## Phase velocity nonuniformity-resulted beam patterns in difference frequency generation

Optics Express, Vol. 15, Issue 8, pp. 5050-5058 (2007)

http://dx.doi.org/10.1364/OE.15.005050

Acrobat PDF (654 KB)

### Abstract

The evolution of the difference frequency generation between a planar pump wave and a focused signal wave has been numerically investigated in this paper. We show that, at the difference frequency wave, various beam patterns such as ring and moon-like, are resulted due to the nonuniform distribution of phase velocity in the focused signal wave. The subluminal and superluminal regions can be identified by the intersection of two generated beam profiles that correspond to a pair of phase-mismatches with equal value but opposite signs.

© 2007 Optical Society of America

## 1. Introduction

1. G. D. Boyd, A. Ashkin, J. M. Dziedzic, and D. A. Kleinman, “Second-harmonic generation of light with double refraction,” Phys. Rev. **137**, A1305–A1320 (1965). [CrossRef]

_{opt}>0) along the beam axis that compromises the PM conditions among on-axis SHG and off-axis frequency mixing with diverging (or converging) portions of the beam [2

2. G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. App. Phys. **39**, 3597–3639 (1968). [CrossRef]

5. A. Piskarskas, V. Smilgevicius, A. Stabinis, V. Jarutis, V. Pasiskevicius, S. Wang, J. Tellefsen, and F. Laurell, “Noncollinear second-harmonic generation in periodically poled KTiOPO4 excited by the Bessel beam,” Opt. Lett. **24**, 1053–1055 (1999). [CrossRef]

6. P. Xu, S. H. Ji, S. N. Zhu, X.Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional chi((2)) photonic crystal: A hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. **93**, 133904 (2004). [CrossRef] [PubMed]

7. M. Goulkov, O. Shinkarenko, L. Ivleva, P. Lykov, T. Granzow, T. Woike, M. Imlau, and M. Wohlecke, “New parametric scattering in photorefractive Sr0.61Ba0.39Nb2O6 : Cr,” Phys. Rev. Lett. **91**, 243903 (2003). [CrossRef] [PubMed]

8. G. Giusfredi, D. Mazzotti, P. Cancio, and P. De Natale, “Spatial mode control of radiation generated by frequency difference in periodically poled crystals,” Phys. Rev. Lett. **87**, 113901 (2001). [CrossRef] [PubMed]

9. X. Y. Xu, V. G. Minogin, K. Lee, Y.Z. Wang, and W. H. Jhe, “Guiding cold atoms in a hollow laser beam,” Phys. Rev. A **60**, 4796–4804 (1999). [CrossRef]

10. Z. Chen, Y. K. Ho, P. X. Wang, Q. Kong, Y. J. Xie, W. Wang, and J. J. Xu, “A formula on phase velocity of waves and pplication,” Appl. Phys. Lett. **88**, 121125 (2006). [CrossRef]

19. M. D. Perry, D. Pennington, B. C. Stuart, G. Tietbohl, J. A. Britten, C. Brown, S. Herman, B. Golick, M. Kartz, J. Miller, H.T. Powell, M. Vergino, and V. Yanovsky, “Petawatt laser pulses,” Opt. Lett. **24**, 160–162 (1999). [CrossRef]

## 2. Preliminaries

### 2.1. Phase-velocity distribution of a focused Gaussian beam

10. Z. Chen, Y. K. Ho, P. X. Wang, Q. Kong, Y. J. Xie, W. Wang, and J. J. Xu, “A formula on phase velocity of waves and pplication,” Appl. Phys. Lett. **88**, 121125 (2006). [CrossRef]

*z*direction:

*ω*and

*k*denote the angular-frequency and the wave-number, respectively. In addition,

*φ*(

*z*) = tan

^{-1}(

*z*/

*z*),

_{R}*R*(

*z*) =

*z*[1 + (

*z*/

_{R}*z*)

^{2}], and

*z*is the Rayleigh length.

