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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 22 — Oct. 24, 2011
  • pp: 21707–21716
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Phase-matched sum frequency generation in strained silicon waveguides using their second-order nonlinear optical susceptibility

Ivan Avrutsky and Richard Soref  »View Author Affiliations


Optics Express, Vol. 19, Issue 22, pp. 21707-21716 (2011)
http://dx.doi.org/10.1364/OE.19.021707


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Abstract

Using analysis and numerical simulation, we have investigated near-infrared and mid-infrared second-harmonic generation (SHG) and sum frequency generation (SFG) in crystal silicon (SOI) waveguides that possess a strong second-order nonlinear susceptibility by virtue of a Si3N4 straining layer applied directly to the top surface of the waveguide. This layer induces anisotropic compressive strain in the waveguide core. Using the technique of TE/TM mode birefringence, we have derived waveguide geometries for both slab and strip channel waveguides that offer perfect phase matching of three lightwaves for SHG/SFG along a uniform waveguide, thereby offering the prospect of efficient wavelength conversion in monolithic silicon photonics.

© 2011 OSA

1. Introduction

There are several applications of silicon-based group IV photonic circuits in which a strong second order susceptibility χ(2) is needed within the Si waveguide core material instead of having the nonlinearity provided by an organic material situated in a slot. Monolithic manufacturing is easier with the first approach. Recent experimental studies indicate that a crystalline silicon channel-waveguide structure will possess a large second-order optical nonlinearity when the waveguide core is anisotropically strained by a dielectric cladding layer (especially a silicon nitride film) applied to the top of the structure. Early experiments by Jacobsen et al [1

1. R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441(7090), 199–202 (2006). [CrossRef] [PubMed]

] and Fage-Pedersen at al [2

2. J. Fage-Pedersen, L. H. Frandsen, A. V. Lavrinenko, and P. I. Borel, “A linear electro-optic effect in silicon,” in IEEE 3rd International Conference on Group IV Photonics, 37-39 (2006).

] studied linear electrooptic (LEO) modulation in a silicon-on-insulator (SOI) strip waveguide initially coated with SiO2 and then covered with a straining Si3N4 layer. The modulation was linked to a second-order nonlinearity found to be χ(2) = 15 pm/V.

The Si3N4 layer was grown directly upon Si in subsequent experiments by other research groups. Bianco et al [3

3. F. Bianco, E. Borga, A. Yeremian, B. Dierre, K. Fedus, P. Bettoni, A. Pitanti, R. Pierbon, M. Ghulinyan, G. Pucker, M. Cazzanelli, and L. Pavesi, “Second-order susceptibility χ(2) in Si waveguides,” paper WB2, IEEE 8th International Conference on Group IV Photonics, London (13 September 2011).

] measured the efficiency of near-infrared second harmonic generation in 10-μm-wide SOI/Si3N4 strip waveguides and evaluated χ(2) to be 20 ± 10 pm/V. Their estimate requires further clarification because measurements were done with multimode waveguides, so it is not clear whether occasional close-to-phase-matching between several pump and harmonic guided modes contributed to the large measured second harmonic output. While the measured value of χ(2) might be somewhat questionable, this must be the first experimental report on second harmonic generation in a uniform silicon waveguide. Hon et al [4

4. N. K. Hon, K. K. Tsia, D. R. Solli, and B. Jalali, “Periodically-poled silicon,” Appl. Phys. Lett. 94(9), 091116 (2009). [CrossRef]

,5

5. N. K. Hon, K. K. Tsia, D. R. Solli, B. Jalali, and J. B. Khurgin, "Stress-induced χ(2) in silicon - comparison between theoretical and experimental values,” in IEEE 6th International Conference on Group IV Photonics, San Francisco, CA (9-11 September 2009).

