## Chirped-quasi-periodic structure for quasi-phase-matching |

Optics Express, Vol. 18, Issue 14, pp. 14717-14723 (2010)

http://dx.doi.org/10.1364/OE.18.014717

Acrobat PDF (834 KB)

### Abstract

We propose in this paper a chirped-quasi-periodic structure using the projection method. This type of new structure combines the advantages of chirped and quasi-periodic structures, and can be used for both multiple quasi-phase-matching and multiple bandwidths control. Numerical simulation of second-harmonic generation performance is in good agreement with the Fourier spectrum of the structure.

© 2010 OSA

## 1. Introduction

^{(2)}processes [1

1. S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science **278**(5339), 843–846 (1997). [CrossRef]

6. M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched LiNbO_{3} wavelength converter with a continuously phase-modulated domain structure,” Opt. Lett. **28**(7), 558–560 (2003). [CrossRef] [PubMed]

^{(2)}process involves the nonlinear mixing of two waves to produce a third wave at the sum or difference frequency. The process proceeds efficiently if the quasi-phase-matching (QPM) condition is satisfied. For example, in a sum frequency generation process

*G*, i.e.

^{(2)}process is desired, it is well known that for obtaining the largest Fourier coefficient the grating need to be periodically poled with period

^{(2)}processes are desired to be cascaded into one grating, the structure of the grating needs to be fabricated by some new means, such as quasi-periodic structures [1

1. S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science **278**(5339), 843–846 (1997). [CrossRef]

2. C. Zhang, H. Wei, Y. Y. Zhu, H. T. Wang, S. N. Zhu, and N. B. Ming, “Third-harmonic generation in a general two-component quasi-periodic optical superlattice,” Opt. Lett. **26**(12), 899–901 (2001). [CrossRef]

3. B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. **75**(15), 2175–2177 (1999). [CrossRef]

4. A. H. Norton and C. M. de Sterke, “Aperiodic 1-dimensional structures for quasi-phase matching,” Opt. Express **12**(5), 841–846 (2004). [CrossRef] [PubMed]

5. Z. W. Liu, Y. Du, J. Liao, S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. T. Wang, J. L. He, C. Zhang, and N. B. Ming, “Engineering of a dual-periodic optical superlattice used in a coupled optical parametric interaction,” J. Opt. Soc. Am. B **19**(7), 1676–1684 (2002). [CrossRef]

6. M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched LiNbO_{3} wavelength converter with a continuously phase-modulated domain structure,” Opt. Lett. **28**(7), 558–560 (2003). [CrossRef] [PubMed]

^{(2)}processes, for effective generation of every process the matching temperatures of all these processes need to be the same. For example, in a third-harmonic generation (THG) process [1

1. S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science **278**(5339), 843–846 (1997). [CrossRef]

7. X. P. Hu, G. Zhao, C. Zhang, Z. D. Xie, J. L. He, and S. N. Zhu, “High-power, blue-light generation in a dual-structure, periodically poled, stoichiometric LiTaO3 crystal,” Appl. Phys. B **87**(1), 91–94 (2007). [CrossRef]

8. Z. D. Gao, S. N. Zhu, S.-Y. Tu, and A. H. Kung, “Monolithic red-green-blue laser light source based on cascaded wavelength conversion in periodically poled stoichiometric lithium tantalite,” Appl. Phys. Lett. **89**(18), 181101 (2006). [CrossRef]

9. G. K. Samanta and M. Ebrahim-Zadeh, “Continuous-wave, single-frequency, solid-state blue source for the 425-489 nm spectral range,” Opt. Lett. **33**(11), 1228–1230 (2008). [CrossRef] [PubMed]

10. M. A. Arbore, O. Marco, and M. M. Fejer, “Pulse compression during second-harmonic generation in aperiodic quasi-phase-matching gratings,” Opt. Lett. **22**(12), 865–867 (1997). [CrossRef] [PubMed]

12. X. J. Lv, Z. Sui, Z. D. Gao, M. Z. Li, Q. H. Deng, and S. N. Zhu, “Bandwidth and stability enhancement of optical parametric amplification using chirped ferroelectric superlattice,” Opt. Laser Technol. **40**(1), 21–29 (2008). [CrossRef]

## 2. Structure design

*A*and block

*B*—are projected by vertical and horizontal spacing, respectively. For dielectric optical superlattice such as periodically-poled lithium niobate (PPLN) or lithium tantalite (PPLT), we set both block

*A*and

*B*consisting of a positive domain and a negative domain, and the lengths of the positive domains in each block have the same value

*l*. The reciprocal vectors of this structure are given in Ref [2

2. C. Zhang, H. Wei, Y. Y. Zhu, H. T. Wang, S. N. Zhu, and N. B. Ming, “Third-harmonic generation in a general two-component quasi-periodic optical superlattice,” Opt. Lett. **26**(12), 899–901 (2001). [CrossRef]

*τ*is equal to the proportion of block number of

*A*,

*N*, to block number of

_{A}*B*,

*N*, i.e.

