## Parametric amplification of Raman-inactive lattice oscillations induced by two-color cross-beam excitation

Optics Express, Vol. 14, Issue 7, pp. 2831-2838 (2006)

http://dx.doi.org/10.1364/OE.14.002831

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### Abstract

The second-order Raman bands of SrTiO_{3} are excited under two-color cross-beam configuration using femto-second laser pulses. Raman-inactive one-phonon waves are generated by the coherently excited large amplitude two-phonon wave. The one-phonon waves are observed as a train of visible light spots, the frequency steps of which are coincident with the frequencies of the one-phonon modes. In order to understand the mechanism, a model of three-wave interaction among one two-phonon wave and two one-phonon waves is proposed.

© 2006 Optical Society of America

1. M. Gühr, M. Bargheer, and N. Schwentner, ”Generation of coherent zone boundary phonons by impulsive excitation of molecules,” Phys. Rev. Lett. **91**, 085504 (2003). [CrossRef] [PubMed]

2. S. Hunsche, K. Wienecke, T. Dekorsy, and H. Kurz, ”Impulsive softening of coherent phonons in tellurium,” Phys. Rev. Lett. **75**, 1815–1818 (1995). [CrossRef] [PubMed]

3. X. Hu and F. Nori, ”Squeezed phonon states: modulating quantum fluctuations of atomic displacements,” Phys. Rev. Lett. **76**, 2294–2297 (1996). [CrossRef] [PubMed]

4. X. Hu and F. Nori, ”Phonon squeezed states generated by second-order Raman scattering,” Phys. Rev. Lett. **79**, 4605–4608 (1997). [CrossRef]

5. G. A. Garrett, A. G. Rojo, A. K. Sood, J. F. Whitaker, and R. Merlin, ”Vacuum squeezing of solids: macroscopic quantum states driven by light pulses,” Science **275**, 1638–1640 (1997). [CrossRef] [PubMed]

6. A. Bartels, T. Dekorsy, and H. Kurz, ”Impulsive excitation of phonon-pair combination states by second-order Raman scattering,” Phys. Rev. Lett. **84**, 2981–2984 (2000). [CrossRef] [PubMed]

7. J. Zhao, A. V. Bragas, D. J. Lockwood, and R. Merlin, ”Magnon squeezing in an antiferromagnet: reducing the spin noise below the standard quantum limit,” Phys. Rev. Lett. **93**, 107203 (2004). [CrossRef] [PubMed]

**k**

_{1}-

**k**

_{2}=

**K**=

**q**

_{1}+

**q**

_{2}, where

**k**

_{1}and

**k**

_{2}are the wavevectors of the incident light,

**K**is that of the two-phonon wave, and

**q**

_{1}and

**q**

_{2}are those of the constituent one-phonon waves. Because

**k**

_{1}and

**k**

_{2}are sufficiently smaller than the inverse of the lattice parameters, the two-phonon wave is excited mostly at the Brillouin zone center. On the other hand, because

**q**

_{1}and

**q**

_{2}themselves can be arbitrary under

**k**

_{1}-

**k**

_{2}=

**q**

_{1}+

**q**

_{2}, the phonons can be excited at whole Brillouin zone. These indicate that excitation of the second-order Raman bands may give a new pathway of coherent control of quantum oscillations which are not accessible by exciting the first-order Raman bands. We have shown that the impulsive excitation using two different color ultrashort light pulses are beneficial for selective and strong excitation of phonons[8

8. J.-i. Takahashi, E. Matsubara, T. Arima, and E. Hanamura, ”Coherent multistep anti-Stokes and stimulated Raman scattering associated with third harmonics in YFeO_{3} crystals,” Phys. Rev. B **68**, 155102 (2003). [CrossRef]

9. J.-i. Takahashi, Y. Kawabe, and E. Hanamura, ”Generation of a broadband spectral comb with multiwave mixing by exchange of an impulsively stimulated phonon,” Optics Express **12**, 1185–1190 (2004). [CrossRef] [PubMed]

10. Y. R. Shen, ”A note on two-phonon coherent anti-Stokes Raman scattering,” J. Raman Spectrosc. **10**, 110–112 (1981). [CrossRef]