] employed a periodic arrangement of stressing films on silicon for quasi-phase matched difference frequency generation. Significant experiments were performed by Chmielak et al [6

6. B. Chmielak, M. Waldow, C. Matheisen, C. Ripperda, J. Bolten, T. Wahlbrink, M. Nagel, F. Merget, and H. Kurz, “Pockels effect based fully integrated, strained silicon electro-optic modulator,” Opt. Express 19(18), 17212–17219 (2011). [CrossRef] [PubMed]

] on 400-nm-wide (100)-SOI/Si3N4 rib waveguides whose strain was asymmetric compression. Using Pockels-effect modulation and other tests, they found a record high value χ(2) = 122 pm/V, and the second-order susceptibility was largest at the side walls of the rib because those walls were pinned at the bottom. Their χ(2) does not apply directly to estimating the efficiency of mixing three lightwaves because one wave in LEO modulation is an RF field; nevertheless χ(2) will be very large in both situations. Knowing that, we can proceed with our investigation of phase-matched Si waveguides.

2. χ(2) and χ(3) techniques

The fabrication of the straining layer for χ(2) is generally easier than constructing the organic-in-slot for χ(2) because the former requires a simple deposition of dielectric while the latter requires e-beam fabrication of a nanoscale slot, followed by insertion of the organic, then followed by DC poling of the polymer in order to orient the molecules in the desired direction [8

8. T. Baehr-Jones, M. Hochberg, G. Wang, R. Lawson, Y. Liao, P. A. Sullivan, L. Dalton, A. K.-Y. Jen, and A. Scherer, “Optical modulation and detection in slotted Silicon waveguides,” Opt. Express 13(14), 5216–5226 (2005). [CrossRef] [PubMed]

]. In addition the organic approach may suffer from thermal stability issues that are generally not present in the straining layer method.

Crystalline silicon also possesses a fairly strong third order nonlinear optical susceptibility χ(3) that is also present in a non-uniformly strained silicon waveguide, although the strain modifies χ(3) somewhat. Thus both χ(2) and χ(3) are simultaneously present in such a waveguide. We can compare the relative contributions of χ(2) and χ(3) to the overall nonlinear optical susceptibility. At moderate levels of infrared pump intensity, we can say that the three-wave process of χ(2) will generally dominate over the four wave process of χ(3). Several research groups have been quite successful in using silicon’s χ(3) [9

9. J. I. Dadap, N. C. Panoiu, X. Chen, I.-W. Hsieh, X. Liu, C.-Y. Chou, E. Dulkeith, S. J. McNab, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. Osgood Jr., “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express 16(2), 1280–1299 (2008). [CrossRef] [PubMed]

] to attain wavelength conversion and optical parametric amplification via four wave mixing, although a special type of waveguide dispersion engineering was needed to attain phase matching in those cases. However, if a strained Si channel waveguide is optimized for SHG by the birefringence method reported in this paper, then the χ(3) phase matching will generally be spoiled (be negligible) since the χ(3) phase engineering is markedly different.

3. Phase matching for second harmonic generation

We shall examine the dispersion characteristics of an SOI waveguide in order to show that perfect phase matching in a uniform waveguide structure is both credible and feasible. To begin with, the waveguide is considered to be planar (a slab) and the cladding is assumed to have infinite thickness. Later in the paper, we examine a more a realistic structure in which both the waveguide width and the cladding thickness are finite (this is known as the strip channel waveguide). We note here that the planar waveguide approximation will be reasonably accurate for the strip if the aspect ratio of the strip’s guiding core remains high and if the guiding mode is localized in the core with little penetration into the cladding. In this section of the paper, we determine the optimal thickness of a slab waveguide at which phase synchronism is achieved. For a strip, the optimal thickness found previously will be somewhat modified when the finite core width and finite cladding thickness are taken into account. The strip channel is investigated in Section 5 by analytic formulae that separate the vertical and horizontal mode solutions. As mentioned, the strip response is indeed linked to the slab waveguide behavior.

