## Robust adiabatic sum frequency conversion

Optics Express, Vol. 17, Issue 15, pp. 12731-12740 (2009)

http://dx.doi.org/10.1364/OE.17.012731

Acrobat PDF (471 KB)

### Abstract

We discuss theoretically and demonstrate experimentally the robustness of the adiabatic sum frequency conversion method. This technique, borrowed from an analogous scheme of robust population transfer in atomic physics and nuclear magnetic resonance, enables the achievement of nearly full frequency conversion in a sum frequency generation process for a bandwidth up to two orders of magnitude wider than in conventional conversion schemes. We show that this scheme is robust to variations in the parameters of both the nonlinear crystal and of the incoming light. These include the crystal temperature, the frequency of the incoming field, the pump intensity, the crystal length and the angle of incidence. Also, we show that this extremely broad bandwidth can be tuned to higher or lower central wavelengths by changing either the pump frequency or the crystal temperature. The detailed study of the properties of this converter is done using the Landau-Zener theory dealing with the adiabatic transitions in two level systems.

© 2009 Optical Society of America

## 1. Introduction

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

2. R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, “Geometrical Representation of the Schrodinger Equation for Solving Maser Problems”, J. Appl. Phys. **28**, 49–52 (1957).
[CrossRef]

3. 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]

*et. al*. [7

7. T. Torosov and N. V. Vitanov, “Exactly soluble two-state quantum models with linear couplings”, J. Phys. A , **41**, 155309 (2008).
[CrossRef]

*et. al*. [8

8. G. Imeshev, M. Fejer, A. Galvanauskas, and D. Harter, “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings”, J. Opt. Soc. Am. B **18**, 534–539 (2001).
[CrossRef]

3. 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]

16. C. Zener, “Non-adiabatic Crossing of Energy Levels”, Proc. Roy. Soc. London A **137**, 696–702 (1932).
[CrossRef]

9. M. Arbore, A. Galvanauskas, D. Harter, M. Chou, and M. Fejer, “Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped-period-poled lithium niobate” Opt. Lett. **22**, 1341–1343 (1997).
[CrossRef]

10. G. Imeshev, M. Arbore, M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, “Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping”, J. Opt. Soc. Am. B , **17**, 304–318 (2000).
[CrossRef]

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

12. M. Charbonneau-Lefort, B. Afeyan, and M. Fejer, “Optical parametric amplifiers using chirped quasi-phasematching gratings. I. Practical design formulas”, J. Opt. Soc. Am. B **25**, 463–480 (2008).
[CrossRef]

13. M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi phase matching in bulk polycrystalline isotropic nonlinear materials”, Nature **432**, 374–376 (2004).
[CrossRef] [PubMed]

14. K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the Phase-Matching Bandwidth in Quasi-Phased-Matched Second Harmonic Generation”, IEEE J. Quantum Electron **30**, 15961604 (1994).
[CrossRef]

*Only*in SFG process, the geometrical visualization on Bloch sphere, and the suggested adiabatic solution are valid.

## 2. Theoretical analysis

### 2.1. Dynamics and Geometrical Representation of Sum Frequency Generation Process

*k*=

*k*

_{1}+

*k*

_{2}-

*k*

_{3}is the phase mismatch, z is the position along the propagation axis,

*ω*

_{1}and

*ω*

_{3}are the frequencies of the signal and idler, respectively,

*k*

_{1}and

*k*

_{3}are their wave numbers,

*v*

_{g1}and

*v*

_{g3}are their group velocities,

*c*is the speed of light in vacuum,

*A*

_{1},

*A*

_{2},

*A*

_{3}are the signal, pump and idler amplitudes, respectively, and

*χ*

^{(2)}is the 2

^{nd}order susceptibility of the crystal (assumed to be frequency independent). In the case where the temporal envelope of the waves are much longer than the length of the crystal (i.e. where we consider monochromatic, quasi-monochromatic laser beams, or stretched ultrashort pulses), one can omit the influence of the waves’ group velocities. In this paper we will deal with quasi-monochromatic laser beams. In the case of ultrashort pulses upconversion, one should first stretch the pulse, in order to minimize the deleterious effect of group velocity mismatch and group velocity dispersion. Typically, a pulse length of more than 1ps would suffice for an interaction in the visible and near infrared. The converted pulse would be re-compressed to a transform-limited pulse after exiting from the nonlinear crystal.

