Abstract

We experimentally measure the optical nonlinearities in hydrogenated-amorphous silicon (a-Si:H) waveguides through the transmission of ultra-short pulses. The measured two-photon absorption coefficient β is 4.1 cm/GW and we obtain a 3.5π nonlinear phase shift at 4.1 W coupled input power corresponding to a nonlinear refractive index n2 of 4.2∙10−13 cm2/W. The measured nonlinear coefficient γ = 2003 (W∙m)−1 is at least 5 times the value in crystalline silicon. The measured free carrier absorption coefficient σ = 1.9∙10−16 cm2 agrees with the values predicted from the Drude-Lorenz model. It is seen that a-Si:H exhibits enhanced nonlinear properties at 1550 nm and is a promising platform for nonlinear silicon photonics.

©2010 Optical Society of America

1. Introduction

Hydrogenated-amorphous silicon (a-Si:H) is emerging as an alternate material for the integration of silicon photonics for on-chip optical interconnects. Hydrogenated amorphous silicon (a-Si:H) can be deposited using CMOS compatible low temperature (~250-400 °C) plasma-enhanced chemical vapor deposition (PECVD). This allows amorphous silicon to be integrated at any point in the fabrication process with minimal complexity enabling vertical stacking of optical interconnects. Low-loss waveguides including horizontal slot waveguides and cavity resonators have been demonstrated using amorphous silicon [15].

The need for higher bandwidth in optical communications necessitates the use of nonlinearities for all-optical processing of information. Nonlinearities in optical waveguides can be utilized to demonstrate ultrafast switches, demultiplexers, amplifiers as well as optical limiters. Nonlinear optical effects in crystalline SOI waveguides including nonlinear two photon absorption (TPA), self-phase modulation (SPM), four-wave mixing (FWM) have been extensively studied over the years [612]. However, studies of the nonlinear characteristics of hydrogenated amorphous silicon are significantly more limited. It has been reported that hydrogenated amorphous silicon (a-Si:H) has an enhanced nonlinear absorption coefficient [13] and the DC Kerr effect is enhanced by an order of magnitude over crystalline silicon [14]. In this paper, we further characterize the optical nonlinearities in a-Si:H through the transmission of ~180 fs pulses through a sub-micron waveguide. The nonlinear processes are modeled using a modified nonlinear Schrödinger equation (NLSE) which takes into account free-carrier effects. We compare the measured nonlinearities with a crystalline SOI waveguide of similar dimensions and observe that amorphous silicon exhibits significantly enhanced nonlinear coefficients. From the simulation fits, we determined the two-photon absorption co-efficient (TPA) is β ~4.1 cm/GW, the free carrier absorption co-efficient (FCA) is σfca ~1.9∙10−16 cm2 and the non-linear refractive index is n2 ~4.2∙10−13 cm2/W. These enhancements may yield high performance nonlinear optical devices on a silicon platform.

2. Fabrication and experimental set-up

The amorphous silicon waveguides were fabricated from a 250 nm thick a-Si:H-on-insulator film deposited using plasma enhanced chemical vapor deposition (PECVD) at 400 °C. The deposition parameters of the film are given in Table 1 . The refractive index of the film was measured to be 3.48 at 1550 nm using a spectroscopic ellipsometer. The strip waveguides, 460 nm wide, were patterned using electron beam lithography, followed by inductively coupled plasma (ICP) chlorine etch. Silicon waveguides of similar dimensions were patterned on a 250 nm thick silicon-on-insulator substrate.

Tables Icon

Table 1. PECVD deposition parameters for a-Si:H

The optical nonlinear properties in amorphous silicon were determined by launching ultra short pulses into sub-micron sized waveguides and measuring the output as a function of the input power as shown in Fig. 1 . The input pulses (~100 fs, 810 nm) from a mode-locked Ti-sapphire laser were used as a pump source to an optical parametric oscillator (OPO) to generate ~180 fs pulses at a repetition rate of 80 MHz at 1550 nm. The input power, controlled using a variable attenuator, passes through a polarizer, a beam splitter and is coupled free space into the waveguides using a 0.25 NA, 12 mm focal length objective lens. The polarizer was adjusted to excite TE (electric field parallel to substrate) mode in the waveguides. Free space coupling was used to launch light into the waveguides in order to eliminate nonlinearities induced in fibers from affecting the transmission spectra through the waveguides. The output from the chip was collected using a tapered lensed fiber and measured using an optical spectrum analyzer (OSA). Adiabatic inverse tapers were used to couple light on and off the chip [15].

 

Fig. 1 Experimental set-up to measure optical nonlinearities in a-si:H waveguides. OPO: Optical Parametric Oscillator, OSA: Optical Spectrum analyzer

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3. Theory

The propagation of optical pulses through a waveguide can be modeled based on the following set of nonlinear differential equations, which are derived from the nonlinear Schrödinger equation [1618],

uz+iβ222ut2=(iωcn2βTPA2)|u|2AeffuNc2(σiωckc)uαl2u
Nct=βTPA2hω[|u|2Aeff]2Ncτc
where u is the slowly varying field amplitude, β2 is second-order dispersion coefficient, n2 is nonlinear Kerr coefficient, βTPA is the two-photon absorption coefficient, Aeff is the effective mode area, Nc is the free carrier density, σ is the free carrier absorption coefficient, kc is free carrier dispersion coefficient, αl is the linear loss parameter and τc is the free carrier lifetime. Equations (1) and (2) are solved using a split-step Fourier technique [19] to model the behavior of the pulses in a-Si:H waveguides. The second order dispersion co-efficient was neglected in the analysis since the waveguide lengths used were smaller than the dispersion lengths given byLD=T02/|β2|, where T0 is the initial pulse width. The carrier lifetime in the crystalline silicon waveguide is measured to be τc = 450 ps [20] and that in a-Si:H waveguide with similar dimensions is τc = 400 ps [21]. Since the carrier lifetimes are much shorter than the repetition rate of the pump pulses, the carrier density accumulation due to prior pulses can be neglected in the analysis. The effective area of the waveguides was determined using a mode solver to be Aeff = 0.085 µm2 for both crystalline and amorphous silicon waveguides. The patterned amorphous silicon (a-Si:H) waveguides are 7 mm long while the crystalline silicon waveguides measure 6 mm in length.

