All-optical wavelength converter based on a ... - OSA Publishing

N. Talebi and M. Shahabadi

Vol. 27, No. 11 / November 2010 / J. Opt. Soc. Am. B

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All-optical wavelength converter based on a heterogeneously integrated GaP on a silicon-on-insulator waveguide Nahid Talebi* and Mahmoud Shahabadi Photonics Research Laboratory, Center of Excellence on Applied Electromagnetic Systems, School of Electrical and Computer Engineering, University of Tehran, North Kargar Avenue, Tehran, Iran *Corresponding author: [email protected] Received April 26, 2010; revised August 11, 2010; accepted August 25, 2010; posted September 10, 2010 (Doc. ID 127509); published October 13, 2010 A micrometer-scaled optical wavelength converter based on a multilayer structure is presented. The proposed structure is comprised of a grating of GaP nano-ribs heterogeneously integrated on the upper layer of a siliconon-insulator waveguide. The GaP grating is used to couple the incident wave into the propagating modes of the waveguide and also to double the frequency of the incident light due to the second-harmonic generation process in GaP. The performance of this structure is numerically investigated using the generalized multipole technique for the linear analysis and the finite-difference time-domain method for the nonlinear analysis. © 2010 Optical Society of America OCIS codes: 130.4310, 130.7405, 130.3120.

1. INTRODUCTION Silicon photonic devices, especially those based on siliconon-insulator (SOI) structures, have recently attracted extensive research attention. The high index contrast between the core layer and its surrounding layers makes confined transport of light or IR electromagnetic fields possible. However, the performance of active SOI devices such as lasers [1] and wavelength converters [2,3] is not comparable with that of devices implemented on III-V semiconductors. This is due to the indirect bandgap and the inversion symmetric lattice that the silicon exhibits. Silicon is not commonly employed in nonlinear optical devices because its nonlinear susceptibility is of the third order such that it reveals its nonlinearity only for higher optical signal levels. On the other hand, the second-order susceptibilities 共␹共2兲兲 of III-V semiconductors such as GaAs, GaP, and InP are even higher than those of the mostly used nonlinear materials such as KDP and LiNbO3 [4,5]. The required phase-matching conditions in nonlinear processes such as second-harmonic generation (SHG) and difference-frequency generation are more easily achieved by utilizing the anisotropy of uni-axial and bi-axial crystals [6]. Phase matching may also be obtained with the help of quasi-phase-matching techniques [7] or built-in artificial birefringence in a composite multilayer comprised of GaAs and oxidized AlAs [8]. Moreover, gratings of Au nano-particles over a multilayer structure comprised of ionic self-assembled films have been used in [9] to both enhance the ␹共2兲 nonlinearity of the structure and to obtain the phase-matched SHG power at all angles of the incident pump power. Heterogeneous structures fabricated by integration of III-V microdisk [10,11] with photonic crystal waveguides [12] on SOI waveguides are used to realize lasers. Bonding InP to silicon waveguides is shown to be a promising 0740-3224/10/112273-6/$15.00

technology in developing new laser devices [11]. However, among the III-V materials, gallium phosphide is readily treated for direct growing on a silicon substrate using molecular beam hetero-epitaxy [13]. This is because of its minor lattice mismatch (0.37% at 300 K) to silicon [13]. GaP has also a remarkable second-order nonlinear susceptibility which makes it suitable for nonlinear optics applications [14]. The electromagnetic modeling of the nonlinear effects, especially SHG, in the aforementioned devices is not thoroughly studied. Analytical modeling such as the coupledmode theory [15] is not appropriate for structures of complex geometries. Numerical methods such as the eigenmode expansion method [16] and the time-domain beampropagation method (TD-BPM) [16] are two alternate methods for the nonlinear processes. The first method is developed under the assumption of the slowly varying envelope approximation, and a time differencing procedure is utilized to solve two coupled equations for the fundamental and the second-harmonic waves. The second method, i.e., TD-BPM, is used under the assumption of the paraxial wave propagation. The finite-difference timedomain (FDTD) method is a versatile method that can be used in modeling various nonlinear processes such as wave mixing and harmonic generation [17,18]. In this paper, we numerically investigate the performance of an all-optical wavelength converter based on a multilayer structure comprised of a grating of GaP nanoribs on a SOI slab waveguide. The GaP grating is exploited to both couple the incident wave into the propagating modes of the slab waveguide and generate the second harmonic of the incident wave. Also the phasematching condition is achieved using the modal dispersion of the slab waveguide. Figure 1 shows the proposed structure. The paper is organized as follows. In the next © 2010 Optical Society of America

