Widely tunable femtosecond optical parametric ... - OSA Publishing

Widely tunable femtosecond optical parametric oscillator based on silicon-on-insulator waveguides Jin Wen, Hongjun Liu,* Nan Huang, Qibing Sun, and Wei Zhao State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Science (CAS), Xi'an, 710119, China * [email protected]

Abstract: A femtosecond optical parametric oscillator (OPO) based on silicon-on-insulator (SOI) waveguide is proposed and analyzed numerically. By utilizing split-step Fourier method (SSFM), it is demonstrated that ultrawide tunable wavelength femtosecond pulse can be realized under the phase matching condition. Due to the interaction between nonlinearity and flexible dispersion design, the output signal wavelength can be tuned from 1645 to 1805 nm and the idler wavelength can be tuned from 1350 to 1456 nm. Moreover, the peak power of the output signal pulse exceeds 10 W from 1700 to 1770 nm with the pump peak power 50 W. The proposed OPO exhibits compact configuration and can find important applications in integrated broadband optical source. ©2012 Optical Society of America OCIS codes: (040.6040) Silicon; (190.4390) Nonlinear optics, integrated optics; (190.4380) Four-wave mixing; (190.4970) Parametric oscillators and amplifiers.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

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Received 21 Nov 2011; accepted 27 Dec 2011; published 30 Jan 2012 13 February 2012 / Vol. 20, No. 4 / OPTICS EXPRESS 3490

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1. Introduction Silicon photonics has attracted much attention because of its important potential applications in the broad spectral regions [1], which enables the fabrication of on-chip, ultrahighbandwidth optical networks that are critical for the future of microelectronics. Compared with silica fiber, SOI waveguides exhibits a higher third-order nonlinearity and better mode confinement, which makes it possible to realize a variety of optical functions on a submicro scale at relatively low power levels [2–9]. Generally, the nonlinear effects in SOI waveguide arise from the large Kerr parameter and Raman gain coefficient, and have been exploited for Raman amplification, FWM, soliton formation, supercontinuum generation, as well as parametric amplifier [10–23]. The OPO has been used in many fields [24, 25], such as pulse generation and optical timedivision multiplexing. Fiber optical parametric oscillators (FOPO) are commonly used to generate the tunable coherent light, and provide the laser from continuous waves to pulses [26–28]. However, the lower nonlinearity of the fiber has limited its applications. Recently, OPO based on high-nonlinear SOI waveguide has attracted a great deal of research interests. In 2008, Lin et al. proposed a novel scheme for continuous-wave pumped OPO inside silicon micro-resonators, which not only requires a relative low lasing threshold, but also exhibits broad tunablity [23]; in 2010, Levy et al. have demonstrated the first monolithically integrated CMOS-compatible source by creating an optical parametric oscillator formed by a silicon nitride ring resonator on silicon, which can generate more than 100 new wavelengths with operating powers below 50 mW [29]. However, the reports of femtosecond OPO based on SOI waveguide with widely tunable wavelength are rare. Actually, it is meaningful to investigate SOI waveguide optical parametric oscillator for the applications in realization of highly integrated optics. In this paper, we numerically investigated the femtosecond OPO based on the high nonlinear ridge SOI waveguide. By utilizing SSFM, it is demonstrated that ultra-wide tunable wavelength can be realized under the phase matching condition. The simulation results show that the ultra-wide tunable range, as wide as 266 nm, can be obtained due to the interaction

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between nonlinearity and flexible dispersion design. Meanwhile, the duration of output signal pulse is as short as 130 fs. The peak power of the signal pulse exceeds 10 W from 1700 to 1770 nm when the peak power of the pump is 50 W. To the best of our knowledge, this is the first time that the ultra-wide tunable femtosecond OPO based on SOI waveguide is demonstrated. 2. Theoretical model 2.1 Background theory of the FWM When a high intense pump travels inside the SOI waveguide, modulation instability (MI) can appear with certain angular frequency bands on either side of the angular frequency of the pump. As a parametric process, FWM is governed by a phase-matching condition. Specifically, degenerate four-wave mixing (DFWM) uses single pump such that, in the correct dispersive conditions, energy from the intense pump wave (at an angular frequency of ωp) is transferred to a low power signal wave (at an angular frequency of ωs) and into the creation of the idler wave (at an angular frequency of ωi = 2ωp-ωs). Efficient FWM requires minimal phase-mismatch of the four interacting waves. Considering a degenerate pump and including the effects of cross- and self-phase modulation, this phase-mismatch ∆k is given by,