11. M. J. Colles and J. A. Giordmaine, ”Generation and detection of large-k-vector phonons,” Phys. Rev. Lett. **27**, 670–674 (1971). [CrossRef]

13. J.-i. Takahashi, K. Mano, and T. Yagi, ”Solid-state anti-Stokes Raman shifter covering extremely broadband tunable range,” Jpn. J. Appl. Phys. (in press). [PubMed]

_{3}plate with a (1,0,0) surface. It has a cubic perovskite structure with a space group

*Pm3m*(

*T*

_{u}acoustic mode and three

*T*

_{1u}and one

*T*

_{2u}optical modes. Each polar

*T*

_{1u}optical mode splits into one LO and two doubly-degenerate TO modes. All the phonon modes are Raman-inactive but there are strong and broad second-order Raman bands, all of which consist of the two-phonon combination bands. The second-order Raman bands of SrTiO

_{3}locate at 300, 700, and 1000 cm

^{-1}. The intensity decreases as the frequency increases [15

15. W. G. Nilsen and J. G. Skinner, ”Raman spectrum of strontium titanate,” J. Chem. Phys. **48**, 2240–2248 (1968). [CrossRef]

*ω*

_{1}) and idler (

*ω*

_{2}) light pulses generated by a Ti:Sapphire-based femto-second optical parametric amplifier (OPA) system. Their pulse widths are 150 fs and the repetition rate is 1 kHz. They are focused nearly normally on to the sample through a lens with a crossing angle of approximately 5 deg. Their polarizations are vertical, which are in the same direction as one of the crystal axis. The size of the focused spots are 50 to 100

*μ*m. The emitted light from the sample is picked up by an optical fiber with a 200-

*μ*m core radius. It is set on an arm which rotates around the sample by an electrically controlled rotational stage. The optical fiber is connected to a multichannel spectrometer. The distance between the sample and the end of fiber is about 40 mm. The angle of the emitted signals is measured from the direction of

*ω*

_{1}light. When photographs are taken, the optical fiber is replaced by a white paper and the bright spots on it are taken by a PC-based CCD camera. In order to measure the power dependence, part of higher-order signals are simultaneously picked-up by an optical fiber after collecting them using a lens. All the experiments are carried out at room temperature.

*ω*

_{1}. At the proper relative delay and for the frequency differences resonant with the second- and the third-strongest Raman peaks, a train of well-separated clear light spots ranging from infrared to blue region appears from the supercontinuum. These signals are sensitive to the relative delay of the incident light pulses. The spots are the clearest when the

*ω*

_{1}pulse arrives a few tens of fs earlier than the

*ω*

_{2}pulse. Trains of the light pulses are emitted symmetrically in both the forward and the backward direction of the sample. It is stressed that only supercontinuum and no clear spots are observed when the frequency difference is tuned to 390 cm

^{-1}, the shoulder of the first strongest peak, which is the lowest for our OPA system, and 500 cm

^{-1}, the valley between the second and the third Raman peaks. Figure 2 show the photographs taken by a PC-based CCD camera against irradiation powers. Because the focuses of the incident light are not optimized, the irradiation powers are higher than those in other measurements. The lower-order signals are invisible to the naked eye because they are infrared light. It is apparent that the power dependence is asymmetric against P

_{1}(signal light) and P

_{2}(idler light). The higher-order signals diminish faster than the lower-order ones as P

_{1}decreases. On the other hand, the number of visible spots does not change against P

_{2}. Spectroscopic plots are shown in Fig. 3. Figures 3(a) and 3(c) are the spectra, the peak intensity of which are scaled. They show the change of the spectral shapes against P

_{1}with fixing P

_{2}to 1.3 mW and those against P

_{2}with fixing P

_{1}to 1.0 mW, respectively. Each spectrum is shifted upward by the value of the varying irradiation powers. The scaling factors are plotted against the powers in Figs. 3(b) and 3(d). One must remember that the absolute shape of the spectra have no meaning because only a part of the emitted light is collected by a lens. It is clearly seen that the change of the spectral shape is mostly independent of P

_{2}and the intensity dependence is linear to it except at very low power. On the other hand, the higher-order signals diminish faster than the lower-order ones as P

_{1}decreases and the intensity dependence has a clear threshold.

