## Investigation of thermally-induced phase mismatching in continuous-wave second harmonic generation: A theoretical model |

Optics Express, Vol. 18, Issue 18, pp. 18732-18743 (2010)

http://dx.doi.org/10.1364/OE.18.018732

Acrobat PDF (1222 KB)

### Abstract

A fraction of the fundamental beam energy deposited into nonlinear crystals to generate second harmonic waves (SHW) causes a temperature gradient within the crystal. This temperature inhomogeneity can alter the refractive index of the medium leading to a well-known effect called thermal dispersion. Therefore, the generated SHW suffers from thermal lensing and a longitudinal thermal phase mismatching. In this work by coupling the heat equation with second harmonic generation (SHG) formalism applied to type-II configuration along with walk-off effect, we investigate the continuous wave (CW) SHW beam profile and conversion efficiency when a non-linear KTP crystal is under induced thermal load. We have demonstrated for average and high powers, the thermal de-phasing lead to considerable reduction in SHG compared to an ideal case in which induced heat is neglected.

© 2010 OSA

## 1. Introduction

1. M. Okada and S. Ieiri, “Influence of self-induced thermal effects on second harmonic generation,” IEEE J. Quantum Electron. **7**(9), 469–470 (1971). [CrossRef]

2. J. D. Barry and C. J. Kennedy, “Thermo-optical effects of intracavity Ba_{2}Na(NbO_{3}) on a frequency-doubled Nd:YAG laser,” IEEE J. Quantum Electron. **11**, 575–579 (1975). [CrossRef]

_{2}Na(NbO

_{3})

_{5}crystal to convert the Nd:YAG laser output. Others discussed different techniques, for instance, Hon tried to compensate self-heating, such as electro-optics effect [3

3. D. T. Hon, “Electro-optical compensation for self-heating in CD*A during second-harmonic generation,” IEEE J. Quantum Electron. **12**(2), 148–151 (1976). [CrossRef]

4. D. T. Hon and H. Bruesselabach, “Beam Shaping to Suppress Phase Mismatch in High Power Second-Harmonic Generation,” IEEE J. Quantum Electron. **16**(12), 1356–1364 (1980). [CrossRef]

6. D. Eimerl, “High Average Power Harmonic Generation,” IEEE J. Quantum Electron. **23**(5), 575–592 (1987). [CrossRef]

7. S. Seidel and G. Mann, “Numerical modeling of thermal effects in nonlinear crystal for high power second harmonic generation,” Proc. SPIE **2989**, 204–214 (1997). [CrossRef]

*et al.*[8

8. J. Zheng, Sh. Zhao, Q. Wang, X. Zhang, and L. Chen, “Influence of thermal effect on KTP type-II phase-matching second-harmonic generation,” Opt. Commun. **199**(1-4), 207–214 (2001). [CrossRef]

*et al.*[9

9. M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. **56**(19), 1831–1833 (1990). [CrossRef]

*et al.*[10

10. S. V. Tovstonog, S. Kurimura, I. Suzuki, K. Takeno, S. Moriwaki, N. Ohmae, N. Mio, and T. Katagai, “Thermal effects in high-power CW second harmonic generation in Mg-doped stoichiometric lithium tantalate,” Opt. Express **16**(15), 11294–11299 (2008). [CrossRef] [PubMed]

*et al.*[11

11. K. H. Hong, C. J. Lai, A. Siddiqui, and F. X. Kärtner, “130-W picosecond green laser based on a frequency-doubled hybrid cryogenic Yb:YAG amplifier,” Opt. Express **17**(19), 16911–16919 (2009). [CrossRef] [PubMed]

12. R. Peng, L. Guo, X. Zhang, F. Li, Q. Cui, Y. Bo, Q. Peng, D. Cui, Z. Xu, and L. Tang, “43 W picosecond laser and second-harmonic generation experiment,” Opt. Commun. **282**(4), 611–613 (2009). [CrossRef]

13. M. Sabaeian and H. Nadgaran, “Bessel-Gauss beams: Investigation of thermal effects on their generation,” Opt. Commun. **281**(4), 672–678 (2008). [CrossRef]

