## Retrieving quasi-phase-matching structure with discrete layer-peeling method |

Optics Express, Vol. 20, Issue 14, pp. 15826-15832 (2012)

http://dx.doi.org/10.1364/OE.20.015826

Acrobat PDF (1272 KB)

### Abstract

An approach to reconstruct a quasi-phase-matching grating by using a discrete layer-peeling algorithm is presented. Experimentally measured output spectra of Šolc-type filters, based on uniform and chirped QPM structures, are used in the discrete layer-peeling algorithm. The reconstructed QPM structures are in agreement with the exact structures used in the experiment and the method is verified to be accurate and efficient in quality inspection on quasi-phase-matching grating.

© 2012 OSA

## 1. Introduction

13. J. Shaar, L. G. Wang, and T. Erdogan, “On the synthesis of fiber bragg gratings by layer peeling,” IEEE J. Quantum Electron. **37**, 165–173 (2001). [CrossRef]

16. Y. Choi, J. Chun, and J. Bae, “Numerically extrapolated discrete layer-peeling algorithm for synthesis of nonuniform fiber Bragg gratings,” Opt. Express **19**, 8254–8266 (2011). [CrossRef] [PubMed]

17. E. C. Levy and M. Horowitz, “Layer-peeling algorithm for reconstructing the birefringence in optical emulators,” J. Opt. Soc. Am. B **23**, 1531–1539 (2006). [CrossRef]

18. X. Chen, J. Shi, Y. Chen, Y. Zhu, Y. Xia, and Y. Chen, “Electro-optic Šolc-type wavelength filter in periodically poled lithium niobate,” Opt. Lett. **28**, 2115–2117 (2003). [CrossRef] [PubMed]

*θ*in successive reversal domains, thus PPLN is similar to so-called folded Šolc filter structure with periodically alternating azimuth angles of the crystal axes [19

19. Y. Q. Lu and Z. L. Wan, “Electro-optic effect of periodically poled optical superlattice LiNbO_{3} and its applications,” Appl. Phys. Lett. **77**, 3719–3721 (2000). [CrossRef]

20. Q. Zhang, X. Zeng, F. Pang, X. Chen, and T. Wang, “Tunable polarization-independent Šolc-type wavelength filter based on periodically poled lithium niobate,” Opt. Laser Technol. **44**, 1992–1994 (2012). [CrossRef]

21. C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Express **15**, 9859–9866 (2007). [CrossRef] [PubMed]

23. C. L. Chang, Y. H. Chen, C. H. Lin, and J. Y. Chang, “Monolithically integrated multi-wavelength filter and second harmonic generator in aperiodically poled lithium niobate,” Opt. Express **16**, 18535–18544 (2008). [CrossRef] [PubMed]

24. Y. L. Lee, Y. C. Noh, C. S. Kee, N. E. Yu, W. Shin, C. Jung, D. K. Ko, and J. Lee, “Bandwidth control of a Ti:PPLN Šolc filter by a temperature-gradient-control technique,” Opt. Express **16**, 13699–13706 (2008). [CrossRef] [PubMed]

25. C. Y. Huang, C. H. Lin, Y. H. Chen, and Y. C. Huang, “Electro-optic Ti:PPLN waveguide as efficient optical wavelength filter and polarization mode converter,” Opt. Express **15**, 2548–2554 (2007). [CrossRef] [PubMed]

26. C. S. Kee, Y. L. Lee, and J. Lee, “Electro- and thermo-optic effects on multi-wavelength Šolc filters based on *χ*^{(2)} nonlinear quasi-periodic photonic crystals,” Opt. Express **16**, 6098–6103 (2008). [CrossRef] [PubMed]

27. J. K. Brenne and J. Skaar, “Design of grating-assisted codirectional couplers with discrete inverse-scattering algorithms,” J. Lightwave. Technol. **21**, 254–263 (2003). [CrossRef]

28. G. Lenz, B. J. Eggleton, and C. R. Giles, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. **34**, 1390–1402 (1998). [CrossRef]

29. L. Wang and T. Erdogan, “Layer peeling algorithm for reconstruction of long-period fibre gratings,” Electron. Lett. **37**, 154–156 (2001). [CrossRef]

## 2. Numerical method and analysis

19. Y. Q. Lu and Z. L. Wan, “Electro-optic effect of periodically poled optical superlattice LiNbO_{3} and its applications,” Appl. Phys. Lett. **77**, 3719–3721 (2000). [CrossRef]

*A*(

_{o}*z*, Δ

*β*) and

*A*(

_{e}*z*, Δ

*β*) represent the normalized amplitudes of ordinary and extraordinary waves, respectively, which can be considered as slowly varying amplitudes of co-directional propagating waves.