_{B}*τ = N*.

_{A}/ N_{B}*D*,

_{A}*D*are the lengths of block

_{B}*A*and

*B*, respectively.

*D*/

_{A}*D*is fixed, and we have

_{B}*τ =*tan

*θ*. However, in a more general situation,

*D*/

_{A}*D*is an adjustable parameter [2

_{B}2. C. Zhang, H. Wei, Y. Y. Zhu, H. T. Wang, S. N. Zhu, and N. B. Ming, “Third-harmonic generation in a general two-component quasi-periodic optical superlattice,” Opt. Lett. **26**(12), 899–901 (2001). [CrossRef]

*τ*still indicates the proportion of block numbers, but the relation

*τ =*tan

*θ*do not hold any more.

*d*and

_{x}*d*indicate the horizontal and the vertical spacing, respectively. Then we can obtain the expression of

_{y}*τ*: Equation (2) indicates the arrangement parameter

*τ*varies with the spacing of the projection lattice.

*r*to describe the chirp rate, see Ref [3

3. B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. **75**(15), 2175–2177 (1999). [CrossRef]

*d*(1),

_{i}*d*(

_{i}*N*)and

*d*represent the length of the first, the last and the nominal spacing of the lattice,

_{i0}*i*=

*x*,

*y*indicates horizontal and vertical, respectively. The

*n*th spacing’s length is:However, chirping is not limited to linear [11]. Indeed, as long as the domain period varies with position, linearly or non-linearly, the grating can be considered as chirped.

*d*=

_{x}*d*(

_{x}*ξ*) and

*d*=

_{y}*d*(

_{y}*ξ*),

*ξ*is the coordinate along the direction of the QP grating. From Fig. 1 we can see that using variable

*ξ*to describe the spacing variation is equivalent to using coordinate

*x*and

*y*since they are proportional. Moreover, here we use continuous functions to describe discrete spacing, which is the same as Eq. (4) in Ref.12. Actually, for a given starting point, using a recursion method the desired structure can be easily obtained.

*A*are projected from vertical spacing while all blocks

*B*are projected from horizontal spacing, the block lengths

*D*and

_{A}*D*will vary with position

_{B}*ξ*as following forms:

*τ*. From Eq. (2) we know that for a CQP structure

*τ*can be expressed as:

*τ*is a localized parameter in a CQP structure, different positions

*ξ*have different quasi-periodic arrangements, which is totally different from the traditional QP structure. Substituting Eq. (5-7) into the expression of reciprocal vectors Eq. (1), we obtain:

*G*of QP structure, which includes arbitrary spacing nonuniform. In Eq. (8) we use two sets of parameters (

_{m,n}*d*,

_{x}*d*,

_{y}*θ*) and (

*D*,

_{A}*D*,

_{B}*θ*) to describe the CQP structure. The former one shows the origin of the structure and is convenient for analyzing in projection method, and the latter one depends directly on the real block lengths

*D*and

_{A}*D*, which is therefore easier to draw structure. In the rest of this paper, we use the latter form to describe the structure. The deriving using the former form will be similar.

_{B}*G*

_{m}_{1,}

_{n}_{1}and

*G*

_{m}_{2,}

_{n}_{2}to stretch with position

*ξ*as arbitrary form

*G*

_{m}_{1,}

_{n}_{1}(

*ξ*) and

*G*

_{m}_{2,}

_{n}_{2}(

*ξ*), then resolving following equations:we obtain:

*ξ*when two desired reciprocal vectors are given. As shown in Eq. (10), the two blocks vary with position as a quite complex form, which in most case is non-linear chirping.