For purposes of description, we introduce a Cartesian coordinate system in which the Z axis is the direction of lightwave propagation (and where Z is also the longitudinal axis of a strip channel waveguide). The waveguide’s TE modes have their E field polarized along the X axis parallel to the plane of a slab waveguide, with X being the “lateral” direction. The TM modes have E field polarized in the Y direction perpendicular to the slab, with Y being the “vertical” direction. For the strained-silicon core region, TE is polarized typically in the Si [110] crystallographic lattice direction, while TM is along the [001] lattice direction [6

6. B. Chmielak, M. Waldow, C. Matheisen, C. Ripperda, J. Bolten, T. Wahlbrink, M. Nagel, F. Merget, and H. Kurz, “Pockels effect based fully integrated, strained silicon electro-optic modulator,” Opt. Express 19(18), 17212–17219 (2011). [CrossRef] [PubMed]

]. The waveguide structure in which a straining layer is placed on top of a thick SiO2 cladding is expected to provide smaller strain and consequently smaller nonlinearity than a structure that has Si3N4 directly on top of the Si core. For this reason, the waveguides studied here are Si3N4/Si/SiO2 asymmetric planar waveguides, where the SiO2 is the buried oxide in SOI.

The refractive indices as a function of wavelength (λ, expressed in micrometers) for Si, SiO2, and Si3N4 are assumed to be as follows:

nSi(λ)=1+10.6684293λ2λ20.3015164852+0.003043475λ2λ21.134751152+1.54133408λ2λ21104.02,
(1)
nSiO2(λ)=1+0.6961663λ2λ20.06840432+0.4079426λ2λ20.11624142+0.8974794λ2λ29.8961612,
(2)
nSi3N4(λ)=4+2.7λ2λ212.02.
(3)

A convenient instrument to study second order nonlinear processes is second harmonic generation. For the asymmetric material system, a suitable pump (subscript p) wavelength could be close to λp = 3μm (for instance, an Er:YAG laser operating at 2.94μm) with second harmonic signal (subscript s) at the wavelength close to λs = 1.5μm. Wavelengths longer than 3.5μm would not be appropriate because of the absorption in silica. Another choice, as it is done in [3

3. F. Bianco, E. Borga, A. Yeremian, B. Dierre, K. Fedus, P. Bettoni, A. Pitanti, R. Pierbon, M. Ghulinyan, G. Pucker, M. Cazzanelli, and L. Pavesi, “Second-order susceptibility χ(2) in Si waveguides,” paper WB2, IEEE 8th International Conference on Group IV Photonics, London (13 September 2011).

] is λp ≈2.3μm with the second harmonic at the short-wavelength edge of silicon’s transparency region, λs = 1.15μm. This defines the wavelength range of interest: 2.3μm < λp < 3.5μm.

In the symmetric SiO2/Si/SiO2 waveguide, compared to the Si3N4/Si/SiO2 structure, similar analysis leads to the optimal core thickness being about 16% larger. In another symmetric structure, Si3N4/Si/Si3N4 with both claddings made of silicon nitride, the optimal core thickness is approximately 17% smaller compared to the Si3N4/Si/SiO2 structure. With both claddings made of silicon nitride, the strain in the core may be larger and the waveguide transparency range is wider because it extends from 1.15 μm to 6.7 μm.

If the core/cladding interfaces are smooth and the cladding materials do not have many defects, then the mode tailing we have described does not necessarily imply high propagation losses for the guided waves.

Accounting for the finite width of a strip waveguide and the finite thickness of the nitride layer will modify the dispersion curves. However, it is appealing that widely used SOI structures with core layer (top Si) thickness close to 200nm, when covered by a thick straining Si3N4 layer, are suitable for perfectly phase-matched second harmonic generation with pump wavelength around 3.0μm.

4. Phase matching for sum-frequency generation

As all the wavelengths need to be within the transparency range of the particular SOI material system (between λa = 1.15μm and λb = 3.5μm), this limits possible values of the parameter a:
λaλb=amin<a<amax=λbλaλb,
(7)
and, for each given value of a, the signal wavelength is limited to

λa<λs<(1a)λb,ifa12,
(8)
λa<λs<aλb,ifa<12
(9)

In this particular example amax ≈0.671 and amin ≈0.329. Limitations for the wavelengths are illustrated schematically in Fig. 3
Fig. 3 Limitations for the choice of wavelength λ1 and λ2 for sum- frequency generation: shadowed area shows allowed wavelengths. The relation between the wavelengths for a given value of the parameter a is illustrated by dashed lines.
.