4. Nonlinear absorption characterization

To characterize the absorption nonlinearities, the output powers from the waveguides were measured as a function of the coupled input powers. The total insertion loss from the a-Si:H waveguides was measured to be 23 dB at low input powers where nonlinear effects from the waveguide are negligible. The linear transmission loss in the waveguide was measured to be 3.5 dB/cm using the cutback method. The overall coupling loss was determined by subtracting the linear loss from the total loss. By observing the difference in the input power required to achieve nonlinearity induced saturation, the free space coupling loss was calculated to be 15 dB and the output fiber coupling loss to be 5 dB. Such a large free-space coupling loss is to be expected with the low numerical aperture (NA) of the lens used. From Fig. 2(a) , it is observed that output power from the waveguide shows a nonlinear behavior as a function of the coupled input power. The optical limiting of the output in a-Si:H waveguide is attributed to the generation of free carriers due to two photon absorption. From the fit to the measured data, the two-photon absorption coefficient β in a-Si:H is estimated to be 4.1 ± 1.5 cm/GW. Such a large TPA coefficient may come as a surprise since the effective bandgap of a-Si:H is ~1.7 eV. However, due to the amorphous nature of the material, there are exponential band tails that exhibit a very high density of states of 1019 - 1020 cm−3 even below 1.6 eV, in turn, allowing for the efficient two-photon absorption observed here [22]. This enhancement in absorption may also be due to two-state absorption (TSA) where mid-gap localized defect states aid absorption as modeled in [13]. We modified Eqs. (1) and (2) to account for mid-gap states but observed a much better fit of the experimental data with TPA alone as observed in Fig. 3 . However, TPA is not the dominating effect in a-Si:H; free-carrier absorption significantly saturates the output. We observe this in Fig. 2(a) where the output power begins to saturate at 3W peak power coupled into the waveguides. We determined that this power corresponds to a free carrier density of 1.1∙1017 cm−3, which is significant enough to induce considerable free-carrier absorption. Through our modeling, we determined the free-carrier absorption coefficient is (1.9 ± 0.4) ∙10−16 cm2. This can be verified by calculating the free-carrier absorption coefficient σ using the Drude-Lorenz model as given by [23]

 

Fig. 2 (a) The output power (black squares) as a function of coupled input power for a-Si:H waveguide. The blue line indicates the fit to the measured data based on solving the nonlinear equations. (b) Similar measurements performed on a SOI waveguide

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Fig. 3 Simulation fit to measured data using the two-photon absorption (TPA) model and two-state absorption (TSA) model in a-Si:H waveguide.

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σ=e3λ24π2c3ε0n0(1me2μe+1mh2μh)

In Eq. (3), e is the electron charge, λ is the probe wavelength, ε0 is permittivity of free space, n0 is the refractive index of the material, me and mh are the effective masses of electrons and holes, µe and µh are the mobilities of the carriers in a-Si:H. Substituting for me = 0.5∙mo, mh = 1.0∙mo, mo = 9.1∙10−31 Kg, µe = 2.0 cm2/V∙s, µh = 0.4 cm2/V∙s [24] in Eq. (3), yields a theoretically estimated value of σ = 1.6∙10−16 cm2. The measured free carrier absorption coefficient is in very good agreement with the value predicted based on the Drude-Lorentz model. It should be noted that the measured enhancement of nonlinear absorption in a-Si:H corroborates the reported enhancement in free-carrier nonlinearities in [13].

Similar measurements were performed on a SOI waveguide to extract the nonlinear parameters. The insertion loss at low powers was measured to be 33 dB with a transmission loss of 13.5 dB/cm. From the fits to the measurements shown in Fig. 2(b), the two photon absorption coefficient and free carrier absorption were estimated to be 1.0 ± 0.1 cm/GW and 1.45 ∙10−17 cm2. These values are consistent with measurements reported in crystalline silicon [7,8,16]. We note that the transmission through SOI waveguides begins to saturate at 4 W peak power coupled in the waveguides which yields a free carrier density of 2.8∙1016 cm−3. In contrast to a-Si:H, we determined that the saturation in the output was dependent more on the two-photon absorption (TPA) effect than from free-carrier absorption (FCA) since not enough carriers are generated in SOI waveguides in order for FCA to become considerable (owing to the considerably smaller free-carrier absorption coefficient). Consequently, on comparing the simulation fit parameters between a-Si:H and SOI waveguides, there is a significant enhancement of the nonlinear absorption in hydrogenated amorphous silicon over crystalline silicon. This could enable high performance all-optical or electro-optic modulators with the material [21].

