SPM and XPM nonlinear effects in plasmonic directional couplers, considering the ponderomotive metal nonlinearity N. Nozhat1,* and N. Granpayeh2 1
Faculty of Electrical Engineering, Shiraz University of Technology, Shiraz 7155713876, Iran 2
Center of Excellence in Electromagnetics, Faculty of Electrical Engineering, K. N. Toosi University of Technology, Tehran 1631714191, Iran *Corresponding author: [email protected] Received 6 January 2014; revised 13 April 2014; accepted 14 April 2014; posted 17 April 2014 (Doc. ID 204173); published 20 May 2014
In this paper, a two-dimensional nonlinear plasmonic directional coupler (2D-NPDC), with 90° waveguide bends, has been numerically analyzed by the finite-difference time-domain (FDTD) method, considering the nonlinear response of metal due to the ponderomotive force. It has been shown that the required switching power of the 2D-NPDC is 0.05% of that when only the dielectric is nonlinear and the nonlinearity of metal is neglected. Also, the cross-phase modulation (XPM) nonlinear effect has been investigated, which the power for switching is decreased significantly compared to the one with the self-phase modulation (SPM) effect. © 2014 Optical Society of America OCIS codes: (060.1810) Buffers, couplers, routers, switches, and multiplexers; (130.4815) Optical switching devices; (190.3270) Kerr effect; (240.6680) Surface plasmons; (250.5403) Plasmonics. http://dx.doi.org/10.1364/AO.53.003328
1. Introduction
In recent years, researches in nanophotonics based on surface plasmon polaritons (SPPs) have been expanded rapidly. SPPs propagate through metal– dielectric interfaces and guide in nanoscale devices with dimensions below the diffraction limit. Therefore, linear and nonlinear plasmonic devices are appropriate candidates for use in photonic integrated circuits (PICs) [1–6]. Several nonlinear plasmonic structures have been studied, so far. However, in most of these structures, nonlinearity is thought to originate from the dielectric, and the nonlinear response of metal has been neglected [7–13]. In general, metal nonlinearity is based on several physical effects, such as interband transition of 1559-128X/14/153328-05$15.00/0 © 2014 Optical Society of America 3328
APPLIED OPTICS / Vol. 53, No. 15 / 20 May 2014
bound electrons and laser induced hot-electron contribution, which have previously been studied. It has been shown that another nonlinear response phenomenon of metals is based on the inherent electron plasma nonlinearity of the metal layers. The ponderomotive force, which is the nonlinear force due to the electromagnetic wave propagation within the plasma, repels electrons from the high field intensity region. Therefore, the local plasma frequency and the absolute value of the permittivity of metal are reduced and the real part of the local refractive index of SPPs is enhanced. Furthermore, the propagation loss is decreased because of the confinement of the field to the low carrier density region. The effect of ponderomotive nonlinearity on SPP propagation, nonlinear effects in plasmonic metal film waveguides, and nonlinear guided plasmonic modes of two coupled thin metallic films by the third-order optical response due to the ponderomotive metal nonlinearities have been investigated [14–19].
Previously, we analyzed and simulated the twodimensional nonlinear plasmonic directional coupler (2D-NPDC) switches with 45° and 90° waveguide bends utilizing Kerr self-phase modulation (SPM) and cross-phase modulation (XPM) nonlinear effects, considering only the Kerr nonlinear effects of the dielectrics [11,12]. It has been shown theoretically and experimentally that, because of the strong nonlinearity of metals at optical wavelengths, the nonlinear response of metallic surfaces is enhanced [16,20–22]. In this paper, we analyze the performance of the same 2D-NPDC by considering the metal nonlinearity due to the ponderomotive force. Both SPM and XPM nonlinear effects are studied and it is shown that the required switching power is significantly reduced compared to the case when the metal nonlinearity is neglected. The paper is organized as follows. In Section 2, the simulation method and in Section 3, the results are described and discussed. The paper is concluded in Section 4. To obtain the intensity dependent dielectric constant of metal εPM , a fluid model is used to describe the electron motion in metals. The nonlinear force acting on an electron in an oscillating electromagnetic field is given by [14,15]: FPM
(1)