*k*value; also, the population of the ground and excited states are analogous to the magnitude of the signal and idler fields, respectively. This analogy is further detailed in Ref. [3

3. 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]

*et. al*. [2

2. R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, “Geometrical Representation of the Schrodinger Equation for Solving Maser Problems”, J. Appl. Phys. **28**, 49–52 (1957).
[CrossRef]

*k*(

*z*) and κ (

*z*), can be visualized as a trajectory on the surface of an upconversion Bloch sphere. The loss-free evolution equations can be written as a single vector precession equation

*ρ*⃗

*SFG*=(

*U,V,W*)=

*A**

_{3}

*A*

_{1}+

*A*

_{3}

*A**

_{1},

*i*(

*A**

_{3}

*A*

_{1}-

*A*

_{3}

*A**

_{1}), |

*A*

_{3}|

^{2}-|

*A*

_{1}|

^{2}, and includes the coherence between the signal and idler amplitudes along the propagation. In particular, its z component (

*W*) gives information about the conversion efficiency. The south pole

_{SFG}*ρ*⃗=(0,0,-1) corresponds to zero conversion (

*A*

_{3}=0), while the north pole

*ρ*⃗=(0,0,1) corresponds to full conversion. In between, the conversion efficiency is defined as:

*η*=(

*W*+1)/2. The rotating vector (also known as the torque vector),

_{SFG}*g*⃗=(

*Re*{κ},

*Im*{κ},Δ

*k*), represents the coupling between the signal and idler frequencies, and the size of the phase mismatch parameter. This analogy was extended to include the semi-phenomenological decay constants

*T*

_{1}and

*T*

_{2}, which appear in the original Bloch equations, and in our context are characteristic decay lengths rather than times [3

**78**, 063821 (2008)
[CrossRef]

*k*(

*z*) and κ, and in most of the cases, no analytical or approximate solution exists. In such cases, the geometrical visualization could be helpful, where the trajectory of the SFG process, for any function of Δ

*k*(

*z*) and κ (

*z*), is guaranteed to be on the surface of the SFG Bloch sphere.

### 2.2. Adiabaticity criteria and application of Landau-Zener theory

*k*(

*z*) should also be very large compared to κ, and should change adiabatically from a large negative value to a large positive value, i.e. |Δ

*k*|≫κ, Δ

*k*(

*z*=0)<0, Δ

*k*(

*z*=

*L*)>0. These adiabatic constraints were derived by following their analogous dynamical counterparts in the RAP mechanism, where a strong chirped excitation pulse scans slowly through the resonance to achieve robust full inversion [7

7. T. Torosov and N. V. Vitanov, “Exactly soluble two-state quantum models with linear couplings”, J. Phys. A , **41**, 155309 (2008).
[CrossRef]

*k*(

*z*) is varied linearly along the crystal, a simple parameter for the degree of adiabaticity appears. In the quantum literature it is known as the Landau-Zener criterion [15, 16

16. C. Zener, “Non-adiabatic Crossing of Energy Levels”, Proc. Roy. Soc. London A **137**, 696–702 (1932).
[CrossRef]

*d*Δ

*k/dz*, and the square of the coupling coefficient, κ

^{2}. Mathematically this represents the ratio between the left hand side and the right hand side of Eq. 3, at the location where Δ

*k*=0.

*Adiabatic*propagation is obtained when

*α*≪1, which is the case where the conversion efficiency reaches unity. This can be achieved either by changing slowly the sweep rate at a given pump intensity, or by applying strong pump for a given sweep rate.