5. Nonlinear refraction characterization

The nonlinear refractive index n2 was determined by modeling the measured spectral broadening due to self-phase modulation (SPM) of the pulses in the waveguides. The measured transmission spectra and the simulation of the spectral broadening of the pulses through the a-Si:H waveguides for different coupled powers is shown in Fig. 4 . It is seen that there is an increase in the spectral broadening along with induced phase shifts with increasing coupled powers. A nonlinear phase shift of 3.5π is obtained at 4.1 W coupled input power. It was observed through simulations that TPA plays a role in limiting the maximum achievable phase shift. It is known that TPA reduces the induced phase shift while leaving the pulse spectrum symmetric while FCA results in an asymmetric pulse spectrum [16]. From the measured spectral data, it can be observed that the symmetric pulse spectrum indicates that TPA is the more dominant effect as compared to FCA. The simulation fits closely match the measured spectrum. From the fits, the nonlinear refractive index of a-Si:H is estimated to be (4.2 ± 1) ∙10−13 cm2/W. We should note that the enhanced nonlinear Kerr index of a-Si:H observed here is in agreement with the enhancement of the DC Kerr effect of a-Si:H previously observed [14]. Based on the measurement of the Kerr index, the nonlinear coefficient γ=ωn2/cAeffin a-Si:H waveguides is (2000 ± 500) (W∙m)−1.

 

Fig. 4 Measured spectral broadening in a-Si:H waveguides at different coupled powers due to self-phase modulation and simulation fits based on solving nonlinear Schrödinger equations.

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We also observed spectral broadening in SOI waveguides as shown in Fig. 5 . A nonlinear phase shift of 1.5π is obtained at 1.4 W coupled power. Based on the simulation fits to the measured spectrum, the nonlinear refractive index n2 is inferred to be (8 ± 2) ∙10−14 cm2/W which is consistent with other values reported in crystalline silicon [8,10,12]. The measured nonlinear Kerr index corresponds to a nonlinear coefficient γ in crystalline silicon of (380 ± 100) (W∙m)−1. Consequently, the nonlinear coefficient of a-Si:H waveguides is at least 5 times that in SOI. We should note here that the uncertainties in the reported values are based on measurements made on a number of different waveguides (> 10) on different chips and an uncertainty in the coupling coefficients of ± 3 dB.

 

Fig. 5 Measured spectral broadening in SOI waveguides at different coupled powers due to self-phase modulation and simulation fits based on solving nonlinear Schrödinger equations.

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Lastly, the nonlinear figure of merit (FOM) is defined as the ratio of the nonlinear index and the two-photon absorption coefficient at a particular wavelength.

FOM=n2βTPAλ

Based on the measured values of the nonlinear parameters, the figure of merit for a-Si:H is 0.66 ± 0.3 while that for SOI is 0.49 ± 0.15. The figure of merit indicates that a-Si:H waveguides are potentially more suitable for nonlinear optical switching applications [25].

6. Conclusion

We experimentally measured the optical nonlinearities of hydrogenated amorphous silicon waveguide through the propagation of ultra-short pulses. On comparing the measured nonlinearities with a SOI waveguide, it is observed that the nonlinear coefficient γ is at least 5 times that in crystalline silicon. The figure of merit indicates that a-Si:H is a promising alternate platform for enabling nonlinear silicon photonics. We should note that very recently the TPA coefficient and nonlinear refractive index of a-Si:H waveguides was also characterized [26]. However, in contrast to our results and the results in [13,14], the nonlinearities are reported to be significantly smaller than crystalline silicon. The reason for the discrepancy is unclear at this point, but may be attributed to H2 being used as the precursor gas during deposition resulting in passivation of the dangling bonds and defects in their waveguides, thereby resulting in different nonlinear properties than reported here. Consequently, further study of the relationship between deposition parameters, film properties and nonlinearities would be helpful in further understanding the nonlinear properties of amorphous silicon.

Acknowledgements

This work is supported in part by the National Science Foundation under grant ECCS-0903448 and by the Semiconductor Research Corporation under contract SRC-2009-HJ-2000. This work was performed in part at the Cornell Nanoscale Facility, a member of the National Nanotechnology Infrastructure Network, which is supported by the National Science Foundation (Grant ECS – 0335765).

References and links

1. M. J. A. de Dood, A. Polman, T. Zijlstra, and E. W. J. M. van der Drift, “Amorphous silicon waveguides for microphotonics,” J. Appl. Phys. 92(2), 649–653 (2002). [CrossRef]  

2. A. Harke, M. Krause, and J. Mueller, “Low-loss singlemode amorphous silicon waveguides,” Electron. Lett. 41(25), 1377–1379 (2005). [CrossRef]  

3. D. K. Sparacin, R. Sun, A. M. Agarwal, M. A. Beals, J. Michel, L. C. Kimerling, T. J. Conway, A. T. Pomerene, D. N. Carothers, M. J. Grove, D. M. Gill, M. S. Rasras, S. S. Patel, and A. E. White, “Low Loss Amorphous Silicon Channel Waveguides for Integrated Photonics,” in Group IV Photonics, 2006. 3rd IEEE International Conference on, pp. 255–257 (2006)

4. R. Sun, J. Cheng, J. Michel, and L. Kimerling, “Transparent amorphous silicon channel waveguides and high-Q resonators using a damascene process,” Opt. Lett. 34(15), 2378–2380 (2009). [CrossRef]   [PubMed]  

5. R. Sun, P. Dong, N. N. Feng, C. Y. Hong, J. Michel, M. Lipson, and L. Kimerling, “Horizontal single and multiple slot waveguides: optical transmission at λ = 1550 nm,” Opt. Express 15(26), 17967–17972 (2007). [CrossRef]   [PubMed]  

6. H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 um wavelength,” Appl. Phys. Lett. 80(3), 416–418 (2002). [CrossRef]  