*ω*=

*ω*

_{1}-

*ω*

_{2}) are 660 and 1015 cm

^{-1}, the values of which agree with the frequencies of the second- and the third-strongest second-order Raman bands. The relative delays are set as the spots become the clearest. For the purpose of making the peaks clear, each spectrum is scaled against the angle by the following method instead of calibrating by sensitivity. Firstly, an envelope of the strongest peak intensity as for the angle is calculated by taking nine points moving average. Then each spectrum is divided by the value of the envelope. There are series of different frequency spacing values, 475 and 530 cm

^{-1}, in the spectrum of ∆

*ω*= 660 cm

^{-1}, and 475, 545, and 625 cm

^{-1}, in the spectrum of ∆

*ω*= 1015 cm

^{-1}.

^{-1}. The filled and the open markers are the data measured by a CCD and an InGaAs multi channel photodiode array, respectively. Several subsidiary peaks are also plotted. It is apparent that the dispersion relation is different from that of the ordinary multi-step CARS,

**k**

_{n}=

**k**

_{1}+

*n*(

**k**

_{1}-

**k**

_{2}) and

*ω*

_{n}=

*ω*

_{1}+

*n*(

*ω*

_{1}-

*ω*

_{2}) (broken line). The line starts from around the CARS and then curves slightly. At the same time, the step of

**k**

_{∥}is half or one third of the

**k**

_{∥}difference between the ω

_{1}and the

*ω*

_{2}beams.

15. W. G. Nilsen and J. G. Skinner, ”Raman spectrum of strontium titanate,” J. Chem. Phys. **48**, 2240–2248 (1968). [CrossRef]

16. W. G. Stirling, ”Neutron inelastic scattering study of the lattice dynamics of strontium titanate: harmonic models,” J. Phys. C: Solid State Phys. **5**, 2711–2730 (1972). [CrossRef]

17. K. Inoue, N. Asai, and T. Sameshima,”Experimental study of the hyper-Raman scattering due to Raman inactive lattice vibration of SrTiO_{3},” J. Phys. Soc. Jpn. **50**, 1291–1300 (1981). [CrossRef]

_{4}+ TO

_{2}and the overtone of the TO

_{3}mode, and the third one (1038) to 2LO

_{2}and 2TO

_{4}. The first-strongest one is assigned to TO

_{4}- TA, TO

_{4}- TO

_{1}, and 2TO

_{2}. This band is mainly made from the former two difference combination bands. The obtained frequencies 475 and 545 (or 530) cm

^{-1}agree well with those of LO

_{2}(480) and TO

_{4}(544). The frequency 625 cm

^{-1}agrees well with the second-order Raman peak of TO

_{4}+ TO

_{1}or TO

_{4}+ TA (629). I believe that the second-strongest Raman peak should include LO

_{2}+ LO

_{1}. Accepting these assignments, all the obtained frequencies in the spectra are assigned to those of the one-phonons and their combinations.

**E**

_{1}|

*exp*(

*iω*

_{1}

*t*) and |

**E**

_{2}|

*exp*(

*iω*

_{2}

*t*), are injected, they interfere with each other and generate a beating electric field, |

**E**

_{1}∙

*exp*(

*i*(

*ω*

_{1}-

*ω*

_{2})

*t*). When

*ω*

_{1}-

*ω*

_{2}is tuned to the second-order Raman bands, the beating electric field excites the two-phonon wave resonantly. When the amplitude of the two-phonon wave becomes sufficiently large, nonlinear wave-wave interaction among two-phonon wave and one-phonon waves comes to the central stage and the power of the two-phonon wave is transferred to the one-phonon waves. In order to model the experimental result, the phonon Hamiltonian must have a product of at least four creation or annihilation operators of phonon. In the case of the product of three terms, because two of them are excited simultaneously by the resonant two-phonon CARS process, the observed one-phonon wave must be generated directly from the two-phonon wave. This is impossible due to energy and momentum conservation law. Similar to the discussion of phonon-phonon scattering by third-order phonon-phonon interaction[18], the fourth-order phonon-phonon interaction is expressed as