∆k = 2γ Pp − ∆klinear

(1)

where γ is the effective nonlinearity, Pp is the pump power, ∆klinear = 2kp-ks-ki is the linear phase-mismatch, and kp, ks, and ki are the pump, signal and idler propagation constants, respectively. Including the effects of dispersion up to sixth-order, the linear phase-mismatch can be given by,

∆klinear = − β 2 (∆ω ) 2 −

1 1 β 4 ( ∆ω ) 4 − β 6 ( ∆ω ) 6 12 360

(2)

where β2 is the GVD parameter, β4 is the fourth-order dispersion parameter, β6 is the sixthorder dispersion parameter, and ∆ω is the frequency detuning between of the pump and signal waves. Only the even-order dispersion terms play a role in the phase-mismatch due to the symmetry of the FWM process. The signal gain Gs can be given by,

 γ Pp  Gs = 1 +  sinh( gL )   g 

2

(3)

where 1

g = γ Pp ∆klinear − (∆klinear / 2) 2  2

(4)

is the parametric gain parameter, and L is the interaction length. The max signal gain occurs under the condition of perfect phase-matching ∆k = 0. In this situation, the relationship of the linear and nonlinear phase-matching condition can be given by,

2γ Pp + β 2 (∆ω ) 2 +

1 1 β 4 ( ∆ω ) 4 + β 6 (∆ω )6 = 0 12 360

(5)

the phase-matching band can be obtained from solving the Eq. (5). It is observed that the phase-matching band is governed by the parameters of the nonlinear coefficient, pump power and dispersion parameters. The previous research work shows that when the pump is operated around the zero-dispersion wavelength (ZDW) of the SOI waveguide, the fourth-order dispersion parameter plays an important role in the phase-mismatch [6]. The balance among the higher-order dispersions can be found to create a wide frequency side-band. As a result, it

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is possible to construct a widely tunable optical source in the wavelength region through the MI near the ZDW of the SOI waveguide. 2.2 Dispersion tailoring of the SOI waveguide and phase-matching The material GVD of silicon is large and normal near 1550 nm, with a value of approximately D = −1000 ps/[nm km]. The large refractive index contrast of SOI waveguide allows for a large waveguide contribution to the dispersion for highly confining waveguides. The waveguide dispersion can be tailored flexibly by optimizing the geometry parameters of the waveguide. The GVD parameter can be varied to any value by finely adjusting the waveguide dimensions. From the previous discussion, the phase-matching condition can be realized easily when the pump wavelength operates near the ZDW of the SOI waveguide. As a result, it is available to tailor the ZDW near 1550 nm.

Fig. 1. (a) Schematic of the SOI waveguide. (b) The fundamental quasi-TE mode field at wavelength of 1550 nm. (c) The fundamental quasi-TM mode field at wavelength of 1550 nm.

The analyzed SOI waveguide in this paper is the ridge SOI waveguide, the cross section geometry character of the waveguide is shown in Fig. 1(a). Here, the refractive index of the silica and silicon are 1.46 and 3.45 at the wavelength of 1550 nm, respectively. The W = 2 µm is the width of the waveguide, the H = 0.7 µm is the height of the waveguide and the h = 0.35 µm is the etch thickness of the ridge. The effective area of this mode was calculated to be Aeff = 0.8 µm2, combined with the nonlinear refractive index n2 = 6 × 10−18 m2/W. Moreover, the fundamental quasi-TE and quasi-TM mode field diagram are shown in Fig. 1(b) and Fig. 1(c), respectively. Half of the waveguide is used for calculation because the waveguide is symmetric along the x = 0 axis. A finite-element solver was used to calculate the effective refractive index and second-order dispersion of the fundamental modes (including the quasiTE and quasi-TM mode) [30], which is shown in Fig. 2(a) and Fig. 2(b), respectively. For the fundamental TM mode, it can be observed that the effective refractive index changes from 3.405 to 3.100 when the wavelength varies from 1200 nm to 2100 nm. The ZDW of the SOI waveguide are found to be λ0 = 1550 nm for the fundamental quasi-TM mode. As a result, for TM polarization in the ridge SOI waveguide, the phase-matching condition can be realized when the pump wavelength is fixed at 1548 nm. Furthermore, we can also obtain the higherorder dispersion parameters using the same method. The relative dispersion parameters at the wavelength of 1550 nm are following: β3 = 0.02504 ps3/km, β4 = −1.0475 × 10−4 ps4/km, β5 = 3.5025 × 10−7 ps5/km, β6 = −1.1508 × 10−10 ps6/km. From Eq. (1), we can obtain the phase matching curve under the condition of Pp = 50 W. From Fig. 2(c), it can be found that when the pump wavelength is tuning from 1545 nm to 1555 nm, the phase-matched signal wavelength continuously tunes from 1715 nm to 3074 nm, while the phase-matched idler wavelength continuously tunes from 1035 nm to 1422 nm. The total phase-matching bandwidth is as wide as 1746 nm. With balance among the high-order dispersions, phase-matching condition can be realized far from the pump wavelength by pumping around the ZDW of the SOI waveguide. When the wavelength of the pump is fixed at 1548 nm with the width of the pump only 7 nm and the peak power of pump is just 50 W, a widely tuning wavelength range

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is generated due to the steep phase-matching curve. As a result, the broad-band phasematching condition plays an important role in constructing a widely tunable optical parametric oscillator.