19. Y. Q. Lu and Z. L. Wan, “Electro-optic effect of periodically poled optical superlattice LiNbO_{3} and its applications,” Appl. Phys. Lett. **77**, 3719–3721 (2000). [CrossRef]

*ϕ*(

*z*) = [2

*π*/Λ(

*z*) − 2

*π*/Λ

_{0}]

*z*denotes the local phase variation induced by QPM grating chirp in ununiformed QPM structures, and

*n*and

_{o}*n*are refractive indices of ordinary and extraordinary waves. The detuning parameter Δ

_{e}*β*for QPM structure is defined as: Δ

*β*= (

*β*−

_{o}*β*) − 2

_{e}*πm*/Λ

_{0},

*β*and

_{o}*β*are wave vectors of ordinary and extraordinary waves and Λ

_{e}_{0}=

*λ*

_{0}/(

*n*−

_{o}*n*) represents the reference grating period and

_{e}*m*is the ordinary QPM order. The polarization coupling between co-directional propagating waves occurs at the resonance frequency

*ω*

_{0}when satifaying the phase matching condition: Δ

*β*(

*ω*

_{0}) = 0. The local coupling coefficients are determined by the difference of refractive indices Δ

*n*=

*n*−

_{o}*n*and QPM grating period Λ(

_{e}*z*) simultaneously. We assume the dispersion of refractive index negligible and keep Δ

*n*constant in the simulation. Now that we need to figure out the period distribution of QPM gratings from the coupling coefficient

*κ*(

*z*).

*N*layers of uniform sections separated by a distance Δ

*z*. The discretized coupling model is illustrated in Fig. 1. Based on the piecewise uniform coupling model [30

30. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. **15**, 1277–1294 (1997). [CrossRef]

*G*

_{Δ}describes the pure propagation between the instantaneous scattering points and the other matrix

*G*describes the coupling in the

_{ρ}*j*-th layer of QPM structure [31

31. R. Feced and M. N. Zervas, “Efficient inverse scattering algorithm for the design of grating-assisted co-directional mode couplers,” J. Opt. Soc. Am. A **17**, 1573–1582 (2000). [CrossRef]

*ρ*is the coupling strength of the instantaneous scattering point

_{j}*z*along the structure. The discretized coupling coefficient

_{j}*κ*(

*z*) can be obtained from the coupling strength

_{j}*ρ*by using the following relation [27

_{j}27. J. K. Brenne and J. Skaar, “Design of grating-assisted codirectional couplers with discrete inverse-scattering algorithms,” J. Lightwave. Technol. **21**, 254–263 (2003). [CrossRef]

*a*(

_{o,j}*τ*) and

*a*(

_{e,j}*τ*) are discrete and correspond to the response of different time delay.

*a*(0) and

_{o,j}*a*(0) denote the two shortest time delay through QPM structure from causality requirements. At the end of QPM structure (layer

_{e,j}*N*), the discrete coupling strength is simply expressed as:

*ρ*is obtained from the output spectra, we can peel this layer off and calculate the impulse responses of layer

_{N}*N*– 1 by using transfer matrix method [27

27. J. K. Brenne and J. Skaar, “Design of grating-assisted codirectional couplers with discrete inverse-scattering algorithms,” J. Lightwave. Technol. **21**, 254–263 (2003). [CrossRef]

*κ*(

*z*) of each layer from Eq. (3). The period chirp distribution of QPM grating can be obtained from the derivative of phase of the coupling coefficient by using Eq. (7).

_{j}## 3. Experimental results and discussions

*μ*m and two chirped gratings (3 mm width) with period change from 19.90 to 20.40

*μ*m linearly and quadratically, respectively. We use a C + L broadband ASE source with an output wavelength range from approximately 1520 nm to 1610 nm and an optical spectrum analyzer (OSA) to detect the output light. The output spectrum of broadband ASE source used in the experiment is shown in Fig. 2(b). The following experimental transmission spectra of Šolc-type filter are normalized by the spectra of ASE source.

32. O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO_{3},” Appl. Phys. B **91**, 343–348 (2008). [CrossRef]

*N*= 200 layers and the reconstructed period distribution by using DLP method is showed in Fig. 3(b). The result shows a good agreement with the theoretical expectation. The nominal chirped QPM gratings are expressed by Λ(

*z*) = Λ

_{0}+ (Λ

*− Λ*

_{N}_{0})(

*z*/

*L*)

*m*, Λ

_{0}, Λ

*is 19.90*

_{N}*μ*m and 20.40

*μ*m, showing the linearly (

*m*= 1) and quadratically (

*m*= 2) chirped structures, respectively. Figures 4(a) and 4(b) show the measured transmission spectra of the linearly and quadratically chirped QPM gratings. We observe band-pass transmission spectra with some fluctuations in the middle wavelength region compared to the theoretical results.