where m and e are the electron mass and charge, respectively, ω is the lightwave angular frequency, jEj2 is the intensity of local field and ΦPM is the ponderomotive potential. The carrier density due to the ponderomotive potential is obtained as nPM
1 2m 3∕2 Ef − ΦPM 3∕2 ; 3π 2 ℏ
εM ω ε∞ −
ω2p ; ω − jγ p ω 2
(2)
where Ef is the Fermi energy and ℏ is the reduced Planck or Dirac constant. Therefore, the intensity-dependent metal relative permittivity related to the ponderomotive force is described by 3∕2 e2 2m e2 jEj2 3∕2 εPM jEj 1 − 2 Ef − : 3π ε0 mω2 ℏ2 2mω2 (3) 2
(5)
where ε∞ and γ p are the relative permittivity at infinite frequency and collision angular frequency, respectively. From Eq. (4), it can be seen that the nonlinear susceptibility of metal is highly dispersive and, for the second and third telecommunication χwindows of, respectively, 1310 and 1550 nm wavelengths, χ PM is in the order of 10−18 m2 ∕V2 [14]. Due to the Kerr nonlinear effect, the refractive index of a material depends on the signal I s , and pump I p intensities as follows [12]: nd nl n2 I s 2I p ;
2. Simulation Method and Discussion
1 e2 ∇jEj2 −∇ΦPM ; − 2m ω2
where ωp is the plasma angular frequency, χ PM is the nonlinear ponderomotive susceptibility, which is a function of ω−4, and εM ω is the linear metal dielectric constant expressed by the Drude model [11]:
(6)
where nl and n2 are the linear refractive index and the nonlinear Kerr coefficient, respectively. n2 I s and 2n2 I p are the refractive index increments, due to SPM and XPM, respectively. Also, according to Eq. (4), the nonlinear response of metal is a dispersive Kerr-like effect. We have employed the two-dimensional finitedifference time-domain (2D-FDTD) numerical method for the simulation of the 2D-NPDC switch. The convolutional perfectly matched layer (CPML) has been used as the absorbing boundary conditions [11]. 100-cell CPMLs are used to ensure all the powers to be absorbed at the boundaries of the simulation region. The step size of the FDTD cell in x and z directions are Δx Δz 1 nm and the time step isp chosen by Courant condition to be Δt 0.95∕c Δx−2 Δz−2 , where c is the free space speed of light. Since SPPs are excited by only TM-polarization, the nonlinear plasmonic coupler of Fig. 1 is excited by a TM-polarized pulse with electromagnetic field components of H y, Ex , and Ez . The signal and the pump sources and the output power monitors are placed at the distance of 150 nm from the 90° bends. We have utilized the MATLAB and C++ programming languages for our analyses and simulations.
By Taylor series expansion, the leading term gives a Kerr-like nonlinearity for the dielectric constant of metal: 2∕3 4 ωp 3 e jEj2 2 3π 2 ε0 ℏme ω4 εM ω χ PM jEj2 ;
εPM jEj2 εM ω
(4)
Fig. 1. Schematic view of the proposed 2D-NPDC switch with 90° waveguide bends. The width of the dielectric waveguides and the gap between the coupler waveguides are h 30 nm and d 10 nm, respectively. Po3 and Po4 are, respectively, bar and cross ports. Signal input is Pi1 . 20 May 2014 / Vol. 53, No. 15 / APPLIED OPTICS
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The procedure for applying the 2D-FDTD method to our structure is described here. Maxwell’s equations are solved to simulate the performance of our nonlinear structure. The electric flux density D, is calculated from the discrete form of the following equation: ∇×H
∂ D; ∂t
(7)
where H is the magnetic field vector. Since the dispersive medium is characterized by the Drude model and the nonlinear regions have Kerr-type nonlinear effects, the relation between the linear PL and nonlinear PNL polarizations, the electric field E, and the electric flux density D, are given as D ε0 ε∞ E PL PNL :
(8)
The linear polarization of PLD related to the Drude NL model and nonlinear polarizations of PNL K and PPM contributed by the dielectric and metal nonlinearities, respectively, are expressed as ∂2 PLD ∂PLD ε0 ω2p E; γ P ∂t ∂t2
(9)
3 3 PNL K t ε0 χ E t;
(10)
2 χ PNL PM t ε0 Et PM t E t;
(11)
where χ 3 is the third-order nonlinear susceptibility and denotes the convolution. An auxiliary variable can be introduced for the convolution of Eq. (11): Gt χ PM t E2 t:
(12)
The Fourier-transform of Eq. (12) leads to Gω χ PM ωIE2 t;
(13)
where, from Eq. (4), χ PM ω 3∕2ωp ∕3π 2 ε0 ℏme2∕3 e4 ∕ω4 . The inverse Fourier-transform of Eq. (13) results in the following equation: 2∕3 ωp ∂4 G 3 e4 E2 : 2 3π 2 ε0 ℏme ∂t4
APPLIED OPTICS / Vol. 53, No. 15 / 20 May 2014
(16)
where aD 4∕γ p Δt 2, bD γ p Δt − 2∕γ p Δt 2, cD 2ε0 ω2p Δt2 ∕γ p Δt 2, Δt is the time step, and Gn1 4Gn − 6Gn−1 4Gn−2 − Gn−3 Δt4 3∕2ωp ∕ 3π 2 ε0 ℏme2∕3 e4 En−1 2 . Finally, the updated discrete magnetic field is derived from Maxwell’s curl equation of ∇ × E −μ
∂ H; ∂t
