## Model of temperature grating relaxation times in distributed feedback dye lasers

Optics Express, Vol. 11, Issue 13, pp. 1520-1530 (2003)

http://dx.doi.org/10.1364/OE.11.001520

Acrobat PDF (560 KB)

### Abstract

A theoretical model and its experimental realization for the temperature grating relaxation time constant and its impact on the temporal and the spectral profiles of a Q-switched and modelocked Nd:YAG laser pumped distributed feedback dye laser (DFDL) is reported. Boundary conditions for different types of excitation pulses have been established to predict the effect of temperature phase gratings on laser gain build-up and temporal elongation of the DFDL pulses. The proposed transient grating method is useful in measuring grating relaxation time constants for lasing dye solutions. The proposed mathematical model is demonstrated by measurement of the relaxation time constant of R6G in ethanol at 10^{-3}M. The measured relaxation time constant of 16±0.2 ns is very close to the tabulated values determined using other techniques.

© 2003 Optical Society of America

## 1. Introduction

2. J. Liang, H. Sun, and Y. Hu et.al, “The observation of lasing wavelength shift from the reflection center of an Ytterbium doped fiber grating laser,” Opt. Commun. **216**, 173 (2003). [CrossRef]

3. A. A. Afanas’ev et al. “Effect of a thermal lattice on the generated line width of a dye laser with distributed feedback,” J. Appl. Spectr. **37**, 899 (1982) [CrossRef]

5. R. Y. Choie, T.H. Barnes, and W.J. Sandle et al, “Observation of a thermal phase grating contribution to diffraction in erythrosine doped gelatin films,” Opt. Commun. **186**, 43 (2000). [CrossRef]

## 2. Theory of temperature gratings

*C*is the specific heat at constant pressure. The temperature dependence of

_{p}*D*can usually be neglected but cannot be ignored on phase transition and at low temperatures. Further, s and

_{th}*D*are second-rank tensors, α is the absorption coefficient at pump frequency, and

_{th}*I(r,t)*is their intensity. In the treatment of thermal gratings, it is assumed that the two pump beams have the same polarization as the opposite polarizations cannot cause interference patterns. The intensity may be given by

*I*=

_{av}*I*+

_{A}*I*and Δ

_{B}*I*=(

*I*)

_{A}I_{B}^{1/2}. Furthermore it is assumed that the pump beam interaction region is larger than the grating wavelength. Under these circumstances, the temperature response to the absorbed pump radiation can be conveniently split into a slowly varying average

*T*and a grating structureΔ

_{av}*T*cos

*qx*. The two contributions are almost uncoupled under most experimental conditions. It is therefore possible to split Eq. (2) into two parts which can be solved independently

8. H. J. Eichler, Ch. Hartig, and J. Knof, Phys Status Solidi (a)45, 433, 1978. [CrossRef]

*w*is the beam waist. Solution to Eq. (6) is simple when the pump beams are rectangular pulses, short spikes or modulated train of pulses. Rectangular pulses with duration t

_{P}

2. J. Liang, H. Sun, and Y. Hu et.al, “The observation of lasing wavelength shift from the reflection center of an Ytterbium doped fiber grating laser,” Opt. Commun. **216**, 173 (2003). [CrossRef]

*t*226A;interpulse period, τ, is

_{p}*π*results in a square root of a Lorentzian for Δ

*T*

*t*≫

_{p}*τ*, on the other hand, can easily be calculated and helps understanding the total temperature development process. For such a short pulse, the heat dissipation by diffusion out of the pumped zone is negligible while the pump pulse is impinging on it. Integration of Eq. (6) thus simply yields

_{w}*w*

_{th}*w*=

_{th}*w*is satisfied at time

*τ*, justifying Eq. (8). Note that the width of the temperature distribution is that of the pump intensity, the width of which is

_{w}*w*/√2. When the time zero is assumed at the beginning of the pump pulses, the resulting expression for the average temperature distribution is

*T*

_{⊥}the above-discussed plane grating solution. The explicit form of

*g*(

*z,t*), a combination of exponentials and Erfc functions, represents the temperature response to absorption of an unmodulated plane wave. Since the diffracted probe light integrates over the irradiated dye solution, in general, the exact shape of

*g*(

*z,t*) is not very important in a Forced Raleigh Scattering (FRS) experiment with heavily absorbing dye solutions. The time profile of ps-ns regime pumping pulses is often studied by pump probe technique and for ns-µs regime can be measured in real time.