7. M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82(18), 2954–2956 (2003). [CrossRef]  

8. G. W. Rieger, K. S. Virk, and J. F. Young, “Nonlinear propagation of ultrafast 1.5um pulses in high-index-contrast silicon-on-insulator waveguides,” Appl. Phys. Lett. 84(6), 900–902 (2004). [CrossRef]  

9. O. Boyraz, T. Indukuri, and B. Jalali, “Self-phase-modulation induced spectral broadening in silicon waveguides,” Opt. Express 12(5), 829–834 (2004). [CrossRef]   [PubMed]  

10. E. Dulkeith, Y. A. Vlasov, X. Chen, N. C. Panoiu, and R. M. Osgood Jr., “Self-phase-modulation in submicron silicon-on-insulator photonic wires,” Opt. Express 14(12), 5524–5534 (2006). [CrossRef]   [PubMed]  

11. H. Yamada, M. Shirane, T. Chu, H. Yokoyama, S. Ishida, and Y. Arakawa, “Nonlinear-Optic Silicon-Nanowire Waveguides,” Jpn. J. Appl. Phys. 44(No. 9A), 6541–6545 (2005). [CrossRef]  

12. H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J. Takahashi, and S. Itabashi, “Four-wave mixing in silicon wire waveguides,” Opt. Express 13(12), 4629–4637 (2005). [CrossRef]   [PubMed]  

13. K. Ikeda, Y. Shen, and Y. Fainman, “Enhanced optical nonlinearity in amorphous silicon and its application to waveguide devices,” Opt. Express 15(26), 17761–17771 (2007). [CrossRef]   [PubMed]  

14. M. Zelikson, J. Salzman, K. Weiser, and J. Kanicki, “Enhanced electro-optic effect in amorphous hydrogenated silicon based waveguides,” Appl. Phys. Lett. 61(14), 1664–1666 (1992). [CrossRef]  

15. V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Opt. Lett. 28(15), 1302–1304 (2003). [CrossRef]   [PubMed]  

16. L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32(14), 2031–2033 (2007). [CrossRef]   [PubMed]  

17. S. Roy, S. K. Bhadra, and G. P. Agrawal, “Femtosecond pulse propagation in silicon waveguides: Variational approach and its advantages,” Opt. Commun. 281(23), 5889–5893 (2008). [CrossRef]  

18. P. Apiratikul, A. M. Rossi, and T. E. Murphy, “Nonlinearities in porous silicon optical waveguides at 1550 nm,” Opt. Express 17(5), 3396–3406 (2009). [CrossRef]   [PubMed]  

19. G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, New York, Third Edition 2001)

20. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004). [CrossRef]   [PubMed]  

21. K. Narayanan, A. W. Elshaari, and S. F. Preble, “Broadband all-optical modulation in hydrogenated-amorphous silicon waveguides,” Submitted for publication.

22. R. A. Street, Hydrogenated Amorphous Silicon, (Cambridge University Press, Cambridge NY 1991)

23. R. Soref and B. Bennett, “Electrooptical effect in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987). [CrossRef]  

24. P. Fauchet, D. Hulin, R. Vanderhaghen, A. Mourchid, and W. Nighan Jr., “The properties of free carriers in amorphous silicon,” J. Non-Cryst. Solids 141, 76–87 (1992). [CrossRef]  

25. G. I. Stegeman, “Material figures of merit and implications to all-optical waveguide switching,” Proc. SPIE 1852, 75 (1993). [CrossRef]  

26. Y. Shoji, T. Ogasawara, T. Kamei, Y. Sakakibara, S. Suda, K. Kintaka, H. Kawashima, M. Okano, T. Hasama, H. Ishikawa, and M. Mori, “Ultrafast nonlinear effects in hydrogenated amorphous silicon wire waveguide,” Opt. Express 18(6), 5668–5673 (2010). [CrossRef]   [PubMed]  