*α*≪1, full frequency conversion is achieved as shown in Fig. 1(a). When, the pump intensity is not high enough or the sweep rate at a given crystal length and pump intensity is not slow enough, then

*α*~1, and the conversion efficiency will drop, as seen in Fig. 1(b). The case of

*α*≫1, is where the pump intensity is small, or the sweep rate is extremely high. This corresponds to the weak coupling regime (the ”unamplified signal approximation”), which results in low conversion efficiency, as shown in Fig. 1(c). These dynamical trajectories can be projected on theW-axis of the sphere, bringing information regarding the conversion efficiency along the propagation, as shown in Fig. 1(d). Also, their calculated Landau-Zener conversion efficiencies are presented to the right of each projected trajectory. Due to the importance of Eq. 4, we decided to present it in more practical parameters:

*c*=3·10

^{10}

*cm/sec*, λ

_{1}and λ

_{3}are measured in

*cm*,

*I*

_{2}is measured in

*MW/cm*

^{2},

*χ*

^{(2)}is measured in

*pm/V*and |

*d*Δ

*k/dz*| is measured in

*cm*

^{-2}.

## 3. Experimental setup and results

19. J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric”, Phys. Rev. **127**, 1918–1939 (1962).
[CrossRef]

*z*) is the local poling period.

*k*

_{Λ}(

*z*) in a power series:

*k*

_{0}=-Δ

*k*(

_{proc}*z*=

*L*/2), and the linear term,

*k*

_{Λ}(

*z*), should take into account the spectral dependence of Δ

*k*(

_{proc}*z*), due to dispersion, as well as the spectral variation of the coupling constant. In practice, in the near-mid IR regime, these higher order corrections are relatively small.

*µm*to 16.2

*µm*along a flux-grown KTiOPO4 crystal (Raicol Crystals Ltd.), with dimensions of 20×2×1 mm. It was poled by low-temperature electric-field poling [20

20. G. Rosenman, A. Skliar, D. Eger, M. Oron, and M. Katz, “Low temperature periodic electrical poling of flux-grown KTiOPO_{4} and isomorphic crystals”, Appl. Phys. Lett. , **73**, 3650–3652 (1998).
[CrossRef]

*µJ*), and a tunable signal varied from 1400 nm to 1700 nm (5 ns, 1

*µJ*). The pump and the signal, both polarized in the extra-ordinary axis were spatially overlapped and focused collinearly into the crystal having waists of 150

*µm*and 120

*µm*, respectively. These values guarantee that the Rayleigh range is larger than the crystal length and thus the plane wave approximation of our simulations holds. The crystal was held in a temperature-controlled mount. We collected both the input wavelength (signal) and output SFG wavelength (idler) after their propagation in the crystal and recorded them using an InGaAs photodiode, and a cooled CCD spectrometer, respectively.

**78**, 063821 (2008)
[CrossRef]

*MW/cm*

^{2}.

*MW/cm*

^{2}. It was shown that an efficient ultra-broadband conversion of over 140nm wide (1470nm to 1610nm) at room temperature was obtained, except for a small region of low efficiency around 1485nm, which was associated with a local fabrication defect, leading to violation of the adiabaticity condition at this wavelength [3

**78**, 063821 (2008)
[CrossRef]

*mm*and 20

*mm*, of the same crystal, the robustness of our design to variations in the crystal length is also demonstrated. In contrast to frequency converter, where the use of thin crystals is required to achieve maximal bandwidth, in the adiabatic design the achieved bandwidth is expected to grow as the crystal length increases, while maintaining the same conversion efficiency. We show this both numerically and experimentally in Fig. 3. As can be seen, the efficient conversion bandwidth is increased almost linearly with the length of the crystals. This is in good agreement with the experimental results of Fig. 3(b), which show a 30nm wider bandwidth response as the increasing of the crystal length in 3 mm. Two factors practically limit the achievable efficient conversion bandwidth. The first is absorption in the nonlinear crystal itself. Another limiting factor arises from the plane wave approximation which holds only if the crystal length is shorter than the apparatus Rayleigh range. The latter is, in practice, limited by the available pump energy.