14. Zh. Ren, Zh. Huang, S. Jia, Y. Ge, and J. Bai, “532 nm laser based on V-type doubly resonant intra-cavity frequency-doubling,” Opt. Commun. **282**(2), 263–266 (2009). [CrossRef]

15. Ch. Liu, Th. Riesbeck, X. Wang, J. Ge, Zh. Xiang, J. Chen, and H. J. Eichler, “Influence of spherical aberrations on the performance of dynamically stable resonators,” Opt. Commun. **281**, 5222–5228 (2008). [CrossRef]

_{4}(KTP) is an excellent nonlinear crystal with high nonlinear conversion coefficient, wide allowable angle, small walking-off angle and relatively high damage threshold [16

16. J. D. Bierlein and H. Vanherzeele, “Potassium titanyl phosphate: Properties and new applications,” J. Opt. Soc. Am. B **6**(4), 622–633 (1989). [CrossRef]

*θ*is the angle between ray propagation direction and the main axis of the crystal (Z), and

*ϕ*is the angle between ray propagation direction and X axis in X-Y plane [17

17. K. Asaumi, “Second-Harmonic Power of KTiOPO_{4} with Double Refraction,” Appl. Phys. B **54**(4), 265–270 (1992). [CrossRef]

18. P. K. Mukhopadhyay, S. K. Sharma, K. Ranganthan, P. K. Gupta, and T. P. S. Nathan, “Efficient and high-power intracavity frequency doubled diode-side-pumped Nd:YAG/KTP continuous wave (CW) green laser,” Opt. Commun. **259**(2), 805–811 (2006). [CrossRef]

20. J. D. Barry and C. J. Kennedly, “Thermooptical effects of intracavity Ba_{2}Na(NbO_{3})_{5} on a frequency doubling NdYAG laser,” IEEE J. Quantum Electron. **11**, 575–579 (1975). [CrossRef]

21. F. Q. Jia, Q. Zheng, Q. H. Xue, Y. K. Bu, and L. S. Qian, “High-power high-repetition-rate UV light at 355 nm generated by a diode-end-pumped passively Q-switched Nd:YAG laser,” Appl. Opt. **46**(15), 2975–2979 (2007). [CrossRef] [PubMed]

## 2. Method

7. S. Seidel and G. Mann, “Numerical modeling of thermal effects in nonlinear crystal for high power second harmonic generation,” Proc. SPIE **2989**, 204–214 (1997). [CrossRef]

### 2.1 Heat-Harmonic generation coupling

17. K. Asaumi, “Second-Harmonic Power of KTiOPO_{4} with Double Refraction,” Appl. Phys. B **54**(4), 265–270 (1992). [CrossRef]

17. K. Asaumi, “Second-Harmonic Power of KTiOPO_{4} with Double Refraction,” Appl. Phys. B **54**(4), 265–270 (1992). [CrossRef]

8. J. Zheng, Sh. Zhao, Q. Wang, X. Zhang, and L. Chen, “Influence of thermal effect on KTP type-II phase-matching second-harmonic generation,” Opt. Commun. **199**(1-4), 207–214 (2001). [CrossRef]

_{F}is their constant beam waist. Suppose at ambient temperature, the crystal is in a complete phased matched situation, the following system of equations can then describe the continuous wave sum frequency generation in MKS unit [22,23

23. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in nonlinear medium,” Phys. Rev. **127**(6), 1918–1939 (1962). [CrossRef]

*c*is the speed of light, and

*ω*, extraordinary wave at frequency

*ω*and extraordinary wave at frequency

24. M. Sabaeian, H. Nadgaran, and L. Mousave, “Analytical solution of the heat equation in a longitudinally pumped cubic solid-state laser,” Appl. Opt. **47**(13), 2317–2325 (2008). [CrossRef] [PubMed]

*T*is temperature,

*S*is heat source density in W/m

^{3}and

*K*is thermal conductivity in W/m/K. The proper boundary conditions for six faces of the crystal are [24