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Fig. 1. (Color online) Schematic of the proposed multilayer optical wavelength converter. The structure is comprised of a grating of GaP nano-ribs on a SOI waveguide.

section, the linear results for the band diagrams of the structure obtained using the generalized multipole technique (GMT) is given and discussed. In Subsection 3.A, the FDTD method used for the nonlinear analysis of the structure is introduced. Then in Subsection 3.B, the performance of the structure is investigated using the introduced FDTD method.

2. LINEAR ANALYSIS The proposed multilayer structure is shown in Fig. 1. The height of the core 共Hco兲 and the period of the grating 共L兲 are determined to ensure propagation of the fundamental mode at the desired frequencies. First, we have computed the band diagram of the waveguide using the GMT with the code developed in our laboratory [19,20] for the following values of the parameters as depicted in Fig. 1: L = 300 nm, Hcl = 800 nm, Hco = 400 nm, h = 200 nm, and d = 200 nm. It should be mentioned that the GMT is a semianalytical frequency-domain method and can lead to highly accurate results. The measured values for the permittivity of bulk GaP, SiO2, and silicon reported in [21] are used in our modeling. The obtained band diagrams are shown in Figs. 2(a) and 2(b) for both TMz and TEz modes. From these diagrams, it is evident that the fundamental mode of this waveguide is the TMz mode. The phase constant of the asymmetric slab waveguide without the GaP grating in Fig. 1 has also been computed analytically and shown in Fig. 2. A comparison between the two results shows that the lower-order TEz and TMz modes with and without GaP grating behave almost similarly. Therefore, terminating the proposed multilayer structure by an asymmetric slab waveguide may result in a lowreflection discontinuity as long as the normalized frequency satisfies ␻L / 2␲c ⬍ 0.1. The mode-field profiles of the proposed structure at different frequencies for the z-components of the electric and magnetic fields have been given in the inset of Fig. 2. It is evident from these results that the mode-field profiles of the lower-order modes are well confined to the silicon core. The GMT has also been utilized to compute the attenuation constant of the multilayer waveguide using the method introduced in [20]. The attenuation constant of the waveguide for the fundamental mode at ␭ = 1800 nm is ⫺9 dB/mm, while for the TMz mode at ␭ = 900 nm, it is ⫺12 dB/mm. Due to the GaP grating of the top layer, the propagating modes of this waveguide may be excited by a plane wave incident at the angle ␪. Figure 3 shows the zero-

Fig. 2. (Color online) Band diagram of the multilayer structure computed using GMT for the (a) TMz modes. The insets visualize the computed Ez field component at the normalized frequencies of ␻L / 2␲c = 0.1 and 0.348. (b) TEz modes. The insets show the computed Hz field component at the normalized frequencies of ␻L / 2␲c = 0.167 and 0.252.

order diffraction efficiency as a function of ␪ at the normalized frequency of ␻L / 2␲c = 1 / 3 共␭ = 900 nm兲 for a TMz incident plane wave. It can be seen from these results that two different modes can be excited at the incident angles of ␪ = 58° and ␪ = 82°. In the inset of Fig. 3, the only nonzero electric field component 共Ez兲 at the incident angle of ␪ = 58° is shown. The coupling loss of the structure can be approximately computed using the diagram of the dif-

