## Nonlinear phase matching locking via optical readout

Optics Express, Vol. 14, Issue 23, pp. 11256-11264 (2006)

http://dx.doi.org/10.1364/OE.14.011256

Acrobat PDF (212 KB)

### Abstract

For optimal *χ*^{(2)} nonlinear interaction the phase matching condition must be satisfied. For type I and type II phase matched materials, this is generally achieved by controlling the temperature of the nonlinear media. We describe a technique to readout the phase-matching condition interferometrically, and experimentally demonstrate feedback control in a degenerate optical parametric amplifier (OPA) which is resonant at both the fundamental and harmonic frequencies. The interferometric readout technique is based on using the cavity resonances at the fundamental and harmonic frequencies to enable the readout of the phase mismatch. We achieve relatively fast temperature feedback using the photothermal effect, by modulating the amplitude of the OPA pump beam.

© 2006 Optical Society of America

## 1. Introduction

*χ*

^{(2)}interaction is responsible for some of the most important devices in nonlinear optics: the second harmonic generator (SHG), the optical parametric oscillator (OPO) and optical parametric amplifier (OPA). These devices are used in a myriad of applications, for example tunable laser sources [1

1. J. A. Giordmaine and R. C. Miller, “Tunable Coherent Parametric Oscillation in LiNbO3 at Optical Frequencies,” Phys. Rev. Lett. **14**, 973–976 (1965). [CrossRef]

2. R. C. Eckardt, C. D. Nabors, W. J. Kozlovsky, and R. L. Byer, “Optical parametric oscillator frequency tuning and control,” J. Opt. Soc. Am. B **8**, 646–667 (1991). [CrossRef]

*χ*

^{(2)}nonlinear interaction is the conservation of momentum, also referred to as

*phase matching*in this context. The phase matching condition imposes the relation on wavevectors;

*k*

_{b}=

*k*

_{a}+

*k*

_{a′}, where

*k*

_{i}=

*n*

_{i}

*ω*

_{i}/

*c*

_{0}and

*n*

_{i}is the refractive index of the

*i*

^{th}photon in the nonlinear medium and

*c*

_{0}is the speed of light in vacuum. For small (optical) phase mismatch nonlinear interaction still occurs, however with reduced efficiency. For optimum nonlinear interaction the phase matching condition must be met precisely. Similarly, it is important that the phase matching condition be precisely controlled for low noise and long term applications, such as the generation of squeezing for gravitational wave detection [7] and to minimize intensity noise in second harmonic generators.

*type I*,

*type II*or

*quasi-phase matching*[10]. Consider a SHG or degenerate OPO that is type I uniaxial crystal phase matched via temperature tuning such as MgO:LiNbO

_{3}. Here, in SHG or degenerate OPO:

*ω*

_{a}=

*ω*

_{a′}and the phase matching condition simplifies to:

*n*

_{b}=

*n*

_{a}. Here, and henceforth, we refer to the field with frequency

*ω*

_{a}as the fundamental field and to

*ω*

_{b}as the (second) harmonic field. To achieve phase matching in this system, the polarization of the fundamental field is set to the ordinary axis, the harmonic to the extraordinary axis, and the crystal temperature is tuned. Temperate tuning changes the ordinary and extraordinary refractive indices differentially according to the respective Sellmeier equations [2

2. R. C. Eckardt, C. D. Nabors, W. J. Kozlovsky, and R. L. Byer, “Optical parametric oscillator frequency tuning and control,” J. Opt. Soc. Am. B **8**, 646–667 (1991). [CrossRef]

5. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. **31**, 97–105 (1983). [CrossRef]

*n*=

*n*

_{a}-

*n*

_{b}). At the phase matching temperature there is no refractive index difference (Δ

*n*=0) and the two cavities are both resonant [6

6. Assuming there is no differential phase shift on reflection between the harmonic and fundamental frequencies on the mirror coatings. In general, there will be a differential phase shift on reflection on the mirror coatings and on transmission through anti-reflective (AR) coatings. In our system we experimentally determine that the sum of the differential phase shifts per round trip of the cavity is close to an integer multiple times p. When the differential phase shift is significantly large we compensate for the dispersion by inserting a dichroic AR coated BK7 substrate placed in the cavity. This substrate is angled such that the dispersion on transmission through the substrate compensates for the differential phase shift on the mirror coatings and AR coatings, see [7].