References

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  1. M. J. A. de Dood, A. Polman, T. Zijlstra, and E. W. J. M. van der Drift, “Amorphous silicon waveguides for microphotonics,” J. Appl. Phys. 92(2), 649–653 (2002).
    [Crossref]
  2. A. Harke, M. Krause, and J. Mueller, “Low-loss singlemode amorphous silicon waveguides,” Electron. Lett. 41(25), 1377–1379 (2005).
    [Crossref]
  3. D. K. Sparacin, R. Sun, A. M. Agarwal, M. A. Beals, J. Michel, L. C. Kimerling, T. J. Conway, A. T. Pomerene, D. N. Carothers, M. J. Grove, D. M. Gill, M. S. Rasras, S. S. Patel, and A. E. White, “Low Loss Amorphous Silicon Channel Waveguides for Integrated Photonics,” in Group IV Photonics, 2006. 3rd IEEE International Conference on, pp. 255–257 (2006)
  4. R. Sun, J. Cheng, J. Michel, and L. Kimerling, “Transparent amorphous silicon channel waveguides and high-Q resonators using a damascene process,” Opt. Lett. 34(15), 2378–2380 (2009).
    [Crossref] [PubMed]
  5. R. Sun, P. Dong, N. N. Feng, C. Y. Hong, J. Michel, M. Lipson, and L. Kimerling, “Horizontal single and multiple slot waveguides: optical transmission at λ = 1550 nm,” Opt. Express 15(26), 17967–17972 (2007).
    [Crossref] [PubMed]
  6. H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 um wavelength,” Appl. Phys. Lett. 80(3), 416–418 (2002).
    [Crossref]
  7. M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82(18), 2954–2956 (2003).
    [Crossref]
  8. G. W. Rieger, K. S. Virk, and J. F. Young, “Nonlinear propagation of ultrafast 1.5um pulses in high-index-contrast silicon-on-insulator waveguides,” Appl. Phys. Lett. 84(6), 900–902 (2004).
    [Crossref]
  9. O. Boyraz, T. Indukuri, and B. Jalali, “Self-phase-modulation induced spectral broadening in silicon waveguides,” Opt. Express 12(5), 829–834 (2004).
    [Crossref] [PubMed]
  10. E. Dulkeith, Y. A. Vlasov, X. Chen, N. C. Panoiu, and R. M. Osgood., “Self-phase-modulation in submicron silicon-on-insulator photonic wires,” Opt. Express 14(12), 5524–5534 (2006).
    [Crossref] [PubMed]
  11. H. Yamada, M. Shirane, T. Chu, H. Yokoyama, S. Ishida, and Y. Arakawa, “Nonlinear-Optic Silicon-Nanowire Waveguides,” Jpn. J. Appl. Phys. 44(No. 9A), 6541–6545 (2005).
    [Crossref]
  12. H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J. Takahashi, and S. Itabashi, “Four-wave mixing in silicon wire waveguides,” Opt. Express 13(12), 4629–4637 (2005).
    [Crossref] [PubMed]
  13. K. Ikeda, Y. Shen, and Y. Fainman, “Enhanced optical nonlinearity in amorphous silicon and its application to waveguide devices,” Opt. Express 15(26), 17761–17771 (2007).
    [Crossref] [PubMed]
  14. M. Zelikson, J. Salzman, K. Weiser, and J. Kanicki, “Enhanced electro-optic effect in amorphous hydrogenated silicon based waveguides,” Appl. Phys. Lett. 61(14), 1664–1666 (1992).
    [Crossref]
  15. V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Opt. Lett. 28(15), 1302–1304 (2003).
    [Crossref] [PubMed]
  16. L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32(14), 2031–2033 (2007).
    [Crossref] [PubMed]
  17. S. Roy, S. K. Bhadra, and G. P. Agrawal, “Femtosecond pulse propagation in silicon waveguides: Variational approach and its advantages,” Opt. Commun. 281(23), 5889–5893 (2008).
    [Crossref]
  18. P. Apiratikul, A. M. Rossi, and T. E. Murphy, “Nonlinearities in porous silicon optical waveguides at 1550 nm,” Opt. Express 17(5), 3396–3406 (2009).
    [Crossref] [PubMed]
  19. G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, New York, Third Edition 2001)
  20. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
    [Crossref] [PubMed]
  21. K. Narayanan, A. W. Elshaari, and S. F. Preble, “Broadband all-optical modulation in hydrogenated-amorphous silicon waveguides,” Submitted for publication.
  22. R. A. Street, Hydrogenated Amorphous Silicon, (Cambridge University Press, Cambridge NY 1991)
  23. R. Soref and B. Bennett, “Electrooptical effect in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
    [Crossref]
  24. P. Fauchet, D. Hulin, R. Vanderhaghen, A. Mourchid, and W. Nighan., “The properties of free carriers in amorphous silicon,” J. Non-Cryst. Solids 141, 76–87 (1992).
    [Crossref]
  25. G. I. Stegeman, “Material figures of merit and implications to all-optical waveguide switching,” Proc. SPIE 1852, 75 (1993).
    [Crossref]
  26. Y. Shoji, T. Ogasawara, T. Kamei, Y. Sakakibara, S. Suda, K. Kintaka, H. Kawashima, M. Okano, T. Hasama, H. Ishikawa, and M. Mori, “Ultrafast nonlinear effects in hydrogenated amorphous silicon wire waveguide,” Opt. Express 18(6), 5668–5673 (2010).
    [Crossref] [PubMed]

2010 (1)

2009 (2)

2008 (1)

S. Roy, S. K. Bhadra, and G. P. Agrawal, “Femtosecond pulse propagation in silicon waveguides: Variational approach and its advantages,” Opt. Commun. 281(23), 5889–5893 (2008).
[Crossref]

2007 (3)

2006 (1)

2005 (3)

H. Yamada, M. Shirane, T. Chu, H. Yokoyama, S. Ishida, and Y. Arakawa, “Nonlinear-Optic Silicon-Nanowire Waveguides,” Jpn. J. Appl. Phys. 44(No. 9A), 6541–6545 (2005).
[Crossref]

H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J. Takahashi, and S. Itabashi, “Four-wave mixing in silicon wire waveguides,” Opt. Express 13(12), 4629–4637 (2005).
[Crossref] [PubMed]

A. Harke, M. Krause, and J. Mueller, “Low-loss singlemode amorphous silicon waveguides,” Electron. Lett. 41(25), 1377–1379 (2005).
[Crossref]

2004 (3)

G. W. Rieger, K. S. Virk, and J. F. Young, “Nonlinear propagation of ultrafast 1.5um pulses in high-index-contrast silicon-on-insulator waveguides,” Appl. Phys. Lett. 84(6), 900–902 (2004).
[Crossref]

O. Boyraz, T. Indukuri, and B. Jalali, “Self-phase-modulation induced spectral broadening in silicon waveguides,” Opt. Express 12(5), 829–834 (2004).
[Crossref] [PubMed]

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
[Crossref] [PubMed]

2003 (2)

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82(18), 2954–2956 (2003).
[Crossref]

V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Opt. Lett. 28(15), 1302–1304 (2003).
[Crossref] [PubMed]

2002 (2)

M. J. A. de Dood, A. Polman, T. Zijlstra, and E. W. J. M. van der Drift, “Amorphous silicon waveguides for microphotonics,” J. Appl. Phys. 92(2), 649–653 (2002).
[Crossref]

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 um wavelength,” Appl. Phys. Lett. 80(3), 416–418 (2002).
[Crossref]