*ω*

_{1}=1550

*nm*as function of the crystal temperature is plotted. Highly efficient conversion is experimentally observed in a temperature range of over 80°C. The simulation, which contains the effect of thermal expansion (taken from Ref. [21

21. S. Emanueli and A. Arie, “Temperature-Dependent Dispersion Equations for KTiOPO_{4} and KTiOAsO_{4}”, Appl. Opt. **42**, 6661–6665 (2003).
[CrossRef] [PubMed]

*ω*

_{1}=1550

*nm*, and crystal length of

*L*=20

*mm*.

*C*and 110°

*C*. As can be seen, the efficient conversion band is redshifted by ≈50

*nm*, which is in good correspondence with the numerical simulations shown,as horizontal cross sections, in Fig. 4(a). The horizontal 1D cross-sections of the 2D numerical simulation would give the expected bandwidth in each crystal temperature. The two experimental values are marked at the left of the figure, and use the same color definition (blue for 50°C and red for 110°C). An even more dramatic effect is presented in Fig. 5(b), where the effect of changing the pump wavelength is simulated. In the figure we compare the response of our adiabatic design pumped at 1064nm (as in the experiments) with the response to a pump wavelength of 1047 nm (pumping with a Nd:YLF laser), and pump wavelength of 1159 nm, which can be readily obtained by raman shifting the 1064nm excitation beam, in a Raman shifter.

## 4. Conclusion

_{3}having the same chirp parameter, and with the same interacting wavelengths, the spectral bandwidth is slightly narrower, 120 nm instead of 140 nm, but owing to the higher nonlinear coefficient in LiNbO

_{3}, lower pump intensity of 310

*MW/cm*

^{2}instead of 360

*MW/cm*

^{2}, would suffice for full conversion.

## References and links

1. | F. Bloch, “Nuclear Induction”, Phys. Rev. |

2. | R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, “Geometrical Representation of the Schrodinger Equation for Solving Maser Problems”, J. Appl. Phys. |

3. | H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical Representation of Sum Frequency Generation and Adiabatic Frequency Conversion”, Phys. Rev. A , |

4. | D. Boyd, |

5. | A. Yariv, |

6. | L. Allen and J. H. Eberly, |

7. | T. Torosov and N. V. Vitanov, “Exactly soluble two-state quantum models with linear couplings”, J. Phys. A , |

8. | G. Imeshev, M. Fejer, A. Galvanauskas, and D. Harter, “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings”, J. Opt. Soc. Am. B |

9. | M. Arbore, A. Galvanauskas, D. Harter, M. Chou, and M. Fejer, “Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped-period-poled lithium niobate” Opt. Lett. |

10. | G. Imeshev, M. Arbore, M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, “Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping”, J. Opt. Soc. Am. B , |

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

12. | M. Charbonneau-Lefort, B. Afeyan, and M. Fejer, “Optical parametric amplifiers using chirped quasi-phasematching gratings. I. Practical design formulas”, J. Opt. Soc. Am. B |

13. | M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi phase matching in bulk polycrystalline isotropic nonlinear materials”, Nature |

14. | K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the Phase-Matching Bandwidth in Quasi-Phased-Matched Second Harmonic Generation”, IEEE J. Quantum Electron |

15. | L. D. Landau, “Zur Theorie der Energieubertragung. II”, Physics of the Soviet Union |

16. | C. Zener, “Non-adiabatic Crossing of Energy Levels”, Proc. Roy. Soc. London A |