Fig. 2. (a) Wavelength dependence of the effective refractive for the fundamental TE mode and TM mode using the waveguide shown in Fig. 1(a). (b) Wavelength dependence of the second order dispersion for the fundamental TE mode and TM mode using the waveguide shown in Fig. 1(a). (c) The phase matching curve of the SOI waveguide. The inset is the frequency shift detuning curve. The pump power used is Pp = 50 W.

2.3 Schematic of the optical parametric oscillator The broad-band phase-matching curve shown in Fig. 2(c) indicates the wideband wavelength tunability that we wish to exploit in the SOI waveguide based OPO. There are two important conditions that need to be satisfied for the oscillator to start oscillating. First, the repetition rate of the pump laser must be matched to the reciprocal of the round trip time of the resonant side-band in the oscillator. Second, the pump power must be sufficiently high so that the parametric gain exceeds the resonant round trip loss [26]. The schematic of the optical parametric oscillator is shown in Fig. 3. The ring cavity is adopted in this scheme, which consists of a pump source, lensed fiber tapers (LFT), SOI waveguide, a polarizing beam splitter (PBS), tuning grating, wave plates and output coupled mirror. The pump source includes passively mode-locked Er-doped ðber laser and amplifier. The peak power of pump is up to 50 W with the duration of 400 fs and 100 MHz repetition rate. The center wavelength of the pump is at 1548 nm, which is located in the normal dispersion regime near the ZDW. P1 and P2 are half-wave plates and P3 is 1/4 wave plate, respectively, which are used to align the state of polarization of the signal. To reduce the couple losses and obtain the low threshold oscillator, a lensed fiber taper (LFT) is used to couple the pulses into the waveguide. A second lensed fiber taper is used at the output end. Additionally, the PBS with a tuning grating is used as a tunable band pass filter with 2 nm bandwidth to select the desire signal wavelength feedback to the oscillator cavity. The output coupled mirror offers the feedback fraction of 0.9. Wavelength tuning can be achieved by adjusting the position of the grating and length of cavity. As shown in Fig. 3, in order to compensate the GVD-induced cavity length changed, the position of the end mirror must be changed accurately. The high repetition, ultra-short pulsed OPO can be obtained from synchronous pumping in the normal dispersion regime of the waveguide.

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Fig. 3. Schematic of the SOI waveguide optical parametric oscillator.

3. Numerical results and discussion The SOI waveguide optical parametric oscillator is based on the third-order nonlinearity χ(3), in which the two pump waves are assumed to be degenerate in frequency domain while the initial signal and idler waves originate from noise. As a result, the numerical model of widely tunable femtosecond optical parametric oscillator in the SOI waveguide can be described by three-wave coupled nonlinear Schrödinger equation (NLSE), which is shown as Eq. (6–8): ∂Ap ∂z

∂As ∂z

∂Ai ∂z

i m −1 β pm ∑ m = 2 m! ∞

+

∞

+

i m −1

m

∂ Ap ∂T

m

∞

m=2

i

m −1

m!

β im

∂ m Ap ∂T

m

1 2

m

∂ As

∑ β sm ∂T m m = 2 m!

+∑

=−

=−

=−

1 2

1 2

(α

(α

(α

li

ls

lp



i ∂



ω p ∂t 

+ α fcp ) Ap + iγ pe  1 +

+ α fcs ) As + iγ se

+ α fci ) Ai + iγ ie

i ∂

2

ω s ∂t

i ∂

ωi ∂t

 A

As As + i

2

Ai Ai + i

2 p

2π

λs

2π

λi

Ap + i

2π

λp

δ nfcp AP + 2iγ p As Ai Ap , (6) *

2

δ nfcs As + 2iγ se Ap As + iγ s Ap2 Ai* , (7)