(17)
where μ is the magnetic permeability [11]. 3. Numerical Results and Discussion
In our previous work, we demonstrated that the loss of the 2D-NPDC with 90° waveguide bends is lower than that of the one with 45° bends. Therefore, as shown in Fig. 1, we have studied the 90° 2D-NPDC [11]. The metal and nonlinear dielectric materials for our nonlinear PDC are chosen to be silver and chalcogenide glass, respectively. The Drude parameters of silver and linear and nonlinear parameters of chalcogenide glass are given in Table 1. The PDC length of the coupling region is chosen to be L 380 nm. A sinusoidal wave as a signal lightwave with the wavelength of 1493 nm is launched to port 1 of the 2D-NPDC of Fig. 1. The selected wavelength enables us to compare the performance of the 2D-NPDC by considering the metal nonlinearity with our previous work [12]. The simulation is valid for other signal wavelengths in the third telecommunication window, but the coupler length should be modified. The output powers ratio versus the input signal power is depicted in Fig. 2. For the case that both the metal and dielectric are nonlinear, it has been shown in Fig. 2(b) that, in the linear state with the input signal level of 1.07 × 10−8 W∕μm, the output powers ratio Po3 ∕Po4 is −4.38 dB and the light exits from port 4, as illustrated in Fig. 3(a). The coupling length is inversely related to the differences between the symmetric and asymmetric propagation constants of SPP modes, defined as a coupling coefficient [12]. By increasing the input signal power, the refractive index of the nonlinear medium and the coupling coefficient is increased and so the coupling length of the PDC is decreased due to the Kerr nonlinear effects of dielectric and
Table 1.
Parameters for Simulation of Proposed PDC [11]
Material
Parameters
Value
Metal (Silver’s Drude model)
ε∞ γ p (Hz) ωp (Hz) nl n2 (cm2 ∕W) χ 3 (m2 ∕V2 )
3.7 2.73 × 1013 1.38 × 1016 2.4 2 × 10−13 4 × 10−19
(15)
By inserting the FDTD update expressions of Eqs. (9), (10), and (15) into Eq. (8), the electric field is derived from the following equation: 3330
n−1 Dn1 − aD PnD − bD PD − c D En ; 3 n1 2 ε0 ε∞ ε0 χ E ε0 Sn1
(14)
From Eqs. (11) and (12) we can write PNL PM t ε0 EtGt:
En1
Insulator (Chalcogenide)
4 3
Po3 /Po4 (dB)
2 1 0 -1 -2 -3 -4 -5 0
10
20
30
40
50
Input Signal Power (W/µm) 4 3
Po3 /Po4 (dB)
2 1
Fig. 3. Magnetic field distributions in the 2D-NPDC with metal nonlinearity and dielectric SPM Kerr nonlinearity, at λs 1493 nm in (a) linear and (b) nonlinear states when the signal input powers are 1.07 × 10−8 W∕μm and 0.028 W∕μm, respectively.
0 -1 -2 -3 -4 -5 0
0.005
0.01
0.015
0.02
0.025
Input Signal Power (W/µm)
the pump light does not switch to port 3, the signal switches from the cross to port. The magnetic field distributions of NPDC performance in these cases are in Fig. 4.
whereas the bar the 2Ddepicted
Fig. 2. Output powers ratio versus the input signal power for the 2D-NPDC switch of Fig. 1. (a) Without metal nonlinearity [12] and (b) with metal nonlinearity [this work].
metal. Therefore, the input signal light is switched to port 3 and exits from it, as shown in Fig. 3(b). In this case, the output power ratio is increased to its maximum value of 3.7 dB with input power of 0.028 W∕μm. For comparison, the output powers ratio versus input power for the 2D-NPDC of Fig. 1 with Kerr nonlinear effect of dielectric, obtained in our previous work, is depicted in Fig. 2(a) [12]. It can be observed from Figs. 2(a) and 2(b) that, by considering the Kerr-like nonlinear response of metal, the required PDC switching power is 0.05% that required in our previous work. Next, we studied the effect of XPM on the performance of 2D-NPDC. From the transmission spectra of the coupler, we found that the wavelength of the pump light can be selected in the range of 1060–1200 nm. For this purpose, the signal light at the wavelengths of λs 1493 nm and a low power pump light at λp 1194 nm are simultaneously launched to ports 1 and 2, respectively. In the linear state, the pump light exits from port 4. At higher pump power, the coupling length decreases and
Fig. 4. Magnetic field distributions of the 2D-NPDC with metal nonlinearity and dielectric XPM Kerr nonlinearity (a) in linear or nonlinear state of pump light at λp 1194 nm and powers of 0 and 4.35 × 10−6 W∕μm, respectively, and (b) signal light with high power pump lightwave. The pump power is omitted to show the switching of signal from cross port to bar port. 20 May 2014 / Vol. 53, No. 15 / APPLIED OPTICS
3331
4
the switching power of the PDC in the XPM case is 1.55 × 10−4 of the case with SPM effect.
