## Two photon frequency conversion |

Optics Express, Vol. 20, Issue 4, pp. 3613-3619 (2012)

http://dx.doi.org/10.1364/OE.20.003613

Acrobat PDF (945 KB)

### Abstract

We study the case of two simultaneous three-wave-mixing processes, where one frequency is converted to another through an intermediate frequency. The common assumption is that these processes can occur only when the material is transparent at all participating frequencies. Here we show experimentally that, under appropriate conditions, the intermediate frequency remains dark throughout the interaction. This means that even if the material is opaque at the intermediate frequency, the conversion will remain efficient. New possibilities of frequency conversion are therefore available, e.g. through absorptive bands in the ultraviolet or mid-infrared. Moreover, though it was hitherto assumed that the phase mismatch value is governed only by dispersion, we show here that phase matching also depends on light intensity. These findings promise novel all optical switching techniques.

© 2012 OSA

## 1. Introduction

*ω*

_{1}and

*ω*

_{p1}are summed to produce a higher frequency

*ω*

_{2}=

*ω*

_{1}+

*ω*

_{p1}. In order for this process to have significant efficiency, momentum conservation needs to be maintained. For the given example, this would require that

*k⃗*

_{2}=

*k⃗*

_{1}+

*k⃗*

_{p1}where

*k⃗*

*is the momentum (wave-vector) of the wave with frequency*

_{i}*ω*, a requirement also known as phase-matching. Three-wave-mixing processes can also be cascaded [1

_{i}1. S. M. Saltiel, A. A. Sukhorukov, and Y. S. Kivshar, “Multistep parametric processes in nonlinear optics,” Prog. Opt. **47**, 1–73 (2005). [CrossRef]

*ω*

_{3}=

*ω*

_{2}+

*ω*

_{p2}=

*ω*

_{1}+

*ω*

_{p1}+

*ω*

_{p2}.

2. H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A **78**, 063821 (2008). [CrossRef]

3. H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express **17**, 12731–12740 (2009). [CrossRef] [PubMed]

*two*STWM processes. This analogy leads to mapping of atomic phenomena such as dark states and Stark shifts to the field of nonlinear optics.

## 2. Theoretical analysis

### 2.1. Dynamical equations and atomic analogy

*ω*

_{p2}, in an additional process, generating a new frequency,

*ω*

_{3}. For the case of two SFG processes, we obtain

*ω*

_{3}=

*ω*

_{2}+

*ω*

_{p2}=

*ω*

_{1}+

*ω*

_{p1}+

*ω*

_{p2}. Generally, two distinct pumps can drive the interaction: the first transfers energy between

*ω*

_{1}and

*ω*

_{2}, while the second connects

*ω*

_{2}with

*ω*

_{3}. We consider the case where the pumps are much more intense than the other waves and thereby are negligibly affected by the interaction (undepleted pumps approximation). The procedure of adiabatic elimination [5, 6

6. D. Grischkowsky, M. M. T. Loy, and P. F. Liao, “Adiabatic following model for two-photon transitions: nonlinear mixing and pulse propagation,” Phys. Rev. A **12**, 2514–2533 (1975). [CrossRef]

7. J. C. Delagnes and L. Canioni, “Third harmonic generation in periodically poled crystals,” Proc. SPIE **7917**, 79171C (2011). [CrossRef]

2. H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A **78**, 063821 (2008). [CrossRef]

3. H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express **17**, 12731–12740 (2009). [CrossRef] [PubMed]

8. C. Trallero-Herrero and T. C. Weinacht, “Transition from weak- to strong-field coherent control,” Phys. Rev. A **75**, 063401 (2007). [CrossRef]

9. F. T. Hioe, “Dynamic symmetries in quantum electronics,” Phys. Rev. A **28**, 879–886 (1983). [CrossRef]

*A*(

_{i}*z*) corresponds to

*a*(

_{i}*t*), the probability amplitude of population in each state. Each effective coupling coefficient

*σ*represents the strength of the dipole interaction between levels i and j, usually described by the Rabi frequency Ω

_{ij}*=*

_{ij}*d*(

_{ij}· ε*t*)

*/h̄*, where

*d*is the dipole moment between states

_{ij}*i*and

*j*,

*ε*(

*t*) is the induced EM field, and

*h̄*is Planck’s constant. Respectively, the terms Δ

*k*

_{1}and Δ

*k*

_{2}correspond to the detunings, Δ

_{1}and Δ

_{2}, with the two atomic transitions.