_{R}*v*of a Gaussian beam is nonuniformly distributed in both transverse and longitudinal directions. In the longitudinal region around the beam waist (|

_{pz}*z*|<

*z*), the phase-velocity decreases rapidly with the radial position. Subluminal phase-velocity occurs approximately in the region

_{R}*z*|<

*z*with

_{R}*r*>

*w*

_{0}, where

*w*

_{0}is the radius of the beam waist. In the application of electron acceleration, when relativistic electrons are injected into this subluminal region, they can be trapped in the acceleration phase for a sufficiently long time and receive considerable account of energy from the strong laser field. This is the so called capture and acceleration scenario (CAS) [20

20. P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. **78**, 2253–2255 (2001). [CrossRef]

*c*.

### 2.2. Theoretical model of DFG between a planar pump wave and a focused signal wave

*k*(corresponding to the case if all the interacting waves are planar), an additional phase-mismatch

*δk*will contribute. This additional phase-mismatch is also nonuniformly distributed and will vary the conversion efficiency transversely which results in a patterned DF beam. In order to reveal the influence of the phase-velocity distribution on the DF beam pattern, we assume that the gain experienced by the signal is small (i.e., the nonlinear length L

_{NL}is assumed to be much large than the crystal length L). In such a case, the signal evolution can be approximated by its linear propagation and the approximation of pump nondepletion can also be adopted.

## 3. Numerical results

### 3.1. Nonlinear evolution of the DF beam with a Gaussian signal beam

_{0s}is the beam waist of the signal in vacuum, z

_{Rc}is the Rayleigh length of the signal in the crystal. In Fig. 2(a), the phase-mismatch is taken as Δ

*kL*= 8 . As expected, the additional phase-mismatch

*δk*plays an important role in generating the beam patterns. At the beginning, the DF beam has a bell-like pattern. With its nonlinear propagation in the crystal, a dip appears in the beam center. Then the central dip gradually spread outwards to form a ringlike pattern. And then a small bell-like pattern emerges at the center and the central dip turns into a dark ring and moves to the beam periphery. Finally, the DF beam returns to a bell-like form again but with a lower intensity. In Fig. 2(b) where the phase-mismatch is taken as Δ

*kL*= -8 , the evolution of the DF beam is opposite to that in Fig. 2(a): it has a bell-like pattern at the beginning, then comes a dark ring at the beam periphery, and then the dark ring moves to the beam center and turns into a central dip, at last the central dip disappears and the beam becomes a bell-like form again.

*r*. The nonuniform distribution of the phase-velocity difference will cause the phase-mismatch to vary with transverse coordinate in the DFG process. As shown in Fig. 2, in the region | Δ

*kL*|< 2

*π*, the pattern of the DF beam is similar to that of the signal beam, then the additional phase mismatch resulted by the nonuniform distribution of the phase velocity is:

*δk*varies with the radial position

*r*especially in the region |

*z*|<

*z*, where it decreases rapidly with

_{R}*r*. In the DFG process, the conversion efficiency is influenced by the total phase mismatch

*Dk*, which can be written as

*δk*.

*δk*with the transverse coordinate

*r*, the evolution of the DF beam generated from a planar pump wave and a focused Gaussian signal wave can be divided into three steps. Firstly, the transverse distribution of the DF beam near the entrance is a Gaussian one, just as that of the signal beam. In the region | Δ

*kl*|< 2

*π*, with the increase of the interaction length, the diffraction of the DF beam is similar to the signal beam, therefore it keeps a bell-like form that is similar to a Gaussian one.

*kl*|= 2

*π*, the distribution of the DF beam is very sensitive to

*δk*since its intensity is proportional to sinc

^{2}(

*Dkl*/2). It is no other than the sensitivity causes the evolution of the dark ring in Fig. 2. In Fig. 2(a) (Fig. 2(b)), since Δ

*k*> 0 (Δ

*k*< 0), the total phase mismatch |

*Dkl*| at the beam center (periphery) reaches 2

*π*first. As a result, a dip (dark ring) emerges at the beam center (periphery). With the increase of

*l*, the dark ring moves to the beam periphery (center) because the transverse position corresponding to |

*Dk*|

*l*= 2

*π*moves to the beam periphery (center).

