## Simultaneous phase matching and internal interference of two second-order nonlinear parametric processes

Optics Express, Vol. 14, Issue 24, pp. 11756-11765 (2006)

http://dx.doi.org/10.1364/OE.14.011756

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### Abstract

We demonstrate the simultaneous generation and internal interference of two second-order parametric processes in a single nonlinear quadratic crystal. The two-frequency doubling processes are Type 0 (two extraordinary fundamental waves generate an extraordinary secondharmonic wave) and Type I (two ordinary fundamental waves generate an extraordinary second-harmonic wave) parametric interactions. The phasematching conditions for both processes are satisfied in a single periodically poled grating in LiNbO_{3} using quasi-phase-matching (QPM) vectors with different orders. We observe an interference of two processes, and compare the results with the theoretical analysis. We suggest several applications of this effect such as polarization-independent frequency doubling and a method for stabilizing the level of the generated second-harmonic signal.

© 2006 Optical Society of America

## 1. Introduction

## 2. Theoretical background

*Δk*

_{1}=0 and

*Δk*

_{0}=0 for the Type I and Type 0, respectively, should be satisfied simultaneously. Lithium niobate (LiNbO

_{3}) is a popular material for implementing quadratic nonlinear processes due to its good nonlinear response, large transparency window covering the visible and near-mid infrared, its availability as a cheap and high quality material, and its associated technologies such as proton exchange waveguides and periodic poling for QPM. For QPM of a DPM process in poled LiNbO

_{3}, two phase-matching conditions must be simultaneously satisfied. The first condition is for SHG with ordinary fundamental waves (y-polarized), which we denote as Y

_{1}Y

_{1}-Z

_{2}. The second condition is for SHG with extraordinary (z-polarized) fundamental waves, which we denote as Z

_{1}Z

_{1}-Z

_{2}. The QPM conditions for these processes is

*Δk*

_{1}=

*k*

_{2z}-2

*k*

_{y1}-

*m*

_{1}

*G*and Δ

*k*

_{o}=

*k*

_{z2}-2

*k*

_{z1}-

*m*

_{2}

*G*, where

*m*

_{1}and

*m*

_{2}are the orders of the QPM vector,

*G*=2

*π*/Λ and Λ is the period of the QPM grating. Both the processes generate extraordinary (z-polarized) SH waves.

_{4}laser used in these experiments, we find two suitable periods in periodically poled LiNbO

_{3}for simultaneous PM of two SHG processes : (a) the period 45.77 µm at 179.53°C with

*m*

_{1}=1 and

*m*

_{2}=7, and (b) the period of 32.03 µm at 242.1°C with

*m*

_{1}=1 and

*m*

_{2}=5. In this paper, we present the experimental results for a 45.77 µm (

*m*

_{1}=1,

*m*

_{2}=7) periodically poled grating such that the DPM process is Y

_{1}Y

_{1}-Z

_{2}(1

^{st}order): Z

_{1}Z

_{1}-Z

_{2}(7

^{th}order).

_{3}), assuming no losses for the fundamental and SH waves and neglecting possible temporal delay effects, have the following form [8

8. G. Assanto, I. Torelli, and S. Trillo, “All-optical processing by means of vectorial interactions in 2nd-order cascading -novel approaches,” Opt. Lett. **19**, 1720–1722. (1994). [CrossRef] [PubMed]

*z*axis. S is the second harmonic field, which in LiNbO

_{3}is polarized along the

*z*axis. Furthermore,

*d*

_{33}=

*d*

_{zzz}and

*d*

_{31}=

*d*

_{yyz}are the LiNbO

_{3}nonlinear coefficients,

*f*

_{1}=2/(

*m*

_{1}

*π*);

*f*

_{2}=2/(

*m*

_{2}

*π*) are the QPM reduction factors. The mismatch parameters Δ

*k*

_{1}and Δ

*k*

_{0}defined above depend strongly on the temperature. For the numerical calculations with this system of Eqs. (1–3), we replace

*f*

_{1}and

*f*

_{2}with

*f*

_{1}=

*f*

_{2}=

*F*(

*x*) (the poling function), describing the actual shape of the QPM grating. The phase mismatches

*Δk*

_{1}and

*Δk*

_{0}have been calculated using Sellmeier data from [12].