*K*Σ

_{q,i,q′,j;q″,k;q‴,l}

*δ*

_{q+q′+q″+q‴}(

*b*

_{-q,i})(

*b*

_{-q′,j})(

*b*†

_{q″,k}-

*b*

_{-q″,k})(

*b*

_{-q‴,l}) , where

*b*

_{q,i}(

**q**and eigen frequency Ω

_{i}and

*K*is the strength of the lattice nonlinearity. In the case of our experimental situation, a two-phonon wave is excited externally by the beating electric field with the frequency of Ω

_{1}+ Ω

_{2}and the wavevector of

**k**

_{1}-

**k**

_{2}. The two-phonon wave is composed of two one-phonon waves with Ω

_{1},

**q**and Ω

_{2},

**k**

_{1}-

**k**

_{2}-

**q**, respectively, where

**q**distributes over whole Brillouin zone. Retaining the terms which satisfy energy and momentum conservation, the dominant nonlinear terms become

*b*

^{†}and

*b*

^{†}, which is rewritten as a single operator 〈

*b*

_{q,1}

*b*

_{k1-k2-q,2}〉 = 〈

*b*

_{1}

*b*

_{2}〉

_{0}

*exp*(-

*i*(Ω

_{1}+ Ω

_{2})

*t*), where

*b*

_{q,1}and

*b*

_{k1-k2-q,2}are abbreviated to

*b*

_{1}and

*b*

_{2}. Then the fourth-order nonlinear term is rewritten as (1\2)(〈

*b*

_{1}

*b*

_{2}+

*b*

^{†}

_{2}〈

*b*

_{1}

*b*

_{2}〉). This formulation stems that oscillation of one-phonon waves are modulated under the existence of an elastic oscillation which is driven externally by two-phonon CARS process. Under some condition, the one-phonon waves are amplified from quantum fluctuation by this modulation, which is well known as the parametric amplification process. Combining the harmonic terms, the equations of motion of the one-phonon waves have a form of a nonlinear three-wave interaction,

*K*Σ

_{q}〈

*b*

_{1}

*b*

_{2}〉 = Λ

_{0}

*exp*(-

*i*(Ω

_{1}+ Ω

_{2})

*t*). Here, it is assumed for the simplicity that the two-phonon wave is a plane wave and its amplitude does not change during the interaction. The equations have solutions of plane waves. The eigen frequencies are

*b*

^{†}

_{1}

*b*

_{2}〉. Then Eq.(1) and (2) are replaced by

*K*Σ

_{q}〈

*b*

_{2}〉 = Λ′

_{0}

*exp*(

*i*(Ω

_{1}- Ω

_{2})

*t*) and the eigen frequencies become

^{-1}excitation, where no clear spots are found.

**K**=

**q**

_{1}+

**q**

_{2}, each constituent phonon wave can have arbitrary wavevector under this constraint. Then each phonon can couple with light with any wavevectors. It should be noted that the multi-colored signals accompany with a supercontinuum which satisfies the same dispersion relation of the higher-order signals. The origin of the supercontinuum is thought to be the cross-phase modulation of the two incident light pulses. The

*ω*

_{1}light modulates the second-order refractive index by the optical Kerr effect. The

*ω*

_{2}light is injected to this modulated region from a different direction of the

*ω*

_{1}light and is scattered from the region. The phase modulation of the

*ω*

_{2}light is ∆

*ϕ*(

*t*) = (

*ω*

_{2}/

*c*)∫

*n*

_{2}|

**E**

_{1}(

*t*)|

^{2}

*dl*and the frequency modulation is ∆

*ω*= -∂(∆

*ϕ*)/∂

*t*, where

*n*

_{2}is the second-order refractive index [19]. The integration is carried out along the path of the

*ω*

_{2}beam. The frequency modulation generates a supercontinuum covering visible regions when femto-second light pulses are used. This mechanism of supercontinuum generation is well known in the self-phase modulation. It is expected from the above equation that the spectrum of the supercontinuum depends only on |