*kl*|> 2

*π*, the newly generated DF beam is not so sensitive to

*δk*, and the pattern formed around the region | Δ

*kl*|~2

*π*is too weak to influence its intensity distribution, therefore the DF beam comes back to a bell-like form.

### 3.2. Evolution of the DF beams with other types of signal beams

10. Z. Chen, Y. K. Ho, P. X. Wang, Q. Kong, Y. J. Xie, W. Wang, and J. J. Xu, “A formula on phase velocity of waves and pplication,” Appl. Phys. Lett. **88**, 121125 (2006). [CrossRef]

*ρ*= 1.2 (the ellipticity is defined as

*ρ*=

*w*

_{0y}/

*w*

_{0x}, where

*w*

_{0x}(

*w*

_{0y}) is the beam waist in the

*x*(

*y*) direction). As expected, the DF beam first keeps an elliptical bell-like form. In the position around Δ

*kl*= 2

*π*, an elliptical dip appears in the beam center; then it comes into a dark elliptical ring and moves toward the beam periphery. At last, the beam comes back to an elliptical bell-like form again.

_{10}Hermite-Gaussian signal beam (HGSB) and a planar pump wave is shown in Fig. 3(b). At the beginning, the DF beam has a pattern similar to TEM

_{10}Hermite Gaussian beam (HGB). With the increase of

*l*, the two side-lobes evolve to the follow patterns: the gibbous moon-like patterns, the half moon-like patterns, and the old moon-like patterns. When the side-lobes evolve to the old moon-like patterns, two small gibbous moon-like patterns appear in the center. Then with the broadening of the interior gibbous moon-like patterns, the exterior old moon-like patterns become narrower and narrower. At last, the exterior old moon-like patterns disappear and the interior gibbous moonlike patterns become a pattern similar to TEM

_{10}mode HGB signal beam again. This is caused by the evolution of the dark ring from the beam center to the periphery, which is induced by the phase-velocity of the HGB which is axially symmetric and decreases with

*r*.

### 3.3. DF beams generated from differently located crystals

*z*= -

*z*(

_{Rc}*z*=

*z*), and the exit plane is

_{Rc}*z*=

*z*(

_{Rc}*z*= -3

*z*) We find that at the exist plane, the two DF beams have the same amplitude and spatial distribution, but their phases are conjugated (i.e., these two beams are phase-conjugated waves), as clearly illustrated in Fig. 4(a). We have compared the DF beams generated for different values of crystal length and phase-mismatch Δ

_{Rc}*k*, the results show that the two DF beams are also phase-conjugated waves under the conditions of symmetric incident planes and equal crystal lengths.

*h*from the plane

*z*= -

*z*

_{0}, according to Eq. (4), the locally generated DF beam at the position

*z*= -

*z*

_{0}+

*h*is the phase-conjugated wave of the signal beam:

*z*=

*z*

_{0}. When the DF wave generated at

*z*= -

*z*

_{0}propagates to the exist plane

*z*= -

*z*

_{0}+

*L*, it is equivalent to the signal beam at the plane

*z*=

*z*

_{0}+

*L*in situation II. Thus, when it propagates to the exist plane

*z*= -

*z*

_{0}+

*L*, the DF wave generated at

*z*= -

*z*

_{0}in situation I is the phase-conjugated wave of that generates at

*z*=

*z*

_{0}+

*L*in situation II, because in situation II the locally generated DF wave is also the phase-conjugated wave of the signal beam. In a similar way, when they propagate to the exist plane, all the DF waves generated at other planes in situation I should have their corresponding phase-conjugated waves in the locally generated DF waves in situation II. Consequently, the overall DF beam at the exit plane in situation I, which is the coherent combination of the DF waves generated at different positions in the crystal, is the phase-conjugated wave of that in situation II.

