## Harmonic generation in silicon nitride ring resonators |

Optics Express, Vol. 19, Issue 12, pp. 11415-11421 (2011)

http://dx.doi.org/10.1364/OE.19.011415

Acrobat PDF (1075 KB)

### Abstract

We demonstrate second- and third-harmonic generation in a centrosymmetric CMOS-compatible material using ring resonators and integrated optical waveguides. The *χ*^{(2)} response is induced by using the nanoscale structure of the waveguide to break the bulk symmetry of silicon nitride (Si_{3}N_{4}) with the silicon dioxide (SiO_{2}) cladding. Using a high-*Q* ring resonator cavity to enhance the efficiency of the process, we detect the second-harmonic output in the visible wavelength range with milliwatt input powers at telecom wavelengths. We also observe third-harmonic generation from the intrinsic *χ*^{(3)} susceptibility of the silicon nitride. Phase matching of the harmonic processes occurs due to the near coincidence of indices of refraction of the fundamental mode at the pump frequency and the corresponding higher-order modes of the harmonic fields.

© 2011 OSA

## 1. Introduction

1. M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature **441**(7096), 960–963 (2006). [CrossRef] [PubMed]

2. B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O'Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics **3**(4), 206–210 (2009). [CrossRef]

3. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. **93**(8), 083904 (2004). [CrossRef] [PubMed]

4. T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third-harmonic generation,” Nat. Phys. **3**(6), 430–435 (2007). [CrossRef]

5. J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics **4**(1), 37–40 (2010). [CrossRef]

6. K. Ikeda, R. E. Saperstein, N. Alic, and Y. Fainman, “Thermal and Kerr nonlinear properties of plasma-deposited silicon nitride/ silicon dioxide waveguides,” Opt. Express **16**(17), 12987–12994 (2008). [CrossRef] [PubMed]

^{(2)}vanishes in these centrosymmetric materials. Generating the lowest-order nonlinearity enables a new array of CMOS-compatible optical devices capable of nonlinear functionalities such as electro-optic modulation, sum frequency up-conversion, and difference frequency generation.

_{3}N

_{4}) is a centrosymmetric, CMOS compatible material shown to be useful for integrated optics and to possess interesting nonlinear properties [5

5. J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics **4**(1), 37–40 (2010). [CrossRef]

7. D. T. H. Tan, K. Ikeda, P. C. Sun, and Y. Fainman, “Group velocity dispersion and self phase modulation in silicon nitride waveguides,” Appl. Phys. Lett. **96**(6), 061101–061103 (2010). [CrossRef]

*χ*

^{(2)}response utilizing the interface between two centrosymmetric materials, the Si

_{3}N

_{4}core and the silicon dioxide (SiO

_{2}) cladding. The waveguide interface breaks the bulk symmetry and a second-order nonlinear response can arise [8

8. N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. **174**(3), 813–822 (1968). [CrossRef]

11. Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature **337**(6207), 519–525 (1989). [CrossRef]

9. H. W. K. Tom, T. F. Heinz, and Y. R. Shen, “Second-Harmonic Reflection from Silicon Surfaces and Its Relation to Structural Symmetry,” Phys. Rev. Lett. **51**(21), 1983–1986 (1983). [CrossRef]

11. Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature **337**(6207), 519–525 (1989). [CrossRef]

12. S. Lettieri, S. D. Finizio, P. Maddalena, V. Ballarini, and F. Giorgis, “Second-harmonic generation in amorphous silicon nitride microcavities,” Appl. Phys. Lett. **81**(25), 4706–4708 (2002). [CrossRef]

13. S. Lettieri, F. Gesuele, P. Maddalena, M. Liscidini, L. C. Andreani, C. Ricciardi, V. Ballarini, and F. Giorgis, “Second-harmonic generation in hydrogenated amorphous-Si _{1 - x}N _{x} doubly resonant microcavities with periodic dielectric mirrors,” Appl. Phys. Lett. **87**(19), 191110 (2005). [CrossRef]

^{(3)}of Si

_{3}N

_{4}, we also measure the generated third-harmonic (TH) in the integrated cavities. Previously silica microtoroids on silicon [4

4. T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third-harmonic generation,” Nat. Phys. **3**(6), 430–435 (2007). [CrossRef]

2. B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O'Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics **3**(4), 206–210 (2009). [CrossRef]