11. V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, “Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae,” Phys. Lett. A **264**, 1–10 (1999). [CrossRef]

12. Y. T. Liu and K. S. Thorne, “Thermoelastic noise and homogeneous thermal noise in finite sized gravitationalwave test masses,” Phys. Rev. D **62**, 122002 (2000). [CrossRef]

13. M. Cerdonio, L. Conti, A. Heidmann, and M. Pinard, “Thermoelastic effects at low temperatures and quantum limits in displacement measurements,” Phys. Rev. D **63**, 082003 (2001). [CrossRef]

14. K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, “Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator,” Phys. Rev. A **72**, 043819 (2005). [CrossRef]

## 2. Theory

*χ*

^{(2)}nonlinear optic equations of motion are [8

8. M. J. Collett and C.W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A **30**, 1386–1391 (1984). [CrossRef]

*a*and

*b*are proportional to the the intra-cavity fundamental and second harmonic fields, respectively;

*κ*

^{a}and

*κ*

^{b}are the total resonator decay rates for each field;

*ε*is the nonlinear coupling parameter; and

*A*

_{in}and

*B*

_{in}are the driving fields with the respective input coupling rates are

^{a}and Δ

^{b}.

*δp*, are given by the following equations [9],

^{j}*j*=

*a*,

*b*}. The free spectral range is given by

*ω*

_{fsr}=2

*πc*

_{0}/

*p*, with the total OPL;

*p*=

*L*+

*nL*

_{c}with

*L*the round trip length in free space and

*L*

_{c}the length of the crystal with refractive index,

*n*and

*λ*

_{j}is the wavelength in vacuum and

*c*

_{0}the speed of light in vacuum.

*δ*

_{p}, can come from change to the free space OPL and from change in the crystal OPL. The crystal OPL is a function of crystal temperature change,

*δT*, arising from two mechanisms, thermal expansion and refractive index change, i.e.;

*dn*

^{j}/

*dT*are the crystal’s photorefractive constants and

*α*

_{j}are the crystal’s thermal expansion constants. We define the total detuning as;

_{fs}is the due to the change in free space OPL and

^{b}→0). If the temperature of the crystal is changed, the control system will have to change the free space OPL to compensate for the change in crystal OPL (Δ

_{fs}=-

*δT*)) to maintain cavity resonance. Substituing this into the total detuning of the fundamental cavity we find;

*crystal OPL*at the fundamental and harmonic cavities due to temperature tuning of the crystal. We can extend Eq. 9 to considered more than one longitudinal mode of the fundamental cavity. Also, for completeness, we add an arbitrary differential phase shift between the fundamental and harmonic fields,

*θ*, which adds an detuning of Δ

^{θ}=

*θω*

_{fsr}. Thus the total detuning then becomes;

*l*is the cavity mode number. The intra-cavity amplitude of an

*l*th mode is:

*ϕ*

_{l}is the combined angle ∠(

*ε**

*b**), which can be varied to control the sign of the parametric gain. Here

*ϕ*

_{l}=±∠(

*κ*

^{a}+

*i*

*k*, in the following form [10];

*ε*

_{0}is a constant. The Sellmeier equation [2] at the fundamental frequency is Δ

*k*=ξ(

*δT*) where ξ is a constant whose value depends on the crystal’s properties, and

*δT*is the crystal’s temperature offset from the phase matched temperature. The nonlinear gain is found when the transmitted power with/without the pump field is compared;

## 3. Experiment

_{3}with 7% doping. The optical surfaces were flat and coated for anti-reflection at both wavelengths. The crystal was placed in a peltier driven oven held at ~63