### 3.4. Division between the subluminal and the superluminal region

*z*|<

*z*of a focused Gaussian beam, the phase-velocity-induced

_{R}*δk*decreases with the transverse coordinate

*r*and is equal to zero at a certain radial position. This property results in an interesting characteristic shown in Fig. 5 where the DFG beam profiles for different values of phase-mismatch Δ

*k*are plotted. The beam profiles for + Δ

*k*and - Δ

_{n}*k*(where Δ

_{n}*k*presents a certain phase mismatch value) intersect each other, and the transverse positions of intersections are identical for all pairs of the phase-mismatch +Δ

_{n}*k*and -Δ

_{n}*k*. This property reflects the fact that, at this transverse position, the intensity of the DF wave in the case of perfect PM (Δ

_{n}*k*= 0) is the largest compared to those in the cases of phase-mismatch, and the decrease of the DF intensity due to phase-mismatch Δ

*k*= +Δ

*k*is equal to that in the case of Δ

_{n}*k*= -Δ

*k*.

_{n}*z*=0), the locally generated DF wave in every plane inside the crystal is a Gaussian beam, and the beam radii of DF waves vary from those of the signal beam between

*z*=0 and

*z*=

*L*planes. Therefore during the DFG process both the signal and the locally generated DF wave are Gaussian (although the overall DF beam at the exit plane deviates from the standard Gaussian beam). And as a result, as shown in Fig. 5, the transverse position where the intersections locate at is in a good agreement with the radial position (

*r*=1.075

*w*

_{0}) where the phase velocity of the Gaussian beam at

*z*=

*L*/2 is equal to the standard light velocity

*c*. This property will be valuable in identifying the subluminal region of the Gaussian beam, which is important in many applications such as electron acceleration.

22. S. M. Saltiel, K. Koynov, B. Agate, and W. Sibbett, “Second-harmonic generation with focused beams under conditions of large group-velocity mismatch,” J. Opt. Soc. Am. B **21**, 591–598 (2004) [CrossRef]

22. S. M. Saltiel, K. Koynov, B. Agate, and W. Sibbett, “Second-harmonic generation with focused beams under conditions of large group-velocity mismatch,” J. Opt. Soc. Am. B **21**, 591–598 (2004) [CrossRef]

## 4. Conclusion

## Acknowledgments

## References and links

1. | G. D. Boyd, A. Ashkin, J. M. Dziedzic, and D. A. Kleinman, “Second-harmonic generation of light with double refraction,” Phys. Rev. |

2. | G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. App. Phys. |

3. | S. Guha, F. J. Wu, and J. Falk, “The effects of focusing on parametric oscillation,” IEEE J. Quantum Electron. |

4. | J. J. Zondy, “The effects of focusing in type-I and type-II difference-frequency generations,” Opt. Commun. |

5. | A. Piskarskas, V. Smilgevicius, A. Stabinis, V. Jarutis, V. Pasiskevicius, S. Wang, J. Tellefsen, and F. Laurell, “Noncollinear second-harmonic generation in periodically poled KTiOPO4 excited by the Bessel beam,” Opt. Lett. |

6. | P. Xu, S. H. Ji, S. N. Zhu, X.Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional chi((2)) photonic crystal: A hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. |

7. | M. Goulkov, O. Shinkarenko, L. Ivleva, P. Lykov, T. Granzow, T. Woike, M. Imlau, and M. Wohlecke, “New parametric scattering in photorefractive Sr0.61Ba0.39Nb2O6 : Cr,” Phys. Rev. Lett. |

8. | G. Giusfredi, D. Mazzotti, P. Cancio, and P. De Natale, “Spatial mode control of radiation generated by frequency difference in periodically poled crystals,” Phys. Rev. Lett. |

