## Two-dimensional poling patterns for 3rd and 4th harmonic generation

Optics Express, Vol. 11, Issue 9, pp. 1008-1014 (2003)

http://dx.doi.org/10.1364/OE.11.001008

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### Abstract

We find globally optimal poling patterns for 2-dimensional χ^{(2)} photonic crystals for 3rd and 4th harmonic generation.

© 2003 Optical Society of America

## 1. Introduction

^{(2)}photonic crystal, pairs of waves interact via the quadratic susceptibility of the material to produce harmonics with sum or difference frequencies. If a wave vector of such a harmonic is phase matched by a reciprocal lattice vector of the photonic crystal then waves generated at different source points throughout the crystal interfere constructively. In a 2-dimensional photonic crystal [1

1. V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. **81**, 4136–4139 (1998). [CrossRef]

2. N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled Lithium Niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. **84**, 4345–4348 (2000). [CrossRef] [PubMed]

3. A. Chowdhury, C. Staus, B. F. Boland, T. F. Kuech, and L. McCaughan, “Experimental demonstration of 1535–1555-nm simultaneous optical wavelength interchange with a nonlinear photonic crystal,” Opt. Lett. **26**, 1353–1355 (2001). [CrossRef]

4. N. G. R. Broderick, R. T. Bratfalean, T. M. Monro, D. J. Richardson, and C. M. de Sterke, “Temperature and wavelength tuning of 2nd. 3rd and 4th harmonic generation in a two dimensional hexagonally poled nonlinear crystal,” J. Opt. Soc. Am. B **19**, 2263 (2002). [CrossRef]

5. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. **28**, 2631–2654 (1992). [CrossRef]

^{(2)}poling pattern of the photonic crystal. The poling function

*p*(

*x, y*)=±1 describes the direction of the optical axis of the material at position (

*x, y*) to be aligned (

*p*=+1) or anti-aligned (

*p*=-1) with the

*x*

^{3}≡

*z*coordinate axis. Since the quadratic susceptibility is a rank 3 tensor field, those components

*i, j, k*are equal to 3 have their sign determined by

*p*(

*x, y*). Thus, χ

^{(2)}

^{333}(

*x, y*)=constant ×

*p*(

*x, y*). Since

*x*-direction with

*z*-polarization, the quadratic processes involve only

*x, y*}-plane defines a photonic crystal. The photonic crystal will support the required QPM processes provided its unit cell geometry (size and shape) satisfies the corresponding QPM relations (Eq. (2) below). These relations are solved by a 2-parameter family of unit cell geometries. Each such QPM unit cell determines a corresponding photonic crystal reciprocal lattice. Since the poling function can be defined by its Fourier coefficients with respect to the reciprocal lattice, the problem of choosing a poling function splits naturally into two parts: (1) selection of a reciprocal lattice basis (or QPM unit cell); and (2) prescription of the poling function Fourier coefficients. Problem (1) has no obvious formulation in terms of harmonic generation efficiency, and is not considered here (we believe the choice of QPM unit cell should be based on fabrication constraints). On the other hand, problem (2) does have a formulation as a THG/FHG efficiency optimization problem and, remarkably, can be solved in full generality for an arbitrary QPM unit cell. For different choices of QPM unit cell, the resulting optimal poling patterns differ only by a linear transformation of the {

*x, y*}-plane.

## 2. Phase matching in two dimensions

6. A. H. Norton and C. M. de Sterke, “Optimal poling of nonlinear photonic crystals for frequency conversion,” Opt. Lett. **28**, 188 (2003). [CrossRef] [PubMed]

*q*ω has wave vector

**k**

_{q}=(cosθ

_{q}, sin θ

_{q}, 0)

*k*

_{q}, where

*k*

_{q}=

*n*

_{q}

*q*ω/

*c*and

*n*

_{q}=

*n*

_{e}(

*q*ω) is the extraordinary refractive index of the crystal at frequency

*q*ω. The fundamental propagates in the

*x*-direction, so θ

_{1}=0. The propagation angles θ

_{q},

*q*=2, 3, 4 are free parameters that may be negative or zero, provided the resulting phase matching vectors

**G**

_{1}and

**G**

_{2}of Fig. 1 remain linearly independent.