2. H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A **78**, 063821 (2008). [CrossRef]

10. H. Suchowski, B. D. Bruner, A. Ganany-Padowicz, I. Juwiler, A. Arie, and Y. Silberberg, “Adiabatic frequency conversion of ultrafast pulses,” Appl. Phys. B **105**, 697–702 (2011). [CrossRef]

### 2.2. Adiabatic elimination

*k*

_{1}|

*L*≫ 1, |Δ

*k*

_{2}|

*L*≫ 1 while |Δ

*k*

_{1}+ Δ

*k*

_{2}|

*L*≪ 1, where L is the length of the nonlinear crystal. The sum of the phase-mismatches, which is a central notion here, will be defined as the

*two-photon phase mismatch*, and denoted Δ

*k*= Δ

_{TP}*k*

_{1}+ Δ

*k*

_{2}. Under these conditions, it is reasonable to assume that the oscillating terms in Eq. (1) vary much faster than the fields amplitudes at

*ω*

_{1}and

*ω*

_{3}, allowing us to write an approximation for the intermediate frequency complex amplitude: As a consequence of this approximation, the amplitude of the intermediate frequency, |

*A*

_{2}|, will remain always low along the propagation in the nonlinear crystal.

*A*

_{1}=

*Ã*

_{1}(

*z*)

*exp*[−

*i*(Δ

*k*/2 –

_{eff}*σ*

_{12}

*σ*

_{21}/Δ

*k*

_{1})

*z*] and

*A*

_{3}=

*Ã*

_{3}(

*z*)

*exp*[

*i*(Δ

*k*/2 –

_{eff}*σ*

_{23}

*σ*

_{32}

*/*Δ

*k*

_{2})

*z*], a reduced two coupled wave equations set for

*ω*

_{1}and

*ω*

_{3}is obtained, where Δ

*k*is the phase-mismatch of the effective process connecting

_{eff}*ω*

_{1}and

*ω*

_{3}, defined as

*A*

_{2}|, will oscillate very rapidly in comparison to variation in the amplitudes of

*ω*

_{1}and

*ω*

_{3}. As a result, |

*A*

_{2}| will never build up to a significant value and thus remain low throughout the interaction. Nevertheless, energy can still be transferred between

*ω*

_{1}and

*ω*

_{3}via

*ω*

_{2}, effectively imitating a four-wave-mixing process with undepleted pumps, where

*ω*

_{3}=

*ω*

_{1}+

*ω*

_{p1}+

*ω*

_{p2}in the case of two simultaneous SFG processes. We emphasize that the efficiency of conversion from

*ω*

_{1}to

*ω*

_{3}does not change even if the material is opaque at

*ω*

_{2}.

*k*. In this mechanism, two additional contributions to the conventional phase mismatch parameters appear. These contributions, which depend on the coupling coefficients (and thus on the intensities of the pumps,

_{eff}*I*

_{p1}and

*I*

_{p2}), are the equivalent of the atomic Stark shift [5]. In the nonlinear optics context, we choose to define it as

*δ*

*k*=

_{S}*σ*

_{12}

*σ*

_{21}/Δ

*k*

_{1}+

*σ*

_{23}

*σ*

_{32}/Δ

*k*

_{2}.