9. | X. Y. Xu, V. G. Minogin, K. Lee, Y.Z. Wang, and W. H. Jhe, “Guiding cold atoms in a hollow laser beam,” Phys. Rev. A |

10. | Z. Chen, Y. K. Ho, P. X. Wang, Q. Kong, Y. J. Xie, W. Wang, and J. J. Xu, “A formula on phase velocity of waves and pplication,” Appl. Phys. Lett. |

11. | M. Born and E. Wolf, |

12. | R. W. Boyd, |

13. | A. M. Weiner, A. M. Kan’an, and D. E. Leaird, “High-efficiency blue generation by frequency doubling of femtosecond pulses in a thick nonlinear crystal,” Opt. Lett. |

14. | S. Yu and A. M. Weiner, “Phase-matching temperature shifts in blue generation by frequency doubling of femtosecond pulses in KNbO3,” J. Opt. Soc. Am. B |

15. | V. Magni, “Optimum beams for efficient frequency mixing in crystals with second order nonlinearity,” Opt. Comm. |

16. | G. D. Xu, T. W. Ren, Y. H. Wang, Y. Y. Zhu, S. N. Zhu, and N. B. Ming, “Third-harmonic generation by use of focused Gaussian beams in an optical superlattice,” J. Opt. Soc. Am. B |

17. | S. M. Saltiel, K. Koynov, B. Agate, and W. Sibbett, “Second-harmonic generation with focused beams under conditions of large group-velocity mismatch,” J. Opt. Soc. Am. B |

18. | D. Q. Lu, L. J. Qian, Y. Z. Li, and D.Y. Fan, “Frequency mixing of off-axis focused Gaussian beams: An approach to measure the phase velocity distribution,” Appl. Phys. Lett. |

19. | M. D. Perry, D. Pennington, B. C. Stuart, G. Tietbohl, J. A. Britten, C. Brown, S. Herman, B. Golick, M. Kartz, J. Miller, H.T. Powell, M. Vergino, and V. Yanovsky, “Petawatt laser pulses,” Opt. Lett. |

20. | P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. |

21. | D. Umstadter, S. Y. Chen, A. Maksimchuk, G. Mourou, and R. Wagner, “Nonlinear optics in relativistic plasmas and laser wake field acceleration of electrons,” Science |

22. | S. M. Saltiel, K. Koynov, B. Agate, and W. Sibbett, “Second-harmonic generation with focused beams under conditions of large group-velocity mismatch,” J. Opt. Soc. Am. B |

**OCIS Codes**

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(350.4990) Other areas of optics : Particles

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: December 20, 2006

Revised Manuscript: March 21, 2007

Manuscript Accepted: March 28, 2007

Published: April 11, 2007

**Citation**

Daquan Lu, Liejia Qian, Yongzhong Li, Hua Yang, Heyuan Zhu, and Dianyuan Fan, "Phase velocity nonuniformity-resulted beam patterns in difference frequency generation," Opt. Express **15**, 5050-5058 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-5050