## 3. Fourier transformation of CQP structure

*d*= 19.1

_{x0}*μm*and vertical spacing

*d*= 17.5

_{y0}*μm*to a line with slope tan

*θ*= 0.414, the initial block lengths are

*D*

_{A}_{0}= 6.69

*μm*and

*D*

_{B}_{0}= 17.64

*μm*and the lengths of positive domain in both blocks are

*l*= 6

*μm*. For a 10-mm-long grating, the Fourier transformation of this QP grating is:

*G*

_{11}and

*G*

_{21}, and let

*G*

_{11}holds its

*δ*–function shape while

*G*

_{21}linearly stretches to ±

*δG*, the two reciprocal vectors will vary with position

*ξ*as following forms:

*L*is the length of the grating,

*ξ*is the position coordinate of the grating, varies from 0 to

*L*. Substituting (11) into (10), we obtain the relation of two block lengths vary with position:

*D*and

_{A}*D*both non-linearly vary with position

_{B}*ξ*. From Eq. (12) we could obtain the structure parameters of the CQP grating. Here we choose

*δG*= 0.01

*μm*

^{−1}. The Fourier transformation of this structure is shown in Fig. 3 and the detail of

*G*

_{21}is shown in inset. From Fig. 3 we can see that

*G*

_{11}holds its original shape and

*G*

_{21}is stretched to ± 0.01

*μm*

^{−1}.

## 4. Second-harmonic generation in CQP structure

*δ*-function shape while the others were stretched. Thus we could expect these two kinds of reciprocal vectors will have different bandwidths in second-harmonic generation (SHG) process due to the bandwidth-broadening effect of chirped gratings [12

12. X. J. Lv, Z. Sui, Z. D. Gao, M. Z. Li, Q. H. Deng, and S. N. Zhu, “Bandwidth and stability enhancement of optical parametric amplification using chirped ferroelectric superlattice,” Opt. Laser Technol. **40**(1), 21–29 (2008). [CrossRef]

*G*

_{21}, which was stretched to ± 0.01

*μm*

^{−1}, and

*G*

_{11}, which was not stretched, for numerical SHG. And we choose lithium tantalate (LT) as the nonlinear crystal and set the working temperature at 180°C. Using Sellmeier equation. of LT, we can estimate that

*G*

_{11}and

*G*

_{21}can quasi-phase-match SHG process around 1334.4 and 1103.9

*nm*, respectively. Under the slow-varying amplitude approximation, a SHG process should satisfy the following couple-wave equations [14] during propagation:where

*E*,

_{i}*ω*,

_{i}*n*, indicate the electric field, angular frequency and refractive index of waves, and the subscript

_{i}*i*= 1,2 represents the fundamental and second-harmonic wave, respectively.

*dk*=

*k*

_{2}

*-2k*

_{1}represents the wave vector mismatch between fundamental and SH wave. Here we choose

*x*, instead of

*ξ*, to describe the propagating direction of waves, and

*f*(

*x*) satisfies:

*δ*-function-shaped reciprocal vector

*G*

_{11}is around 0.3

*nm*while the FWHM of the linearly-stretched reciprocal vector

*G*

_{21}is about 9

*nm*, which is 30 times that of

*G*

_{11}. The two SHG bandwidths are in good agreement with the Fourier spectrum in Fig. 3.

12. X. J. Lv, Z. Sui, Z. D. Gao, M. Z. Li, Q. H. Deng, and S. N. Zhu, “Bandwidth and stability enhancement of optical parametric amplification using chirped ferroelectric superlattice,” Opt. Laser Technol. **40**(1), 21–29 (2008). [CrossRef]

*G*

_{11}the 0.3

*nm*wavelength FWHM equals 2.4°C temperature FWHM, and for the stretched

*G*

_{21}the 9

*nm*wavelength FWHM equals 134.3°C-222.3°C, which is 88°C temperature FWHM.

## 5. Conclusion

^{(2)}processes, ultrashort multi-wavelength pulse-compression, et. al.

## Acknowledgements:

## References and links

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

2. | C. Zhang, H. Wei, Y. Y. Zhu, H. T. Wang, S. N. Zhu, and N. B. Ming, “Third-harmonic generation in a general two-component quasi-periodic optical superlattice,” Opt. Lett. |

3. | B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. |

4. | A. H. Norton and C. M. de Sterke, “Aperiodic 1-dimensional structures for quasi-phase matching,” Opt. Express |

5. | Z. W. Liu, Y. Du, J. Liao, S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. T. Wang, J. L. He, C. Zhang, and N. B. Ming, “Engineering of a dual-periodic optical superlattice used in a coupled optical parametric interaction,” J. Opt. Soc. Am. B |

6. | M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched LiNbO |

7. | X. P. Hu, G. Zhao, C. Zhang, Z. D. Xie, J. L. He, and S. N. Zhu, “High-power, blue-light generation in a dual-structure, periodically poled, stoichiometric LiTaO3 crystal,” Appl. Phys. B |

8. | Z. D. Gao, S. N. Zhu, S.-Y. Tu, and A. H. Kung, “Monolithic red-green-blue laser light source based on cascaded wavelength conversion in periodically poled stoichiometric lithium tantalite,” Appl. Phys. Lett. |

