## Aperiodic 1-dimensional structures for quasi-phase matching

Optics Express, Vol. 12, Issue 5, pp. 841-846 (2004)

http://dx.doi.org/10.1364/OPEX.12.000841

Acrobat PDF (711 KB)

### Abstract

We describe a method for designing 1-dimensional aperiodic poled grating structures of finite length that quasi-phase match multiple *χ*^{(2)} processes. The poling functions for such gratings are best aligned, in terms of the dot product in Fourier space, with a design target. No restrictions are placed on the quasi-phase matching wave numbers. A grating designed for third harmonic generation is simulated.

© 2004 Optical Society of America

## 1. Introduction

*χ*

^{(2)}processes [1

1. K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quantum Electron. **35**, 1649–1656 (1999). [CrossRef]

7. S. M. Saltiel, A. A. Sukhorukov, and Y. S. Kivshar, “Multistep parametric processes in nonlinear optics,” http://arxiv.org/abs/nlin.PS/0311013 (2003).

*χ*

^{(2)}grating structure entails the specification of a

*poling function, p*:

**R**→

**R**, of the general form

*L*is the grating length. The poling function specifies the sign of the

*χ*

^{(2)}nonlinearity. For ferro-electrics such as LiNbO

_{3}, the sign of

*χ*

^{(2)}, as a function of position, can be engineered using the fabrication technique of electric field poling.

*χ*

^{(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 three wave vectors are quasi-phase matched by a reciprocal lattice vector associated with a strong Fourier coefficient of the grating. For example, the frequency sum process

*ω*

_{A}+

*ω*

_{B}=

*ω*

_{C}in general requires quasi-phase matching (QPM) by a reciprocal lattice vector

**G=k**

_{C}-

**k**

_{A}-

**k**

_{B}. The Fourier transform of the poling function is therefore required to have a strong component

*p̂*(

**G**).

*χ*

^{(2)}process is to be quasi-phase matched then the efficiency of the process is highest for the poling function that has the largest QPM Fourier coefficient. This is well known to be a square wave with period 2

*π*/|

**G**|. If, however, QPM is required for several different

*χ*

^{(2)}processes then it is not obvious how best to choose the poling function. At least four different grating design approaches can be found in the literature. These include approaches based on Fibonacci (or quasi-crystal) structures [1

1. K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quantum Electron. **35**, 1649–1656 (1999). [CrossRef]

2. O. Bang, C. B. Clausen, P. L. Christiansen, and L. Torner, “Engineering competing nonlinearities,” Opt. Lett. **24**, 1413–1415 (1999). [CrossRef]

3. 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**, 558–560 (2003). [CrossRef] [PubMed]

5. M. H. Chou, K. R. Parameswaran, M. M. Fejer, and I. Brener, “Multiple-channel wavelength conversion by use of engineered quasi-phase-matching structures in LiNbO_{3} waveguides,” Opt. Lett. **24**, 1157–1159 (1999). [CrossRef]

6. 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**, 1676–1684 (2002). [CrossRef]

*p*(

*x*)=

*p*

_{1}(

*x*)…

*p*

_{n}(

*x*), where each

*p*

_{j}(

*x*) is a square-wave function.

*p̂*(

*k*)|

^{2}

*dk*=2

*π*∫

*p*(

*x*)

^{2}

*dx*=2

*πL*. Thus, unnecessary peaks in

*p̂*(

*k*) at other than the required QPM wave vectors can only reduce the efficiency of the grating.

*p̂*(

*k*) are important for QPM, so it may have some advantage over previous design methods. We formulate the design problem as that of finding a poling function that is

*best aligned*with a prescribed target transform. We define the property of being best aligned in terms of the inner product in Fourier space. This leads immediately to an elegant and explicit expression for the poling function

*p*(

*x*) that is best aligned with a prescribed target transform

*n̂*(

*k*).

*χ*

^{(2)}poling function for third harmonic generation (THG) based on the two nonlinear processes [7

7. S. M. Saltiel, A. A. Sukhorukov, and Y. S. Kivshar, “Multistep parametric processes in nonlinear optics,” http://arxiv.org/abs/nlin.PS/0311013 (2003).