3 2
Po3/Po4 (dB)
References 1 0 -1 -2 -3 -4 -5
0
0.5
1
1.5
2
2.5
3
3.5
4
Input Pump Power (W/µm)
4.5 -6
x 10
Fig. 5. Output powers ratio versus input pump power for the 2D-NPDC switch of Fig. 1 with metal nonlinearity for the signal power and wavelength of 1.07 × 10−8 W∕μm and 1493 nm, respectively, and a pump wavelength of 1194 nm.
Since the pump power is higher than the signal power, in Fig. 4(b) the pump power is omitted to show clearly the lightwave signal switching performance in the PDC. It can be seen from Eq. (6) that by utilizing the XPM Kerr effect, the refractive index variation of the nonlinear medium is higher than that of SPM. Therefore, the required switching power in the 2D-NPDC using XPM is much lower than the signal power of the SPM state, as depicted in Fig. 5. By increasing the pump power from 0 to 4.35 × 10−6 W∕μm, the signal lightwave switches from the cross port to the bar port and the output powers ratio increases from −4.24 dB to 3.95 dB, respectively. By further increasing the input power, the output power ratio decreases, due to the coupling length change through the variation of the refractive index by XPM. By comparison of Figs. 2 and 5, it is clear that, by employing XPM nonlinear effect, the 2D-NPDC switching power is reduced significantly. By considering the nonlinearity of metal, the switching powers obtained in our simulations are reasonable for using the nonlinear 2D-NPDC in the telecommunication systems. 4. Conclusion
In this paper, the performance of the twodimensional nonlinear plasmonic directional coupler (2D-NPDC) with 90° waveguide bends is analyzed by the finite-difference time-domain (FDTD) numerical method, considering the ponderomotive nonlinear response of metal and dielectric Kerr nonlinear effect. Both SPM and XPM nonlinear effects have been considered. We have shown that, in the SPM case, the required switching input signal power is 0.05% that of when only the structure dielectric Kerr nonlinear effect is considered. In addition, we have shown that
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1. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007). 2. W. Xue, Y. Guo, J. Zhang, and W. Zhang, “Propagation properties of a modified slot surface plasmonic waveguides,” J. Lightwave Technol. 27, 2634–2641 (2009). 3. N. Nozhat, R. C. McPhedran, C. M. de Sterke, and N. Granpayeh, “The plasmonic folded directional coupler,” Photon. Nanostr. Fundam. Appl. 9, 308–314 (2011). 4. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988). 5. J. Gosciniak, T. Holmgaard, and S. I. Bozhevolnyi, “Theoretical analysis of long-range dielectric-loaded surface plasmon polariton waveguides,” J. Lightwave Technol. 29, 1473–1481 (2011). 6. M. Talafi Noghani and M. H. Vadjed Samiei, “Ultrashort hybrid metal-insulator plasmonic directional coupler,” Appl. Opt. 52, 7498–7503 (2013). 7. C. Min, P. Wang, C. Chen, Y. Deng, Y. Lu, H. Ming, T. Ning, Y. Zhou, and G. Yang, “All-optical switching in subwavelength metallic grating structure containing nonlinear optical materials,” Opt. Lett. 33, 869–871 (2008). 8. Z.-J. Zhong, Y. Xu, S. Lan, Q.-F. Dai, and L.-J. Wu, “Sharp and asymmetric transmission response in metal-dielectric-metal plasmonic waveguides containing Kerr nonlinear media,” Opt. Express 18, 79–86 (2010). 9. J. R. Salguiro and Y. S. Kivshar, “Nonlinear plasmonic directional couplers,” Appl. Phys. Lett. 97, 081106 (2010). 10. M. Pu, N. Yao, C. Hu, X. Xin, Z. Zhao, C. Wang, and X. Luo, “Directional coupler and nonlinear Mach–Zehnder interferometer based on metal-insulator-metal plasmonic waveguide,” Opt. Express 18, 21030–21037 (2010). 11. N. Nozhat and N. Granpayeh, “Switching power reduction in the ultra-compact Kerr nonlinear plasmonic directional coupler,” Opt. Commun. 285, 1555–1559 (2012). 12. N. Nozhat and N. Granpayeh, “Switching power reduction of the plasmonic directional coupler by XPM nonlinear effect,” IEEE Photon. Technol. Lett. 24, 1154–1156 (2012). 13. G. Wang, H. Lu, X. Liu, Y. Gong, and L. Wang, “Optical bistability in metal-insulator-metal plasmonic waveguide with nanodisk resonator containing Kerr nonlinear medium,” Appl. Opt. 50, 5287–5290 (2011). 14. P. Ginzburg, A. Hayat, N. Berkovitch, and M. Orenstein, “Nonlocal ponderomotive nonlinearity in plasmonics,” Opt. Lett. 35, 1551–1553 (2010). 