Fig. 3. (Color online) Zero-order reflection r0 and transmission t0 at ␭ = 900 nm. The inset shows the z-component of the electric field at ␪ = 58°.



fraction efficiency. The sum of the different orders of the diffraction efficiencies for a lossless medium is equal to 1, which should be also the case for the angles at which the anomalies occur. However, for a lossy medium, this sum can be smaller than 1. For example, it is obvious from Fig. 3 that the sum of the reflection and transmission efficiencies is smaller than 1, at the angles at which the anomalies occur, so that the ratio of the dissipated power to the incident power can be expressed as 1 − r0 − t0, which is equal to 0.0848 at ␪ = 58°.

3. NONLINEAR ANALYSIS A. Nonlinear FDTD Method In order to include the nonlinear response of the structure, we have employed the nonlinear two-dimensional FDTD method [17]. For modeling the GaP grating the relation between the polarization vector and the electric field is assumed to be of second order under the limit of the electric-dipole approximation, i.e., P⬀共r,t兲 = ␧0

␹␣␤␥E␤共r,t兲E␥共r,t兲, 兺␤ ␹␣␤E␤共r,t兲 + ␧ 兺 ␤␥ 共1兲

共2兲

0

共1兲共2兲 in which ␹共1兲 ␣␤ is the dielectric tensor of GaP, and ␹␣␤␥ is the second-order nonlinear susceptibility tensor. The nonlinear polarization vector can be simplified as

P共NL兲 = ␧0

冤

共2兲 ␹xyz

0

0

0

共2兲 ␹yxz

0

0

共2兲 ␹zxy

0

冥冤冥

共2兲

E xE y

共2兲共2兲共2兲 = ␹yxz = ␹zxy = ␹共2兲 for semiconductors having in which ␹xyz 43m point symmetry (like GaP) when the Kleinman symmetry condition is valid. The permittivity of GaP exhibits a resonance at ␭ = 340 nm. The dispersion of ␹共2兲共␻兲 is mostly governed by the joint density of states between GaP atomic bands. As an approximation it can be attributed to a weighted sum over ␹共L兲共␻兲 and ␹共L兲共2␻兲, in which the weights are the permuted electric-dipole transition moments over different Cartesian indices [22], and ␹共L兲 stands for the linear susceptibility. Hence, the resonant wavelengths of ␹共2兲共␻兲 are at ␭ = 340 nm and ␭ = 170 nm. Moreover, the absorption spectrum of III-V semiconductors shows significant peaks in the Reststrahl band which is associated with longitudinal- and transverse-optical phonons [23]. The wavelengths of these resonances for GaP are located at ␭ = 27.24 ␮m for transverse-optical phonons and ␭ = 24.81 ␮m for longitudinal-optical phonons. For our simulations, the wavelength of the incident wave is assumed to be far from these resonances so that the permittivity and nonlinear susceptibility are considered to be constant. For relative permittivity of the different layers of the structure, we have assumed the following values: ␧r,GaP = 9.2, ␧r,Si = 12, and ␧r,SiO2 = 1.31. For the nonlinear susceptibility of GaP, we assume ␹共2兲 = 100 pm/ V [24]. Considering these simplifications, the constitutive relation can be written as

共2兲

D = ␧ 0E + P = ␧ 0 ␹ E z 0

0

␹ 共2兲E y

␧r

0

共2兲

␹ Ex

␧r

冥冤冥 Ex

Ey .