*°*C to approximately 5mK accuracy. The DROPA bow-tie cavity configuration consisted of three high reflectivity (HR) mirrors at both wavelengths and the input/output coupler with transmission of 10% and 3% at 1064nm and 532nm, respectively. The incident pump power was 100mW, giving a circulating pump power of ~2.7W, corresponding to a parametric gain of just under 3dB. The OPA was seeded through a HR mirror with ~ 10mW. The input harmonic field was phase modulated at 70MHz. The harmonic field reflected from the OPA was demodulated to produce an error signal for the cavity length. This error signal was fed back to a piezo-electric tranducer (PZT1) bonded to a OPA cavity mirror. The PZT1 also had a modulation signal added at 30kHz to produce phase modulation on the intra-cavity fields. The transmission at the seed was detected and demodulated (at 30kHz) to produce an error signal also using the PDH technique. Figure 3 shows the transmission of the cavity at the fundamental frequency, plot (a), and the associated error signal, plot (b), as the temperature of the crystal was swept across the phase matching temperature. These data was taken with the harmonic cavity locked on resonance. The vertical axis of the Figure 3(a) is normalized to the resonant power transmitted through the OPA without parametric gain. Thus on resonance, below 1 indicates parametric de-amplification and above 1 parametric amplification. The relative phase of the harmonic and fundamental frequencies was swept rapidly by dithering PZT2 at 1kHz in order to sample amplification and de-amplification, to show the nonlinear gain envelope. The error signal here has been low pass filtered to remove any component associated with the nonlinear gain at 1kHz.1 This (phase matching) error signal was sent to an amplitude modulator (AM) in the input harmonic fields’ path to actuate on the crystal temperature via the photothermal effect. Photothermal actuation was proven to be very effective since most of the harmonic field was absorbed in the crystal, as the cavity is nearly impedance matched and the crystal represents the dominant loss mechanism.

## 4. Discussion

13. M. Cerdonio, L. Conti, A. Heidmann, and M. Pinard, “Thermoelastic effects at low temperatures and quantum limits in displacement measurements,” Phys. Rev. D **63**, 082003 (2001). [CrossRef]

*χ*

^{(2)}effects.

## 5. Conclusions

## Footnotes

1 | When using this technique for applications one may lock the phase of the fundamental/harmonic fields to amplification or deamplification. |

## References and links

1. | J. A. Giordmaine and R. C. Miller, “Tunable Coherent Parametric Oscillation in LiNbO3 at Optical Frequencies,” Phys. Rev. Lett. |

2. | R. C. Eckardt, C. D. Nabors, W. J. Kozlovsky, and R. L. Byer, “Optical parametric oscillator frequency tuning and control,” J. Opt. Soc. Am. B |

3. | For Example: D. F. Walls and G. J. Milburn, Quantum Optics, Springer-Verlag, Berlin, 1st ed. (1994). |

4. | We define doubly resonant to be resonant at both the fundamental and harmonic frequencies. |

5. | R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. |

6. | Assuming there is no differential phase shift on reflection between the harmonic and fundamental frequencies on the mirror coatings. In general, there will be a differential phase shift on reflection on the mirror coatings and on transmission through anti-reflective (AR) coatings. In our system we experimentally determine that the sum of the differential phase shifts per round trip of the cavity is close to an integer multiple times p. When the differential phase shift is significantly large we compensate for the dispersion by inserting a dichroic AR coated BK7 substrate placed in the cavity. This substrate is angled such that the dispersion on transmission through the substrate compensates for the differential phase shift on the mirror coatings and AR coatings, see [7]. |

7. | K. McKenzie, M. B. Gray, S. Goβler, P. K. Lam, and D. E. McClelland, “Squeezed State Generation for Interferometric Gravitational-Wave Detection,” Class. Quant. Grav. 23, S245–S250 (2006). |

8. | M. J. Collett and C.W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A |

9. | A.E. Siegman, Lasers, University Science Books (1986). |

10. | A. Yariv, Optical Electonics in Modern Communications, Fifth Edition, Oxford University Press (1997). |