## 4. Conclusion

## Acknowledgments

## References and links

1. | K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the Phase-Matching Bandwidth in Quasi-Phase-Matched Second-Harmoic Generation,” IEEE J. Quantum Electron. |

2. | S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science |

3. | M. Charbonneau-Lefort, M. M. Fejer, and B. Afeyan,“Tandem chirped quasi-phase-matching grating optical parametric amplifier design for simultaneous group delay and gain control,” Opt. Lett. |

4. | J. Huang, X. P. Xie, C. Langrock, R. V. Roussev, D. S. Hum, and M. M. Fejer, “Amplitude modulation and apodization of quasiphase-matched interactions,” Opt. Lett. |

5. | T. Umeki, M. Asobe, T. Yanagawa, O. Tadanaga, Y. Nishida, K. Magari, and H. Suzuki, “Broadband wavelength conversion based on apodized |

6. | X. Zeng, S. Ashihara, Z. Wang, T. Wang, Y. Chen, and M. Cha, “Excitation of two-colored temporal solitons in a segmented quasi-phase-matching structure, ” Opt. Express |

7. | Y. W. Lee, F. C. Fan, Y. C. Huang, B. Y. Gu, B. Z. Dong, and M. H. Chou, “Nonlinear multiwavelength conversion based on an aperiodic optical superlattice in lithium niobate,” Opt. Lett. |

8. | X. Zeng, X. Chen, F. Wu, Y. Chen, Y. Xia, and Y. Chen, “Second-harmonic generation with broadened flattop bandwidth in aperiodic domain-inverted gratings,” Opt. Commun. |

9. | A. M. Schober, G. Imeshev, and M. M. Fejer, “Tunable-chirp pulse compression in quasi-phase-matched second-harmonic generation,” Opt. Lett. |

10. | K. Beckwitt, F. Ö. Ilday, and F. W. Wise, “Frequency shifting with local nonlinearity management in nonuniformly poled quadratic nonlinear materials,” Opt. Lett. |

11. | X. Zeng, S. Ashihara, X. Chen, T. Shimura, and K. Kuroda, “Two-color pulse compression in aperiodically poled lithium niobate,” Opt. Commun. |

12. | H. Miao, S. Yang, C. Langrock, R. V. Roussev, M. M. Fejer, and A. M. Weiner, “Ultralow-power second-harmonic generation frequency-resolved optical gating using aperiodically poled lithium niobate waveguides,” J. Opt. Soc. Am. B |

13. | J. Shaar, L. G. Wang, and T. Erdogan, “On the synthesis of fiber bragg gratings by layer peeling,” IEEE J. Quantum Electron. |

14. | J. Zhang, P. Shum, S. Y. Li, N. Q. Ngo, X. P. Cheng, and J. H. Ng, “Design and fabrication of flat-band long-period grating,” IEEE Photon. Technol. Lett. |

15. | H. Li, T. Kumagai, and K. Ogusu, “Advanced design of a multichannel fiber Bragg grating based on a layer-peeling method,” J. Opt. Soc. Am. B |

16. | Y. Choi, J. Chun, and J. Bae, “Numerically extrapolated discrete layer-peeling algorithm for synthesis of nonuniform fiber Bragg gratings,” Opt. Express |

17. | E. C. Levy and M. Horowitz, “Layer-peeling algorithm for reconstructing the birefringence in optical emulators,” J. Opt. Soc. Am. B |

18. | X. Chen, J. Shi, Y. Chen, Y. Zhu, Y. Xia, and Y. Chen, “Electro-optic Šolc-type wavelength filter in periodically poled lithium niobate,” Opt. Lett. |

19. | Y. Q. Lu and Z. L. Wan, “Electro-optic effect of periodically poled optical superlattice LiNbO |

20. | Q. Zhang, X. Zeng, F. Pang, X. Chen, and T. Wang, “Tunable polarization-independent Šolc-type wavelength filter based on periodically poled lithium niobate,” Opt. Laser Technol. |

21. | C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Express |

22. | X. Gu, X. Chen, Y. Chen, X. Zeng, Y. Xia, and Y. Chen, “Narrowband multiple wavelengths filter in aperiodic optical superlattice,” Opt. Commun. |

23. | C. L. Chang, Y. H. Chen, C. H. Lin, and J. Y. Chang, “Monolithically integrated multi-wavelength filter and second harmonic generator in aperiodically poled lithium niobate,” Opt. Express |