## 3. Modeling thermal gain accumulation

_{c}/c is very short. The thermal heat transfer to the subsequent pulses depends entirely on the comparison of the thermal constant τ and the interpulse period 2L

_{c}/c, which in turn is determined by the cavity length of the passively Q-switched and modelocked pump laser.

- It is assumed that some r% amplitude of the previous temperature grating is retained in period 2L
_{c}/c and is added up in the newly created temperature gratings by next pulse provided the interpulse period is less than the thermal relaxation time of the constant of the temperature gratings. - Gradual increase in the amplitude of the temperature grating results in higher reflectivity and lower threshold. This may reinforce a weak pump pulse at the end of the envelope to exceed the threshold, which otherwise it would not.
- Computer simulation based on this model can create the experimentally observed delay between the peak pulses to lase at shifted central peaks. It does not take into consideration gain of the medium. It provides with percentage value of the overall increase in amplitude of the temperature gratings by creating a peak pulse delay and envelope profile as observed. The maximum reduction in threshold value may cause 1, 2 or 3 extra pulses to appear depending upon the interpulse period or pulse power in q.switched lasers [9].
9. Zs. Bor, A. Muller, and B. Racz, “UV and blue ps pulse generation by a N-laser pumped DFDL,” Optic. Commun.

**40**, 294 (1982). [CrossRef]

### 3.1 Model for modelocked laser

_{q}’ be thermal relaxation time of grating and 2L

_{c}/c=τ

_{1}. If τ

_{q}≪τ

_{1}then the DFDL output pulses will follow the excitation pulses in time and profile. On the other hand if τ

_{q}≥τ

_{1}then the amplitude of temperature gratings will increase gradually resulting in nonuniform pulse train output from the DFDL. Suppose some r% of the amplitude of temperature gratings remain undiffused before the induction of next gratings. It can be expressed by [10].

_{1}is the amplitude of single temperature grating. If distribution of amplitude in the temperature fringe is Gaussian i.e.

_{o}is the peak amplitude of the temperature fringe with Gaussian distribution and z

_{od}=Δz/2 and z

_{a}is a constant. In CW modelocked lasers “n” is very large. Increase in amplitude of the temperature fringe represents the percentage of the amplitude of a previously produced temperature fringe. Let us assume it tends to be infinite to be generalized as

*et al*[9

9. Zs. Bor, A. Muller, and B. Racz, “UV and blue ps pulse generation by a N-laser pumped DFDL,” Optic. Commun. **40**, 294 (1982). [CrossRef]

### 3.2 Model for Q-switched and modelocked laser

_{c}/c. If the period 2L

_{c}/c is more than τ (thermal relaxation time constant) then thermal relaxation of the medium does not affect the steady state operation of the DFDL. On the other hand, if the interpulse time is less than the relaxation time, the situation becomes complex. Let the amplitude of a fringe in the temperature grating be n

_{1}and the accumulated amplitude (n

_{a})

_{k}where k=1, 2, 3…..n. The accumulation of temperature grating in the dye medium will enhanced due to the rising intensities of the excitation pulses in the early part of the envelope. This will tend to increase the effective amplitude of a temperature grating and the subsequent intensity of the DFDL output pulses. The effective increase for the nth pulse will be given by [10].

_{1}is the amplitude of the temperature grating.

_{c}/c, temperature relaxation time constant τ

_{q}and pulse length τ

_{p}. These cases are

**Case 1**

_{op}=[(t

_{e}/2)+(τ

_{p}/2)], t

_{e}=(n2L

_{c}/c+nτ

_{p}) and n

_{op}is the peak amplitude of the q.switching envelope. The total accumulated amplitude can be written as

_{k}=[(n-1)2L

_{c}/c+nτ

_{p}] varies from 0 to the envelope length which is usually from 50 to 200 ns. The generalized equation in terms of pulse and the envelope profile shapes can be expressed as

