## Annular symmetry nonlinear frequency converters

Optics Express, Vol. 14, Issue 20, pp. 9371-9376 (2006)

http://dx.doi.org/10.1364/OE.14.009371

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### Abstract

We present a new type of two-dimensional nonlinear structure for quasi-phase matching. This structure has continuous rotational symmetry, and in contrary to the commonly used periodic structures, is not lattice shaped and has no translation symmetry. It is shown that this annular symmetry structure possesses interesting phase matching attributes that are significantly different than those of periodic structures. In particular, it enables simultaneous phase-matched frequency doubling of the same pump into several different directions. Moreover, it has extremely wide phase-mismatch tolerance, since a change in the phase matching conditions does not change the second harmonic power, but only changes its propagation direction. Several structures were fabricated using either the indirect e-beam method in LiNbO3 or the electric field poling method in stoichiometric LiTaO3, and their conversion efficiencies, as well as angular and thermal dependencies, were characterized by second harmonic generation.

© 2006 Optical Society of America

^{1}(1D), or in two dimensions

^{2,3,4}(2D) is widely used nowadays for quasi-phase matched (QPM) frequency conversion. These structures are lattice shaped, and have discrete rotational symmetry, i.e., efficient frequency conversion is achieved only for specific input and output angles. On the other hand, they usually have continuous translation symmetry: translating the frequency converters in a direction perpendicular to the input pump wave usually does not change the power of the generated waves. We present here a new type of nonlinear structure, one that possesses only continuous rotational symmetry and no translation symmetry.

^{1}. The Fourier transform of an infinite structure with a period of Λ consists of concentric impulse rings

^{5}with a period of 2

*π*/Λ. Figure 1 shows the structure in real space and its Fourier transform. Figure 1(b) also shows an example of the phase matching diagram for second harmonic generation (SHG), in the case where the pump passes exactly at the center of the structure. One can see that phase matching occurs symmetrically from both sides of the pump and that simultaneous phase matching at different orders and angles can occur. The different walk-off angles (angles between the fundamental and the second harmonic) at which we get phase matching are obtained by the law of cosines:

*G*=

_{n}*n*·2

*π*/Λ, and

*k*the wave-vectors of the fundamental and second harmonic, respectively.

^{ω},k^{2ω}^{6}in LiNbO

_{3}and electric field poling

^{7}in stoichiometirc LiTaO

_{3}(SLT). Two structures were used in second harmonic generation experiments:

_{3}crystal of congruent composition was used. A Shipley-1818 photo-resist of 2 µm thickness was spin coated over the crystal C- face. The electron exposure was performed by using a commercial electron beam lithography system (ELPHY Plus) adapted to a JEOL JSM 6400 scanning electron microscope. The acceleration voltage was selected to be 15kV corresponding with the photo-resist dielectric layer thickness so that the majority of incident electrons would stay trapped in the coating layer, therefore inducing high electric field that causes the inversion. The adaption of the electron energy and photoresist thickness was performed using Monte-Carlo simulations to calculate the effective penetration depth of electrons into the layer. The beam current was set to 0.5nA and the deposited surface charge density 300 µC/cm

^{2}. After exposure the resist layer was removed and the crystal was etched for 120 min using hydrofluoric (HF) acid at room temperature to reveal the formed domain structures. The duty cycle of this structure was kept at a relatively low value, around 10%, in order to reduce Coulomb deflection of the electron-beam, caused by the charging of the photo-resist

^{6}.

^{8}, using our standard electric field poling technique as described previously for 1D structures

^{9}. First, the annular structured pattern of photoresist was contact printed on the C+ face of the wafer from a lithographic mask. Then, the surface was coated by a uniform metallic layer.

^{2}. After poling, the photoresist and the metallic coatings were removed and the domain structure was revealed by HF etching. Most of the structure of the SLT crystal had a duty factor of 70% at the C+ surface and close to 80% at the C- surface, while the period length of 7.5 µm was precisely kept along the crystal. The end faces of the two crystals were polished for the optical measurements. Fig. 2 shows optical and AFM pictures of the surface morphology for the two different crystals.