*k*

_{1},

*k*

_{2},

*k*

_{3}are wave vectors for the fundamental, second harmonic and third harmonic respectively, and

*G*

_{1},

*G*

_{2}are quasi-phase matching wave vectors. In this example the

*χ*

^{(2)}grating structure is required to have a Fourier transform with strong

*G*

_{1},

*G*

_{2}components. So the target transform

*n̂*(

*k*) was chosen to be the piecewise constant function shown in Fig. 1, having two square peaks centered at

*G*

_{1},

*G*

_{2}. Also shown in Fig. 1 is the Fourier transform

*p̂*(

*k*) for the aperiodic grating of length

*L*=1cm that is best aligned with the prescribed target. The discrepancy between

*n̂*(

*k*) and

*p̂*(

*k*) is of course due to the constraint that the poling function be of the form (1).

## 2. Notation

*V*(

*x*) be the vector space with complex coefficients and basis vectors {

*u*

_{k}(

*x*)},

*V*(

*x*) are complex valued functions of the real variable

*x*. Let

*f*∈

*V*(

*x*) have components

*f̂*(

*k*) with respect to the basis (3). Then

*f̂*(

*k*) define the Fourier transform of

*f*(

*x*). The complex valued inner product 〈,〉 is defined for

*f*

_{1},

*f*

_{2}∈

*V*(

*x*) by

*u*

_{k}(

*x*)} is orthonormal with respect to Eq. (5), 〈

*u*

_{k}

*, u*

_{k}′〉=

*δ*(

*k-k*′). The Fourier transform of f is then

*V*(

*x*) may also be considered as a vector space

*V*′(

*x*) with real coefficients and basis vectors {

*u*

_{k}

*, iu*

_{k}}, since for any

*f*∈

*V*(

*x*) one may also write

*f*∈

*V*′(

*x*) as

*V*′(

*x*) can be defined using the complex valued inner product in

*V*(

*x*),

*u*

_{k}

*, iu*

_{k}} of

*V*′(

*x*) is orthonormal with respect to the dot product (8). Thus, in terms of the Fourier coefficients of

*f*

_{1}and

*f*

_{2}, one has

## 3. The best aligned poling function for a given target

*n̂*(

*k*) be given. We say the poling function

*p*(

*x*) is

*best aligned*with

*n̂*(

*k*) if the dot product

*p·n*is maximal amongst all possible poling functions of the form Eq. (1). From expression (9) for the dot product, it is clear that the property that

*p*be best aligned with

*n*is a natural way to specify that the peaks in

*p̂*(

*k*) appear at the values of

*k*required for QPM—one need only specify a suitable target. Moreover, Eqs. (1) and (8) lead to an explicit expression for the best aligned poling function. Since,

*n̂*(

*k*) is chosen so that its inverse Fourier transform

*n*(

*x*) can be easily calculated, then Eq. (11) provides a simple expression for the poling function best aligned with

*n*. In the following section we evaluate Eq. (11) in the case that the target

*n̂*(

*k*) is a piecewise constant function with an arbitrary number of QPM peaks.

*p*(

*x*) generally has no simple expression. However,

*p̂*(

*k*) is easily calculated numerically by solving for the positions of the sign changes in Eq. (11). Let

*x*

_{j}

*, j*=2,…,

*r*-1 be those roots of Re{

*n*(

*x*)} that lie in the interval [-

*L*/2,

*L*/2], and let

*x*

_{1}=-

*L*/2,

*x*

_{r}

*=L*/2. Then

*p*

_{j}

*=p*((

*x*

_{j}+

*x*

_{j+1})/2) is the sign of the

*j*th poled region and

*ĥ*

_{j}(

*k*) is the Fourier transform of the unit rectangular function with support on the

*j*th poled region,

## 4. Piecewise constant target

*n̂*(

*k*) is a piecewise constant function with

*N*different QPM peaks. We suppose QPM peaks are required at wave numbers

*G*

_{j}

*, j*=1,…,

*N*, with corresponding bandwidths 2Δ

*G*

_{j}

*, j*=1,…,

*N*. A target Fourier transform

*n̂*(

*k*) that encodes these design requirements in a simple manner is given by a weighted sum of rectangular functions