## 2. Design

_{3}N

_{4}ring resonators with both high

*Q*and high modal confinement [14

14. A. Gondarenko, J. S. Levy, and M. Lipson, “High confinement micron-scale silicon nitride high Q ring resonator,” Opt. Express **17**(14), 11366–11370 (2009). [CrossRef] [PubMed]

5. J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics **4**(1), 37–40 (2010). [CrossRef]

15. A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express **16**(7), 4881–4887 (2008). [CrossRef] [PubMed]

16. M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photonics **2**(12), 737–740 (2008). [CrossRef]

**4**(1), 37–40 (2010). [CrossRef]

14. A. Gondarenko, J. S. Levy, and M. Lipson, “High confinement micron-scale silicon nitride high Q ring resonator,” Opt. Express **17**(14), 11366–11370 (2009). [CrossRef] [PubMed]

_{2}, in both microtoroids [4

4. T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third-harmonic generation,” Nat. Phys. **3**(6), 430–435 (2007). [CrossRef]

17. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Optical properties of high-delta air silica microstructure optical fibers,” Opt. Lett. **25**(11), 796–798 (2000). [CrossRef]

18. F. G. Omenetto, A. J. Taylor, M. D. Moores, J. Arriaga, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, “Simultaneous generation of spectrally distinct third harmonics in a photonic crystal fiber,” Opt. Lett. **26**(15), 1158–1160 (2001). [CrossRef]

## 3. Experimental results

### 3.1 Second-harmonic conversion

*λ*is tuned into the resonance of the ring cavity near 1554 nm and, with suitable power levels, we are able to observe generation of the second harmonic (Fig. 2(a) ). We use a spectrometer to measure the wavelength of the visible emitted light. Figure 2(b) shows the output with the generated SH wavelength

_{P}*λ*measured to be 777.1 nm which, as expected, is

_{SH}*λ*/2. Since only the SH wavelength is detected, we conclude we are not generating broadband photoluminescence which has been previously described in silicon nitride [19

_{P}19. A. Serpengüzel and S. Tanriseven, “Controlled photoluminescence in amorphous-silicon-nitride microcavities,” Appl. Phys. Lett. **78**(10), 1388–1390 (2001). [CrossRef]

20. M. Barth, N. Nusse, J. Stingl, B. Lochel, and O. Benson, “Emission properties of high-Q silicon nitride photonic crystal heterostructure cavities,” Appl. Phys. Lett. **93**(2), 021112–021113 (2008). [CrossRef]

### 3.2 Modal profile image

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

### 3.3 Third-harmonic generation

*χ*

^{(3)}nonlinearity of the silicon nitride induces the third order process in which three pump photons generate a single photon at the third-harmonic frequency. Efficient THG requires meeting the same phase-matching condition as SHG: the effective index of the pump must equal the effective index at the TH wavelength. By performing similar simulations as before we find the fundamental mode of the pump is closest in refractive index to the 18th order mode at the third-harmonic wavelength.

2. B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O'Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics **3**(4), 206–210 (2009). [CrossRef]

## 4. Analysis

^{(2)}to be as large as 4 x 10

^{−14}m/V from the conversion efficiency observed in the ring. For SHG with an undepleted pump, the expected signal power may be calculated for a given pump intensity and propagation distance by solving the coupled amplitude equations [22]. Since we are using a resonator, the intensity of the pump and SH are increased by the respective cavity enhancement effects of the ring. In order to accurately model the nonlinear susceptibility, we take into account the finesse of the cavity, the simulated modal field overlap and phase-mismatch, and the radius of the ring. The power

*P*for second harmonic wave is then given by:

_{sh}*ω*is the pump frequency,

_{p}*ε*

_{0}is the permittivity of free space,

*c*is the speed of light in vacuum,

*n*is the effective index for the modes,

_{i}*A*

_{i}is the mode area, Δ

*k*is the phase mismatch,

*L*is the ring circumference and

*P*is the pump power in the waveguide. The modal overlap integral between the fundamental and second-harmonic fields is accounted for by the function

_{p}*f*(

*A*,

_{p}*A*).