24. M. Sabaeian, H. Nadgaran, and L. Mousave, “Analytical solution of the heat equation in a longitudinally pumped cubic solid-state laser,” Appl. Opt. **47**(13), 2317–2325 (2008). [CrossRef] [PubMed]

^{2}/K is the heat transfer coefficient,

*a*,

*b*and

*c*are the dimensions of crystal as

8. J. Zheng, Sh. Zhao, Q. Wang, X. Zhang, and L. Chen, “Influence of thermal effect on KTP type-II phase-matching second-harmonic generation,” Opt. Commun. **199**(1-4), 207–214 (2001). [CrossRef]

25. K. Kato, “Parametric oscillation at 3.2 μm in KTP pumped at 1.064 μm,” IEEE J. Quantum Electron. **27**(5), 1137–1140 (1991). [CrossRef]

*λ*is wavelength in

26. D. Zhang, J. Lu, B. Feng, and J. Zhang, “Increased temperature bandwith of second harmonic generator using two KTiOPO_{4} crystals cut at different angles,” Opt. Commun. **281**(10), 2918–2922 (2008). [CrossRef]

27. Y. Bi, R. Li, Y. Feng, X. Lin, D. Cui, and Z. Xu, “Walk-off compensation of second harmonic generation in type-II phase-matched configuration with controled temperature,” Opt. Commun. **218**(1-3), 183–187 (2003). [CrossRef]

27. Y. Bi, R. Li, Y. Feng, X. Lin, D. Cui, and Z. Xu, “Walk-off compensation of second harmonic generation in type-II phase-matched configuration with controled temperature,” Opt. Commun. **218**(1-3), 183–187 (2003). [CrossRef]

27. Y. Bi, R. Li, Y. Feng, X. Lin, D. Cui, and Z. Xu, “Walk-off compensation of second harmonic generation in type-II phase-matched configuration with controled temperature,” Opt. Commun. **218**(1-3), 183–187 (2003). [CrossRef]

*T*, then convert them to

_{4} with Double Refraction,” Appl. Phys. B **54**(4), 265–270 (1992). [CrossRef]

26. D. Zhang, J. Lu, B. Feng, and J. Zhang, “Increased temperature bandwith of second harmonic generator using two KTiOPO_{4} crystals cut at different angles,” Opt. Commun. **281**(10), 2918–2922 (2008). [CrossRef]

*l*is also defined as

## 3. Results and Discussions

^{−1}and at 532nm is 100 m

^{−1}[7

7. S. Seidel and G. Mann, “Numerical modeling of thermal effects in nonlinear crystal for high power second harmonic generation,” Proc. SPIE **2989**, 204–214 (1997). [CrossRef]

_{0}pm/V and its ordinary and extraordinary refractive indices are

### 3.1 Thermal phase mismatching

_{F}= 0.2 mm and 0.3mm, respectively. The figure clearly shows that at low pump power, the amount of the thermal phase mismatching is rather small, whereas it increases considerably when the crystal is irradiated by more powerful pumps.

_{F}= 0.2mm this quantity for P = 240W is nearly 2.2 rad (in the midle of the crystal) whereas in Fig. (2-b) for ω

_{F}= 0.3mm is halved. Figures (3-a) -(3-c) can be very useful in addressing the serious impact of induced heat on SHG. The figures show phase mismatching increase by increasing pump power along z-axis of the crystal. What we can extract from the these figures is that not only the induced thermal load is a serious matter in the SHG systems, particularly for average and high input powers and small spot size, but also its magnitude is not that negligible so that a sophisticated cooling mechanism must be designed to expect considerable efficiency for SHG.

### 3-2 Efficiency

_{3}, versus z along the crystal axis (x = 0 and y = 0) for various ω

_{F}and total powers of fundamental waves P = P

^{o}+ P

^{e}are depicted. We compare these figures with that of non-thermal case, Fig. (4-d), to highlight the effects of thermal load, and to point out that the maximum efficiency for thermal case reaches about 57% for ω

_{F}= 0.1 mm whereas this value would be almost 100% for non-thermal case. For small spot sizes, the efficiency tends to revive in shorter lengths, severely (Fig. 4-a). Close attention to these figures, reveals that for small spot sizes we can obtain higher efficiencies. The variation of the maximum efficiency against the input power is investigated more in Fig. (5-a) .

