## Phase-matched sum frequency generation in strained silicon waveguides using their second-order nonlinear optical susceptibility |

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 Si_{3}N_{4} 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

*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]

*at al*[2] studied linear electrooptic (LEO) modulation in a silicon-on-insulator (SOI) strip waveguide initially coated with SiO

_{2}and then covered with a straining Si

_{3}N

_{4}layer. The modulation was linked to a second-order nonlinearity found to be

*χ*

^{(2)}= 15 pm/V.

_{3}N

_{4}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 8 ^{th} International Conference on Group IV Photonics*, London (13 September 2011).

_{3}N

_{4}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]

*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]

_{3}N

_{4}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.

*TE*and

*TM*modes (the mode birefringence) to attain the desired synchrony of waves. In a previous publication [7

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]

## 2. *χ*^{(2)} and *χ*^{(3)} techniques

*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]

*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]

*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

*χ*

^{(3)}-induced phenomena such as self-phase modulation and cross-phase modulation. They effectively change the modal indices for the waves involved in the

*χ*

^{(2)}interactions and thus affect the phase-matching condition. At present, we ignore the corrections to the phase-matching conditions due to self-phase and cross-phase modulation. Most likely, these corrections are tiny and would only be needed for fine tuning of the model when system operates with extremely high power densities.

*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]

_{2}cladding is expected to provide smaller strain and consequently smaller nonlinearity than a structure that has Si

_{3}N

_{4}directly on top of the Si core. For this reason, the waveguides studied here are Si

_{3}N

_{4}/Si/SiO

_{2}asymmetric planar waveguides, where the SiO

_{2}is the buried oxide in SOI.

*λ*, expressed in micrometers) for Si, SiO

_{2}, and Si

_{3}N

_{4}are assumed to be as follows:

*p*) wavelength could be close to

*λ*= 3μm (for instance, an Er:YAG laser operating at 2.94μm) with second harmonic signal (subscript

_{p}*s*) at the wavelength close to

*λ*= 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

_{s}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 8 ^{th} International Conference on Group IV Photonics*, London (13 September 2011).

*λ*≈2.3μm with the second harmonic at the short-wavelength edge of silicon’s transparency region,

_{p}*λ*= 1.15μm. This defines the wavelength range of interest: 2.3μm <

_{s}*λ*< 3.5μm.

_{p}*TE*and

*TM*fundamental modes at the fundamental (

*λ*= 2.94μm) and second harmonic wavelengths (

_{p}*λ*= 1.47μm) as a function of the silicon core layer thickness

_{s}*d*in the Si

_{3}N

_{4}/Si/SiO

_{2}planar waveguide structure. In the absence of material dispersion, the modal indices would be functions of

*d*/

*λ*, so that the dispersion curves at different wavelengths would be appropriately scaled copies of one another.

*TE*mode always has a higher modal index, so that the dispersion curves never intersect. However, dispersion curves for the

*TM*mode at a shorter wavelength and the

*TE*mode at a longer wavelength may indeed intersect. In this particular case, intersection happens at the value of core thickness of

*d*= 212nm.

_{2}/Si/SiO

_{2}waveguide, compared to the Si

_{3}N

_{4}/Si/SiO

_{2}structure, similar analysis leads to the optimal core thickness being about 16% larger. In another symmetric structure, Si

_{3}N

_{4}/Si/Si

_{3}N

_{4}with both claddings made of silicon nitride, the optimal core thickness is approximately 17% smaller compared to the Si

_{3}N

_{4}/Si/SiO

_{2}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.

*TE*-polarized and the second harmonic wave

*TM*-polarized. The corresponding field profiles are illustrated in Fig. 2 . While the modal fields show penetration into the claddings, a significant fraction of the profiles remains within the Si core. The core confines 46.4% of

*χ*

^{(2)}that can be effectively accessed in SHG. However, for the process of guided wave SHG, the appropriately defined overlap integral is not dimensionless and its interpretation requires some additional considerations (see, section 5).

_{3}N

_{4}layer, are suitable for perfectly phase-matched second harmonic generation with pump wavelength around 3.0μm.

## 4. Phase matching for sum-frequency generation

*λ*

_{1}and

*λ*

_{2}in this case generate the signal at wavelength

*λ*to satisfy the energy conservationand the momentum conservation that could be written in a formwhereand

_{s}*n*(

*λ*) is the modal index at wavelength

*λ*. The parameter

*a*, similar to the case of optical parametric oscillator considered in [7

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]

*ℰ*

_{1}and

*ℰ*

_{2}of light quanta for the waves with wavelengths

*λ*

_{1}and

*λ*

_{2}, namely

*ℰ*

_{1}=

*aℰ*and

_{s}*ℰ*

_{2}= (1 –

*a*)ℰ

*, where*

_{s}*ℰ*is the quantum energy for the signal at the sum frequency. Polarizations of the waves participating in the sum frequency generation are not shown explicitly here.

_{s}*λ*= 1.15μm and

_{a}*λ*= 3.5μm), this limits possible values of the parameter

_{b}*a*:and, for each given value of

*a*, the signal wavelength is limited to

*a*

_{max}≈0.671 and

*a*

_{min}≈0.329. Limitations for the wavelengths are illustrated schematically in Fig. 3 .

*a*and the signal wavelength

*λ*(Fig. 4 ). Here we assume

_{s}*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.