**E**

_{1}(

*t*)|

^{2}and its shortest wavelength becomes longer as |

**E**

_{1}(

*t*)|

^{2}

*max*decreases. This is consistent with the experimental results as shown in Fig. 2 and Fig. 3. The shortest wavelength of the multi-colored signals becomes longer as P

_{1}decreases and is independent of P

_{2}. If the signals come from the multi-step CARS, i.e. the multiple diffraction from the dynamic grating generated by the incident two light, the dispersion relation must be linear [8

8. J.-i. Takahashi, E. Matsubara, T. Arima, and E. Hanamura, ”Coherent multistep anti-Stokes and stimulated Raman scattering associated with third harmonics in YFeO_{3} crystals,” Phys. Rev. B **68**, 155102 (2003). [CrossRef]

## References and links

1. | M. Gühr, M. Bargheer, and N. Schwentner, ”Generation of coherent zone boundary phonons by impulsive excitation of molecules,” Phys. Rev. Lett. |

2. | S. Hunsche, K. Wienecke, T. Dekorsy, and H. Kurz, ”Impulsive softening of coherent phonons in tellurium,” Phys. Rev. Lett. |

3. | X. Hu and F. Nori, ”Squeezed phonon states: modulating quantum fluctuations of atomic displacements,” Phys. Rev. Lett. |

4. | X. Hu and F. Nori, ”Phonon squeezed states generated by second-order Raman scattering,” Phys. Rev. Lett. |

5. | G. A. Garrett, A. G. Rojo, A. K. Sood, J. F. Whitaker, and R. Merlin, ”Vacuum squeezing of solids: macroscopic quantum states driven by light pulses,” Science |

6. | A. Bartels, T. Dekorsy, and H. Kurz, ”Impulsive excitation of phonon-pair combination states by second-order Raman scattering,” Phys. Rev. Lett. |

7. | J. Zhao, A. V. Bragas, D. J. Lockwood, and R. Merlin, ”Magnon squeezing in an antiferromagnet: reducing the spin noise below the standard quantum limit,” Phys. Rev. Lett. |

8. | J.-i. Takahashi, E. Matsubara, T. Arima, and E. Hanamura, ”Coherent multistep anti-Stokes and stimulated Raman scattering associated with third harmonics in YFeO |

9. | J.-i. Takahashi, Y. Kawabe, and E. Hanamura, ”Generation of a broadband spectral comb with multiwave mixing by exchange of an impulsively stimulated phonon,” Optics Express |

10. | Y. R. Shen, ”A note on two-phonon coherent anti-Stokes Raman scattering,” J. Raman Spectrosc. |

11. | M. J. Colles and J. A. Giordmaine, ”Generation and detection of large-k-vector phonons,” Phys. Rev. Lett. |

12. | J.-i. Takahashi, K. Mano, and T. Yagi, ”Raman lasing and cascaded coherent anti-Stokes Raman scattering of two-phonon Raman band,” Opt. Lett. doc. ID 64854 (posted 2 March 2006, in press). |

13. | J.-i. Takahashi, K. Mano, and T. Yagi, ”Solid-state anti-Stokes Raman shifter covering extremely broadband tunable range,” Jpn. J. Appl. Phys. (in press). [PubMed] |

14. | M. A. Nielsen and I. L. Chuang, |

15. | W. G. Nilsen and J. G. Skinner, ”Raman spectrum of strontium titanate,” J. Chem. Phys. |

16. | W. G. Stirling, ”Neutron inelastic scattering study of the lattice dynamics of strontium titanate: harmonic models,” J. Phys. C: Solid State Phys. |

17. | K. Inoue, N. Asai, and T. Sameshima,”Experimental study of the hyper-Raman scattering due to Raman inactive lattice vibration of SrTiO |

18. | J. M. Ziman, in |

19. | Y. R. Shen, in |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.4160) Nonlinear optics : Multiharmonic generation

