## Two-dimensional modeling of transient gain gratings in saturable gain media |

Optics Express, Vol. 20, Issue 7, pp. 6887-6896 (2012)

http://dx.doi.org/10.1364/OE.20.006887

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### Abstract

A transient two-dimensional model describing degenerate four-wave mixing inside saturable gain media is presented. The new model is compared to existing one-dimensional models with their qualitative results confirmed. Large quantitative differences with respect to peak reflectivity and optimum pump fluence are observed. Furthermore, the influence of the beam focus size, the transverse position and the crossing angle on the reflectivity of the grating is investigated using the improved model. It is demonstrated that the phase conjugate reflectivity depends sensitively on the transverse features of the interacting beams with a transverse shift in the position of the pump beams yielding a threefold improvement in reflectivity.

© 2012 OSA

## 1. Introduction

1. R. P. M. Green, S. Camacho-Lopez, and M. J. Damzen, “Experimental investigation of vector phase conjugation in Nd3+:YAG,” Opt. Lett. **21**, 1214–1216 (1996). [CrossRef] [PubMed]

8. S. Lam and M. Damzen, “Self-adaptive Nd:YLF holographic laser with selectable wavelength operation,” Appl. Phys. B. **76**, 237–240 (2003). [CrossRef]

9. M. J. Damzen, Y. Matsumoto, G. J. Crofts, and R. P. M. Green, “Bragg-selectivity of a volume gain grating,” Opt. Commun. **123**, 182–188 (1996). [CrossRef]

10. A. Minassian, G. Crofts, and M. Damzen, “Spectral filtering of gain gratings and spectral evolution of holographic laser oscillators,” IEEE J. Quantum Electron. **36**, 802–809 (2000). [CrossRef]

11. P. C. Shardlow and M. J. Damzen, “Phase conjugate self-organized coherent beam combination: a passive technique for laser power scaling,” Opt. Lett. **35**, 1082–1084 (2010). [CrossRef] [PubMed]

9. M. J. Damzen, Y. Matsumoto, G. J. Crofts, and R. P. M. Green, “Bragg-selectivity of a volume gain grating,” Opt. Commun. **123**, 182–188 (1996). [CrossRef]

10. A. Minassian, G. Crofts, and M. Damzen, “Spectral filtering of gain gratings and spectral evolution of holographic laser oscillators,” IEEE J. Quantum Electron. **36**, 802–809 (2000). [CrossRef]

12. K. S. Syed, G. J. Crofts, and M. J. Damzen, “Transient modelling of a self-starting holographic laser oscillator,” Opt. Commun. **146**, 181–185 (1998). [CrossRef]

15. M. J. Damzen, R. P. M. Green, and G. J. Crofts, “Reflectivity and oscillation conditions of a gain medium in a self-conjugating loop geometry,” Opt. Lett. **19**, 34–36 (1994). [CrossRef] [PubMed]

6. G. Smith and M. J. Damzen, “Quasi-CW diode-pumped self-starting adaptive laser with self-Q-switched output,” Opt. Express **15**, 6458–6463 (2007). [CrossRef] [PubMed]

*χ*

^{(3)}media.

## 2. Model

16. K. Syed, R. Green, G. Crofts, and M. Damzen, “Transient modeling of pulsed phase conjugation experiments in a saturable Nd:YAG amplifier,” Opt. Commun. **112**, 175–180 (1994). [CrossRef]

17. K. S. Syed, G. J. Crofts, R. P. M. Green, and M. J. Damzen, “Vectorial phase conjugation via four-wave mixing in isotropic saturable-gain media,” J. Opt. Soc. B **14**, 2067–2078 (1997). [CrossRef]

*μ*s. Therefore the much slower dynamics of spontaneous emission and inversion build up due to pumping are neglected in this treatment.

*A*

_{1}to

*A*

_{3}enter a saturable gain medium with a given population inversion. Depending on the polarization states of the beams

*A*

_{1}to

*A*

_{4}, any combination of the gratings

*τ*,

*ρ*, Δ,

*δ*might be present simultaneously. Generally the beams

*A*

_{1}and

*A*

_{2}are called the

*pump*beams with

*A*

_{3}and

*A*

_{4}the

*probe*and

*conjugate*respectively. The positive z-direction is along the direction of propagation of the

*A*

_{1}pump beam.