共3兲

Ez

The magnetic field and the electric displacement 共D兲 are updated in every iteration. For updating the electric field, a Newton iteration procedure is utilized to compute the electric field from the previously updated electric displacement using Eq. (3) [18]. B. Numerical Results for the Second-Harmonic Generation A total number of 30 GaP nano-ribs are assumed in our FDTD simulation. The structure is excited by a TEz picosecond optical pulse with a hyperbolic secant envelop function at the carrier wavelength of ␭ = 1800 nm 共␻L / 2␲c = 1 / 6兲. The envelop contains approximately 30 cycles of the optical carrier. The spatial distribution of the incident optical beam is assumed to be a Gaussian function with W = ␭c, in which W is the beam-waist of the Gaussian beam, and ␭c is the wavelength of the optical carrier. The solution domain is discretized using 20 nm ⫻ 20 nm rectangular grids and terminated using a second-order absorbing boundary condition. First, we consider a perpendicularly incident wave as shown in the left inset of Fig. 4. The average incident power 共Pi兲 and the guided power in the waveguide 共Pg兲 are calculated and shown in Fig. 4, where Pi and Pg are calculated as follows:

再冕

1 Pi,g共␻兲 = Re 2

E yE z

E xE z ,

冤

␧r

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Si,Sg

冎

ជ 共␻兲 ⫻ H ជ 共␻兲 · dS ជ i,g , E

共4兲

ជ 共␻兲 and H ជ 共␻兲 are Fourier transforms of E ជ 共t兲 in which E ជ 共t兲, and Si and Sg are the cross sections shown in and H Fig. 4. It can be seen that the overall power coupled to the waveguide at the fundamental frequency is small. However, at the second harmonic, the magnitude of the guided power in the waveguide is comparable to the guided power at the fundamental frequency. The conversion efficiency is defined as the ratio of the average incident power at ␭ = 1800 nm to the SHG power at ␭ = 900 nm.

Fig. 4. (Color online) Incident power 共Pi兲 and guided power 共Pg兲 in the proposed structure with perpendicular excitation. The left inset shows the z-component of the magnetic field at t = 1.6 fs. The right inset shows the z-component of the electric field at the same time.

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This is approximately 1%. Note that this value is obtained under the assumption of large amplitude for the incident field. The double peak visible in Fig. 4 for the guided power is because of the excitation of two modes (the fundamental and second-order modes) which are mutually coupled while propagating along the waveguide. The generated second-harmonic signal is illustrated using the distribution of the z-component of the electric field shown in the right inset of Fig. 4. Because of the small cross section of the GaP nano-ribs, they act as cylindrical electric dipoles radiating their power at the second-harmonic frequency. In order to increase the SHG power, one makes effort to increase the amount of the power coupled into the waveguide at the fundamental wavelength. This can be achieved by an oblique excitation. Figure 5 shows the conversion efficiency versus the incident angle ␪. The incident beam is a Gaussian one with the beam-waist of W = ␭c. It is visible from the results shown in Fig. 5 that the conversion efficiency has two peaks at ␪ = 30° and ␪ = 80°. At ␪ = 30° mostly the second-order mode is propagating in the waveguide, while at ␪ = 80° the fundamental mode is excited. The diffraction efficiency of the TEz mode at ␭ = 1800 nm shows no anomaly. However, the excitation is in the form of a Gaussian beam which can be expanded in terms of plane waves, where some of them are evanescent, according to E共x,y兲 =

冕

+⬁

˜ 共k ,y兲exp共− jk x − jk y兲dk , E x x y x

共5兲

−⬁

in which ky =

再

冑k02 − k2x , − j冑k2x − k02 ,

兩kx兩 ⱕ k0 兩kx兩 ⬎ k0 .

冎

共6兲

The propagating modes of the structure may be excited by the x-component of the wave-vector satisfying the relation kx ⬎ k0, using the phase-matching condition ␤共␻兲 = kx共␻兲. When a tilted excitation is introduced, the x-component of the wave-vector for which the mentioned phase-matching condition is satisfied carries a larger portion of the incident power than the case of normal excitation, so that the coupled power into the waveguide is get increased at an oblique incidence. However, not all of the incident angles result in the guided second-harmonic beams. The generated second harmonic is propagating in the structure at

Fig. 5. (Color online) Conversion efficiency versus the incident angle. The inset shows the spatial distribution of Hz at t = 3.5 fs. The unit of the color bar is ampere per meter.