*ω*

_{1}and

*ω*

_{3}is desired, which occurs when the effective phase-matching parameter is equal to zero, i.e. when Δ

*k*= Δ

_{eff}*k*

_{1}+ Δ

*k*

_{2}+

*δ*

*k*= 0. Since

_{S}*δ*

*k*∝

_{S}*I*it introduces a pump intensity dependence into the phase-matching condition. This is contrary to the conventional wave-mixing phase mismatch, which is considered to depend solely on the dispersion properties of the nonlinear material. Such dependence exists since the rate of each of the two STWM processes depends on pump intensity. To obtain a net flow of energy from one frequency to another, these rates must be compatible over a significant interaction length. The mechanism here is thus fundamentally different from phase-modulation in four-wave-mixing processes, which stems from intensity dependent refractive index.

_{p}*A*

_{3}(0) = 0, is where

*K*

_{1}= −

*i*

*σ*

_{12}

*σ*

_{32}/Δ

*k*

_{2}and

*K*

_{3}=

*i*

*σ*

_{21}

*σ*

_{32}/Δ

*k*

_{1}. It is interesting to note that having reduced the system to an effective two-states system, it can now be represented geometrically using the method introduced by Bloch [11

11. F. Bloch, “Nuclear induction,” Phys. Rev. **70**, 460–474 (1946). [CrossRef]

12. R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, “Geometrical representation of the Schrödinger equation for solving maser problems,” J. Appl. Phys. **28**, 49–52 (1957). [CrossRef]

13. D. S. Hum and M. M. Fejer, “Quasi-phasematching,” C. R. Physique **8**, 180–198 (2007). [CrossRef]

*k̃*

_{1}= Δ

*k*

_{1}+ 2

*π*/Λ and Δ

*k̃*

_{2}= Δ

*k*

_{2}+ 2

*π*/Λ. Substituting these expressions into the phase-matching condition produces a cubic equation with one solution that adheres to the conditions of adiabatic elimination.

## 3. Numerical simulation

*λ*

_{p1}=

*λ*

_{p2}= 1064

*nm*was used to drive both processes. We have used a periodically poled

*KTiOPO*

_{4}(PPKTP) crystal with a poling period of Λ = 8.6

*μ*

*m*, which was calculated to phase-match conversion from

*λ*

_{1}= 3010

*nm*to

*λ*

_{3}= 452

*nm*, via the two implicit SFG processes 3010

*nm*+ 1064

*nm*→ 786

*nm*and 786

*nm*+ 1064

*nm*→ 452

*nm*, when the crystal temperature is 125°

*C*. The simulation does not assume undepleted pump or adiabatic elimination.

*I*= 10

_{p}*GW/cm*

^{2}, is plotted in Fig. 1(a). Energy oscillations between the

*λ*

_{1}= 3010

*nm*input and the

*λ*

_{3}= 452

*nm*output are clearly seen. Since each photon converted from

*λ*

_{1}to

*λ*

_{3}is paired with two pump photons, the peak output intensity at

*λ*

_{3}is higher than that at

*λ*

_{1}. The inset shows the fast oscillations of the low intensity at the adiabatically eliminated intermediate wavelength

*λ*

_{2}= 786

*nm*, which are over 3 orders of magnitude lower than the peak output intensity.

*λ*

_{1}to

*λ*

_{3}is calculated for two cases: low pump intensity of 1

*GW/cm*

^{2}and high pump intensity of 12.4

*GW/cm*

^{2}. The results are plotted as a function of the two-photon phase-mismatch Δ

*k*= Δ

_{TP}*k*

_{1}+ Δ

*k*

_{2}. For low pump intensity the Stark shift is negligible, so the peak efficiency is obtained near the two-photon resonance condition Δ

*k*= 0. However, when the pump intensity is high, the Stark shift can no longer be neglected and the efficiency peak shifts by

_{TP}*δ*

*k*to Δ

_{S}*k*= −

_{TP}*δk*, which is the exact shift amount predicted analytically using Eq. (4).