Equation (5) now can be used to find the core layer thickness to provide the phase matching in the sum frequency generation process for a given combination of the parameter a and the signal wavelength λs (Fig. 4
Fig. 4 Optimal core thickness d versus SFG signal wavelength λs for several representative values of a = 0.35, 0.37, 0.40, 0.44, 0.50. Polarizations: TE for both pump waves and TM for the signal at sum frequency.
). Here we assume TE polarizations for the pump waves and TM for the signal. Sum-frequency generation includes the above-considered second-harmonic generation as a particular case in which a = ½.

With both pump waves having identical polarizations, there is degeneration between the cases corresponding to values a and (1 – a). For this reason, Fig. 4 shows only the cases of a ≤ ½.

5. Corrections for the finite width of the waveguide

For the strip width greater than 3μm the squared overlap integral is scaled approximately as 1/w. The overlap reaches a maximal value of I2 = 0.193μm−2 at approximately w = 2.5μm. At this strip width the optimal core thickness is d = 196nm, so that wdI2 = 0.095.

6. Conclusions

In conclusion, we show that a uniform waveguide in the Si3N4/Si/SiO2 material system can be designed to provide the perfect phase matching condition for second harmonic generation and sum frequency generation. In the presence of strain-induced second-order nonlinearity, such waveguides can be used in various on-chip integrated silicon photonic devices. The results here provide a foundation for second-order optical parametric oscillation in silicon.

Acknowledgments

This work is supported in part by the Air Force Office of Scientific Research, Gernot Pomrenke, Program Manager, under Grant Number FA9550-10-1-0417.

References and links

1.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441(7090), 199–202 (2006). [CrossRef] [PubMed]

2.

J. Fage-Pedersen, L. H. Frandsen, A. V. Lavrinenko, and P. I. Borel, “A linear electro-optic effect in silicon,” in IEEE 3rd International Conference on Group IV Photonics, 37-39 (2006).

3.

F. Bianco, E. Borga, A. Yeremian, B. Dierre, K. Fedus, P. Bettoni, A. Pitanti, R. Pierbon, M. Ghulinyan, G. Pucker, M. Cazzanelli, and L. Pavesi, “Second-order susceptibility χ(2) in Si waveguides,” paper WB2, IEEE 8th International Conference on Group IV Photonics, London (13 September 2011).

4.

N. K. Hon, K. K. Tsia, D. R. Solli, and B. Jalali, “Periodically-poled silicon,” Appl. Phys. Lett. 94(9), 091116 (2009). [CrossRef]

5.

N. K. Hon, K. K. Tsia, D. R. Solli, B. Jalali, and J. B. Khurgin, "Stress-induced χ(2) in silicon - comparison between theoretical and experimental values,” in IEEE 6th International Conference on Group IV Photonics, San Francisco, CA (9-11 September 2009).

6.

B. Chmielak, M. Waldow, C. Matheisen, C. Ripperda, J. Bolten, T. Wahlbrink, M. Nagel, F. Merget, and H. Kurz, “Pockels effect based fully integrated, strained silicon electro-optic modulator,” Opt. Express 19(18), 17212–17219 (2011). [CrossRef] [PubMed]

7.

I. Avrutsky, R. Soref, and W. Buchwald, “Mid-infrared optical parametric oscillators based on uniform GaP waveguides,” Opt. Express 18(19), 20370–20383 (2010). [CrossRef] [PubMed]

8.

T. Baehr-Jones, M. Hochberg, G. Wang, R. Lawson, Y. Liao, P. A. Sullivan, L. Dalton, A. K.-Y. Jen, and A. Scherer, “Optical modulation and detection in slotted Silicon waveguides,” Opt. Express 13(14), 5216–5226 (2005). [CrossRef] [PubMed]

9.