δ nfci Ai + 2iγ ie Ap

2

Ai + iγ i Ap2 As* , (8)

where the p, s and i denote the pump, signal and idler waves, respectively; the Ap, As and Ai are the amplitudes of the electric field at the pump, signal and idler waves, respectively; the T = t-z/vgp is the time measured in a reference frame moving with the pump pulse; the vgp is the group velocity of the pump pulse; the βjm = dmβ/dωm|ω = ωj (j = p, s, i; m = 2, 3,…) is the dispersion coefficient, here we consider the dispersion to the sixth order. The nonlinear coefficient γje is given by [17]; γ je = γ j + i

β TPA

(9)

,

2 Aeff

where γj = ωjn2/cAeff is the effective nonlinearity of the SOI waveguide, n2 = 12π2χ(3)/n0c is the nonlinear index coefficient, c is the speed of light in vacuum, n0 is the linear refractive index, Aeff is the effective area of the propagating mode and βTPA is the coefficient of two photon absorption (TPA). Here n2 = 6 × 10−18 m2W−1 and βTPA = 5 × 10−12 mW−1 in the 1550 nm regime [13]. In Eqs. (6–8), the αlj (j = p, s, i) represents the linear losses coefficient of the SOI waveguide for the pump, signal and idler waves, respectively. The αfcj = σjNc represents the nonlinear loss from free carrier absorption (FCA), where σj is the FCA cross section and Nc is the free-carrier density. The free-carrier induced index change is δnfcj = ζjNc. These freecarrier parameters can be written as [17] σ j = 1.45 × 10 −21 ( λ j λref

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)

2

m 2 , ζ j = −1.35 × 10−27 ( λ j λref

)

2

m3 ,

(10)


∂N c ( z , t ) ∂t

=

πβ TPA

4

2hω p Aeff 2

Ap ( z , t ) −

Nc ( z, t )

τc

,

(11)

where λj is the wavelength, λref = 1550 nm, h is Planck’s constant, and the carrier lifetime is τc≈1 ns. The first term of the right side of the equations represents the linear loss and nonlinear loss caused by FCA; the second term of the right side of the equations describes the nonlinear loss caused by TPA, which can be obtained from the image part of the nonlinear coefficient γje; Furthermore, the right side of the equations represents the nonlinear effects: include self-phase modulation (SPM), cross-phase modulation (XPM) and FWM term. The three-wave coupled nonlinear Schrödinger equation can be solved numerically by using the SSFM in this paper [31]. In the process of simulation, the peak power of the pump pulse is 50 W at the center wavelength of 1548 nm and the length of the SOI waveguide is 5 cm. From the quantum-mechanical viewpoint, the signal and idler waves originate from the noise. In the simulation, the injected noise has the power level of noise limit, which can be amplified to the signal or idler waves under the phase matching condition and then output. The three-wave coupled nonlinear Schrödinger equations can be used to describe the formation of the signal and idler waves in the optical parametric oscillator cavity under the phase matching condition. The signal and idler waves can be output through generation, amplification, oscillation and output. At the same time, the pump power depleted as the oscillation reaches to a stable state. Through solving the three-wave coupled nonlinear Schrödinger equations using SSFM, the output pulse and spectra of the signal or idler waves can be displayed as the function of the time measured in a reference frame and the length of the SOI waveguide. In this simulation process, the relative parameters of the optical parametric oscillator are following: the linear loss coefficient of the SOI waveguide is 0.22 dB/cm, the peak power of the pump is 50 W and the duration of the pump pulse is 400 fs. The FWM term of the three-wave coupled equation changes the amplitudes of the pulses in cavity, and the terms of the TPA, FCA, dispersion, the SPM and the XPM affect the spectra. The Fig. 4(a) shows the output signal pulses of the optical parametric oscillator while Fig. 4(b) shows the output idler pulses of the optical parametric oscillator. Figure 5 represents the spectra of the signal and idler waves. As shown in Fig. 4 and Fig. 5, it can be observed that the signal wavelength can be tuned from 1645 nm to 1805 nm on the longer wavelength side of the pump, while the idler wavelength is varied from 1350 nm to 1456 nm simultaneously on the shorter wavelength side of the pump by fixing the pump wavelength at 1548 nm. As a result, the signal wavelength range can be tuned reaches 160 nm, while the idler covers 106 nm, the total tuning wavelength range is 266 nm.

Fig. 4. (a) The output signal pulse of the optical parametric oscillator based on SOI waveguide. (b) The output idler pulse of the optical parametric oscillator based on SOI waveguide.

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Fig. 5. (a) The output signal spectra. (b) The output idler spectra. The signal wavelength is tuned on the longer wavelength side of the pump while the idler wavelength is tuned on the shorter wavelength side of the pump. T0 is the duration of the pump pulse and ν0 is the center frequency of the pump pulse.