*A*

_{1},

*A*

_{2}and

*A*

_{3},

*A*

_{4}are counter-propagating with the wave vectors given by

*α*in the transient regime and exploiting the slowly-varying envelope approximation where appropriate, yields a system of four coupled wave Eq. (5). To arrive at that result, the periodic modulation of the amplitude gain coefficient was conveniently expanded into a Fourier cosine series [17

17. K. S. Syed, G. J. Crofts, R. P. M. Green, and M. J. Damzen, “Vectorial phase conjugation via four-wave mixing in isotropic saturable-gain media,” J. Opt. Soc. B **14**, 2067–2078 (1997). [CrossRef]

*α*and (3) is readily established via the complex refractive index

*n*=

*n*′ +

*in*″ with

*n*″ = −

*α*/|

*k*

_{0}| [18

18. M. Chi, J. Huignard, and P. M. Petersen, “A general theory of two-wave mixing in nonlinear media,” J. Opt. Soc. B **26**, 1578–1584 (2009). [CrossRef]

*j*= 1...4 and the plus sign for the waves

*A*

_{1},

*A*

_{3}.

*I*is the total intensity,

_{T}*I*the saturation intensity,

_{S}*τ*the upper state lifetime of the gain medium and

_{l}*∂*and

_{x}*∂*terms describing advection and a nonlinear source term

_{z}*F*(

_{j}**A**,

*t*). The right-hand sides

*F*(

_{j}**A**,

*t*) are given by where

*γ*can be interpreted as the average gain and

*κ*,

_{τ}*κ*,

_{ρ}*κ*

_{Δ},

*κ*as the grating strength of the respective

_{δ}*τ*,

*ρ*, Δ,

*δ*gratings. Note that these equations describe only the right-hand sides of the two-dimensional Eq. (5) with

*A*=

_{i}*A*(

_{i}*x, z,t*). In the limit

*θ*= 0 and neglecting the second order diffusion terms, this expression reduces to the well-known one-dimensional model [17

17. K. S. Syed, G. J. Crofts, R. P. M. Green, and M. J. Damzen, “Vectorial phase conjugation via four-wave mixing in isotropic saturable-gain media,” J. Opt. Soc. B **14**, 2067–2078 (1997). [CrossRef]

*A*=

_{i}*A*(

_{i}*z,t*). For the sake of brevity, the complex definition of the coupling coefficients

*γ*,

*κ*has been omitted. A detailed derivation is given in [17

_{i}**14**, 2067–2078 (1997). [CrossRef]

*A*(

_{i}*x, z,t*) automatically results in the correct coupling coefficient

*γ*=

*γ*(

*x,z,t*) and

*κ*=

_{i}*κ*(

_{i}*x, z,t*).

*τ*= 230

_{l}*μs*. The rod has a homogenous gain distribution with a small-signal amplitude gain coefficient

*α*

_{0}= 14.6. The saturation fluence used for the calculations in this paper is

*U*= 5835 J/m

_{sat}^{2}.

## 3. Sensitivity to spatial features

*w*

_{0}at the focus, the crossing angle

*θ*between pump and probe and the transverse position Δ

*x*of the pump beams with respect to the center of the rod. This model describes the physical situation in which three laser pulses

*A*

_{1}–

*A*

_{3}enter the gain medium simultaneously, focused at the rod’s center

*x*= 0

*,z*=

*L*/2. The pulses have a Gaussian temporal intensity profile with a FWHM width of 10ns. Unless noted otherwise, all simulations are performed with respect to a transmission (

*τ*) grating with the default values

*w*

_{0}= 0.5

*mm*,

*θ*= 1° and Δ

*x*= 0

*mm*. The normalized pump fluence

*Û*=

_{P}*Û*

_{1}=

*Û*

_{2}is fixed to

*Û*= 10

_{P}^{−1}and the normalized probe fluence to

*Û*

_{3}= 10

^{−3}. Normalization is with respect to the saturation fluence of Nd:YAG with

*Û*=

*U/U*. The definition given in Eq. (12) is used for the two-dimensional fluence. The colored contour plots showing the spatial distribution of the

_{sat}*κ*coefficient (Figs. 3, 4, and 5) are temporal snapshots taken at the end of the simulation run. Because the model neglects spontaneous emission and pumping, these images show the final state of the

_{τ}*κ*distribution after the pulses have passed through the active medium.