(270.6570) Quantum optics : Squeezed states

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: November 29, 2005

Revised Manuscript: March 23, 2006

Manuscript Accepted: March 25, 2006

Published: April 3, 2006

**Citation**

Jun-ichi Takahashi, "Parametric amplification of Raman-inactive lattice oscillations induced by two-color cross-beam excitation," Opt. Express **14**, 2831-2838 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-2831

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### References

- M. Gühr, M. Bargheer, and N. Schwentner, "Generation of coherent zone boundary phonons by impulsive excitationof molecules," Phys. Rev. Lett. 91, 085504 (2003). [CrossRef] [PubMed]
- S. Hunsche, K. Wienecke, T. Dekorsy and H. Kurz, "Impulsive softening of coherent phonons in tellurium," Phys. Rev. Lett. 75, 1815-1818 (1995). [CrossRef] [PubMed]
- X. Hu and F. Nori, "Squeezed phonon states: modulating quantum fluctuations of atomic displacements," Phys. Rev. Lett. 76, 2294-2297 (1996). [CrossRef] [PubMed]
- X. Hu and F. Nori, "Phonon squeezed states generated by second-order Raman scattering," Phys. Rev. Lett. 79, 4605-4608 (1997). [CrossRef]
- G. A. Garrett, A. G. Rojo, A. K. Sood, J. F. Whitaker and R. Merlin, "Vacuum squeezing of solids: macroscopic quantum states driven by light pulses," Science 275, 1638-1640 (1997). [CrossRef] [PubMed]
- A. Bartels, T. Dekorsy and H. Kurz, "Impulsive excitation of phonon-pair combination states by second-order Raman scattering," Phys. Rev. Lett. 84, 2981-2984 (2000). [CrossRef] [PubMed]
- J. Zhao, A. V. Bragas, D. J. Lockwood and R. Merlin, "Magnon squeezing in an antiferromagnet: reducing the spin noise below the standard quantum limit," Phys. Rev. Lett. 93, 107203 (2004). [CrossRef] [PubMed]
- J.-i. Takahashi, E. Matsubara, T. Arima and E. Hanamura, "Coherent multistep anti-Stokes and stimulated Raman scattering associated with third harmonics in YFeO3 crystals," Phys. Rev. B 68, 155102 (2003). [CrossRef]
- J.-i. Takahashi, Y. Kawabe and E. Hanamura, "Generation of a broadband spectral comb with multiwave mixing by exchange of an impulsively stimulated phonon," Optics Express 12, 1185-1190 (2004). [CrossRef] [PubMed]
- Y. R. Shen, "A note on two-phonon coherent anti-Stokes Raman scattering," J. Raman Spectrosc. 10, 110-112 (1981).Q1 [CrossRef]
- M. J. Colles and J. A. Giordmaine, "Generation and detection of large-k-vector phonons," Phys. Rev. Lett. 27, 670-674 (1971). [CrossRef]
- J.-i. Takahashi, K. Mano, and T. Yagi, "Raman lasing and cascaded coherent anti-Stokes Raman scattering of two-phonon Raman band," Opt. Lett. (in press).
- J.-i. Takahashi, K. Mano, and T. Yagi, "Solid-state anti-Stokes Raman shifter covering extremely broadband tunable range," Jpn. J. Appl. Phys. (in press). [PubMed]
- M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, Cambridge, 2000).
- W. G. Nilsen and J. G. Skinner, "Raman spectrum of strontium titanate," J. Chem. Phys. 48, 2240-2248 (1968). [CrossRef]
- W. G. Stirling, "Neutron inelastic scattering study of the lattice dynamics of strontium titanate: harmonic models," J. Phys. C: Solid State Phys. 5, 2711-2730 (1972).Q2 [CrossRef]
- K. Inoue, N. Asai, and T. Sameshima, "Experimental study of the hyper-Raman scattering due to Raman inactive lattice vibration of SrTiO3," J. Phys. Soc. Jpn. 50, 1291-1300 (1981). [CrossRef]
- J. M. Ziman, Electrons and Phonons, (Oxford University Press, 1963) pp. 130.
- Y. R. Shen, The Principles of Nonlinear Optics, (John Wiley & Sons, New York, 1984) pp. 303.

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