_{S}## 4. Experiment

*λ*= 1064

_{p}*nm*, which served as the strong pumps, was passed through a temperature controlled PPKTP crystal with a Λ = 8.6

*μ*

*m*period. A laser with

*λ*

_{1}= 3010

*nm*and an estimated 10

*nm*FWHM bandwidth, which was produced from the same pump through an optical parametric oscillator, served as our input. The emerging power at the

*λ*

_{3}= 452

*nm*output and the intermediate wavelength

*λ*

_{2}= 786

*nm*were independently measured, using power detectors or a spectrometer, and peaks of both wavelengths were observed, corresponding to the expected output and intermediate wavelengths. When either the pump or the input were prevented from entering the crystal, both peaks would vanish.

*C*to 175°

*C*, affecting the refractive index [14

14. S. Emanueli and A. Arie, “Temperature-dependent dispersion equations for *KTiOPO*_{4} and *KTiOAsO*_{4},” Appl. Opt. **42**, 6661–6665 (2003). [CrossRef] [PubMed]

*kW/cm*

^{2}and 44.6

*MW/cm*

^{2}, respectively. Figure 3(a) shows the experimentally measured 452

*nm*output power vs. the PPKTP crystal temperature, after decrementing the background noise, and accounting for the transmission and reflection of the optical components in the apparatus. The detected noise, defined as the power measured when only the pump beam was incident on the PPKTP crystal, was found to be around 152nW and thus negligible compared to the measured output power which reaches 38

*μ*

*W*. As seen, optimum two-photon phase matching was obtained at crystal temperature of 133°

*C*. The small deviation from the expected value of 125°

*C*can be accounted for by the inaccuracy in the Sellemeier equations that were used to calculate the wavelength and temperature dependence of the crystal’s refractive index [14

14. S. Emanueli and A. Arie, “Temperature-dependent dispersion equations for *KTiOPO*_{4} and *KTiOAsO*_{4},” Appl. Opt. **42**, 6661–6665 (2003). [CrossRef] [PubMed]

15. K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled *KTiOPO*_{4},” Appl. Phys. Lett. **74**, 914–916 (1999). [CrossRef]

*λ*

_{2}was below the noise level and therefore could not be measured. Its upper bound was found to be 51.8nW, which is 733 times lower than the peak 452

*nm*output power, agreeing with the theoretical prediction of adiabatic elimination.

*nm*output power on pump power was experimentally checked. A half wave plate was placed in the path of the pump beam before it entered the PPKTP crystal, and gradually rotated through 45 degrees. Due to the fact that

*KTiOPO*

_{4}is a biaxial birefringent crystal, its phase-matching condition is polarization-dependent: only the pump power polarized along the PPKTP c axis contributes to the STWM processes. Figure 3(b) shows the experimentally measured 452

*nm*output power obtained with various values of c-polarized pump power. In this experiment the noise (defined above) was measured at each wave plate angle and decremented from the detected power. The experimental measurements fit well on the analytically calculated output power (using Eq. (5)), showing a quadratic nature, where the nonlinear coefficient

*χ*

^{(2)}was taken to be 9% higher than the value reported for other processes and wavelengths [16

16. 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]

## 5. Conclusion

17. N. V. Vitanov, B. W. Shore, and K. Bergmann, “Adiabatic population transfer in multistate chains via dressed intermediate states,” Eur. Phys. J. D **4**, 15–29 (1998). [CrossRef]

## Acknowledgments

## References and links

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

2. | H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A |

3. | H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express |

4. | R. W. Boyd, |

5. | D. Tannor, |

6. | D. Grischkowsky, M. M. T. Loy, and P. F. Liao, “Adiabatic following model for two-photon transitions: nonlinear mixing and pulse propagation,” Phys. Rev. A |

7. | J. C. Delagnes and L. Canioni, “Third harmonic generation in periodically poled crystals,” Proc. SPIE |

8. | C. Trallero-Herrero and T. C. Weinacht, “Transition from weak- to strong-field coherent control,” Phys. Rev. A |