*z*-polarized and have amplitudes varying slowly with position,

*A*

_{q}(

*x*) follow from a perturbation analysis of Maxwell’s equations [7]. Each term in the equations is associated with a QPM relation,

*q=m+n*, and the phase matching vector

**G**belongs to the reciprocal lattice of the photonic crystal, so appears in the Fourier expansion of the poling function,

**x**=(

*x, y*),

**G**

_{ab}=

*a*

**G**

_{10}+

*b*

**G**

_{01}, and {

**G**

_{10},

**G**

_{01}} is the reciprocal lattice basis. The equations for

*A*

_{q}(

*x*) depend on the poling pattern only through a few of its Fourier coefficients, each being associated with a phase matching vector. In the following we suppose the two phase matching vectors are

## 3. Standard THG solutions

^{(2)}

_{333}=2

*d*

_{33}denotes the quadratic nonlinearity, assumed here to be independent of frequency (Kleinman symmetry). For lithium niobate we use χ=82×10

^{-12}m/V [8]. The perturbation theory leading to Eqs. (4)–(6) relies on expansion in the dimensionless small parameter ∊=χ

*E*

_{max}, where

*E*

_{max}is a typical strong electric field.

*A*

_{2}(0)=

*A*

_{3}(0)=0 and

*A*

_{1}(0) arbitrary. Our task is to find, amongst all THG solutions of all possible THG systems (differing by system parameters σ

_{j}), a solution and corresponding poling pattern, that are in some sense optimal. To do this we first establish the following result: Every THG solution can be obtained from some member of a 1-parameter space of

*standard solutions*, using symmetry transformations of the THG system.

*standard solutions*is defined as follows. The standard solution at parameter value ϕ has

*standard initial data*

*A*

_{2}(0)=

*A*

_{3}(0)=0 and

*U*=30 MW/cm

^{2}, and solves the THG system for which the Fourier parameters are given by

_{1}, σ

_{2}being Fourier coefficients of some poling function). The function σ(ϕ) clearly exists, since by Parseval’s theorem σ(ϕ)

^{2}≤∫

*p*(

*x*)

^{2}=1. An efficient numerical method for evaluating σ(ϕ) is given in [6

6. A. H. Norton and C. M. de Sterke, “Optimal poling of nonlinear photonic crystals for frequency conversion,” Opt. Lett. **28**, 188 (2003). [CrossRef] [PubMed]

*q*th harmonic through a plane perpendicular to the

*x*direction is

*U*

_{q}=

*n*

_{q}cos(θ

_{q})

*A**

_{q}

*A*

_{q}/(2

*c*µ

_{0}). By Eqs. (4)–(6) the total energy flux

*U*≡

*U*

_{1}+

*U*

_{2}+

*U*

_{3}, satisfies

*dU/dx*=0, so is uniform throughout the crystal.

_{j}real. To see this, note that a translation of the poling pattern introduces phase factors into its Fourier coefficients. If

*p*(x) ↦

*p*(x-x

_{0}) then

**x**

_{0}such that σ

_{j}are real. THG systems with translated poling patterns are physically equivalent. The associated phase symmetry is (10), by which solutions of complex σ

_{j}systems with standard initial data can be transformed into solutions of real σ

_{j}systems with standard initial data. For the latter, the solutions are such that

*A*

_{1}(

*x*) and

*A*

_{3}(

*x*) are real, and

*A*

_{2}(

*x*) is imaginary.

_{1}, σ

_{2}}-plane we need only consider one value of σ(ϕ)≡(

^{1/2}. Solutions for different σ(ϕ) can be obtained by rescaling in

*x*. Since we want the frequency conversion process to take place over the shortest length of crystal, we suppose σ(ϕ) is chosen as large as possible.

## 4. Σ-patterns

6. A. H. Norton and C. M. de Sterke, “Optimal poling of nonlinear photonic crystals for frequency conversion,” Opt. Lett. **28**, 188 (2003). [CrossRef] [PubMed]

**28**, 188 (2003). [CrossRef] [PubMed]

*n*is the unit normal to the hypersurface Σ, and δ

_{x}is a Fourier coefficient vector for the periodic extension of the Dirac delta function translated to position

**x**(see [6

**28**, 188 (2003). [CrossRef] [PubMed]

_{1}, σ

_{2}}-plane be

*n*=(cosψ, sin ψ). Then Eq. (13) evaluates to,

*p*(

**x**) is clearly 2π-periodic in the arguments

**G**

_{10}and

**G**

_{01}.

## 5. Optimal THG

*n*

_{1}=2.1434,

*n*

_{2}=2.1857 and

*n*

_{3}=2.2510 [9

9. D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, n_{e}, in congruent lithium niobate,” Opt. Lett. **22**, 1553 (1997). [CrossRef]

_{q}=1). One finds a critical value ψ

_{crit}≈ 41.42° for which lim

_{x→∞}

*U*

_{q}(

*x*)=0 for

*q*=1, 2. The critical solution is clearly optimal for THG. For a crystal of length

*x*=0.5 cm almost all (99.995%) input power is converted to the 3rd harmonic. Fig. 4 shows the corresponding amplitudes. The Σ-pattern for ψ=41.42° is shown in Fig. 2(b), and has Fourier coefficients σ

_{1}=0.4834 and σ

_{2}=0.3228.