1993 (1)

G. I. Stegeman, “Material figures of merit and implications to all-optical waveguide switching,” Proc. SPIE 1852, 75 (1993).
[Crossref]

1992 (2)

P. Fauchet, D. Hulin, R. Vanderhaghen, A. Mourchid, and W. Nighan., “The properties of free carriers in amorphous silicon,” J. Non-Cryst. Solids 141, 76–87 (1992).
[Crossref]

M. Zelikson, J. Salzman, K. Weiser, and J. Kanicki, “Enhanced electro-optic effect in amorphous hydrogenated silicon based waveguides,” Appl. Phys. Lett. 61(14), 1664–1666 (1992).
[Crossref]

1987 (1)

R. Soref and B. Bennett, “Electrooptical effect in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[Crossref]

Agrawal, G. P.

S. Roy, S. K. Bhadra, and G. P. Agrawal, “Femtosecond pulse propagation in silicon waveguides: Variational approach and its advantages,” Opt. Commun. 281(23), 5889–5893 (2008).
[Crossref]

L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32(14), 2031–2033 (2007).
[Crossref] [PubMed]

Almeida, V. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
[Crossref] [PubMed]

V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Opt. Lett. 28(15), 1302–1304 (2003).
[Crossref] [PubMed]

Apiratikul, P.

Arakawa, Y.

H. Yamada, M. Shirane, T. Chu, H. Yokoyama, S. Ishida, and Y. Arakawa, “Nonlinear-Optic Silicon-Nanowire Waveguides,” Jpn. J. Appl. Phys. 44(No. 9A), 6541–6545 (2005).
[Crossref]

Asghari, M.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 um wavelength,” Appl. Phys. Lett. 80(3), 416–418 (2002).
[Crossref]

Barrios, C. A.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
[Crossref] [PubMed]

Bennett, B.

R. Soref and B. Bennett, “Electrooptical effect in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[Crossref]

Bhadra, S. K.

S. Roy, S. K. Bhadra, and G. P. Agrawal, “Femtosecond pulse propagation in silicon waveguides: Variational approach and its advantages,” Opt. Commun. 281(23), 5889–5893 (2008).
[Crossref]

Boyraz, O.

Chen, X.

Cheng, J.

Chu, T.

H. Yamada, M. Shirane, T. Chu, H. Yokoyama, S. Ishida, and Y. Arakawa, “Nonlinear-Optic Silicon-Nanowire Waveguides,” Jpn. J. Appl. Phys. 44(No. 9A), 6541–6545 (2005).
[Crossref]

Day, I. E.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 um wavelength,” Appl. Phys. Lett. 80(3), 416–418 (2002).
[Crossref]

de Dood, M. J. A.

M. J. A. de Dood, A. Polman, T. Zijlstra, and E. W. J. M. van der Drift, “Amorphous silicon waveguides for microphotonics,” J. Appl. Phys. 92(2), 649–653 (2002).
[Crossref]

Dinu, M.

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82(18), 2954–2956 (2003).
[Crossref]

Dong, P.

Drake, J.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 um wavelength,” Appl. Phys. Lett. 80(3), 416–418 (2002).
[Crossref]

Dulkeith, E.

Elshaari, A. W.

K. Narayanan, A. W. Elshaari, and S. F. Preble, “Broadband all-optical modulation in hydrogenated-amorphous silicon waveguides,” Submitted for publication.

Fainman, Y.

Fauchet, P.

P. Fauchet, D. Hulin, R. Vanderhaghen, A. Mourchid, and W. Nighan., “The properties of free carriers in amorphous silicon,” J. Non-Cryst. Solids 141, 76–87 (1992).
[Crossref]

Feng, N. N.

Fukuda, H.

Garcia, H.

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82(18), 2954–2956 (2003).
[Crossref]

Harke, A.

A. Harke, M. Krause, and J. Mueller, “Low-loss singlemode amorphous silicon waveguides,” Electron. Lett. 41(25), 1377–1379 (2005).
[Crossref]

Harpin, A.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 um wavelength,” Appl. Phys. Lett. 80(3), 416–418 (2002).
[Crossref]

Hasama, T.

Hong, C. Y.

Hulin, D.

P. Fauchet, D. Hulin, R. Vanderhaghen, A. Mourchid, and W. Nighan., “The properties of free carriers in amorphous silicon,” J. Non-Cryst. Solids 141, 76–87 (1992).
[Crossref]

Ikeda, K.

Indukuri, T.

Ishida, S.

H. Yamada, M. Shirane, T. Chu, H. Yokoyama, S. Ishida, and Y. Arakawa, “Nonlinear-Optic Silicon-Nanowire Waveguides,” Jpn. J. Appl. Phys. 44(No. 9A), 6541–6545 (2005).
[Crossref]

Ishikawa, H.

Itabashi, S.

Jalali, B.

Kamei, T.

Kanicki, J.

M. Zelikson, J. Salzman, K. Weiser, and J. Kanicki, “Enhanced electro-optic effect in amorphous hydrogenated silicon based waveguides,” Appl. Phys. Lett. 61(14), 1664–1666 (1992).
[Crossref]

Kawashima, H.

Kimerling, L.

Kintaka, K.

Krause, M.

A. Harke, M. Krause, and J. Mueller, “Low-loss singlemode amorphous silicon waveguides,” Electron. Lett. 41(25), 1377–1379 (2005).
[Crossref]

Liang, T. K.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 um wavelength,” Appl. Phys. Lett. 80(3), 416–418 (2002).
[Crossref]

Lipson, M.

Michel, J.

Mori, M.