**Case 2**

_{c}/c is of the order of 5–9 ns. Further it has been notice from the computer program that for these ranges the peak of DFDL always starts declining. So there was no point to use the full limits but one can use the full length of envelope that is in this case 110 ns. Let us ignore the remaining pulses as the first half can exhibit the maximum delay τ

_{D}between the peaks of the DFDL and the excitation laser envelopes. The effective energy of the nth pulse may be given by

_{o})

_{x}” is the fixed amount of amplitude being added by each pulse to the subsequently induced temperature fringe. This assumption was made to simplify the formulation leading to the sum of geometrical and arithmetical progressions [6]. The peak energy of the pulses increases gradually. Let us assume that the modelocking and Q-switching envelope is of Gaussian shape. The peak value of the amplitude may be given by

_{k}=t

_{l}→t

_{n}. Substitution of Eq. (28) into Eq. (27) leads to the following generalized expression.

_{a}and t

_{b}are constant, t

_{x}=(t

_{k}-t

_{oe}) and z

_{y}=(z

_{1}-z

_{2})/2. The limits have been written in the above equation up to half of the envelope, however they can be extended up to the end of the envelope too. The shift in intensity profile of pump and DFDL lasers may be explained as shown in Fig. 2.

**Case 3**

_{p}or τ

_{q}. This is the most general version of the accumulative model and can be applied to the q-switched lasers. The derivations are same as for mode-locked lasers. However, the value of r% would remain zero in this case. It will not cause any delay at all.

## 4. Experimental results

^{-3}M. Experimental setup is shown in Fig. 3. A microdensitometer scanned streak record is shown in figure 4a showing both the pump laser streak record and the DFDL streak record. A half period delay was introduced deliberately in these results for clarity and it was subtracted for data processing.

_{c}/c=4.43 ns, 12.8±0.5 ns for 2L

_{c}/c=5.93 ns and 10.9±0.5 ns for 2L

_{c}/c=8.31 ns correspond to computer calculated percentage increase of amplitudes of the temperature gratings of 74.2%, 67.5% and 57.6% respectively assuming Gaussian pulses and envelopes profiles. If the exact shape of the pulses and envelopes is known then the above values are 100% correct otherwise the accuracy depends upon the nearness of the actual envelopes and pulses to those assumed in the computer program.

## 5. Temperature grating time constant

_{q}is thermal decay in liquids then it can be expressed by

_{o}is the maximum amplitude of the grating and for different pulses of the q-switched and modelocked Nd:YAG laser (assuming Gaussian profile) this varies as

_{o})

_{m}is the peak amplitude of the central pump pulse of the envelope and to is equal t

_{o}the half of the envelope length. At t=2L

_{c}/c the Eq. (33) reduces to the amplitude retained by the medium, is in fact the energy which is not yet diffused when next pulse comes. Let r=n

_{1}/n

_{o}then we can write

_{c}/c versus r gives the value of τ

_{q}as 15.8 ns. The same results can also be achieved in terms of the delay between the peaks of the envelopes as follows. The experimentally measured delay between the peaks of the envelopes was plotted against the percentage energy retained by the medium for similar delay in the computer program. The curve fitting program gave the following relation between r and τ

_{D}.

_{q}was also calculated using

11. A. Penzkofer and W. Falkenstein et al, Chem. Phys. Lett, 44, 82 (1976). [CrossRef]

_{q}=12 ns for water and τ

_{q}=20 ns for ethanol at some other concentration of Rh6G.

## 6. Conclusions

## References

1. | V. Yu. Kurstak and S. S. Anufrick “Influence of thermal phase lattice on ultrashort pulses characteristics generated by DFDL”, LFNM, Kharkiv, Ukraine, 3–5 (2002). |

2. | J. Liang, H. Sun, and Y. Hu et.al, “The observation of lasing wavelength shift from the reflection center of an Ytterbium doped fiber grating laser,” Opt. Commun. |

3. | A. A. Afanas’ev et al. “Effect of a thermal lattice on the generated line width of a dye laser with distributed feedback,” J. Appl. Spectr. |

4. | A. W. Broerman, D.C. Venerus, and J. D. Schiebler, “Evidence of the stress thermal rule in an elastomer subjected to simple elongation,” J. Chem. Phys. |