10. 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 (2000). [CrossRef]

_{2}/Κ

_{1}≈0.651780083√3).

## 6. Optimal FHG

^{2}incident intensity, the minimum crystal length is

*l*=0.698 cm at ψ=30.7°. For 95% conversion,

*l*=0.465 cm at ψ=36.6°.

## 7. Conclusion

*x*=∞) is predicted. This solution is also optimal for finite length crystals, and there corresponds a unique poling pattern (the Σ-pattern in Fig. 2(b)) for which the THG process takes place most rapidly. The solutions for FHG and THG are qualitatively different. For FHG we suggest using the optimality criteria that 99% (or 95%) conversion is attained in the shortest length of crystal.

*x*=∞) is predicted, however, the conversion process is most rapid for the Σ-pattern. For all these patterns the Fourier coefficients have the critical ratio

## References and links

1. | V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. |

2. | N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled Lithium Niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. |

3. | A. Chowdhury, C. Staus, B. F. Boland, T. F. Kuech, and L. McCaughan, “Experimental demonstration of 1535–1555-nm simultaneous optical wavelength interchange with a nonlinear photonic crystal,” Opt. Lett. |

4. | N. G. R. Broderick, R. T. Bratfalean, T. M. Monro, D. J. Richardson, and C. M. de Sterke, “Temperature and wavelength tuning of 2nd. 3rd and 4th harmonic generation in a two dimensional hexagonally poled nonlinear crystal,” J. Opt. Soc. Am. B |

5. | M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. |

6. | A. H. Norton and C. M. de Sterke, “Optimal poling of nonlinear photonic crystals for frequency conversion,” Opt. Lett. |

7. | J. Kevorkian and J. D. Cole, |

8. | R. W. Boyd, |

9. | D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, n |

10. | 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. |

**OCIS Codes**

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4400) Nonlinear optics : Nonlinear optics, materials

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 7, 2003

Revised Manuscript: April 20, 2003

Published: May 5, 2003

**Citation**

Andrew Norton and C. de Sterke, "Two-dimensional poling patterns for 3rd and 4th harmonic generation," Opt. Express **11**, 1008-1014 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-9-1008

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### References

- V. Berger,�??Nonlinear photonic crystals,�?? Phys. Rev. Lett. 81, 4136-4139 (1998). [CrossRef]
- N.G.R. Broderick, G.W. Ross, H.L. Offerhaus, D.J. Richardson, and D.C. Hanna,�??Hexagonally poled Lithium Niobate:a two-dimensional nonlinear photonic crystal,�?? Phys. Rev. Lett. 84, 4345 �?? 4348 (2000). [CrossRef] [PubMed]
- A. Chowdhury, C.Staus, B.F. Boland, T.F. Kuech and L. McCaughan, �??Experimental demonstration of 1535 �??1555-nm simultaneous optical wavelength interchange with a nonlinear photonic crystal,�?? Opt. Lett. 26, 1353 �??1355 (2001). [CrossRef]
- A. Chowdhury, C.Staus, B.F. Boland, T.F. Kuech and L. McCaughan, �??Experimental demonstration of 1535 �??1555-nm simultaneous optical wavelength interchange with a nonlinear photonic crystal,�?? Opt. Lett. 26, 1353 �??1355 (2001). [CrossRef]
- M.M. Fejer, G.A. Magel, D.H. Jundt, and R.L. Byer, �??Quasi-phase-matched second harmonic generation:tuning and tolerances,�?? IEEE J. Quantum Electron. 28, 2631 �??2654 (1992). [CrossRef]
- A.H. Norton and C.M. de Sterke, �??Optimal poling of nonlinear photonic crystals for frequency conversion,�?? Opt. Lett. 28, 188 (2003). [CrossRef] [PubMed]
- J. Kevorkian and J.D. Cole, Perturbation methods in applied mathematics (Springer-Verlag, New York, 1981).
- R.W. Boyd, Nonlinear Optics (Academic Press, San Diego,1992).
- D.H. Jundt, �??Temperature-dependent Sellmeier equation for the index of refraction,ne incongruent lithium niobate,�?? Opt. Lett. 22, 1553 (1997). [CrossRef]
- C. Zhang,Y. Zhu, S. Yang, Y. Qin, S. Zhu, Y. Chen, H. Liu, and N. Ming, �??Crucial effects of coupling coeficients on quasi-phase-matched harmonic generation in an optical superlattice,�?? Opt. Lett. 25, 436 (2000). [CrossRef]

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