Mourchid, A.

P. Fauchet, D. Hulin, R. Vanderhaghen, A. Mourchid, and W. Nighan., “The properties of free carriers in amorphous silicon,” J. Non-Cryst. Solids 141, 76–87 (1992).
[Crossref]

Mueller, J.

A. Harke, M. Krause, and J. Mueller, “Low-loss singlemode amorphous silicon waveguides,” Electron. Lett. 41(25), 1377–1379 (2005).
[Crossref]

Murphy, T. E.

Narayanan, K.

K. Narayanan, A. W. Elshaari, and S. F. Preble, “Broadband all-optical modulation in hydrogenated-amorphous silicon waveguides,” Submitted for publication.

Nighan, W.

P. Fauchet, D. Hulin, R. Vanderhaghen, A. Mourchid, and W. Nighan., “The properties of free carriers in amorphous silicon,” J. Non-Cryst. Solids 141, 76–87 (1992).
[Crossref]

Ogasawara, T.

Okano, M.

Osgood, R. M.

Panepucci, R. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
[Crossref] [PubMed]

V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Opt. Lett. 28(15), 1302–1304 (2003).
[Crossref] [PubMed]

Panoiu, N. C.

Polman, A.

M. J. A. de Dood, A. Polman, T. Zijlstra, and E. W. J. M. van der Drift, “Amorphous silicon waveguides for microphotonics,” J. Appl. Phys. 92(2), 649–653 (2002).
[Crossref]

Preble, S. F.

K. Narayanan, A. W. Elshaari, and S. F. Preble, “Broadband all-optical modulation in hydrogenated-amorphous silicon waveguides,” Submitted for publication.

Quochi, F.

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82(18), 2954–2956 (2003).
[Crossref]

Rieger, G. W.

G. W. Rieger, K. S. Virk, and J. F. Young, “Nonlinear propagation of ultrafast 1.5um pulses in high-index-contrast silicon-on-insulator waveguides,” Appl. Phys. Lett. 84(6), 900–902 (2004).
[Crossref]

Roberts, S. W.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 um wavelength,” Appl. Phys. Lett. 80(3), 416–418 (2002).
[Crossref]

Rossi, A. M.

Roy, S.

S. Roy, S. K. Bhadra, and G. P. Agrawal, “Femtosecond pulse propagation in silicon waveguides: Variational approach and its advantages,” Opt. Commun. 281(23), 5889–5893 (2008).
[Crossref]

Sakakibara, Y.

Salzman, J.

M. Zelikson, J. Salzman, K. Weiser, and J. Kanicki, “Enhanced electro-optic effect in amorphous hydrogenated silicon based waveguides,” Appl. Phys. Lett. 61(14), 1664–1666 (1992).
[Crossref]

Shen, Y.

Shirane, M.

H. Yamada, M. Shirane, T. Chu, H. Yokoyama, S. Ishida, and Y. Arakawa, “Nonlinear-Optic Silicon-Nanowire Waveguides,” Jpn. J. Appl. Phys. 44(No. 9A), 6541–6545 (2005).
[Crossref]

Shoji, T.

Shoji, Y.

Soref, R.

R. Soref and B. Bennett, “Electrooptical effect in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[Crossref]

Stegeman, G. I.

G. I. Stegeman, “Material figures of merit and implications to all-optical waveguide switching,” Proc. SPIE 1852, 75 (1993).
[Crossref]

Suda, S.

Sun, R.

Takahashi, J.

Takahashi, M.

Tsang, H. K.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 um wavelength,” Appl. Phys. Lett. 80(3), 416–418 (2002).
[Crossref]

Tsuchizawa, T.

van der Drift, E. W. J. M.

M. J. A. de Dood, A. Polman, T. Zijlstra, and E. W. J. M. van der Drift, “Amorphous silicon waveguides for microphotonics,” J. Appl. Phys. 92(2), 649–653 (2002).
[Crossref]

Vanderhaghen, R.

P. Fauchet, D. Hulin, R. Vanderhaghen, A. Mourchid, and W. Nighan., “The properties of free carriers in amorphous silicon,” J. Non-Cryst. Solids 141, 76–87 (1992).
[Crossref]

Virk, K. S.

G. W. Rieger, K. S. Virk, and J. F. Young, “Nonlinear propagation of ultrafast 1.5um pulses in high-index-contrast silicon-on-insulator waveguides,” Appl. Phys. Lett. 84(6), 900–902 (2004).
[Crossref]

Vlasov, Y. A.

Watanabe, T.

Weiser, K.

M. Zelikson, J. Salzman, K. Weiser, and J. Kanicki, “Enhanced electro-optic effect in amorphous hydrogenated silicon based waveguides,” Appl. Phys. Lett. 61(14), 1664–1666 (1992).
[Crossref]

Wong, C. S.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 um wavelength,” Appl. Phys. Lett. 80(3), 416–418 (2002).
[Crossref]

Yamada, H.

H. Yamada, M. Shirane, T. Chu, H. Yokoyama, S. Ishida, and Y. Arakawa, “Nonlinear-Optic Silicon-Nanowire Waveguides,” Jpn. J. Appl. Phys. 44(No. 9A), 6541–6545 (2005).
[Crossref]

Yamada, K.

Yin, L.

Yokoyama, H.

H. Yamada, M. Shirane, T. Chu, H. Yokoyama, S. Ishida, and Y. Arakawa, “Nonlinear-Optic Silicon-Nanowire Waveguides,” Jpn. J. Appl. Phys. 44(No. 9A), 6541–6545 (2005).
[Crossref]

Young, J. F.