5. | R. Y. Choie, T.H. Barnes, and W.J. Sandle et al, “Observation of a thermal phase grating contribution to diffraction in erythrosine doped gelatin films,” Opt. Commun. |

6. | Laser induced dynamic gratings, edited by H.J Eichler, P. Gunter, and D.W. Pohl, Springer Verlag Series on Opt. Sciences, 50, 17 (1986). |

7. | A. Bosh, M. Brodin, N. Orchair, S. Odulov, and S. Soskin, Soviet Physics, JEPT Lett.18, 397 (1973). |

8. | H. J. Eichler, Ch. Hartig, and J. Knof, Phys Status Solidi (a)45, 433, 1978. [CrossRef] |

9. | Zs. Bor, A. Muller, and B. Racz, “UV and blue ps pulse generation by a N-laser pumped DFDL,” Optic. Commun. |

10. | M. Fogiel, ‘Handbook of Mathematical Scientific and Engineering Formulas, Tables, Functions, Graphs and Transforms,’ TEA New York, 304, 7 (1986). |

11. | A. Penzkofer and W. Falkenstein et al, Chem. Phys. Lett, 44, 82 (1976). [CrossRef] |

12. | D.Y. Key, ‘The scattering of light from light induced structures in liquids,’ Ph.D. Thesis, London University, (1977). |

13. | P.Y. Key and R.G. Harrison, IEEE. J. Quant. Electron.QE-6, 645 (1970). |

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(140.2050) Lasers and laser optics : Dye lasers

**ToC Category:**

Research Papers

**History**

Original Manuscript: April 10, 2003

Revised Manuscript: June 12, 2003

Published: June 30, 2003

**Citation**

Nasrullah Khan, Tom Hall, and Norman Mariun, "Model of temperature grating relaxation times in distributed feedback dye lasers," Opt. Express **11**, 1520-1530 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-13-1520

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### References

- V. Yu. Kurstak and S. S. Anufrick�?? Influence of thermal phase lattice on ultrashort pulses characteristics generated by DFDL�??, LFNM, Kharkiv, Ukraine, 3-5 (2002).
- J. Liang, H. Sun and Y. Hu et.al, �??The observation of lasing wavelength shift from the reflection center of an Ytterbium doped fiber grating laser,�?? Opt. Commun. 216, 173 (2003). [CrossRef]
- A. A. Afanas'ev et al."Effect of a thermal lattice on the generated line width of a dye laser with distributed feedback," J. Appl. Spectr. 37, 899 (1982) [CrossRef]
- A. W. Broerman, D.C. Venerus and J. D. Schiebler,�?? Evidence of the stress thermal rule in an elastomer subjected to simple elongation,�?? J. Chem. Phys. 11, 6955 (1999).
- R. Y. Choie, T.H. Barnes and W.J. Sandle et al,�?? Observation of a thermal phase grating contribution to diffraction in erythrosine doped gelatin films,�?? Opt. Commun. 186, 43 (2000). [CrossRef]
- Laser induced dynamic gratings, edited by H.J.Eichler, P. Gunter and D.W. Pohl, Springer Verlag Series on Opt. Sciences, 50, 17 (1986).
- A. Bosh, M. Brodin, N. Orchair, S. Odulov and S. Soskin, Soviet Physics, JEPT Lett. 18, 397 (1973).
- H. J. Eichler, Ch. Hartig, J. Knof, Phys Status Solidi (a) 45, 433, 1978. [CrossRef]
- Zs. Bor, A. Muller and B. Racz,�??UV and blue ps pulse generation by a N-laser pumped DFDL,�?? Opt. Commun. 40, 294 (1982). [CrossRef]
- M. Fogiel,�??Handbook of Mathematical Scientific and Engineering Formulas, Tables, Functions, Graphs and Transforms,�?? TEA New York, 304, 7 (1986).
- A. Penzkofer and W. Falkenstein et al, Chem. Phys. Lett., 44, 82 (1976). [CrossRef]
- D.Y. Key, �??The scattering of light from light induced structures in liquids,�?? Ph.D. Thesis, London University, (1977).
- P.Y. Key and R.G. Harrison, IEEE. J. Quant. Electron. QE-6, 645 (1970).

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