*λ*=532

*nm*). The resulting diffraction pattern can be seen in Fig. 3. This pattern was highly periodic which indicates a stable period. The angle to the first circle of diffraction was

*θ*≈0.0705

*rad*. By using the well known law of diffraction (

*λ*=Λsinθ) we calculated the period of the structure to be Λ=7.55

*µm*, which agrees well with the design period of 7.5

*µm*.

^{10}. We used the Green’s function approach, which in our case is a simple spherical wave

^{11}:

*G*(

*R*)=

*e*(4

^{ikR}*πR*),

*R*=

*r-r*′. The second harmonic wave is obtained by convolution of the Green’s function with the source distribution — the non linear polarization

*E*is the input field (Gaussian beam with 80

^{ω}*µm*beam waist),

*ω*is the angular frequency of the input, c is the velocity of light, and d is, as defined earlier, a tensor element of the nonlinear susceptibility.

^{nd}order behaves very differently and is much less temperature dependent.

## Acknowledgments

## References and Links

1. | 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. |

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

3. | 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. |

4. | S. M. Saltiel, A. A. Sukhorukov, and Y. S. Kivshar, “Multistep parametric processes in nonlinear optics,” Prog. Optics |

5. | Isaac Amidror, “Fourier spectrum of radially periodic images,” J. Opt. Soc. Am. A |

6. | Y. Glickman, E. Winebrand, A. Arie, and G. Rosenman, “Electron-beam-induced domain poling in LiNbO |

7. | M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, “First-order quasi-phase-matched LiNbO |

8. | Y. Furukawa, K. Kitamura, E. Suzuki, and K. Niwa, “Stoichiometric LiTaO |

9. | A. Bruner, D. Eger, and S. Ruschin, “Second harmonic generation of green light in periodically-poled stoichiometric LiTaO |

10. | R.W. Boyd, |

11. | F.W. Byron and R.W. Fuller, |

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

**OCIS Codes**

(160.4330) Materials : Nonlinear optical materials

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4360) Nonlinear optics : Nonlinear optics, devices

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: July 3, 2006

Revised Manuscript: September 14, 2006

Manuscript Accepted: September 14, 2006

Published: October 2, 2006

**Citation**

Dror Kasimov, Ady Arie, Emil Winebrand, Gil Rosenman, Ariel Bruner, Pnina Shaier, and David Eger, "Annular symmetry nonlinear frequency converters," Opt. Express **14**, 9371-9376 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9371

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### References

- 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]
- 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]
- S. M. Saltiel, A. A. Sukhorukov and Y. S. Kivshar, "Multistep parametric processes in nonlinear optics," Prog. Opt. 47, 1-73 (2005). [CrossRef]
- I. Amidror, "Fourier spectrum of radially periodic images," J. Opt. Soc. Am. A 14, 816-826 (1997). [CrossRef]
- Y. Glickman, E. Winebrand, A. Arie, and G. Rosenman, "Electron-beam-induced domain poling in LiNbO3 for two-dimensional nonlinear frequency conversion," Appl. Phys. Lett. 88, 011103 (2006). [CrossRef]
- M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, "First-order quasi-phase-matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second harmonic generation," Appl. Phys. Lett. 62, 435-436 (1993). [CrossRef]
- Y. Furukawa, K. Kitamura,E. Suzuki and K. Niwa, "Stoichiometric LiTaO3 single crystal growth by double-crucible Czochralski method using automatic powder supply system," J. Cryst. Growth 197, 889-895 (1999). [CrossRef]
- A. Bruner, D. Eger and S. Ruschin, "Second harmonic generation of green light in periodically-poled stoichiometric LiTaO3 doped with MgO," J. Appl. Phys. 96, 7445-7449 (2004). [CrossRef]
- R.W. Boyd, Nonlinear Optics, second edition, (Academic Press, 2002), Chap. 2.
- F. W. Byron and R. W. Fuller, Mathematics of classical and quantum physics, (Dover Publications, 1992), Chap. 7.
- D. H. Jundt, "Temperature-dependent Sellmeier equations for the index of refraction ne, in congruent lithium niobate," Opt. Lett. 22, 1553-1555 (1997). [CrossRef]

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