_{sh}*C*takes into account the circulating power in the ring [15

_{i}15. A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express **16**(7), 4881–4887 (2008). [CrossRef] [PubMed]

*κ*and

*τ*represent the coupling parameters from the waveguide to the ring,

*α*is the propagation loss in the ring and

*k*is the wavenumber. We can directly measure

_{i}*C*from the transmission spectrum of the pump resonance and calculate this to be approximately 156. For the SH resonance we are unable to directly characterize the intrinsic and coupling

_{p}*Q*’s. We estimate that

*C*is at most

_{sh}*C*and at least unity. From this approximation we come up with a range of potential values for the effective χ

_{p}^{(2)}3 x 10

^{−15}to 4 x 10

^{−14}m/V. We believe the actual enhancement at the SH wavelength is closer to unity than

*C*and therefore the effective χ

_{p}^{(2)}to be on the larger side of the approximation range. From Eq. (1), we see that the SH wave has a quadratic dependence on the pump power and nonlinear susceptibility

*χ*

^{(2)}. In order to clearly demonstrate the theoretical square dependence of the SH process, we plot the dropped pump power against the generated SH on a log-log scale (Fig. 5 ) and calculate a best fit slope of 1.9745 ± 0.0225. From the measured output power values we estimate our induced χ

^{(2)}. The integrated high finesse resonator in silicon nitride increases the efficiency of the SHG significantly when the pump and SH are both resonant.

*χ*

^{(3)}response in silicon nitride has been previously examined [5

**4**(1), 37–40 (2010). [CrossRef]

7. D. T. H. Tan, K. Ikeda, P. C. Sun, and Y. Fainman, “Group velocity dispersion and self phase modulation in silicon nitride waveguides,” Appl. Phys. Lett. **96**(6), 061101–061103 (2010). [CrossRef]

*n*

_{2}, through the Kerr-shift, self-phase modulation and four-wave mixing respectively. Like previous work involving SiO

_{2}resonators [4

**3**(6), 430–435 (2007). [CrossRef]

*χ*

^{(2)}could be a lower bound since we assume ideal phase-matching, as suggested by our simulations, in this approximation.

## 5. Conclusions

23. S. F. Pereira, M. Xiao, H. J. Kimble, and J. L. Hall, “Generation of squeezed light by intracavity frequency doubling,” Phys. Rev. A **38**(9), 4931–4934 (1988). [CrossRef] [PubMed]

24. N. K. Hon, K. K. Tsia, D. R. Solli, and B. Jalali, “Periodically poled silicon,” Appl. Phys. Lett. **94**(9), 091116–091113 (2009). [CrossRef]

## Acknowledgements

## References and links

1. | M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature |

2. | B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O'Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics |

3. | T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. |

4. | T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third-harmonic generation,” Nat. Phys. |

5. | J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics |

6. | K. Ikeda, R. E. Saperstein, N. Alic, and Y. Fainman, “Thermal and Kerr nonlinear properties of plasma-deposited silicon nitride/ silicon dioxide waveguides,” Opt. Express |

7. | D. T. H. Tan, K. Ikeda, P. C. Sun, and Y. Fainman, “Group velocity dispersion and self phase modulation in silicon nitride waveguides,” Appl. Phys. Lett. |

8. | N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. |

9. | H. W. K. Tom, T. F. Heinz, and Y. R. Shen, “Second-Harmonic Reflection from Silicon Surfaces and Its Relation to Structural Symmetry,” Phys. Rev. Lett. |

10. | J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B Condens. Matter |

11. | Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature |

12. | S. Lettieri, S. D. Finizio, P. Maddalena, V. Ballarini, and F. Giorgis, “Second-harmonic generation in amorphous silicon nitride microcavities,” Appl. Phys. Lett. |

13. | S. Lettieri, F. Gesuele, P. Maddalena, M. Liscidini, L. C. Andreani, C. Ricciardi, V. Ballarini, and F. Giorgis, “Second-harmonic generation in hydrogenated amorphous-Si |

14. | A. Gondarenko, J. S. Levy, and M. Lipson, “High confinement micron-scale silicon nitride high Q ring resonator,” Opt. Express |

15. | A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express |

16. | M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photonics |

17. | J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Optical properties of high-delta air silica microstructure optical fibers,” Opt. Lett. |

18. | F. G. Omenetto, A. J. Taylor, M. D. Moores, J. Arriaga, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, “Simultaneous generation of spectrally distinct third harmonics in a photonic crystal fiber,” Opt. Lett. |

19. | A. Serpengüzel and S. Tanriseven, “Controlled photoluminescence in amorphous-silicon-nitride microcavities,” Appl. Phys. Lett. |

20. | M. Barth, N. Nusse, J. Stingl, B. Lochel, and O. Benson, “Emission properties of high-Q silicon nitride photonic crystal heterostructure cavities,” Appl. Phys. Lett. |

21. | V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Opt. Lett. |

22. | R. W. Boyd, |

23. | S. F. Pereira, M. Xiao, H. J. Kimble, and J. L. Hall, “Generation of squeezed light by intracavity frequency doubling,” Phys. Rev. A |