_{F}to achieve acceptable efficiency.

**2989**, 204–214 (1997). [CrossRef]

_{F}= 0.1mm the higher efficiency is seen with some oscillations compared to ω

_{F}= 0.2mm and ω

_{F}= 0.3mm where show a smooth curve with smaller efficiencies. Our calculations show that for each spot size, the efficiency reaches to maxima for some specified input powers and then falls. The oscillatory behavior of efficiency becomes sever for smaller spot size. However, for non-thermal case, the efficiency reaches to a asymptotic value, i.e. to nearly 100% .

### 3-3 Temperature distribution

_{F}= 0.2mm and ω

_{F}= 0.3mm, respectively. In Figs. (8) we see that from the head of the crystal to the point that the SHW efficiency has its way towards maximum efficiency, i.e. from 0 to nearly 7 mm in Fig. (4-b) and (4-c), the temperature increase shows a positive gradient. This is well attributed to the absorption of SHW that is one-order of magnitude larger than that of fundamental wave and therefore generates an appreciable amount of heat. This effect is also observed by Seidel and Mann [7

**2989**, 204–214 (1997). [CrossRef]

## 4. Conclusion

## References and links

1. | M. Okada and S. Ieiri, “Influence of self-induced thermal effects on second harmonic generation,” IEEE J. Quantum Electron. |

2. | J. D. Barry and C. J. Kennedy, “Thermo-optical effects of intracavity Ba |

3. | D. T. Hon, “Electro-optical compensation for self-heating in CD*A during second-harmonic generation,” IEEE J. Quantum Electron. |

4. | D. T. Hon and H. Bruesselabach, “Beam Shaping to Suppress Phase Mismatch in High Power Second-Harmonic Generation,” IEEE J. Quantum Electron. |

5. | E. Moses, H. Brusselbach, D. Stovall, and D. T. Hon, in |

6. | D. Eimerl, “High Average Power Harmonic Generation,” IEEE J. Quantum Electron. |

7. | S. Seidel and G. Mann, “Numerical modeling of thermal effects in nonlinear crystal for high power second harmonic generation,” Proc. SPIE |

8. | J. Zheng, Sh. Zhao, Q. Wang, X. Zhang, and L. Chen, “Influence of thermal effect on KTP type-II phase-matching second-harmonic generation,” Opt. Commun. |

9. | M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. |

10. | S. V. Tovstonog, S. Kurimura, I. Suzuki, K. Takeno, S. Moriwaki, N. Ohmae, N. Mio, and T. Katagai, “Thermal effects in high-power CW second harmonic generation in Mg-doped stoichiometric lithium tantalate,” Opt. Express |

11. | K. H. Hong, C. J. Lai, A. Siddiqui, and F. X. Kärtner, “130-W picosecond green laser based on a frequency-doubled hybrid cryogenic Yb:YAG amplifier,” Opt. Express |

12. | R. Peng, L. Guo, X. Zhang, F. Li, Q. Cui, Y. Bo, Q. Peng, D. Cui, Z. Xu, and L. Tang, “43 W picosecond laser and second-harmonic generation experiment,” Opt. Commun. |

13. | M. Sabaeian and H. Nadgaran, “Bessel-Gauss beams: Investigation of thermal effects on their generation,” Opt. Commun. |

14. | Zh. Ren, Zh. Huang, S. Jia, Y. Ge, and J. Bai, “532 nm laser based on V-type doubly resonant intra-cavity frequency-doubling,” Opt. Commun. |

15. | Ch. Liu, Th. Riesbeck, X. Wang, J. Ge, Zh. Xiang, J. Chen, and H. J. Eichler, “Influence of spherical aberrations on the performance of dynamically stable resonators,” Opt. Commun. |

16. | J. D. Bierlein and H. Vanherzeele, “Potassium titanyl phosphate: Properties and new applications,” J. Opt. Soc. Am. B |