*H*

_{j}(

*k*) centered at

*k*=

*G*

_{j},

*n*(

*x*) to be real, in which case Re{

*n̂*(

*k*)} is an even function of

*k*and Im{

*n̂*(

*k*)} is an odd function of

*k*. Thus, we take

*a*

_{j}

*, b*

_{j}∈

**R**are weights that should be chosen optimally for the particular QPM problem (see Section 5 for an example). Applying the inverse transform (4) to (16) gives

*w*

_{j}=(

^{1/2}and cos

*ϕ*

_{j}

*=a*

_{j}

*/w*

_{j}, sin

*ϕ*

_{j}

*=b*

_{j}

*/w*

_{j}. The poling function that is best aligned with the target transform (16) is therefore

*G*

_{j}is too large then the QPM peaks in

*p̂*(

*k*) simply split rather than become wider. The most satisfactory results seem to be obtained for Δ

*G*

_{j}=2

*π/L*, in which case the first zeros of the sinc factor in (18) are at

*x*=±

*L*/2. Expression (18) then simplifies to

## 5. An aperiodic poled grating for THG

*χ*

^{(2)}processes (2). The grating is designed to operate at 140ΔC with a fundamental wavelength of λ=1550nm. We suppose the grating material is LiNbO

_{3}with quadratic susceptibility

*χ*

^{(2)}(

*x*)=

*χp*(

*x*), where

*χ*=41×10

^{-12}m/V. The refractive indices at the first, second and third harmonics are respectively

*n*

_{1}=2.1430,

*n*

_{2}=2.1848 and

*n*

_{3}=2.2487 [8

8. D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate,” Opt. Lett. **22**, 1553–1555 (1997). [CrossRef]

*k*

_{q}

*=qωn*

_{q}

*/c*, for

*q*=1,2, 3. Using (2) one finds that

*G*

_{1}=0.33865×10

^{6}m

^{-1}and

*G*

_{2}=0.94639×10

^{6}m

^{-1}. Taking

*ϕ*

_{j}=0 in (19) we set

*p*(

*x*)=sign(

*w*

_{1}cos(

*G*

_{1}

*x*)+

*w*

_{2}cos(

*G*

_{2}

*x*)) for

*x*∈(-

*L*/2,

*L*/2) and 0 otherwise, for grating length

*L*=1cm. The weights

*w*

_{1}=5.2874×10

^{-3}and

*w*

_{2}=4.6951×10

^{-3}are normalized so

*L*

^{2}/2. The ratio

*w*

_{1}/

*w*

_{2}was adjusted so as to maximize THG efficiency (for THG there exists a critical ratio of QPM Fourier coefficients [9

9. C. Zhang, Y. Zhu, S. Yang, Y. Qin, S. Zhu, Y. Chen, H. Liu, and N. Ming, “Crucial effects of coupling coefficients on quasi-phase-matched harmonic generation in an optical superlattice,” Opt. Lett. **25**, 436–438 (2000). [CrossRef]

10. A. H. Norton and C. M. de Sterke, “Two-dimensional poling patterns for 3rd and 4th harmonic generation,” Opt. Express **11**, 1008–1014 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1008. [CrossRef] [PubMed]

*n̂*(

*k*) and

*p̂*(

*k*) and Fig. 2 shows

*n*(

*x*) and

*p*(

*x*).

*p*(

*x*), one has

*a*

_{q}(

*x*) is the complex amplitude of the

*q*-harmonic,

*q*=1, 2, 3. Figure 3 shows the relative energy fluxes for the three waves. The boundary conditions were

*a*

_{1}(0)=1×10

^{7}V/m, and

*a*

_{2}(0)=

*a*

_{3}(0)=0. Equations (20)–(22) were integrated using a 4th order Runge-Kutta (RK) method with the step size dynamically adjusted so that the boundaries of the poled domains (discontinuities in

*p*(

*x*)) coincided with an RK step boundary. The THG curves obtained for grating phases

*ϕ*

_{j}≠0, were visually indistinguishable from those of Fig. 3 for

*ϕ*

_{j}=0. The removal of small grating features (e.g., domains of width <1

*µ*m) had a small effect on the THG curves, that could be compensated for by small adjustments to the ratio

*w*

_{1}/

*w*

_{2}.