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

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4400) Nonlinear optics : Nonlinear optics, materials

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: April 28, 2011

Revised Manuscript: May 22, 2011

Manuscript Accepted: May 23, 2011

Published: May 26, 2011

**Citation**

Jacob S. Levy, Mark A. Foster, Alexander L. Gaeta, and Michal Lipson, "Harmonic generation in silicon nitride ring resonators," Opt. Express **19**, 11415-11421 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-12-11415

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### References

- M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441(7096), 960–963 (2006). [CrossRef] [PubMed]
- B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O'Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3(4), 206–210 (2009). [CrossRef]
- T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93(8), 083904 (2004). [CrossRef] [PubMed]
- T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third-harmonic generation,” Nat. Phys. 3(6), 430–435 (2007). [CrossRef]
- J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010). [CrossRef]
- K. Ikeda, R. E. Saperstein, N. Alic, and Y. Fainman, “Thermal and Kerr nonlinear properties of plasma-deposited silicon nitride/ silicon dioxide waveguides,” Opt. Express 16(17), 12987–12994 (2008). [CrossRef] [PubMed]
- D. T. H. Tan, K. Ikeda, P. C. Sun, and Y. Fainman, “Group velocity dispersion and self phase modulation in silicon nitride waveguides,” Appl. Phys. Lett. 96(6), 061101–061103 (2010). [CrossRef]
- N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. 174(3), 813–822 (1968). [CrossRef]
- H. W. K. Tom, T. F. Heinz, and Y. R. Shen, “Second-Harmonic Reflection from Silicon Surfaces and Its Relation to Structural Symmetry,” Phys. Rev. Lett. 51(21), 1983–1986 (1983). [CrossRef]
- J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B Condens. Matter 35(3), 1129–1141 (1987). [CrossRef] [PubMed]
- Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337(6207), 519–525 (1989). [CrossRef]
- S. Lettieri, S. D. Finizio, P. Maddalena, V. Ballarini, and F. Giorgis, “Second-harmonic generation in amorphous silicon nitride microcavities,” Appl. Phys. Lett. 81(25), 4706–4708 (2002). [CrossRef]
- S. Lettieri, F. Gesuele, P. Maddalena, M. Liscidini, L. C. Andreani, C. Ricciardi, V. Ballarini, and F. Giorgis, “Second-harmonic generation in hydrogenated amorphous-Si 1 - xN x doubly resonant microcavities with periodic dielectric mirrors,” Appl. Phys. Lett. 87(19), 191110 (2005). [CrossRef]
- A. Gondarenko, J. S. Levy, and M. Lipson, “High confinement micron-scale silicon nitride high Q ring resonator,” Opt. Express 17(14), 11366–11370 (2009). [CrossRef] [PubMed]
- A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express 16(7), 4881–4887 (2008). [CrossRef] [PubMed]
- M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photonics 2(12), 737–740 (2008). [CrossRef]
- J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Optical properties of high-delta air silica microstructure optical fibers,” Opt. Lett. 25(11), 796–798 (2000). [CrossRef]
- F. G. Omenetto, A. J. Taylor, M. D. Moores, J. Arriaga, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, “Simultaneous generation of spectrally distinct third harmonics in a photonic crystal fiber,” Opt. Lett. 26(15), 1158–1160 (2001). [CrossRef]
- A. Serpengüzel and S. Tanriseven, “Controlled photoluminescence in amorphous-silicon-nitride microcavities,” Appl. Phys. Lett. 78(10), 1388–1390 (2001). [CrossRef]
- M. Barth, N. Nusse, J. Stingl, B. Lochel, and O. Benson, “Emission properties of high-Q silicon nitride photonic crystal heterostructure cavities,” Appl. Phys. Lett. 93(2), 021112–021113 (2008). [CrossRef]
- V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Opt. Lett. 28(15), 1302–1304 (2003). [CrossRef] [PubMed]
- R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, San Diego, 2008).
- S. F. Pereira, M. Xiao, H. J. Kimble, and J. L. Hall, “Generation of squeezed light by intracavity frequency doubling,” Phys. Rev. A 38(9), 4931–4934 (1988). [CrossRef] [PubMed]
- N. K. Hon, K. K. Tsia, D. R. Solli, and B. Jalali, “Periodically poled silicon,” Appl. Phys. Lett. 94(9), 091116–091113 (2009). [CrossRef]

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