Additionally, the output spectra corresponding to the output pulse is shown in the Fig. 5. Figure 5(a) shows the output signal spectra of the optical parametric oscillator while Fig. 5(b) shows the output idler spectra of the optical parametric oscillator. As is discussed above, the terms of the FWM have no effects on the spectra. The spectra of the signal and idler pulse are only governed by the contributions of TPA, FCA, SPM and XPM. The effects of SPM and XPM induced chirps broaden the signal and idler spectra symmetrically, which can be observed in Fig. 5. Furthermore, we consider the output peak power of the signal pulse. The numerical results show that the peak power of the output signal pulse can keep at about 10 W in widely tunable wavelength range. As shown in Fig. 6(a), the output peak power of the signal pulse can exceed 10 W in the wavelength range of 1700 nm to 1770 nm. The maximum output signal peak power of the OPO can reach 12.5 W when the signal wavelength is 1735 nm. However, when the wavelength is shorter than 1665 nm or longer than 1785 nm, the peak power of the signal pulse decreased remarkably, which caused by the phase-mismatching. The comparison between the output signal pulses and pump pulse is shown in Fig. 6(b). When the signal wavelength is 1705 nm, the duration of the signal pulse is about 130 fs, which is nearly 1/3 of the duration of the initial pump pulse. This phenomenon can be explained that the pump operates in the normal dispersion regime of the SOI waveguide, which cooperates with SPM and causes the pulse compression effect [31]. Moreover, as shown in Fig. 6(b), the output pump pulse has the multi-peak structure as the peak power decreased to 26 W, which is the effect of the FWM among the pump, signal and idler pulses.

Fig. 6. (a) The peak power of the output signal pulse. (b) Comparison among the signal pulse, the initial pump pulse and output pump pulse of the optical parametric oscillator based on SOI waveguide. The signal wavelength is 1705 nm.

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In addition to the phase-matching curve the dispersion of the SOI waveguide also governs the walk-off among the three waves. For the ultra-short pump pulse used in this paper walkoff is an important parameter for the operation of the OPO and sets upper limit for the tuning range. As the side-band tuning is increased the net parametric gain drops due to the increased walk-off among the pump, signal and idler waves. In Fig. 7, we plot the walk-off delay of the SOI waveguide as a function of tuning wavelength. It shows that the walk-off delay is increased as the tuning wavelength aparts from the pump wavelength. When the tuning wavelength is near 1300 nm, the walk-off delay is 0.5 ps/m. However, the value of the walkoff delay can be reached to be 0.45 ps/m if the tuning wavelength is located at 1800 nm. The numerical results show that the walk-off effect must be considered when the tuning wavelength is far away from the pump. In order to reduce the walk-off effect, we can adopt the short waveguide. In this paper, the length of the SOI waveguide used is 5 cm, which is choose to support the sufficient parametric gain and reduce the walk-off effect. The further widely tuning of the OPO is limited by the phase-matching condition. Using the high nonlinearity and flexible dispersion engineering of the SOI waveguide, widely tunable wavelength and ultra-short pulse from the optical parametric oscillator based on short rib SOI waveguide can be realized under the phase-matching condition, which can be used to generate pulses with the duration as short as 130 fs. Meanwhile, the output peak power of the signal pulses can keep at 10 W in the broad spectral range with the peak power of the pump 50 W. With the two important outstanding features, this optical parametric oscillator can play a dominated role in the generation of the ultra-short pulse and integrated optics.

Fig. 7. Walk-off delay of the SOI waveguide, when the wavelength is tuned from the ZDW of the waveguide 1550 nm.

4. Conclusions In conclusion, the widely tunable femtosecond optical parametric oscillator based on FWM in the ridge SOI waveguide is investigated. The high nonlinearity and flexible dispersion engineering allows broad-band phase-matching condition through tailoring the dimensions of the SOI waveguide. By solving the three coupled nonlinear wave equations with SSFM, the output signal and idler waves with a large tunable wavelength range of 266 nm can be obtained from the OPO. When the suitable pump power and length of SOI waveguide are set, the wavelength of signal from 1645 nm to 1805 nm and the idler wavelength from 1350 nm to 1456 nm can be achieved simultaneously. The output peak power of the signal pulse can exceed 10 W in the range of 1700 nm to 1770 nm while the peak power of the pump pulse used is just 50 W. At the same time, the maximum output signal peak power of the OPO can reach 12.5 W when the signal wavelength is 1735 nm. This work will make some contribution to the silicon photonics and nonlinear optics. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant 60878060. #158490 - $15.00 USD (C) 2012 OSA