15. P. Kaw, G. Schmidt, and T. Wilcox, “Filamentation and trapping of electromagnetic radiation in plasmas,” Phys. Fluids 16, 1522–1525 (1973). 16. P. Ginzburg, A. Hayat, E. Feigenbaum, N. Berkovitch, and M. Orenstein, “Nonlinear surface plasmon polaritons and the ponderomotive force,” in Lasers and Electro-Optics Society (LEOS) (2007), pp. 624–625. 17. M. Raynaud and J. Kupersztych, “Ponderomotive effects in the femtosecond plasmon-assisted photoelectric effect in bulk metals: evidence for coupling between surface and interface plasmons,” Phys. Rev. B 76, 241402 (2007). 18. A. R. Davoyan, “Plasmonic couplers with metal nonlinearities,” Phys. Lett. A 375, 1615–1618 (2011). 19. A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Symmetry breaking in plasmonic waveguides with metal nonlinearities,” Opt. Lett. 36, 930–932 (2011). 20. J. Renger, R. Quidant, and L. Novotny, “Enhanced nonlinear response from metal surfaces,” Opt. Express 19, 1777–1785 (2011). 21. M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6, 737–748 (2012). 22. S. Palomba, H. Harutyunyan, J. Renger, R. Quidant, N. F. V. Hulst, and L. Novotny, “Nonlinear plasmonics at planar metal surfaces,” Phil. Trans. R. Soc. A 369, 3497–3509 (2011).
Faculty of Electrical Engineering, Shiraz University of Technology, Shiraz 7155713876, Iran 2
Center of Excellence in Electromagnetics, Faculty of Electrical Engineering, K. N. Toosi University of Technology, Tehran 1631714191, Iran *Corresponding author: [email protected] Received 6 January 2014; revised 13 April 2014; accepted 14 April 2014; posted 17 April 2014 (Doc. ID 204173); published 20 May 2014
In this paper, a two-dimensional nonlinear plasmonic directional coupler (2D-NPDC), with 90° waveguide bends, has been numerically analyzed by the finite-difference time-domain (FDTD) method, considering the nonlinear response of metal due to the ponderomotive force. It has been shown that the required switching power of the 2D-NPDC is 0.05% of that when only the dielectric is nonlinear and the nonlinearity of metal is neglected. Also, the cross-phase modulation (XPM) nonlinear effect has been investigated, which the power for switching is decreased significantly compared to the one with the self-phase modulation (SPM) effect. © 2014 Optical Society of America OCIS codes: (060.1810) Buffers, couplers, routers, switches, and multiplexers; (130.4815) Optical switching devices; (190.3270) Kerr effect; (240.6680) Surface plasmons; (250.5403) Plasmonics. http://dx.doi.org/10.1364/AO.53.003328
1. Introduction
In recent years, researches in nanophotonics based on surface plasmon polaritons (SPPs) have been expanded rapidly. SPPs propagate through metal– dielectric interfaces and guide in nanoscale devices with dimensions below the diffraction limit. Therefore, linear and nonlinear plasmonic devices are appropriate candidates for use in photonic integrated circuits (PICs) [1–6]. Several nonlinear plasmonic structures have been studied, so far. However, in most of these structures, nonlinearity is thought to originate from the dielectric, and the nonlinear response of metal has been neglected [7–13]. In general, metal nonlinearity is based on several physical effects, such as interband transition of 1559-128X/14/153328-05$15.00/0 © 2014 Optical Society of America 3328
APPLIED OPTICS / Vol. 53, No. 15 / 20 May 2014
bound electrons and laser induced hot-electron contribution, which have previously been studied. It has been shown that another nonlinear response phenomenon of metals is based on the inherent electron plasma nonlinearity of the metal layers. The ponderomotive force, which is the nonlinear force due to the electromagnetic wave propagation within the plasma, repels electrons from the high field intensity region. Therefore, the local plasma frequency and the absolute value of the permittivity of metal are reduced and the real part of the local refractive index of SPPs is enhanced. Furthermore, the propagation loss is decreased because of the confinement of the field to the low carrier density region. The effect of ponderomotive nonlinearity on SPP propagation, nonlinear effects in plasmonic metal film waveguides, and nonlinear guided plasmonic modes of two coupled thin metallic films by the third-order optical response due to the ponderomotive metal nonlinearities have been investigated [14–19].