## 6. Conclusion

## References and links

1. | K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quantum Electron. |

2. | O. Bang, C. B. Clausen, P. L. Christiansen, and L. Torner, “Engineering competing nonlinearities,” Opt. Lett. |

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

4. | M. L. Bortz, “Quasi-Phasematched Optical Frequency Conversion in Lithium NiobateWaveguides,” Ph.D. thesis, Stanford University (1994). |

5. | M. H. Chou, K. R. Parameswaran, M. M. Fejer, and I. Brener, “Multiple-channel wavelength conversion by use of engineered quasi-phase-matching structures in LiNbO |

6. | 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 |

7. | S. M. Saltiel, A. A. Sukhorukov, and Y. S. Kivshar, “Multistep parametric processes in nonlinear optics,” http://arxiv.org/abs/nlin.PS/0311013 (2003). |

8. | D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate,” Opt. Lett. |

9. | C. Zhang, Y. Zhu, S. Yang, Y. Qin, S. Zhu, Y. Chen, H. Liu, and N. Ming, “Crucial effects of coupling coefficients on quasi-phase-matched harmonic generation in an optical superlattice,” Opt. Lett. |

10. | A. H. Norton and C. M. de Sterke, “Two-dimensional poling patterns for 3rd and 4th harmonic generation,” Opt. Express |

**OCIS Codes**

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4400) Nonlinear optics : Nonlinear optics, materials

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 21, 2004

Revised Manuscript: February 19, 2004

Published: March 8, 2004

**Citation**

Andrew Norton and C. de Sterke, "Aperiodic 1-dimensional structures for quasi-phase matching," Opt. Express **12**, 841-846 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-5-841

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### References

- K. Fradkin-Kashi and A. Arie, �??Multiple-wavelength quasi-phase-matched nonlinear interactions,�?? IEEE J. Quantum Electron. 35, 1649�??1656 (1999). [CrossRef]
- O. Bang, C. B. Clausen, P. L. Christiansen, and L. Torner, �??Engineering competing nonlinearities,�?? Opt. Lett. 24, 1413�??1415 (1999). [CrossRef]
- M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, �??Multiple quasi-phase-matched LiNbO3 wave length converter with a continuously phase-modulated domain structure,�?? Opt. Lett. 28, 558�??560 (2003). [CrossRef] [PubMed]
- M. L. Bortz, �??Quasi-Phasematched Optical Frequency Conversion in Lithium Niobate Waveguides,�?? Ph.D. thesis, Stanford University (1994).
- M. H. Chou, K. R. Parameswaran, M. M. Fejer, and I. Brener, �??Multiple-channel wavelength conversion by use of engineered quasi-phase-matching structures in LiNbO3 waveguides,�?? Opt. Lett. 24, 1157�??1159 (1999). [CrossRef]
- 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, 1676�??1684 (2002). [CrossRef]
- S. M. Saltiel, A. A. Sukhorukov, and Y. S. Kivshar, �??Multistep parametric processes in nonlinear optics,�?? <a href="http://arxiv.org/abs/nlin.PS/0311013"> (2003).http://arxiv.org/abs/nlin.PS/0311013</a>
- D. H. Jundt, �??Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate,�?? Opt. Lett. 22, 1553�??1555 (1997). [CrossRef]
- C. Zhang, Y. Zhu, S. Yang, Y. Qin, S. Zhu, Y. Chen, H. Liu, and N. Ming, �??Crucial effects of coupling coefficients on quasi-phase-matched harmonic generation in an optical superlattice,�?? Opt. Lett. 25, 436�??438 (2000). [CrossRef]
- A. H. Norton and C. M. de Sterke, �??Two-dimensional poling patterns for 3rd and 4th harmonic generation,�?? Opt. Express 11, 1008�??1014 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1008">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1008</a>. [CrossRef] [PubMed]

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