J. I. Dadap, N. C. Panoiu, X. Chen, I.-W. Hsieh, X. Liu, C.-Y. Chou, E. Dulkeith, S. J. McNab, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. Osgood Jr., “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express 16(2), 1280–1299 (2008). [CrossRef] [PubMed]

10.

H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuits (McGraw-Hill, 1989), p. 31.

11.

T. K. Lim and H. J. Melchior, “Effective index method for generalized waveguide dispersion characteristics analysis of optical channel waveguides,” Electron. Lett. 27(11), 917–918 (1991). [CrossRef]

OCIS Codes
(130.3130) Integrated optics : Integrated optics materials
(190.0190) Nonlinear optics : Nonlinear optics

ToC Category:
Nonlinear Optics

History
Original Manuscript: August 25, 2011
Revised Manuscript: September 29, 2011
Manuscript Accepted: September 29, 2011
Published: October 19, 2011

Citation
Ivan Avrutsky and Richard Soref, "Phase-matched sum frequency generation in strained silicon waveguides using their second-order nonlinear optical susceptibility," Opt. Express 19, 21707-21716 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-21707


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References

  1. R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature441(7090), 199–202 (2006). [CrossRef] [PubMed]
  2. J. Fage-Pedersen, L. H. Frandsen, A. V. Lavrinenko, and P. I. Borel, “A linear electro-optic effect in silicon,” in IEEE 3rd International Conference on Group IV Photonics, 37-39 (2006).
  3. F. Bianco, E. Borga, A. Yeremian, B. Dierre, K. Fedus, P. Bettoni, A. Pitanti, R. Pierbon, M. Ghulinyan, G. Pucker, M. Cazzanelli, and L. Pavesi, “Second-order susceptibility χ(2) in Si waveguides,” paper WB2, IEEE 8th International Conference on Group IV Photonics, London (13 September 2011).
  4. N. K. Hon, K. K. Tsia, D. R. Solli, and B. Jalali, “Periodically-poled silicon,” Appl. Phys. Lett.94(9), 091116 (2009). [CrossRef]
  5. N. K. Hon, K. K. Tsia, D. R. Solli, B. Jalali, and J. B. Khurgin, "Stress-induced χ(2) in silicon - comparison between theoretical and experimental values,” in IEEE 6th International Conference on Group IV Photonics, San Francisco, CA (9-11 September 2009).
  6. B. Chmielak, M. Waldow, C. Matheisen, C. Ripperda, J. Bolten, T. Wahlbrink, M. Nagel, F. Merget, and H. Kurz, “Pockels effect based fully integrated, strained silicon electro-optic modulator,” Opt. Express19(18), 17212–17219 (2011). [CrossRef] [PubMed]
  7. I. Avrutsky, R. Soref, and W. Buchwald, “Mid-infrared optical parametric oscillators based on uniform GaP waveguides,” Opt. Express18(19), 20370–20383 (2010). [CrossRef] [PubMed]
  8. T. Baehr-Jones, M. Hochberg, G. Wang, R. Lawson, Y. Liao, P. A. Sullivan, L. Dalton, A. K.-Y. Jen, and A. Scherer, “Optical modulation and detection in slotted Silicon waveguides,” Opt. Express13(14), 5216–5226 (2005). [CrossRef] [PubMed]
  9. J. I. Dadap, N. C. Panoiu, X. Chen, I.-W. Hsieh, X. Liu, C.-Y. Chou, E. Dulkeith, S. J. McNab, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. Osgood., “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express16(2), 1280–1299 (2008). [CrossRef] [PubMed]
  10. H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuits (McGraw-Hill, 1989), p. 31.
  11. T. K. Lim and H. J. Melchior, “Effective index method for generalized waveguide dispersion characteristics analysis of optical channel waveguides,” Electron. Lett.27(11), 917–918 (1991). [CrossRef]

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