Previously, we analyzed and simulated the twodimensional nonlinear plasmonic directional coupler (2D-NPDC) switches with 45° and 90° waveguide bends utilizing Kerr self-phase modulation (SPM) and cross-phase modulation (XPM) nonlinear effects, considering only the Kerr nonlinear effects of the dielectrics [11,12]. It has been shown theoretically and experimentally that, because of the strong nonlinearity of metals at optical wavelengths, the nonlinear response of metallic surfaces is enhanced [16,20–22]. In this paper, we analyze the performance of the same 2D-NPDC by considering the metal nonlinearity due to the ponderomotive force. Both SPM and XPM nonlinear effects are studied and it is shown that the required switching power is significantly reduced compared to the case when the metal nonlinearity is neglected. The paper is organized as follows. In Section 2, the simulation method and in Section 3, the results are described and discussed. The paper is concluded in Section 4. To obtain the intensity dependent dielectric constant of metal εPM , a fluid model is used to describe the electron motion in metals. The nonlinear force acting on an electron in an oscillating electromagnetic field is given by [14,15]: FPM
(1)
where m and e are the electron mass and charge, respectively, ω is the lightwave angular frequency, jEj2 is the intensity of local field and ΦPM is the ponderomotive potential. The carrier density due to the ponderomotive potential is obtained as nPM
1 2m 3∕2 Ef − ΦPM 3∕2 ; 3π 2 ℏ
εM ω ε∞ −
ω2p ; ω − jγ p ω 2
(2)
where Ef is the Fermi energy and ℏ is the reduced Planck or Dirac constant. Therefore, the intensity-dependent metal relative permittivity related to the ponderomotive force is described by 3∕2 e2 2m e2 jEj2 3∕2 εPM jEj 1 − 2 Ef − : 3π ε0 mω2 ℏ2 2mω2 (3) 2
(5)
where ε∞ and γ p are the relative permittivity at infinite frequency and collision angular frequency, respectively. From Eq. (4), it can be seen that the nonlinear susceptibility of metal is highly dispersive and, for the second and third telecommunication χwindows of, respectively, 1310 and 1550 nm wavelengths, χ PM is in the order of 10−18 m2 ∕V2 [14]. Due to the Kerr nonlinear effect, the refractive index of a material depends on the signal I s , and pump I p intensities as follows [12]: nd nl n2 I s 2I p ;
2. Simulation Method and Discussion
1 e2 ∇jEj2 −∇ΦPM ; − 2m ω2
where ωp is the plasma angular frequency, χ PM is the nonlinear ponderomotive susceptibility, which is a function of ω−4, and εM ω is the linear metal dielectric constant expressed by the Drude model [11]:
(6)
where nl and n2 are the linear refractive index and the nonlinear Kerr coefficient, respectively. n2 I s and 2n2 I p are the refractive index increments, due to SPM and XPM, respectively. Also, according to Eq. (4), the nonlinear response of metal is a dispersive Kerr-like effect. We have employed the two-dimensional finitedifference time-domain (2D-FDTD) numerical method for the simulation of the 2D-NPDC switch. The convolutional perfectly matched layer (CPML) has been used as the absorbing boundary conditions [11]. 100-cell CPMLs are used to ensure all the powers to be absorbed at the boundaries of the simulation region. The step size of the FDTD cell in x and z directions are Δx Δz 1 nm and the time step isp chosen by Courant condition to be Δt 0.95∕c Δx−2 Δz−2 , where c is the free space speed of light. Since SPPs are excited by only TM-polarization, the nonlinear plasmonic coupler of Fig. 1 is excited by a TM-polarized pulse with electromagnetic field components of H y, Ex , and Ez . The signal and the pump sources and the output power monitors are placed at the distance of 150 nm from the 90° bends. We have utilized the MATLAB and C++ programming languages for our analyses and simulations.