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### References

- G. D. Boyd, A. Ashkin, J. M. Dziedzic, and D. A. Kleinman, "Second-harmonic generation of light with double refraction," Phys. Rev. 137, A1305-A1320 (1965). [CrossRef]
- G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. App. Phys. 39, 3597-3639 (1968). [CrossRef]
- S. Guha, F. J. Wu, and J. Falk, "The effects of focusing on parametric oscillation," IEEE J. Quantum Electron. QE-18, 907-912 (1982). [CrossRef]
- J. J. Zondy, "The effects of focusing in type-I and type-II difference-frequency generations," Opt. Commun. 149, 181-206 (1998). [CrossRef]
- A. Piskarskas, V. Smilgevicius, A. Stabinis, V. Jarutis, V. Pasiskevicius, S. Wang, J. Tellefsen, and F. Laurell, "Noncollinear second-harmonic generation in periodically poled KTiOPO4 excited by the Bessel beam," Opt. Lett. 24, 1053-1055 (1999). [CrossRef]
- P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, "Conical second harmonic generation in a two-dimensional chi((2)) photonic crystal: A hexagonally poled LiTaO3 crystal," Phys. Rev. Lett. 93, 133904 (2004). [CrossRef] [PubMed]
- M. Goulkov, O. Shinkarenko, L. Ivleva, P. Lykov, T. Granzow, T. Woike, M. Imlau, and M. Wohlecke, "New parametric scattering in photorefractive Sr0.61Ba0.39Nb2O6 : Cr," Phys. Rev. Lett. 91, 243903 (2003). [CrossRef] [PubMed]
- G. Giusfredi, D. Mazzotti, P. Cancio, and P. De Natale, "Spatial mode control of radiation generated by frequency difference in periodically poled crystals, " Phys. Rev. Lett. 87, 113901 (2001). [CrossRef] [PubMed]
- X. Y. Xu, V. G. Minogin, K. Lee, Y.Z. Wang, and W. H. Jhe, "Guiding cold atoms in a hollow laser beam," Phys. Rev. A 60, 4796-4804 (1999). [CrossRef]
- Z. Chen, Y. K. Ho, P. X. Wang, Q. Kong, Y. J. Xie, W. Wang, and J. J. Xu, "A formula on phase velocity of waves and application," Appl. Phys. Lett. 88, 121125 (2006). [CrossRef]
- M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999).
- R. W. Boyd, Nonlinear Optics (Academic, New York, 1992).
- A. M. Weiner, A. M. Kan'an, and D. E. Leaird, "High-efficiency blue generation by frequency doubling of femtosecond pulses in a thick nonlinear crystal," Opt. Lett. 23, 1441-1443 (1998). [CrossRef]
- S. Yu and A. M. Weiner, "Phase-matching temperature shifts in blue generation by frequency doubling of femtosecond pulses in KNbO3," J. Opt. Soc. Am. B 16, 1300-1304 (1999). [CrossRef]
- V. Magni, "Optimum beams for efficient frequency mixing in crystals with second order nonlinearity," Opt. Comm. 184, 245-255 (2000). [CrossRef]
- G. D. Xu, T. W. Ren, Y. H. Wang, Y. Y. Zhu, S. N. Zhu, and N. B. Ming, "Third-harmonic generation by use of focused Gaussian beams in an optical superlattice," J. Opt. Soc. Am. B 20,360-365 (2003). [CrossRef]
- S. M. Saltiel, K. Koynov, B. Agate, and W. Sibbett, "Second-harmonic generation with focused beams under conditions of large group-velocity mismatch," J. Opt. Soc. Am. B 21, 591-598 (2004). [CrossRef]
- D. Q. Lu, L. J. Qian, Y. Z. Li, and D.Y. Fan, "Frequency mixing of off-axis focused Gaussian beams: An approach to measure the phase velocity distribution," Appl. Phys. Lett. 88, 261112 (2006). [CrossRef]
- M. D. Perry, D. Pennington, B. C. Stuart, G. Tietbohl, J. A. Britten, C. Brown, S. Herman, B. Golick, M. Kartz, J. Miller, H.T. Powell, M. Vergino, and V. Yanovsky, "Petawatt laser pulses," Opt. Lett. 24, 160-162 (1999). [CrossRef]
- P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, "Vacuum electron acceleration by an intense laser," Appl. Phys. Lett. 78, 2253-2255 (2001). [CrossRef]
- D. Umstadter, S. Y. Chen, A. Maksimchuk, G. Mourou, and R. Wagner, "Nonlinear optics in relativistic plasmas and laser wake field acceleration of electrons," Science 273, 472-475 (1996). [CrossRef] [PubMed]
- S. M. Saltiel, K. Koynov, B. Agate and W. Sibbett, "Second-harmonic generation with focused beams under conditions of large group-velocity mismatch," J. Opt. Soc. Am. B 21, 591-598 (2004) [CrossRef]

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