*ϕ*(

*T*)=Δ

*k*

_{1}-Δ

*k*

_{0}=2

*π*[2(

*n*

_{o}-

*n*

_{e})/

*λ*+

*m*

_{2}-

*m*

_{1}/Λ]. At the QPM temperature required for DPM,

*ϕ*(

*T*)=0 since

*Δk*

_{0}=0 and

*Δk*

_{1}=0. If A and B are obtained with a half wave plate (HWP) producing a linearly polarized fundamental rotated at angle

*θ*, then the amplitude of the two orthogonal fundamental components will be

*A*=

*A*

_{0}

*cos2θ*and

*B*=

*A*

_{0}

*sin2θ*for an initial amplitude of

*A*

_{0}. A quarter wave plate (QWP) situated after the HWP can introduce a

*π*/2 phase shift between the two fundamental waves. This is introduced in Eq. (4) by the parameter

*g*which is

*g*=- 1 with the QWP and

*g*=+1 without the QWP. In Fig. 1(a) the dependences of the second harmonic intensity on the HWP angle are shown for the cases of

*g*=±1, and for two different ratios of the contributions from both fundamental waves based on the upper and lower limits for the published values of the nonlinear coefficients (

*d*

_{33}/

*7d*

_{31}=0.5 and

*d*

_{33}/

*7d*

_{31}=0.9). It is clearly seen how the internal interference of the two SH waves after the introduction of QWP causes the generated SH signal to be reduced to zero when both processes are generating a SH waves with equal amplitude but out of phase with π. Assuming a 50 % duty cycle the HWP the angle at which the SH signal will be reduce to zero will be

*ϕ*(

*T*)=0. If the processes are not perfectly phase matched such that

*Δk*

_{1},

*Δk*

_{0}≠0 then

*ϕ*(

*T*)=0 and will have a finite value and affect both the phase and the efficiency of the process described by Eq. (4). In Fig. 1(b) the second harmonic intensity is plotted as a function of temperature tuning with the HWP rotation angle fixed at 22.5°. The two different ratios of the contributions from both fundamentals are shown. The interference effects are present and lead to an increase or decrease of the net SH signal at the peak of the temperature curve. In the high power regime it is no longer possible to neglect the depletion of the fundamental waves. In this case the set of Eqs. (1–3) is solved numerically with the poling function on

*F*(

*x*)=

*signum*(sin(2

*πx*/Λ)). Depending on the signs of the two nonlinearities (or the phase shift between the two fundamental fields) one can observe a collective contribution of both fundamentals to the SH efficiency, or the exchange of energy between the fundamental fields. The resulting dependence of the intensity of the two fundamental waves and the SH signal is shown in Fig. 2 as a function of the normalized crystal length

*L*/

*L*

_{nl}(with

*L*

_{nl}=1/(

*σ*

_{1}

*A*

_{0})). Figure 2(a) shows the case of the same signed nonlinearities with π/2 shifted fundamentals such that g=- 1 (or equivalently differently signed nonlinearties with fundamentals in phase) and Fig. 2(b) shows the same sign nonlinearities with fundamentals in phase such that g=+1 (or equivalently different signed nonlinearities with π/2 shifted fundamentals). The power exchange between the fundamental waves in Fig. 2(a) is the result of a third order process due to

*χ*

^{(2)}:

*χ*

^{(2)}cascading. This cascaded third order process is analogous to a four wave mixing process. The equivalent process can be symbolically marked as B:AB*A. To illustrate this interpretation let us assume that one of the pumps (B) is much weaker than the other and neglect the depletion only for the stronger fundamental wave. Then for the fundamental wave B at exact temperature for DPM we have:

*σ*

_{3}

*σ*

_{2}is proportional to the product,

*χ*

^{(2)}:

*χ*

^{(2)}cascading. For the case where B is not weak compared to A we can numerically integrate the system of Eq. (1–3) to gain insight into the parametric process taking place. Figure 2 demonstrates the case where A and B are initially of equal amplitude.