By Taylor series expansion, the leading term gives a Kerr-like nonlinearity for the dielectric constant of metal: 2∕3 4 ωp 3 e jEj2 2 3π 2 ε0 ℏme ω4 εM ω χ PM jEj2 ;
εPM jEj2 εM ω
(4)
Fig. 1. Schematic view of the proposed 2D-NPDC switch with 90° waveguide bends. The width of the dielectric waveguides and the gap between the coupler waveguides are h 30 nm and d 10 nm, respectively. Po3 and Po4 are, respectively, bar and cross ports. Signal input is Pi1 . 20 May 2014 / Vol. 53, No. 15 / APPLIED OPTICS
3329
The procedure for applying the 2D-FDTD method to our structure is described here. Maxwell’s equations are solved to simulate the performance of our nonlinear structure. The electric flux density D, is calculated from the discrete form of the following equation: ∇×H
∂ D; ∂t
(7)
where H is the magnetic field vector. Since the dispersive medium is characterized by the Drude model and the nonlinear regions have Kerr-type nonlinear effects, the relation between the linear PL and nonlinear PNL polarizations, the electric field E, and the electric flux density D, are given as D ε0 ε∞ E PL PNL :
(8)
The linear polarization of PLD related to the Drude NL model and nonlinear polarizations of PNL K and PPM contributed by the dielectric and metal nonlinearities, respectively, are expressed as ∂2 PLD ∂PLD ε0 ω2p E; γ P ∂t ∂t2
(9)
3 3 PNL K t ε0 χ E t;
(10)
2 χ PNL PM t ε0 Et PM t E t;
(11)
where χ 3 is the third-order nonlinear susceptibility and denotes the convolution. An auxiliary variable can be introduced for the convolution of Eq. (11): Gt χ PM t E2 t:
(12)
The Fourier-transform of Eq. (12) leads to Gω χ PM ωIE2 t;
(13)
where, from Eq. (4), χ PM ω 3∕2ωp ∕3π 2 ε0 ℏme2∕3 e4 ∕ω4 . The inverse Fourier-transform of Eq. (13) results in the following equation: 2∕3 ωp ∂4 G 3 e4 E2 : 2 3π 2 ε0 ℏme ∂t4
APPLIED OPTICS / Vol. 53, No. 15 / 20 May 2014
(16)
where aD 4∕γ p Δt 2, bD γ p Δt − 2∕γ p Δt 2, cD 2ε0 ω2p Δt2 ∕γ p Δt 2, Δt is the time step, and Gn1 4Gn − 6Gn−1 4Gn−2 − Gn−3 Δt4 3∕2ωp ∕ 3π 2 ε0 ℏme2∕3 e4 En−1 2 . Finally, the updated discrete magnetic field is derived from Maxwell’s curl equation of ∇ × E −μ
∂ H; ∂t
(17)
where μ is the magnetic permeability [11]. 3. Numerical Results and Discussion
In our previous work, we demonstrated that the loss of the 2D-NPDC with 90° waveguide bends is lower than that of the one with 45° bends. Therefore, as shown in Fig. 1, we have studied the 90° 2D-NPDC [11]. The metal and nonlinear dielectric materials for our nonlinear PDC are chosen to be silver and chalcogenide glass, respectively. The Drude parameters of silver and linear and nonlinear parameters of chalcogenide glass are given in Table 1. The PDC length of the coupling region is chosen to be L 380 nm. A sinusoidal wave as a signal lightwave with the wavelength of 1493 nm is launched to port 1 of the 2D-NPDC of Fig. 1. The selected wavelength enables us to compare the performance of the 2D-NPDC by considering the metal nonlinearity with our previous work [12]. The simulation is valid for other signal wavelengths in the third telecommunication window, but the coupler length should be modified. The output powers ratio versus the input signal power is depicted in Fig. 2. For the case that both the metal and dielectric are nonlinear, it has been shown in Fig. 2(b) that, in the linear state with the input signal level of 1.07 × 10−8 W∕μm, the output powers ratio Po3 ∕Po4 is −4.38 dB and the light exits from port 4, as illustrated in Fig. 3(a). The coupling length is inversely related to the differences between the symmetric and asymmetric propagation constants of SPP modes, defined as a coupling coefficient [12]. By increasing the input signal power, the refractive index of the nonlinear medium and the coupling coefficient is increased and so the coupling length of the PDC is decreased due to the Kerr nonlinear effects of dielectric and
Table 1.
Parameters for Simulation of Proposed PDC [11]
Material
Parameters
Value
Metal (Silver’s Drude model)
ε∞ γ p (Hz) ωp (Hz) nl n2 (cm2 ∕W) χ 3 (m2 ∕V2 )
3.7 2.73 × 1013 1.38 × 1016 2.4 2 × 10−13 4 × 10−19
(15)
By inserting the FDTD update expressions of Eqs. (9), (10), and (15) into Eq. (8), the electric field is derived from the following equation: 3330
n−1 Dn1 − aD PnD − bD PD − c D En ; 3 n1 2 ε0 ε∞ ε0 χ E ε0 Sn1
(14)
From Eqs. (11) and (12) we can write PNL PM t ε0 EtGt:
En1
Insulator (Chalcogenide)
4 3
Po3 /Po4 (dB)
2 1 0 -1 -2 -3 -4 -5 0
10
20
30
40
50
Input Signal Power (W/µm) 4 3
Po3 /Po4 (dB)
2 1
Fig. 3. Magnetic field distributions in the 2D-NPDC with metal nonlinearity and dielectric SPM Kerr nonlinearity, at λs 1493 nm in (a) linear and (b) nonlinear states when the signal input powers are 1.07 × 10−8 W∕μm and 0.028 W∕μm, respectively.