9. | G. K. Samanta and M. Ebrahim-Zadeh, “Continuous-wave, single-frequency, solid-state blue source for the 425-489 nm spectral range,” Opt. Lett. |

10. | M. A. Arbore, O. Marco, and M. M. Fejer, “Pulse compression during second-harmonic generation in aperiodic quasi-phase-matching gratings,” Opt. Lett. |

11. | G. Imeshev, “Tailoring of Ultrafast Frequency Conversion with Quasi-Phase-Matching Gratings,” Ph.D. dissertation (Stanford University, 2000). |

12. | X. J. Lv, Z. Sui, Z. D. Gao, M. Z. Li, Q. H. Deng, and S. N. Zhu, “Bandwidth and stability enhancement of optical parametric amplification using chirped ferroelectric superlattice,” Opt. Laser Technol. |

13. | R. K. P. Zia and W. J. Dallas, “A simple derivation of quasi-crystalline spectra,” J. Phys. Math. Gen. |

14. | A. Yariv, and P. Yeh, |

**OCIS Codes**

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(130.7405) Integrated optics : Wavelength conversion devices

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: April 27, 2010

Revised Manuscript: June 11, 2010

Manuscript Accepted: June 12, 2010

Published: June 24, 2010

**Citation**

J. Yang, X. P. Hu, P. Xu, X. J. Lv, C. Zhang, G. Zhao, H. J. Zhou, and S. N. Zhu, "Chirped-quasi-periodic structure for quasi-phase-matching," Opt. Express **18**, 14717-14723 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-14717

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### References

- S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278(5339), 843–846 (1997). [CrossRef]
- C. Zhang, H. Wei, Y. Y. Zhu, H. T. Wang, S. N. Zhu, and N. B. Ming, “Third-harmonic generation in a general two-component quasi-periodic optical superlattice,” Opt. Lett. 26(12), 899–901 (2001). [CrossRef]
- B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 75(15), 2175–2177 (1999). [CrossRef]
- A. H. Norton and C. M. de Sterke, “Aperiodic 1-dimensional structures for quasi-phase matching,” Opt. Express 12(5), 841–846 (2004). [CrossRef] [PubMed]
- Z. W. Liu, Y. Du, J. Liao, S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. T. Wang, J. L. He, C. Zhang, and N. B. Ming, “Engineering of a dual-periodic optical superlattice used in a coupled optical parametric interaction,” J. Opt. Soc. Am. B 19(7), 1676–1684 (2002). [CrossRef]
- M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched LiNbO3 wavelength converter with a continuously phase-modulated domain structure,” Opt. Lett. 28(7), 558–560 (2003). [CrossRef] [PubMed]
- X. P. Hu, G. Zhao, C. Zhang, Z. D. Xie, J. L. He, and S. N. Zhu, “High-power, blue-light generation in a dual-structure, periodically poled, stoichiometric LiTaO3 crystal,” Appl. Phys. B 87(1), 91–94 (2007). [CrossRef]
- Z. D. Gao, S. N. Zhu, S.-Y. Tu, and A. H. Kung, “Monolithic red-green-blue laser light source based on cascaded wavelength conversion in periodically poled stoichiometric lithium tantalite,” Appl. Phys. Lett. 89(18), 181101 (2006). [CrossRef]
- G. K. Samanta and M. Ebrahim-Zadeh, “Continuous-wave, single-frequency, solid-state blue source for the 425-489 nm spectral range,” Opt. Lett. 33(11), 1228–1230 (2008). [CrossRef] [PubMed]
- M. A. Arbore, O. Marco, and M. M. Fejer, “Pulse compression during second-harmonic generation in aperiodic quasi-phase-matching gratings,” Opt. Lett. 22(12), 865–867 (1997). [CrossRef] [PubMed]
- G. Imeshev, “Tailoring of Ultrafast Frequency Conversion with Quasi-Phase-Matching Gratings,” Ph.D. dissertation (Stanford University, 2000).
- X. J. Lv, Z. Sui, Z. D. Gao, M. Z. Li, Q. H. Deng, and S. N. Zhu, “Bandwidth and stability enhancement of optical parametric amplification using chirped ferroelectric superlattice,” Opt. Laser Technol. 40(1), 21–29 (2008). [CrossRef]
- R. K. P. Zia and W. J. Dallas, “A simple derivation of quasi-crystalline spectra,” J. Phys. Math. Gen. 18(7), L341–L345 (1985). [CrossRef]
- A. Yariv, and P. Yeh, Optical Waves in Crystal (Wiley, 1984).

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