*a*and (1 –

*a*). For this reason, Fig. 4 shows only the cases of

*a*≤ ½.

*TE*polarized or only one of them could be

*TE*and the other

*TM*, while the sum frequency waves in both cases are

*TM*polarized. Other combinations of polarizations are difficult to achieve because both material and waveguide dispersion as a rule result in lower index for longer wavelengths, and the modal index for a

*TM*guided mode in a slab waveguide is lower than the index for a

*TE*mode of the same order. Some additional efforts in engineering the guided wave dispersion may help overcome this limitation, but this issue is beyond the scope of the article.

## 5. Corrections for the finite width of the waveguide

*X*leads to some corrections for the modal indices and thus for the optimal

*Y*thickness of the core required to achieve the phase matching. Rigorous calculation will require numerical methods that might be computationally intensive. Qualitatively, the corrections due to the finite width of the waveguide can be estimated using the effective index method [10, 11

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]

_{3}N

_{4}/Si/SiO

_{2}material system so that the straining Si

_{3}N

_{4}layer extends far enough beyond the Si ridge. This nitride arrangement implies – within the effective index model – that in the lateral direction the claddings have the index of silicon nitride. Thus, in the

*X*direction, we assume a Si

_{3}N

_{4}/Effective_Medium/Si

_{3}N

_{4}system, whereas in the

*Y*direction we have an effective medium whose effective index is that of our previously studied Si

_{3}N

_{4}/Si/SiO

_{2}slab waveguide. Therefore to evaluate the modal index in the strip channel structure for the mode polarized parallel (or orthogonal) to the plane of the wafer, formulas for

*TM*(or

*TE*) modes are applied to the in-plane Si

_{3}N

_{4}/Effective_Medium/Si

_{3}N

_{4}system, where the effective medium is assumed to have the refractive index equal to the effective index of the

*TE*(or

*TM*) mode in the Si

_{3}N

_{4}/Si/SiO

_{2}slab waveguide. We have done this and have investigated SHG for the Si strip by deriving dispersion curves for the finite structure.

*TE*-polarized pump at the wavelength of 2.94μm and

*TM*-polarized second harmonic are both equal to 2.26.

*ε*

_{0}is vacuum permittivity,

*c*is speed of light,

*x*and

*y*are the transverse coordinates in the directions parallel to and orthogonal to the substrate, and the

*E*-fields of harmonic and pump are employed. The overlap integral, by definition, is determined by the field distributions only (11). It takes into account that nonlinearity is present only in the core. The core nonlinearity and some other physical parameters of the system under consideration then define the normalized efficiency of SHG (10). With the

*E*-fields normalized so thatthe fields are measured in units inverse to the unit of length, say, 1/μm, and the squared overlap integral

*I*

^{2}in Eq. (10) is measured in units of 1/μm

^{2}. In an unrealistic case of modal fields completely confined within the core and uniformly filling the core cross section, the squared overlap integral

*I*would be equal to 1/

^{2}*wd.*Thus in real-world structures, the product

*wdI*

^{2}characterizes the degree of mode confinement in the guided-wave second harmonic generation. The overlap integral squared for the case of second harmonic generation with pump at

*λ*= 2.94μm, is shown in Fig. 6 at the left.

_{p}*w*. The overlap reaches a maximal value of

*I*

^{2}= 0.193μm

^{−2}at approximately

*w*= 2.5μm. At this strip width the optimal core thickness is

*d*= 196nm, so that

*wdI*

^{2}= 0.095.

*w*= 2.5μm and

*d*= 196nm are shown in Fig. 6 at the right. Field penetration beyond the core is responsible for the product

*wdI*

^{2}being well below unity. Nevertheless, the core cross-section in this case is a small fraction of the pump wavelength squared,

*wd*≈0.057

*λ*

_{p}^{2}, which indicates strong mode confinement. This and high expected valued of the core nonlinearity

*χ*

^{(2)}= 122 pm/V, in accordance with (10), lead to a quite impressive normalized efficiency of second harmonic generation of 4,850% per Watt⋅cm

^{2}.

## 6. Conclusions

_{3}N

_{4}/Si/SiO

_{2}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

## 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 |

2. | J. Fage-Pedersen, L. H. Frandsen, A. V. Lavrinenko, and P. I. Borel, “A linear electro-optic effect in silicon,” in |

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 |

4. | N. K. Hon, K. K. Tsia, D. R. Solli, and B. Jalali, “Periodically-poled silicon,” Appl. Phys. Lett. |

5. | N. K. Hon, K. K. Tsia, D. R. Solli, B. Jalali, and J. B. Khurgin, "Stress-induced |

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 |

7. | I. Avrutsky, R. Soref, and W. Buchwald, “Mid-infrared optical parametric oscillators based on uniform GaP waveguides,” Opt. Express |

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 |

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 |

10. | H. Nishihara, M. Haruna, and T. Suhara, |

11. | T. K. Lim and H. J. Melchior, “Effective index method for generalized waveguide dispersion characteristics analysis of optical channel waveguides,” Electron. Lett. |

**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

- 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]
- 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).
- 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).
- N. K. Hon, K. K. Tsia, D. R. Solli, and B. Jalali, “Periodically-poled silicon,” Appl. Phys. Lett.94(9), 091116 (2009). [CrossRef]
- 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).
- 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]
- 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]
- 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]
- 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]
- H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuits (McGraw-Hill, 1989), p. 31.
- 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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