_{τ}2. R. P. M. Green, D. Udaiyan, G. J. Crofts, D. H. Kim, and M. J. Damzen, “Holographic laser oscillator which adaptively corrects for polarization and phase distortions,” Phys. Rev. Lett. **77**, 3533–3536 (1996). [CrossRef] [PubMed]

3. P. Sillard, A. Brignon, and J. Huignard, “Gain-grating analysis of a self-starting self-pumped phase-conjugate Nd:YAG loop resonator,” IEEE J. Quantum Electron. **34**, 465–472 (1998). [CrossRef]

*A*

_{4}and

*A*

_{3}at the left face (z=0) of the gain medium. This definition of the fluence is unequivocal for a one-dimensional model because those models imply a flat-top transverse intensity profile of the beam with a constant fluence anywhere within the beam. In contrast this definition is not unequivocal in the two-dimensional case, because a two-dimensional simulation generates one fluence value for each grid point on the x-axis. Again the two-dimensional treatment implies a constant intensity profile of the beam in the y-direction which is the neglected spatial dimension. To be able to compare the fluences of the one- and two-dimensional simulations, it is necessary to introduce an equivalent quantity for the two-dimensional case. With the 1/

*e*beam radius

*w*(

_{spot}*z*) in x-direction given at position z by the results of the simulation, the fluence of a flat-top beam with the same spot size and power is This corresponds to an equivalent physical situation as described by the one-dimensional model, allowing for a meaningful comparison of

*U*(

*z*) and

*U*

_{2}

*(*

_{D}*z*). Equation (10) can then be written as

13. R. Green, G. Crofts, and M. Damzen, “Phase conjugate reflectivity and diffraction efficiency of gain gratings in Nd:YAG,” Opt. Commun. **102**, 288–292 (1993). [CrossRef]

16. K. Syed, R. Green, G. Crofts, and M. Damzen, “Transient modeling of pulsed phase conjugation experiments in a saturable Nd:YAG amplifier,” Opt. Commun. **112**, 175–180 (1994). [CrossRef]

*θ*to approximately 1.5 degrees assuming a waist radius of the beams

*A*

_{3},

*A*

_{4}of

*w*

_{0}= 0.5mm. At the same time a sufficient beam separation behind the laser rod requires a minimum angle not smaller than 0.5 degrees. This essentially fixes the paths of the beams

*A*

_{3}and

*A*

_{4}while the beams

*A*

_{1}and

*A*

_{2}enjoy a relative freedom with respect to their transverse position within the laser rod. This freedom can be exploited to optimize the grating efficiency by shifting the interaction region to a favorable area inside the gain medium. Figure 3 shows the position of the interaction volume for three different transverse positions of

*A*

_{1}and

*A*

_{2}. The reflectivity changes by a factor of four between the neutral position Δ

*x*= 0 and the optimal position Δ

*x*≈ −0.9

*mm*(Fig. 3).

*z*=

*L*, both the intensities of

*A*

_{1}and

*A*

_{3}increase due to a longer path through the gain medium, resulting in a deeper modulation of the grating. This in turn leads to an improved reflectivity. Eventually the interaction region is clipped at the rod’s end faces, counterbalancing the positional improvement and resulting in an overall decrease of the reflectivity. Additionally the stronger writing beams at the far end of the rod might contribute significantly to the depletion of the available gain and thus erasing the grating.

*G*as efficiently as possible, it is advantageous to employ an aperture-limited spot-size. On the other hand, with four-wave mixing being a third-order nonlinear effect, it is not obvious if the reduced beam peak intensities at bigger spot sizes lead to a reduced reflectivity or not. A simulation with all three beams having identical waist sizes

_{A}*w*

_{0}ranging from 0.1mm to 1mm at a constant average fluence is shown in Fig. 4. A small focus size leads to a small grating volume and the reduced overlap between the beams

*A*

_{1}and

*A*

_{3}shifts the center of gravity of the grating towards the center of the rod with

*z*=

*L*/2, decreasing reflectivity as explained above. Simultaneously the higher transverse peak intensities quickly deplete the available gain, erasing the grating in the process. Consequently, the grating itself is comparably weak. The processes leading to the increased reflectivity are identical to those above, emphasized by the qualitative similarity between plots 3 and 4. Therefore for a given beam fluence, a bigger beam spot size improves reflectivity. The situation might be different if the pulse energy is kept constant. Nonetheless the same competing processes are bound to limit the reflectivity of the grating to the extent that a small focus does not imply a maximum reflectivity.