## 3. Experimental setup and samples

_{3}with a rapid prototyping method utilizing laser micro-machined topographical electrode geometries [13

13. B. F. Johnston and M. J. Withford, “Dynamics of domain inversion in LiNbO_{3} poled using topographic electrode geometries,” Appl. Phys. Lett. **86**, 262901 (2005). [CrossRef]

15. M. Reich, F. Korte, C. Fallnich, H. Welling, and A. Tunnermann, “Electrode geometries for periodic poling of ferroelectric materials,” Opt. Lett. **23**, 1817–1819 (1998). [CrossRef]

^{th}order QPM (36%), and causes a decrease in the efficiency of the Type I SHG process of about 15 %.

## 4. Experimental results and discussions

_{1}Y

_{1}-Z

_{2}(1st order) and (ii) Type 0 - Z

_{1}Z

_{1}-Z

_{2}(7th order). The theoretical curves are rescaled to match the maximum of the Type I SHG process. In the calculations we use the data for

*d*

_{33}=- 25.3 pm/V and

*d*

_{31}=- 4.6 pm/V from [14

14. I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B **14**, 2268–2294 (1997). [CrossRef]

_{1}Z

_{1}-Z

_{2}frequency doubling process is much more sensitive to the imperfections in the grating, because of the 7

^{th}order QPM used. Figure 4 shows the experimental phase-matching temperature curves, taken separately launching only one of the fundamental polarizations in the crystal for each measurement. In the grating with the 45.75 µm period the peak PM temperatures for the Type I and Type 0 processes are separated by ~ 3.5°C, which corresponds to a deviation of about 50 nm from the exact period required for DPM. The 45.79 µm period grating deviates from the exact DPM period by only 6 nm. The measured SH power for the Type I process for 350 mW input power reaches 21 mW, which corresponds to 6 % external (measured net efficiency) and 8.2 % internal efficiency for this process, (where the internal efficiency is calculated from the external efficiency, allowing for the Fresnel loses at the surfaces which may be recovered if anti-reflection coatings are used). The sufficient overlap in the PM curves demonstrated in these measurements allowed us to study the effects of simultaneous action of two frequency doubling processes by launching both components of the fundamental beam (Y and Z polarized) into the sample. Both fundamental fields are launched simultaneously into the crystal by using a HWP in front of the linearly polarized laser source. Changing the relative amplitudes of the two fundamental polarization components was realized by rotating the HWP. With the addition of a QWP, inserted after the HWP, we were able to introduce a π/2 phase shift between the two fundamentals. Consequently both the Type I and Type 0 SHG process were active in the PPLN, producing second harmonic fields that can interfere constructively or destructively. The sample temperature for the HWP rotation experiments was chosen to correspond to the peak conversion of the Type I process. The results of these measurements together with the theoretical curves are shown in Fig. 5.

*ϕ*~0), and the two generated SH fields interfere constructively when the two fundamental fields are in phase and destructively for the case of π/2 phase shifted fundamentals. This behavior indicates that the two nonlinear coefficients

*d*

_{33}and

*d*

_{31}have the same sign, in agreement with previous measurements [12,14

14. I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B **14**, 2268–2294 (1997). [CrossRef]

*Δk*

_{0}

*L*=- 1.14π at the point of exact PM for the Type I process that is equivalent to introducing phase shift between the two SH fields equal to

*ϕ*=- 1.14π [Eq. (4)]. Existence of this phase shift causes reverse behavior of the two SH interference curves. The theoretical curves derived using the phase mismatch indicated by the temperature tuning curves plotted in Fig. 4 show good agreement with the experimental ones.