G. W. Rieger, K. S. Virk, and J. F. Young, “Nonlinear propagation of ultrafast 1.5um pulses in high-index-contrast silicon-on-insulator waveguides,” Appl. Phys. Lett. 84(6), 900–902 (2004).
[Crossref]

Zelikson, M.

M. Zelikson, J. Salzman, K. Weiser, and J. Kanicki, “Enhanced electro-optic effect in amorphous hydrogenated silicon based waveguides,” Appl. Phys. Lett. 61(14), 1664–1666 (1992).
[Crossref]

Zijlstra, T.

M. J. A. de Dood, A. Polman, T. Zijlstra, and E. W. J. M. van der Drift, “Amorphous silicon waveguides for microphotonics,” J. Appl. Phys. 92(2), 649–653 (2002).
[Crossref]

Appl. Phys. Lett. (4)

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 um wavelength,” Appl. Phys. Lett. 80(3), 416–418 (2002).
[Crossref]

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82(18), 2954–2956 (2003).
[Crossref]

G. W. Rieger, K. S. Virk, and J. F. Young, “Nonlinear propagation of ultrafast 1.5um pulses in high-index-contrast silicon-on-insulator waveguides,” Appl. Phys. Lett. 84(6), 900–902 (2004).
[Crossref]

M. Zelikson, J. Salzman, K. Weiser, and J. Kanicki, “Enhanced electro-optic effect in amorphous hydrogenated silicon based waveguides,” Appl. Phys. Lett. 61(14), 1664–1666 (1992).
[Crossref]

Electron. Lett. (1)

A. Harke, M. Krause, and J. Mueller, “Low-loss singlemode amorphous silicon waveguides,” Electron. Lett. 41(25), 1377–1379 (2005).
[Crossref]

IEEE J. Quantum Electron. (1)

R. Soref and B. Bennett, “Electrooptical effect in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[Crossref]

J. Appl. Phys. (1)

M. J. A. de Dood, A. Polman, T. Zijlstra, and E. W. J. M. van der Drift, “Amorphous silicon waveguides for microphotonics,” J. Appl. Phys. 92(2), 649–653 (2002).
[Crossref]

J. Non-Cryst. Solids (1)

P. Fauchet, D. Hulin, R. Vanderhaghen, A. Mourchid, and W. Nighan., “The properties of free carriers in amorphous silicon,” J. Non-Cryst. Solids 141, 76–87 (1992).
[Crossref]

Jpn. J. Appl. Phys. (1)

H. Yamada, M. Shirane, T. Chu, H. Yokoyama, S. Ishida, and Y. Arakawa, “Nonlinear-Optic Silicon-Nanowire Waveguides,” Jpn. J. Appl. Phys. 44(No. 9A), 6541–6545 (2005).
[Crossref]

Nature (1)

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004).
[Crossref] [PubMed]

Opt. Commun. (1)

S. Roy, S. K. Bhadra, and G. P. Agrawal, “Femtosecond pulse propagation in silicon waveguides: Variational approach and its advantages,” Opt. Commun. 281(23), 5889–5893 (2008).
[Crossref]

Opt. Express (7)

Opt. Lett. (3)

Proc. SPIE (1)

G. I. Stegeman, “Material figures of merit and implications to all-optical waveguide switching,” Proc. SPIE 1852, 75 (1993).
[Crossref]

Other (4)

G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, New York, Third Edition 2001)

K. Narayanan, A. W. Elshaari, and S. F. Preble, “Broadband all-optical modulation in hydrogenated-amorphous silicon waveguides,” Submitted for publication.

R. A. Street, Hydrogenated Amorphous Silicon, (Cambridge University Press, Cambridge NY 1991)

D. K. Sparacin, R. Sun, A. M. Agarwal, M. A. Beals, J. Michel, L. C. Kimerling, T. J. Conway, A. T. Pomerene, D. N. Carothers, M. J. Grove, D. M. Gill, M. S. Rasras, S. S. Patel, and A. E. White, “Low Loss Amorphous Silicon Channel Waveguides for Integrated Photonics,” in Group IV Photonics, 2006. 3rd IEEE International Conference on, pp. 255–257 (2006)

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Figures (5)

Fig. 1
Fig. 1 Experimental set-up to measure optical nonlinearities in a-si:H waveguides. OPO: Optical Parametric Oscillator, OSA: Optical Spectrum analyzer
Fig. 2
Fig. 2 (a) The output power (black squares) as a function of coupled input power for a-Si:H waveguide. The blue line indicates the fit to the measured data based on solving the nonlinear equations. (b) Similar measurements performed on a SOI waveguide
Fig. 3
Fig. 3 Simulation fit to measured data using the two-photon absorption (TPA) model and two-state absorption (TSA) model in a-Si:H waveguide.
Fig. 4
Fig. 4 Measured spectral broadening in a-Si:H waveguides at different coupled powers due to self-phase modulation and simulation fits based on solving nonlinear Schrödinger equations.
Fig. 5
Fig. 5 Measured spectral broadening in SOI waveguides at different coupled powers due to self-phase modulation and simulation fits based on solving nonlinear Schrödinger equations.

Tables (1)

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Table 1 PECVD deposition parameters for a-Si:H

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

u z + i β 2 2 2 u t 2 = ( i ω c n 2 β T P A 2 ) | u | 2 A e f f u N c 2 ( σ i ω c k c ) u α l 2 u
N c t = β T P A 2 h ω [ | u | 2 A e f f ] 2 N c τ c
σ = e 3 λ 2 4 π 2 c 3 ε 0 n 0 ( 1 m e 2 μ e + 1 m h 2 μ h )
F O M = n 2 β T P A λ

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