9. | F. T. Hioe, “Dynamic symmetries in quantum electronics,” Phys. Rev. A |

10. | H. Suchowski, B. D. Bruner, A. Ganany-Padowicz, I. Juwiler, A. Arie, and Y. Silberberg, “Adiabatic frequency conversion of ultrafast pulses,” Appl. Phys. B |

11. | F. Bloch, “Nuclear induction,” Phys. Rev. |

12. | R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, “Geometrical representation of the Schrödinger equation for solving maser problems,” J. Appl. Phys. |

13. | D. S. Hum and M. M. Fejer, “Quasi-phasematching,” C. R. Physique |

14. | S. Emanueli and A. Arie, “Temperature-dependent dispersion equations for |

15. | K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled |

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

17. | N. V. Vitanov, B. W. Shore, and K. Bergmann, “Adiabatic population transfer in multistate chains via dressed intermediate states,” Eur. Phys. J. D |

18. | J. Keeler, |

**OCIS Codes**

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(190.4223) Nonlinear optics : Nonlinear wave mixing

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: December 15, 2011

Revised Manuscript: January 5, 2012

Manuscript Accepted: January 5, 2012

Published: January 30, 2012

**Citation**

Gil Porat, Yaron Silberberg, Ady Arie, and Haim Suchowski, "Two photon frequency conversion," Opt. Express **20**, 3613-3619 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-3613

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### References

- S. M. Saltiel, A. A. Sukhorukov, and Y. S. Kivshar, “Multistep parametric processes in nonlinear optics,” Prog. Opt.47, 1–73 (2005). [CrossRef]
- H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A78, 063821 (2008). [CrossRef]
- H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express17, 12731–12740 (2009). [CrossRef] [PubMed]
- R. W. Boyd, Nonlinear Optics, 3rd ed.(Academic Press, 2008).
- D. Tannor, Introduction to Quantum Mechanics: A Time-Dependent Perspective (University Science Books, 2007).
- D. Grischkowsky, M. M. T. Loy, and P. F. Liao, “Adiabatic following model for two-photon transitions: nonlinear mixing and pulse propagation,” Phys. Rev. A12, 2514–2533 (1975). [CrossRef]
- J. C. Delagnes and L. Canioni, “Third harmonic generation in periodically poled crystals,” Proc. SPIE7917, 79171C (2011). [CrossRef]
- C. Trallero-Herrero and T. C. Weinacht, “Transition from weak- to strong-field coherent control,” Phys. Rev. A75, 063401 (2007). [CrossRef]
- F. T. Hioe, “Dynamic symmetries in quantum electronics,” Phys. Rev. A28, 879–886 (1983). [CrossRef]
- H. Suchowski, B. D. Bruner, A. Ganany-Padowicz, I. Juwiler, A. Arie, and Y. Silberberg, “Adiabatic frequency conversion of ultrafast pulses,” Appl. Phys. B105, 697–702 (2011). [CrossRef]
- F. Bloch, “Nuclear induction,” Phys. Rev.70, 460–474 (1946). [CrossRef]
- R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, “Geometrical representation of the Schrödinger equation for solving maser problems,” J. Appl. Phys.28, 49–52 (1957). [CrossRef]
- D. S. Hum and M. M. Fejer, “Quasi-phasematching,” C. R. Physique8, 180–198 (2007). [CrossRef]
- S. Emanueli and A. Arie, “Temperature-dependent dispersion equations for KTiOPO4 and KTiOAsO4,” Appl. Opt.42, 6661–6665 (2003). [CrossRef] [PubMed]
- K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4,” Appl. Phys. Lett.74, 914–916 (1999). [CrossRef]
- I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B14, 2268–2294 (1997). [CrossRef]
- N. V. Vitanov, B. W. Shore, and K. Bergmann, “Adiabatic population transfer in multistate chains via dressed intermediate states,” Eur. Phys. J. D4, 15–29 (1998). [CrossRef]
- J. Keeler, Understanding NMR Spectroscopy (John Wiley & Sons, 2005).

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