17. | K. Asaumi, “Second-Harmonic Power of KTiOPO |

18. | P. K. Mukhopadhyay, S. K. Sharma, K. Ranganthan, P. K. Gupta, and T. P. S. Nathan, “Efficient and high-power intracavity frequency doubled diode-side-pumped Nd:YAG/KTP continuous wave (CW) green laser,” Opt. Commun. |

19. | R. G. Smith, “Theory of intraccavity optical second harmonic generation,” IEEE J. Quantum Electron. |

20. | J. D. Barry and C. J. Kennedly, “Thermooptical effects of intracavity Ba |

21. | F. Q. Jia, Q. Zheng, Q. H. Xue, Y. K. Bu, and L. S. Qian, “High-power high-repetition-rate UV light at 355 nm generated by a diode-end-pumped passively Q-switched Nd:YAG laser,” Appl. Opt. |

22. | R. W. Boyd, |

23. | J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in nonlinear medium,” Phys. Rev. |

24. | M. Sabaeian, H. Nadgaran, and L. Mousave, “Analytical solution of the heat equation in a longitudinally pumped cubic solid-state laser,” Appl. Opt. |

25. | K. Kato, “Parametric oscillation at 3.2 μm in KTP pumped at 1.064 μm,” IEEE J. Quantum Electron. |

26. | D. Zhang, J. Lu, B. Feng, and J. Zhang, “Increased temperature bandwith of second harmonic generator using two KTiOPO |

27. | Y. Bi, R. Li, Y. Feng, X. Lin, D. Cui, and Z. Xu, “Walk-off compensation of second harmonic generation in type-II phase-matched configuration with controled temperature,” Opt. Commun. |

**OCIS Codes**

(140.6810) Lasers and laser optics : Thermal effects

(190.0190) Nonlinear optics : Nonlinear optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: May 26, 2010

Revised Manuscript: July 10, 2010

Manuscript Accepted: July 11, 2010

Published: August 18, 2010

**Citation**

Mohammad Sabaeian, Laleh Mousave, and Hamid Nadgaran, "Investigation of thermally-induced phase mismatching in continuous-wave second
harmonic generation: a theoretical model," Opt. Express **18**, 18732-18743 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-18-18732