*θ*→ 0 due to the bigger effective interaction volume. Simulations show, that this is not the case. The reflectivity depends critically on the size and the position of the interaction volume. In turn the optimal size and position depends sensitively on the beam fluences and the gain factor. Figure 5 shows a steady increase in reflectivity with decreasing angle up to

*θ*= 0.55°. At small angles - limited by the necessity of separating the beams behind the laser rod - of around

*θ*= 0.5° the reflectivity remains constant or even decreases slightly. Figure 5 shows that both the volume and the z-position of the grating changes with

*θ*. A bigger grating volume at comparable grating strengths equals a higher reflectivity while the effects of a change in position has been elaborated above. This is a typical example - for the chosen beam parameters - where the interaction volume gets shifted to an unfavorable region with decreasing crossing angles smaller than

*θ*= 0.5°. This is due to a premature extinction of the grating because of gain depletion caused by the respective intensity distributions. Therefore the smallest possible crossing angle obtainable in the laboratory might not necessarily yield an optimal reflectivity of the grating.

## 4. Conclusions

1. R. P. M. Green, S. Camacho-Lopez, and M. J. Damzen, “Experimental investigation of vector phase conjugation in Nd3+:YAG,” Opt. Lett. **21**, 1214–1216 (1996). [CrossRef] [PubMed]

2. R. P. M. Green, D. Udaiyan, G. J. Crofts, D. H. Kim, and M. J. Damzen, “Holographic laser oscillator which adaptively corrects for polarization and phase distortions,” Phys. Rev. Lett. **77**, 3533–3536 (1996). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | R. P. M. Green, S. Camacho-Lopez, and M. J. Damzen, “Experimental investigation of vector phase conjugation in Nd3+:YAG,” Opt. Lett. |

2. | R. P. M. Green, D. Udaiyan, G. J. Crofts, D. H. Kim, and M. J. Damzen, “Holographic laser oscillator which adaptively corrects for polarization and phase distortions,” Phys. Rev. Lett. |

3. | P. Sillard, A. Brignon, and J. Huignard, “Gain-grating analysis of a self-starting self-pumped phase-conjugate Nd:YAG loop resonator,” IEEE J. Quantum Electron. |

4. | A. Brignon and J. Huignard, “Transient analysis of degenerate four-wave mixing with orthogonally polarized pump beams in a saturable Nd:YAG amplifier,” IEEE J. Quantum Electron. |

5. | M. Damzen, R. Green, and G. Crofts, “Spatial characteristics of a laser oscillator formed by optically-written holographic gain-grating,” Opt. Commun. |

6. | G. Smith and M. J. Damzen, “Quasi-CW diode-pumped self-starting adaptive laser with self-Q-switched output,” Opt. Express |

7. | B. A. Thompson, A. Minassian, and M. J. Damzen, “Operation of a 33-W, continuous-wave, self-adaptive, solid-state laser oscillator,” J. Opt. Soc. B |

8. | S. Lam and M. Damzen, “Self-adaptive Nd:YLF holographic laser with selectable wavelength operation,” Appl. Phys. B. |

9. | M. J. Damzen, Y. Matsumoto, G. J. Crofts, and R. P. M. Green, “Bragg-selectivity of a volume gain grating,” Opt. Commun. |

10. | A. Minassian, G. Crofts, and M. Damzen, “Spectral filtering of gain gratings and spectral evolution of holographic laser oscillators,” IEEE J. Quantum Electron. |

11. | P. C. Shardlow and M. J. Damzen, “Phase conjugate self-organized coherent beam combination: a passive technique for laser power scaling,” Opt. Lett. |

12. | K. S. Syed, G. J. Crofts, and M. J. Damzen, “Transient modelling of a self-starting holographic laser oscillator,” Opt. Commun. |

13. | R. Green, G. Crofts, and M. Damzen, “Phase conjugate reflectivity and diffraction efficiency of gain gratings in Nd:YAG,” Opt. Commun. |

14. | G. J. Crofts and M. J. Damzen, “Numerical modelling of continuous-wave holographic laser oscillators,” Opt. Commun. |

15. | M. J. Damzen, R. P. M. Green, and G. J. Crofts, “Reflectivity and oscillation conditions of a gain medium in a self-conjugating loop geometry,” Opt. Lett. |

16. | K. Syed, R. Green, G. Crofts, and M. Damzen, “Transient modeling of pulsed phase conjugation experiments in a saturable Nd:YAG amplifier,” Opt. Commun. |

17. | K. S. Syed, G. J. Crofts, R. P. M. Green, and M. J. Damzen, “Vectorial phase conjugation via four-wave mixing in isotropic saturable-gain media,” J. Opt. Soc. B |

18. | M. Chi, J. Huignard, and P. M. Petersen, “A general theory of two-wave mixing in nonlinear media,” J. Opt. Soc. B |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(090.7330) Holography : Volume gratings

(140.3540) Lasers and laser optics : Lasers, Q-switched

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(190.2055) Nonlinear optics : Dynamic gratings