24. | Y. L. Lee, Y. C. Noh, C. S. Kee, N. E. Yu, W. Shin, C. Jung, D. K. Ko, and J. Lee, “Bandwidth control of a Ti:PPLN Šolc filter by a temperature-gradient-control technique,” Opt. Express |

25. | C. Y. Huang, C. H. Lin, Y. H. Chen, and Y. C. Huang, “Electro-optic Ti:PPLN waveguide as efficient optical wavelength filter and polarization mode converter,” Opt. Express |

26. | C. S. Kee, Y. L. Lee, and J. Lee, “Electro- and thermo-optic effects on multi-wavelength Šolc filters based on |

27. | J. K. Brenne and J. Skaar, “Design of grating-assisted codirectional couplers with discrete inverse-scattering algorithms,” J. Lightwave. Technol. |

28. | G. Lenz, B. J. Eggleton, and C. R. Giles, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. |

29. | L. Wang and T. Erdogan, “Layer peeling algorithm for reconstruction of long-period fibre gratings,” Electron. Lett. |

30. | T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. |

31. | R. Feced and M. N. Zervas, “Efficient inverse scattering algorithm for the design of grating-assisted co-directional mode couplers,” J. Opt. Soc. Am. A |

32. | O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO |

**OCIS Codes**

(120.2440) Instrumentation, measurement, and metrology : Filters

(190.4360) Nonlinear optics : Nonlinear optics, devices

(230.2090) Optical devices : Electro-optical devices

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: April 23, 2012

Revised Manuscript: June 19, 2012

Manuscript Accepted: June 21, 2012

Published: June 27, 2012

**Citation**

Q. W. Zhang, X. L. Zeng, M. Wang, T. Y. Wang, and X. F. Chen, "Retrieving quasi-phase-matching structure with discrete layer-peeling method," Opt. Express **20**, 15826-15832 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15826