0 -1 -2 -3 -4 -5 0
0.005
0.01
0.015
0.02
0.025
Input Signal Power (W/µm)
the pump light does not switch to port 3, the signal switches from the cross to port. The magnetic field distributions of NPDC performance in these cases are in Fig. 4.
whereas the bar the 2Ddepicted
Fig. 2. Output powers ratio versus the input signal power for the 2D-NPDC switch of Fig. 1. (a) Without metal nonlinearity [12] and (b) with metal nonlinearity [this work].
metal. Therefore, the input signal light is switched to port 3 and exits from it, as shown in Fig. 3(b). In this case, the output power ratio is increased to its maximum value of 3.7 dB with input power of 0.028 W∕μm. For comparison, the output powers ratio versus input power for the 2D-NPDC of Fig. 1 with Kerr nonlinear effect of dielectric, obtained in our previous work, is depicted in Fig. 2(a) [12]. It can be observed from Figs. 2(a) and 2(b) that, by considering the Kerr-like nonlinear response of metal, the required PDC switching power is 0.05% that required in our previous work. Next, we studied the effect of XPM on the performance of 2D-NPDC. From the transmission spectra of the coupler, we found that the wavelength of the pump light can be selected in the range of 1060–1200 nm. For this purpose, the signal light at the wavelengths of λs 1493 nm and a low power pump light at λp 1194 nm are simultaneously launched to ports 1 and 2, respectively. In the linear state, the pump light exits from port 4. At higher pump power, the coupling length decreases and
Fig. 4. Magnetic field distributions of the 2D-NPDC with metal nonlinearity and dielectric XPM Kerr nonlinearity (a) in linear or nonlinear state of pump light at λp 1194 nm and powers of 0 and 4.35 × 10−6 W∕μm, respectively, and (b) signal light with high power pump lightwave. The pump power is omitted to show the switching of signal from cross port to bar port. 20 May 2014 / Vol. 53, No. 15 / APPLIED OPTICS
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4
the switching power of the PDC in the XPM case is 1.55 × 10−4 of the case with SPM effect.
3 2
Po3/Po4 (dB)
References 1 0 -1 -2 -3 -4 -5
0
0.5
1
1.5
2
2.5
3
3.5
4
Input Pump Power (W/µm)
4.5 -6
x 10
Fig. 5. Output powers ratio versus input pump power for the 2D-NPDC switch of Fig. 1 with metal nonlinearity for the signal power and wavelength of 1.07 × 10−8 W∕μm and 1493 nm, respectively, and a pump wavelength of 1194 nm.
Since the pump power is higher than the signal power, in Fig. 4(b) the pump power is omitted to show clearly the lightwave signal switching performance in the PDC. It can be seen from Eq. (6) that by utilizing the XPM Kerr effect, the refractive index variation of the nonlinear medium is higher than that of SPM. Therefore, the required switching power in the 2D-NPDC using XPM is much lower than the signal power of the SPM state, as depicted in Fig. 5. By increasing the pump power from 0 to 4.35 × 10−6 W∕μm, the signal lightwave switches from the cross port to the bar port and the output powers ratio increases from −4.24 dB to 3.95 dB, respectively. By further increasing the input power, the output power ratio decreases, due to the coupling length change through the variation of the refractive index by XPM. By comparison of Figs. 2 and 5, it is clear that, by employing XPM nonlinear effect, the 2D-NPDC switching power is reduced significantly. By considering the nonlinearity of metal, the switching powers obtained in our simulations are reasonable for using the nonlinear 2D-NPDC in the telecommunication systems. 4. Conclusion
In this paper, the performance of the twodimensional nonlinear plasmonic directional coupler (2D-NPDC) with 90° waveguide bends is analyzed by the finite-difference time-domain (FDTD) numerical method, considering the ponderomotive nonlinear response of metal and dielectric Kerr nonlinear effect. Both SPM and XPM nonlinear effects have been considered. We have shown that, in the SPM case, the required switching input signal power is 0.05% that of when only the structure dielectric Kerr nonlinear effect is considered. In addition, we have shown that
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