(140.3535) Lasers and laser optics : Lasers, phase conjugate

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: November 7, 2011

Revised Manuscript: December 14, 2011

Manuscript Accepted: December 15, 2011

Published: March 12, 2012

**Citation**

Robert Elsner, Roland Ullmann, Axel Heuer, Ralf Menzel, and Martin Ostermeyer, "Two-dimensional modeling of transient gain gratings in saturable gain media," Opt. Express **20**, 6887-6896 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-6887

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### References

- R. P. M. Green, S. Camacho-Lopez, M. J. Damzen, “Experimental investigation of vector phase conjugation in Nd3+:YAG,” Opt. Lett. 21, 1214–1216 (1996). [CrossRef] [PubMed]
- R. P. M. Green, D. Udaiyan, G. J. Crofts, D. H. Kim, M. J. Damzen, “Holographic laser oscillator which adaptively corrects for polarization and phase distortions,” Phys. Rev. Lett. 77, 3533–3536 (1996). [CrossRef] [PubMed]
- P. Sillard, A. Brignon, J. Huignard, “Gain-grating analysis of a self-starting self-pumped phase-conjugate Nd:YAG loop resonator,” IEEE J. Quantum Electron. 34, 465–472 (1998). [CrossRef]
- A. Brignon, J. Huignard, “Transient analysis of degenerate four-wave mixing with orthogonally polarized pump beams in a saturable Nd:YAG amplifier,” IEEE J. Quantum Electron. 30, 2203–2210 (1994). [CrossRef]
- M. Damzen, R. Green, G. Crofts, “Spatial characteristics of a laser oscillator formed by optically-written holographic gain-grating,” Opt. Commun. 110, 152–156 (1994). [CrossRef]
- G. Smith, M. J. Damzen, “Quasi-CW diode-pumped self-starting adaptive laser with self-Q-switched output,” Opt. Express 15, 6458–6463 (2007). [CrossRef] [PubMed]
- B. A. Thompson, A. Minassian, M. J. Damzen, “Operation of a 33-W, continuous-wave, self-adaptive, solid-state laser oscillator,” J. Opt. Soc. B 20, 857–862 (2003). [CrossRef]
- S. Lam, M. Damzen, “Self-adaptive Nd:YLF holographic laser with selectable wavelength operation,” Appl. Phys. B. 76, 237–240 (2003). [CrossRef]
- M. J. Damzen, Y. Matsumoto, G. J. Crofts, R. P. M. Green, “Bragg-selectivity of a volume gain grating,” Opt. Commun. 123, 182–188 (1996). [CrossRef]
- A. Minassian, G. Crofts, M. Damzen, “Spectral filtering of gain gratings and spectral evolution of holographic laser oscillators,” IEEE J. Quantum Electron. 36, 802–809 (2000). [CrossRef]
- P. C. Shardlow, M. J. Damzen, “Phase conjugate self-organized coherent beam combination: a passive technique for laser power scaling,” Opt. Lett. 35, 1082–1084 (2010). [CrossRef] [PubMed]
- K. S. Syed, G. J. Crofts, M. J. Damzen, “Transient modelling of a self-starting holographic laser oscillator,” Opt. Commun. 146, 181–185 (1998). [CrossRef]
- R. Green, G. Crofts, M. Damzen, “Phase conjugate reflectivity and diffraction efficiency of gain gratings in Nd:YAG,” Opt. Commun. 102, 288–292 (1993). [CrossRef]
- G. J. Crofts, M. J. Damzen, “Numerical modelling of continuous-wave holographic laser oscillators,” Opt. Commun. 175, 397–408 (2000). [CrossRef]
- M. J. Damzen, R. P. M. Green, G. J. Crofts, “Reflectivity and oscillation conditions of a gain medium in a self-conjugating loop geometry,” Opt. Lett. 19, 34–36 (1994). [CrossRef] [PubMed]
- K. Syed, R. Green, G. Crofts, M. Damzen, “Transient modeling of pulsed phase conjugation experiments in a saturable Nd:YAG amplifier,” Opt. Commun. 112, 175–180 (1994). [CrossRef]
- K. S. Syed, G. J. Crofts, R. P. M. Green, M. J. Damzen, “Vectorial phase conjugation via four-wave mixing in isotropic saturable-gain media,” J. Opt. Soc. B 14, 2067–2078 (1997). [CrossRef]
- M. Chi, J. Huignard, P. M. Petersen, “A general theory of two-wave mixing in nonlinear media,” J. Opt. Soc. B 26, 1578–1584 (2009). [CrossRef]

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