17. | A. Massiach, |

18. | D. J. Tannor, |

19. | J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric”, Phys. Rev. |

20. | G. Rosenman, A. Skliar, D. Eger, M. Oron, and M. Katz, “Low temperature periodic electrical poling of flux-grown KTiOPO |

21. | S. Emanueli and A. Arie, “Temperature-Dependent Dispersion Equations for KTiOPO |

**OCIS Codes**

(190.4360) Nonlinear optics : Nonlinear optics, devices

(230.4320) Optical devices : Nonlinear optical devices

(140.3613) Lasers and laser optics : Lasers, upconversion

**History**

Original Manuscript: March 11, 2009

Revised Manuscript: May 14, 2009

Manuscript Accepted: May 26, 2009

Published: July 13, 2009

**Citation**

Haim Suchowski, Vaibhav Prabhudesai, Dan Oron, Ady Arie, and Yaron Silberberg, "Robust adiabatic sum frequency
conversion," Opt. Express **17**, 12731-12740 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-12731

Sort: Year | Journal | Reset

### References

- 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 Schrodinger Equation for Solving Maser Problems", J. Appl. Phys. 28, 49-52 (1957). [CrossRef]
- 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]
- D. Boyd, Nonlinear Optics, 2nd ed. (Academic, New York, 2003).
- A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1989).
- L. Allen and J. H. Eberly, Optical Resonance and Two Level Atoms (Dover, New York, 1987).
- T. Torosov and N. V. Vitanov, "Exactly soluble two-state quantum models with linear couplings", J. Phys. A, 41, 155309 (2008). [CrossRef]
- G. Imeshev, M. Fejer, A. Galvanauskas, and D. Harter, "Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings", J. Opt. Soc. Am. B 18, 534-539 (2001). [CrossRef]
- M. Arbore, A. Galvanauskas, D. Harter, M. Chou, and M. Fejer, "Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped-period-poled lithium niobate" Opt. Lett. 22, 1341-1343 (1997). [CrossRef]
- G. Imeshev, M. Arbore, M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, "Ultrashort-pulse secondharmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping", J. Opt. Soc. Am. B, 17, 304-318 (2000). [CrossRef]
- D. S. Hum, and M. Fejer, "Quasi-phasematching", C. R. Physique 8, 180-198 (2007). [CrossRef]
- M. Charbonneau-Lefort, B. Afeyan, and M. Fejer, "Optical parametric amplifiers using chirped quasiphasematching gratings. I. Practical design formulas", J. Opt. Soc. Am. B 25, 463-480 (2008). [CrossRef]
- M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, "Random quasi phase matching in bulk polycrystalline isotropic nonlinear materials", Nature 432, 374-376 (2004). [CrossRef] [PubMed]
- K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, "Broadening of the Phase-Matching Bandwidth in Quasi-Phased-Matched Second Harmonic Generation", IEEE J. Quantum Electron 30, 15961604 (1994). [CrossRef]
- L. D. Landau, "Zur Theorie der Energieubertragung. II", Physics of the Soviet Union 2, 46-51 (1932).
- C. Zener, "Non-adiabatic Crossing of Energy Levels", Proc. Roy. Soc. London A 137, 696-702 (1932). [CrossRef]
- A. Massiach, Quantum Mechanics (N. Holland, Amsterdam, 1962), Vol II.
- D. J. Tannor, Introduction to Quantum Mechanics: A Time-dependent Perspective (University Science Books, Sausalito, California, 2007).
- J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric", Phys. Rev. 127, 1918-1939 (1962). [CrossRef]
- G. Rosenman, A. Skliar and D. Eger, M. Oron, and M. Katz, "Low temperature periodic electrical poling of flux-grown KTiOPO4 and isomorphic crystals", Appl. Phys. Lett., 73, 3650-3652 (1998). [CrossRef]
- S. Emanueli, and A. Arie, "Temperature-Dependent Dispersion Equations for KTiOPO4 and KTiOAsO4", Appl. Opt. 42, 6661-6665 (2003). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.