11. | V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, “Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae,” Phys. Lett. A |

12. | Y. T. Liu and K. S. Thorne, “Thermoelastic noise and homogeneous thermal noise in finite sized gravitationalwave test masses,” Phys. Rev. D |

13. | M. Cerdonio, L. Conti, A. Heidmann, and M. Pinard, “Thermoelastic effects at low temperatures and quantum limits in displacement measurements,” Phys. Rev. D |

14. | K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, “Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator,” Phys. Rev. A |

15. | The SHG is a custom Diabolo model developed by Innolight GmbH. |

16. | S. P. Tewari and G. S. Agarwal, “Control of phase matching and nonlinear generation in dense media by resonant fields,” Phys. Rev. Lett. 17 1811–1814 (1986). |

**OCIS Codes**

(190.4360) Nonlinear optics : Nonlinear optics, devices

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: June 9, 2006

Revised Manuscript: October 11, 2006

Manuscript Accepted: October 11, 2006

Published: November 13, 2006

**Citation**

Kirk McKenzie, Malcolm B. Gray, Ping Koy Lam, and David E. McClelland, "Nonlinear phase matching locking via optical readout," Opt. Express **14**, 11256-11264 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-23-11256

Sort: Year | Journal | Reset

### References

- J. A. Giordmaine and R. C. Miller, "Tunable coherent parametric oscillation in LiNbO3 at optical frequencies," Phys. Rev. Lett. 14, 973-976 (1965). [CrossRef]
- R. C. Eckardt, C. D. Nabors, W. J. Kozlovsky and R. L. Byer, "Optical parametric oscillator frequency tuning and control," J. Opt. Soc. Am. B 8, 646-667 (1991). [CrossRef]
- For Example: D. F. Walls and G. J. Milburn, Quantum Optics, (Springer-Verlag, Berlin, 1st ed., 1994).
- We define doubly resonant to be resonant at both the fundamental and harmonic frequencies.
- R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983). [CrossRef]
- Assuming there is no differential phase shift on reflection between the harmonic and fundamental frequencies on the mirror coatings. In general, there will be a differential phase shift on reflection on the mirror coatings and on transmission through anti-reflective (AR) coatings. In our system we experimentally determine that the sum of the differential phase shifts per round trip of the cavity is close to an integer multiple times π. When the differential phase shift is significantly large we compensate for the dispersion by inserting a dichroic AR coated BK7 substrate placed in the cavity. This substrate is angled such that the dispersion on transmission through the substrate compensates for the differential phase shift on the mirror coatings and AR coatings, see Ref. [7].
- K. McKenzie, M. B. Gray, S. Goßler, P. K. Lam, and D. E. McClelland, "Squeezed State Generation for Interferometric Gravitational-Wave Detection," Class. Quantum Grav. 23, S245-S250 (2006).
- M. J. Collett and C.W. Gardiner, "Squeezing of intracavity and traveling-wave light fields produced in parametric amplification," Phys. Rev. A 30, 1386-1391 (1984). [CrossRef]
- A. E. Siegman, Lasers, (University Science Books, 1986).
- A. Yariv, Optical Electonics in Modern Communications, Fifth Edition, (Oxford University Press 1997).
- V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, "Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae," Phys. Lett. A 264, 1-10 (1999). [CrossRef]
- Y. T. Liu and K. S. Thorne, "Thermoelastic noise and homogeneous thermal noise in finite sized gravitationalwave test masses," Phys. Rev. D 62, 122002 (2000). [CrossRef]
- M. Cerdonio, L. Conti, A. Heidmann, and M. Pinard, "Thermoelastic effects at low temperatures and quantum limits in displacement measurements," Phys. Rev. D 63, 082003 (2001). [CrossRef]
- K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, "Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 72, 043819 (2005) [CrossRef]
- The SHG is a custom Diabolo model developed by Innolight GmbH.
- S. P. Tewari and G. S. Agarwal, "Control of phase matching and nonlinear generation in dense media by resonant fields," Phys. Rev. Lett. 171811-1814 (1986).

## 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.