## 5. Possible applications

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

*Type I*)=

*Type*0), which will require the use of a duty cycle factor that is found from the numerical solution of the equation

*φ*(

*V*) which can partially reduce the efficiency of the generated SH signal to be at the desired application level. If for some reason the SH signal increases, the feedback signal changes the voltage

*V*of the Pockels cell in the direction to reduce the level of the generated SH and vice versa. In the light of this application and assuming that both processes are phase-matched simultaneously Eq. (4) will be:

_{3}Pockels cell, should be oriented such that its axes coincide with those of the frequency doubling PPLN. In LiNbO

_{3}the electrooptic coefficients are of substantial different magnitudes, r

_{33}≫r

_{31}, such that any voltage applied to the Pockels cell will predominantly change the phase of one of the polarizations, leading to a voltagedependent phase difference between the two fundamental waves. The phase induced by the PC will then control the SH power through the dependence in Eq. (8). Also, as we have already noted, another application for the internal interference of the SH fields is to measure the relative sign and relative magnitude of the χ

^{(2)}tensor components in the QPM crystal.

## 6. Conclusions

_{3}crystal, the so-called

*internal second-harmonic interference*. The same effect is expected to occur in other types of ferroelectric crystals such as KTP and LiTaO

_{3}. We have suggested several applications of this novel parametric effect such as the measurement of the sign and relative magnitudes of the second-order nonlinearity, polarization-independent frequency doubling, and stabilized generation of the second harmonic fields.

## Acknowledgments

## References and links

1. | R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, “Selffocusing and self-defocusing by cascaded 2nd-order effects in KTP,” Opt. Lett. |

2. | S M Saltiel, A. A. Sukhorukov, and Yu S Kivshar, “Multistep parametric processes in nonlinear optics,” Progress in Optics |

3. | V. Pasiskevicius, S. J. Holmgren, S. Wang, and F. Laurell, “Simultaneous second-harmonic generation with two orthogonal polarization states in periodically poled KTP,” Opt. Lett. |

4. | C. G. Trevino-Palacios, G. I. Stegeman, M. P. Demicheli, P. Baldi, S. Nouh, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Intensity-dependent mode competition in 2nd-harmonic generation in multimode wave-guides,” Appl. Phys. Lett. |

5. | G. I. Petrov, O. Albert, J. Etchepare, and S. M. Saltiel, “Cross-polarized wave generation by effective cubic nonlinear optical interaction,” Opt. Lett. |

6. | S. Saltiel and Y. Deyanova, “Polarization switching as a result of cascading of two simultaneously phasematched quadratic processes,” Opt. Lett. |

7. | A. DeRossi, C. Conti, and G. Assanto, “Mode interplay via quadratic cascading in a lithium niobate waveguide for all-optical processing,” Opt. Quantum Electron. |

8. | G. Assanto, I. Torelli, and S. Trillo, “All-optical processing by means of vectorial interactions in 2nd-order cascading -novel approaches,” Opt. Lett. |

9. | Yu. S Kivshar, A. A. Sukhorukov, and S. M. Saltiel, “Two-color multistep cascading and parametric soliton-induced waveguides,” Phys. Rev. E |

10. | S. G Grechin, V. G. Dmitriev, and Yu. V. Yur’ev, “Second-harmonic generation under conditions of simultaneous phase-matched and quasi-phase-matched interactions in nonlinear crystals with a regular domain structure,” Kvant. Elektron.26, 155–157 (1999) in Russian [English translation: Quantum Electron. 29, 155–157 (1999). [CrossRef] |

11. | Y. Chen, R. Wu, X. Zeng, Y. Xia, and X. Chen, “Type I quasi-phase-matched blue second-harmonic generation with different polarizations in periodically poled LiNbO |

12. | G. G. Gurzadian, V. G. Dmitriev, and D. N. Nikogosian, “Handbook of Nonlinear Optical Crystals,” 3rd ed., Vol. 64 of Springer Series in |

13. | B. F. Johnston and M. J. Withford, “Dynamics of domain inversion in LiNbO |

14. | I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B |

15. | M. Reich, F. Korte, C. Fallnich, H. Welling, and A. Tunnermann, “Electrode geometries for periodic poling of ferroelectric materials,” Opt. Lett. |