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### References

- K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the Phase-Matching Bandwidth in Quasi-Phase-Matched Second-Harmoic Generation,” IEEE J. Quantum Electron.30, 1596–1604 (1994). [CrossRef]
- S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science278, 843–846 (1997). [CrossRef]
- M. Charbonneau-Lefort, M. M. Fejer, and B. Afeyan,“Tandem chirped quasi-phase-matching grating optical parametric amplifier design for simultaneous group delay and gain control,” Opt. Lett.30, 634–636 (2005). [CrossRef] [PubMed]
- J. Huang, X. P. Xie, C. Langrock, R. V. Roussev, D. S. Hum, and M. M. Fejer, “Amplitude modulation and apodization of quasiphase-matched interactions,” Opt. Lett.31, 604–606 (2006). [CrossRef] [PubMed]
- T. Umeki, M. Asobe, T. Yanagawa, O. Tadanaga, Y. Nishida, K. Magari, and H. Suzuki, “Broadband wavelength conversion based on apodized χ(2) grating,” J. Opt. Soc. Am. B26, 2315–2322 (2009). [CrossRef]
- X. Zeng, S. Ashihara, Z. Wang, T. Wang, Y. Chen, and M. Cha, “Excitation of two-colored temporal solitons in a segmented quasi-phase-matching structure, ” Opt. Express17, 16877–16884 (2009). [CrossRef] [PubMed]
- Y. W. Lee, F. C. Fan, Y. C. Huang, B. Y. Gu, B. Z. Dong, and M. H. Chou, “Nonlinear multiwavelength conversion based on an aperiodic optical superlattice in lithium niobate,” Opt. Lett.27, 2191–2193 (2002). [CrossRef]
- X. Zeng, X. Chen, F. Wu, Y. Chen, Y. Xia, and Y. Chen, “Second-harmonic generation with broadened flattop bandwidth in aperiodic domain-inverted gratings,” Opt. Commun.204, 407–411 (2002). [CrossRef]
- A. M. Schober, G. Imeshev, and M. M. Fejer, “Tunable-chirp pulse compression in quasi-phase-matched second-harmonic generation,” Opt. Lett.27, 1129–1131 (2002). [CrossRef]
- K. Beckwitt, F. Ö. Ilday, and F. W. Wise, “Frequency shifting with local nonlinearity management in nonuniformly poled quadratic nonlinear materials,” Opt. Lett.29, 763–765 (2004). [CrossRef] [PubMed]
- X. Zeng, S. Ashihara, X. Chen, T. Shimura, and K. Kuroda, “Two-color pulse compression in aperiodically poled lithium niobate,” Opt. Commun.281, 4499–4503 (2008). [CrossRef]
- H. Miao, S. Yang, C. Langrock, R. V. Roussev, M. M. Fejer, and A. M. Weiner, “Ultralow-power second-harmonic generation frequency-resolved optical gating using aperiodically poled lithium niobate waveguides,” J. Opt. Soc. Am. B25, A41–A53 (2008). [CrossRef]
- J. Shaar, L. G. Wang, and T. Erdogan, “On the synthesis of fiber bragg gratings by layer peeling,” IEEE J. Quantum Electron.37, 165–173 (2001). [CrossRef]
- J. Zhang, P. Shum, S. Y. Li, N. Q. Ngo, X. P. Cheng, and J. H. Ng, “Design and fabrication of flat-band long-period grating,” IEEE Photon. Technol. Lett.15, 1558–1560 (2003). [CrossRef]
- H. Li, T. Kumagai, and K. Ogusu, “Advanced design of a multichannel fiber Bragg grating based on a layer-peeling method,” J. Opt. Soc. Am. B21, 1929–1938 (2004). [CrossRef]
- Y. Choi, J. Chun, and J. Bae, “Numerically extrapolated discrete layer-peeling algorithm for synthesis of nonuniform fiber Bragg gratings,” Opt. Express19, 8254–8266 (2011). [CrossRef] [PubMed]
- E. C. Levy and M. Horowitz, “Layer-peeling algorithm for reconstructing the birefringence in optical emulators,” J. Opt. Soc. Am. B23, 1531–1539 (2006). [CrossRef]
- X. Chen, J. Shi, Y. Chen, Y. Zhu, Y. Xia, and Y. Chen, “Electro-optic Šolc-type wavelength filter in periodically poled lithium niobate,” Opt. Lett.28, 2115–2117 (2003). [CrossRef] [PubMed]
- Y. Q. Lu and Z. L. Wan, “Electro-optic effect of periodically poled optical superlattice LiNbO3 and its applications,” Appl. Phys. Lett.77, 3719–3721 (2000). [CrossRef]
- Q. Zhang, X. Zeng, F. Pang, X. Chen, and T. Wang, “Tunable polarization-independent Šolc-type wavelength filter based on periodically poled lithium niobate,” Opt. Laser Technol.44, 1992–1994 (2012). [CrossRef]
- C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Express15, 9859–9866 (2007). [CrossRef] [PubMed]
- X. Gu, X. Chen, Y. Chen, X. Zeng, Y. Xia, and Y. Chen, “Narrowband multiple wavelengths filter in aperiodic optical superlattice,” Opt. Commun.237, 53–58 (2004). [CrossRef]
- C. L. Chang, Y. H. Chen, C. H. Lin, and J. Y. Chang, “Monolithically integrated multi-wavelength filter and second harmonic generator in aperiodically poled lithium niobate,” Opt. Express16, 18535–18544 (2008). [CrossRef] [PubMed]
- Y. L. Lee, Y. C. Noh, C. S. Kee, N. E. Yu, W. Shin, C. Jung, D. K. Ko, and J. Lee, “Bandwidth control of a Ti:PPLN Šolc filter by a temperature-gradient-control technique,” Opt. Express16, 13699–13706 (2008). [CrossRef] [PubMed]
- C. Y. Huang, C. H. Lin, Y. H. Chen, and Y. C. Huang, “Electro-optic Ti:PPLN waveguide as efficient optical wavelength filter and polarization mode converter,” Opt. Express15, 2548–2554 (2007). [CrossRef] [PubMed]
- C. S. Kee, Y. L. Lee, and J. Lee, “Electro- and thermo-optic effects on multi-wavelength Šolc filters based on χ(2) nonlinear quasi-periodic photonic crystals,” Opt. Express16, 6098–6103 (2008). [CrossRef] [PubMed]
- J. K. Brenne and J. Skaar, “Design of grating-assisted codirectional couplers with discrete inverse-scattering algorithms,” J. Lightwave. Technol.21, 254–263 (2003). [CrossRef]
- G. Lenz, B. J. Eggleton, and C. R. Giles, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron.34, 1390–1402 (1998). [CrossRef]
- L. Wang and T. Erdogan, “Layer peeling algorithm for reconstruction of long-period fibre gratings,” Electron. Lett.37, 154–156 (2001). [CrossRef]
- T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol.15, 1277–1294 (1997). [CrossRef]
- R. Feced and M. N. Zervas, “Efficient inverse scattering algorithm for the design of grating-assisted co-directional mode couplers,” J. Opt. Soc. Am. A17, 1573–1582 (2000). [CrossRef]
- O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B91, 343–348 (2008). [CrossRef]

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