance matching to free-space. In Fig. 3, we show the group velocity vg ∂ω∕∂β of a pulse propagating along one of the channels. As expected [12], ultraslow pulse propagation can be achieved close to the ENZ operation. In this case a group velocity of c∕26 is achieved at 279 THz, where c is the velocity of light in free-space. The group velocity dispersion (GVD) is given by β2 d1∕vg ∕dω, and it measures the strength of dispersion of the metamaterial, determining the threshold intensity needed to launch temporal solitons. The GVD parameter, normalized to 1∕cω0 , ω0 2πf 0 , is plotted in Fig. 4. The normalized GVD parameter at our design frequency (279 THz) is β2 −1820.
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Fig. 3. Normalized group velocity vg versus frequency for a lossless rectangular plasmonic channel. In this case we operate at 279 THz close to the cut-off frequency (278 THz) where vg c∕26.
The fundamental soliton solution can be found by solving the NLSE as discussed in [12]. Two types of solitons can be supported: (i) bright solitons, obtained when β2 < 0 combined with a focusing nonlinearity χ 3 > 0; and (ii) dark solitons, formed when β2 > 0 and a defocusing nonlinearity χ 3 < 0. In our case, bright solitons are supported at the ENZ operation β2 < 0; χ 3 > 0. The threshold intensity to launch solitons is given by [12] S thr
jβ2 jε ; 2 μT 0 ω0 jχ 3 eff j
(4)
where ε, μ are the relative permittivity and permeability of the rectangular plasmonic waveguide, and T 0 is the input pulse duration. In our case, μ 1 and ε βrect ∕k0 2 , which is equal to ε 0.027 at the fre2 3 quency of interest, while χ 3 1600χ 3 . eff ba∕wt χ The GVD parameter of a standard optical fiber at telecommunication wavelengths λ 1.55 μm expressed in units of 1∕cω0 (similar to Fig. 4) is approximately ≃ −0.01∕cω0 and the threshold intensity βfiber 2 3 2 2 S fiber thr ≃ 0.03∕T 0 ω0 cjχ fiber j, where the fiber’s refractive index is n ≃ 1.4 [12]. The ratio between the power required to launch solitons in the plasmonic ENZ structure S ENZ thr and the optical is computed at the frequency of interest fiber S fiber thr
Fig. 4. Normalized GVD parameter β2 of a lossless rectangular plasmonic channel. In the figure, the value of β2 at the design frequency (279 THz) is reported.
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Fig. 5. Ratio of threshold intensities required to launch solitons in the plasmonic waveguide and the optical fiber ENZ S fiber thr versus the frequency S thr of operation. In the figure, it fiber at the operational is highlighted the value of S ENZ thr ∕S thr frequency (279 THz). fiber f 279 THz and is equal to S ENZ thr ∕S thr 1.57. Interestingly, the threshold intensity to launch these slow-group velocity solitons in the proposed plasmonic metamaterial is comparable to the intensity required to launch standard solitons in a conventional fiber. The high dispersion of the plasmonic waveguide is translated to high values of β2 at the ENZ frequency. However, this large dispersion is compensated by the low effective permittivity and by the enhanced effective nonlinearity of the plasmonic ENZ waveguide. Finally, we plot in Fig. 5 the ratio fiber of threshold intensities S ENZ thr ∕S thr for different frequencies, assuming no frequency dispersion for the optical fiber and constant enhancement of nonlinearity for the plasmonic waveguide. It is interesting that even lower intensities are needed to launch solitons in the plasmonic metamaterial compared to a fiber over a broad frequency range, in addition to the remarkable advantages of large power coupling into the slits, due to impedance matching, and largely enhanced soliton-matter interaction due to the strong field enhancement in the slits (here solitons are launched in slits that may be orders of magnitude smaller than the wavelength). Note that in the aforementioned comparison we have made the fair assumption of considering the nonlinearity of the material filling the plasmonic channel to be the same as the one of a glass fiber. Clearly, the inherent losses of silver, or of other metals, at optical frequencies may play a detrimental role to the creation of temporal and spatial solitons inside the channels. Nevertheless, material scientists are currently in the pursuit of realizing lower loss plasmonic materials [18]. Losses can also be reduced, or even totally disappear, when gain media are appropriately introduced in plasmonic devices [19]. Furthermore, alternative designs based on superconducting metamaterials have been proposed to alleviate losses from DC to the terahertz range [20]. Even considering realistic losses of plasmonic metals, the proposed structure can realistically support soliton propagation in the low-frequency regime, e.g., in the 100 GHz frequency range, considering a damping term in the Drude model γ 4.35 [14]. The design dimensions in this case need to be retuned to: w 1 mm, t 0.4 μm, a 4 μm, b 2 mm, and l 160 mm. The wavenumber
Fig. 6. Real part of dispersion of the normalized guided wavenumber β (normalized to the free-space wavevector k0 ) versus frequency in a lossy rectangular waveguide with a cut-off frequency at approximately 96 GHz.
is computed using Eq. (1), and the corresponding real part of β is now plotted in Fig. 6. For this new geometry, the cut-off frequency has a lower value, approximately equal to f 0 96 GHz. Again, ENZ operation is possible and, when nonlinearities are included, the effective permittivity is given by Eq. (3). As a result, temporal solitons arise despite the presence of losses. The group velocity, normalized to the speed of light, is plotted in Fig. 7 (blue line) for the lossy rectangular channels. Again, we choose to operate slightly off the ENZ frequency at f > f 0 98 GHz, in order to avoid high dispersion. The obtained group velocity is vg c∕7, larger than the previous example at optical frequencies, but still quite small. The GVD parameter β2 is also shown in Fig. 7 (red line). At the frequency of operation the normalized β2 −104.6, which leads to bright soliton formation for χ 3 > 0. The calculated propagation length at the operation frequency is La 1∕2 × Imβ 16.04 cm, and the second order dispersion length Ld T 20 ∕jβ2 j 15.6 cm, where the pulse duration is chosen to be T 0 300 ps. We emphasize that the soliton dynamics within the metamaterial channels are similar to the ones in conventional guided structures, and are therefore not explicitly reported in this Letter. We instead focused here on showing the large nonlinear response and key parameters to
Fig. 7. Normalized group velocity vg (blue line) versus frequency and normalized GVD parameter β2 (red line) of the lossy rectangular waveguide. In this case, we operate at 98 GHz close to the cut-off frequency (96 GHz) where vg c∕7 and β2 −104.6.
October 1, 2014 / Vol. 39, No. 19 / OPTICS LETTERS
support temporal soliton formation in ENZ metamaterials. To summarize, we have studied temporal slow-group velocity solitons sustained by an array of plasmonic channels at the ENZ regime. We believe that our results may inspire further experimental and theoretical efforts in the emerging field of nonlinear plasmonics combined with slow light applications, such as all-optical buffering and switching. This work was partially supported by the National Science Foundation with grant no. ECCS-1348049, by DARPA SBIR project Nonlinear Plasmonic Devices, by the Army Research Office with grant no. W911NF-111-0447, and by the Air Force Office of Scientific Research with grant no. FA9550-13-1-0204. References 1. A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973). 2. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1995). 3. N. A. Zharova, I. Shadrivov, A. A. Zharov, and Y. S. Kivshar, Opt. Express 13, 1291 (2005). 4. Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, Phys. Rev. Lett. 99, 153901 (2007). 5. R. Driben and J. Herrmann, Opt. Lett. 35, 2529 (2010).
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