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

**OCIS Codes**

(190.4360) Nonlinear optics : Nonlinear optics, devices

(230.4320) Optical devices : Nonlinear optical devices

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: September 19, 2006

Revised Manuscript: November 2, 2006

Manuscript Accepted: November 3, 2006

Published: November 27, 2006

**Citation**

Benjamin F. Johnston, Peter Dekker, Michael J. Withford, Solomon M. Saltiel, and Yuri S. Kivshar, "Simultaneous phase matching and internal interference of two second-order nonlinear parametric processes," Opt. Express **14**, 11756-11765 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-24-11756

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### References

- R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, "Self-focusing and self-defocusing by cascaded 2nd-order effects in KTP," Opt. Lett. 17, 28 - 30 (1992). [CrossRef] [PubMed]
- S. M. Saltiel, A. A. Sukhorukov, and Yu. S. Kivshar, "Multistep parametric processes in nonlinear optics," in Progress in Optics, E. Wolf, ed., 47, 1-73 (2005). (Elsevier, Amsterdam, 2005), pp. 1-73. [CrossRef]
- V. Pasiskevicius, S. J. Holmgren, S. Wang, and F. Laurell, "Simultaneous second-harmonic generation with two orthogonal polarization states in periodically poled KTP," Opt. Lett. 27, 1628-1630 (2002). [CrossRef]
- C. G. Trevino-Palacios, G. I. Stegeman, M. P. Demicheli, P. Baldi, S. Nouh, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, "Intensity-dependent mode competition in 2nd-harmonic generation in multimode wave-guides," Appl. Phys. Lett. 67, 170 - 172 (1995). [CrossRef]
- G. I. Petrov, O. Albert, J. Etchepare, and S. M. Saltiel, "Cross-polarized wave generation by effective cubic nonlinear optical interaction," Opt. Lett. 26,355 - 357 (2001). [CrossRef]
- S. Saltiel, and Y. Deyanova, "Polarization switching as a result of cascading of two simultaneously phase-matched quadratic processes," Opt. Lett. 24, 1296 - 1298 (1999). [CrossRef]
- A. DeRossi, C. Conti, and G. Assanto, "Mode interplay via quadratic cascading in a lithium niobate waveguide for all-optical processing," Opt. Quantum Electron. 29, 53 - 63 (1997). [CrossRef]
- G. Assanto, I. Torelli, and S. Trillo, "All-optical processing by means of vectorial interactions in 2nd-order cascading -novel approaches," Opt. Lett. 19, 1720-1722. (1994). [CrossRef] [PubMed]
- Yu. S Kivshar, A. A. Sukhorukov, and S. M. Saltiel, "Two-color multistep cascading and parametric soliton-induced waveguides," Phys. Rev. E 60, R5056 - R5059. (1999). [CrossRef]
- S. G Grechin, V. G. Dmitriev, and Yu. V. Yur’ev, "Second-harmonic generation under conditions of simultaneous phase-matched and quasi-phase-matched interactions in nonlinear crystals with a regular domain structure," Kvant. Elektron. 26, 155 - 157 (1999) in Russian [English translation: Quantum Electron. 29, 155 - 157 (1999)]. [CrossRef]
- Y. Chen, R. Wu, X. Zeng, Y. Xia, and X. Chen, "Type I quasi-phase-matched blue second-harmonic generation with different polarizations in periodically poled LiNbO3," Opt. Laser Technol. 38, 19 - 22 (2006). [CrossRef]
- G. G. Gurzadian, V. G. Dmitriev, and D. N. Nikogosian, "Handbook of Nonlinear Optical Crystals," 3rd ed., Vol. 64 of Springer Series in Optical Sciences (Springer-Verlag, New York, 1999).
- B. F. Johnston and M. J. Withford, "Dynamics of domain inversion in LiNbO3 poled using topographic electrode geometries," Appl. Phys. Lett. 86, 262901 (2005). [CrossRef]
- I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, "Absolute scale of second-order nonlinear-optical coefficients," J. Opt. Soc. Am. B 14, 2268-2294 (1997). [CrossRef]
- M. Reich, F. Korte, C. Fallnich, H. Welling, and A. Tunnermann, "Electrode geometries for periodic poling of ferroelectric materials," Opt. Lett. 23, 1817-1819 (1998). [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]

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