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### References

- M. Okada and S. Ieiri, “Influence of self-induced thermal effects on second harmonic generation,” IEEE J. Quantum Electron. 7(9), 469–470 (1971). [CrossRef]
- J. D. Barry and C. J. Kennedy, “Thermo-optical effects of intracavity Ba2Na(NbO3) on a frequency-doubled Nd:YAG laser,” IEEE J. Quantum Electron. 11, 575–579 (1975). [CrossRef]
- D. T. Hon, “Electro-optical compensation for self-heating in CD*A during second-harmonic generation,” IEEE J. Quantum Electron. 12(2), 148–151 (1976). [CrossRef]
- D. T. Hon and H. Bruesselabach, “Beam Shaping to Suppress Phase Mismatch in High Power Second-Harmonic Generation,” IEEE J. Quantum Electron. 16(12), 1356–1364 (1980). [CrossRef]
- E. Moses, H. Brusselbach, D. Stovall, and D. T. Hon, in Proceeding of Soc. Opt. Quantum Electron. Conf. Lasers, Appl., (Orlando, FL. 1978).
- D. Eimerl, “High Average Power Harmonic Generation,” IEEE J. Quantum Electron. 23(5), 575–592 (1987). [CrossRef]
- S. Seidel and G. Mann, “Numerical modeling of thermal effects in nonlinear crystal for high power second harmonic generation,” Proc. SPIE 2989, 204–214 (1997). [CrossRef]
- J. Zheng, Sh. Zhao, Q. Wang, X. Zhang, and L. Chen, “Influence of thermal effect on KTP type-II phase-matching second-harmonic generation,” Opt. Commun. 199(1-4), 207–214 (2001). [CrossRef]
- M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990). [CrossRef]
- S. V. Tovstonog, S. Kurimura, I. Suzuki, K. Takeno, S. Moriwaki, N. Ohmae, N. Mio, and T. Katagai, “Thermal effects in high-power CW second harmonic generation in Mg-doped stoichiometric lithium tantalate,” Opt. Express 16(15), 11294–11299 (2008). [CrossRef] [PubMed]
- K. H. Hong, C. J. Lai, A. Siddiqui, and F. X. Kärtner, “130-W picosecond green laser based on a frequency-doubled hybrid cryogenic Yb:YAG amplifier,” Opt. Express 17(19), 16911–16919 (2009). [CrossRef] [PubMed]
- R. Peng, L. Guo, X. Zhang, F. Li, Q. Cui, Y. Bo, Q. Peng, D. Cui, Z. Xu, and L. Tang, “43 W picosecond laser and second-harmonic generation experiment,” Opt. Commun. 282(4), 611–613 (2009). [CrossRef]
- M. Sabaeian and H. Nadgaran, “Bessel-Gauss beams: Investigation of thermal effects on their generation,” Opt. Commun. 281(4), 672–678 (2008). [CrossRef]
- Zh. Ren, Zh. Huang, S. Jia, Y. Ge, and J. Bai, “532 nm laser based on V-type doubly resonant intra-cavity frequency-doubling,” Opt. Commun. 282(2), 263–266 (2009). [CrossRef]
- Ch. Liu, Th. Riesbeck, X. Wang, J. Ge, Zh. Xiang, J. Chen, and H. J. Eichler, “Influence of spherical aberrations on the performance of dynamically stable resonators,” Opt. Commun. 281, 5222–5228 (2008). [CrossRef]
- J. D. Bierlein and H. Vanherzeele, “Potassium titanyl phosphate: Properties and new applications,” J. Opt. Soc. Am. B 6(4), 622–633 (1989). [CrossRef]
- K. Asaumi, “Second-Harmonic Power of KTiOPO4 with Double Refraction,” Appl. Phys. B 54(4), 265–270 (1992). [CrossRef]
- P. K. Mukhopadhyay, S. K. Sharma, K. Ranganthan, P. K. Gupta, and T. P. S. Nathan, “Efficient and high-power intracavity frequency doubled diode-side-pumped Nd:YAG/KTP continuous wave (CW) green laser,” Opt. Commun. 259(2), 805–811 (2006). [CrossRef]
- R. G. Smith, “Theory of intraccavity optical second harmonic generation,” IEEE J. Quantum Electron. 6(4), 215–223 (1970). [CrossRef]
- J. D. Barry and C. J. Kennedly, “Thermooptical effects of intracavity Ba2Na(NbO3)5 on a frequency doubling NdYAG laser,” IEEE J. Quantum Electron. 11, 575–579 (1975). [CrossRef]
- F. Q. Jia, Q. Zheng, Q. H. Xue, Y. K. Bu, and L. S. Qian, “High-power high-repetition-rate UV light at 355 nm generated by a diode-end-pumped passively Q-switched Nd:YAG laser,” Appl. Opt. 46(15), 2975–2979 (2007). [CrossRef] [PubMed]
- R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press 2008), Chapt. 2.
- J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in nonlinear medium,” Phys. Rev. 127(6), 1918–1939 (1962). [CrossRef]
- M. Sabaeian, H. Nadgaran, and L. Mousave, “Analytical solution of the heat equation in a longitudinally pumped cubic solid-state laser,” Appl. Opt. 47(13), 2317–2325 (2008). [CrossRef] [PubMed]
- K. Kato, “Parametric oscillation at 3.2 μm in KTP pumped at 1.064 μm,” IEEE J. Quantum Electron. 27(5), 1137–1140 (1991). [CrossRef]
- D. Zhang, J. Lu, B. Feng, and J. Zhang, “Increased temperature bandwith of second harmonic generator using two KTiOPO4 crystals cut at different angles,” Opt. Commun. 281(10), 2918–2922 (2008). [CrossRef]
- Y. Bi, R. Li, Y. Feng, X. Lin, D. Cui, and Z. Xu, “Walk-off compensation of second harmonic generation in type-II phase-matched configuration with controled temperature,” Opt. Commun. 218(1-3), 183–187 (2003). [CrossRef]

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