Fig. 6. (Color online) Average guided power for the incident angles of ␪ = 30° and ␪ = 80°. The incident beam is similar to the one depicted in the inset of Fig. 4. The right inset shows the field pattern at ␪ = 30° and t = 0.35 ps. The left inset shows the field pattern at ␪ = 80° and t = 0.35 ps.

the angles for which the following equation is satisfied: 2m␲ ␻ , ␤共2␻兲 = 2 sin ␪ + c L

共7兲

in which ␤共2␻兲 is the propagation constant at the secondharmonic frequency, and ␪ is the angle of the incident wave shown in the inset of Fig. 5. Figure 6 shows the guided powers at the incident angles of ␪ = 30° and ␪ = 80°. The incident field is the same as in the previous one. The spatial distribution of the electric field is also depicted in the inset. It is seen that the produced second-harmonic power is propagating in the slab waveguide. Note that terminating the proposed multilayer structure with a slab waveguide results in a low-reflection discontinuity at the second-harmonic wavelength. The total number of the GaP nano-ribs has been assumed to be N = 30 so that the length of the introduced coupler is approximately five times the spot size of the incident beam. However, to make clearer the impact of N on the conversion efficiency, the variation of the conversion efficiency at ␪ = 30° versus the number of the GaP nanoribs has been computed and shown in Fig. 7, while the spot size of the incident beam has been kept constant at W = 1800 nm. It is evident from these results that the introduced number of GaP nano-ribs in the previous simulations was not the optimum number, and it is possible to obtain a more efficient performance by simply choosing N = 25 or N = 17.

Fig. 7. (Color online) Conversion efficiency versus the total number of GaP nano-ribs.



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these results that the introduced wavelength converter can efficiently convert the fundamental wavelength at ␭ = 1500 nm to the wavelength of 750 nm.

4. CONCLUSION

Fig. 8. (Color online) Conversion efficiency versus the incident power at the fundamental frequency for the incident angle of ␪ = 30° and N = 25. The inset shows the guided power at Pin = −6.5 dB.

It is evident that the conversion efficiency is dependent on the incident power at the fundamental frequency. Figure 8 shows the results for the variation of the conversion efficiency at the incident angle of ␪ = 30° versus the incident power, for the total number of GaP nano-ribs being equal to N = 25. It can be seen that the conversion efficiency is a linear function of the incident power. The inset of Fig. 8 shows the guided power at the incident power of Pin = −6.5 dB. For the sake of simplicity in numerical simulations, we have only considered SHG. However, it is possible to use other second-order nonlinear phenomena such as sumfrequency generation [25] or difference-frequency generation [26] as well as cascaded second-order nonlinearities [27], as long as the produced sub-harmonic wave becomes phase-matched with the propagation constant of the waveguide at the desired sub-harmonic frequency. Even higher-order nonlinearities such as four-wave mixing [28] may be used upon the phase-matched condition. The periodicity of the coupler integrated upon the SOI can be tuned to satisfy the phase-matching condition for a variety of generated harmonics or sub-harmonics. The mentioned geometrical parameters of the structure and the incident wavelength have been introduced here only in a way to present the idea. For the structure to work in another wavelength, one should tune the geometrical parameters to have the phase-matching condition at the desired wavelength. For example, by setting the geometrical parameters as Hco = 300 nm, L = 300 nm, d = 200 nm, and h = 200 nm, the phase-matching condition can be achieved at the fundamental wavelength being equal to ␭ = 1500 nm. Figure 9, shows the conversion efficiency versus the incident angle for this structure at the incident wavelength of ␭ = 1500 nm. The inset of this figure shows the computed guided power. It is obvious from

In conclusion, a multilayer optical wavelength converter comprised of an array of GaP nano-ribs on a SOI waveguide was proposed and numerically investigated using the GMT and FDTD method. It was shown that with only a small number of nano-ribs, a high conversion efficiency can be achieved.

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