## Short pulses in optical resonators

Optics Express, Vol. 11, Issue 22, pp. 2975-2981 (2003)

http://dx.doi.org/10.1364/OE.11.002975

Acrobat PDF (78 KB)

### Abstract

We model the behavior of short and ultrashort laser pulses in high-finesse Fabry-Perot resonators, examining, in particular, the influence of cavity mirror reflectance and dispersion. The total coupling, peak power enhancement and temporal broadening of circulating pulses are characterized a function of the duration of the incident pulses.We show that there is an optimal input pulse duration which maximizes peak power for a given set of mirror characteristics.

© 2003 Optical Society of America

## 1. Introduction

1. 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 **31**, 97 (1983). [CrossRef]

2. R.J. Jones and J.-C. Diels, “Stabilization of femtosecond lasers for optical frequency metrology and direct optical to radio frequency synthesis,” Phys. Rev. Lett. **86**, 3288 (2001). [CrossRef] [PubMed]

3. S. Link, H.A. Dürr, and W. Eberhardt, “Femtosecond spectroscopy,” J. Phys.: Condens. Matter **13**7873–7884 (2001). [CrossRef]

3. S. Link, H.A. Dürr, and W. Eberhardt, “Femtosecond spectroscopy,” J. Phys.: Condens. Matter **13**7873–7884 (2001). [CrossRef]

5. R.J. Jones and J. Ye, “Femtosecond pulse amplification by coherent addition in a passive optical cavity,” Opt. Lett. **27**, 1848 (2002). [CrossRef]

## 2. Optical frequency combs and resonators

*ν*. The frequency of each comb component can thus be expressed as

*n*Δ

*ν*+

*ν*

_{0}, where

*ν*

_{0}is termed the offset frequency. Current MLL technology enables the generation of optical frequency combs with extraordinarily uniform frequency distribution (to within 3 parts in 10

^{17}[6

6. T. Udem, J. Reichert, R. Holzwarth, M. Niering, M. Weitz, and T.W. Hänsch, “Measuring the Frequency of Light with Mode-Locked Lasers,” in A.N. Luiten (Ed.), *Frequency Measurement and Control*, Topics Appl. Phys.79275 (Springer-Verlag, Berlin, 2001). [CrossRef]

*ν*and

*ν*

_{0}.

*f*=Δ

_{rep}*ν*. The complex amplitude envelope

*ℰ*(

*t*) of each individual pulse is the Fourier transform of the complex amplitude spectrum

*(*ℰ ˜

*ω*) of the comb (where the tilde denotes representation in the frequency domain). It is convenient to characterise these pulses in terms of their instantaneous power

*P*(

*t*), so for simplicity of notation, we define

*ℰ*(

*t*) such that

*P*(

*t*)=

*E*|

_{pulse}*ℰ*(

*t*)|

^{2}, where

*E*is the total energy per pulse. This requires that ∫|

_{pulse}*ℰ*(

*t*)|

^{2}d

*t*=1. Unchirped, Gaussian pulses of length

*τ*at half their maximum power then have the following amplitude envelope in time:

*ω*

_{0}of the comb spectrum to zero frequency, so the complex amplitude spectrum corresponding to (1) must be given not in terms of

*ω*but in terms of the offset frequency Ω=

*ω*-

*ω*

_{0}. In addition, (1) describes only a single pulse and ignores the periodic nature of the pulse train, which amounts to discarding the comb structure in favor of a continuous amplitude spectrum given by the following expression:

*L*and optical length

*L*. If intracavity dispersion is neglected, the resonance frequencies must satisfy

_{opt}*nλ*=2

*L*; the

_{opt}*n*resonance frequency is then

^{th}*ϕ*(

*ω*) to be the single-pass phase shift due to interaction with optical elements in the cavity, so that the optical length of the cavity is

*L*=

_{opt}*L*+Δ

*ϕ*(

*ω*)

*λ*/(2

*π*). Then the resonance frequencies of the cavity must satisfy

*ϕ*(

*ω*)=

*ϕ*

_{0}+

*ωϕ*′

_{0}+

*δϕ*(

*ω*) such that

*δϕ*contains only second- and higher-order variation with

*ω*, and such that

*δϕ*(

*ω*

_{0})=0. Substituting this into (4) and rearranging, one obtains the following expression for the

*n*

^{th}resonance frequency:

*ϕ*

_{0}, has the effect of shifting each resonance frequency by a constant offset. The first-order frequency dependent phase delay

*ϕ*′

_{0}changes the optical length of the cavity (and consequently the mode spacing) by an equal amount for all frequencies. This corresponds to the group delay of the pulse on reflection [7]. It is only the second- and higher-order terms contained in the dispersive phase delay

*δϕ*(

*ω*) that prevent perfect uniformity of mode spacing.

*ν*=

*c*/(2

*L*+2

*cϕ*′

_{0}) will ensure that each comb component is detuned equally from a resonance frequency1; appropriate control of the comb offset frequency

*ν*

_{0}can then set this detuning to zero. Active control of Δ

*ν*and

*ν*

_{0}is required to maintain this resonant condition. An effective technique for this purpose has been demonstrated by Jones and Diels [2

2. R.J. Jones and J.-C. Diels, “Stabilization of femtosecond lasers for optical frequency metrology and direct optical to radio frequency synthesis,” Phys. Rev. Lett. **86**, 3288 (2001). [CrossRef] [PubMed]

1. 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 **31**, 97 (1983). [CrossRef]

*e*

_{1}and

*e*

_{2}which give information about the detuning from two subsets of resonator modes at opposite ends of the comb spectrum. The average of

*e*

_{1}and

*e*

_{2}, which is approximately proportional to the average detuning across the comb, can be used as an error signal for feedback control of

*ν*

_{0}. The difference between

*e*

_{1}and

*e*

_{2}can be used as an error signal for feedback control of Δ

*ν*.

*ν*and

*ν*

_{0}are supressed in an active control system as explained above. Nonetheless, the ability of an OFC to couple to a resonator is fundamentally limited by the intracavity dispersion, to an extent which will be determined in the following sections.

## 3. Circulating comb characteristics

*δϕ*(

*ω*)=0 for all frequencies, each component of an OFC can be made exactly resonant as described in the previous section. The circulating pulse power

*P*(

_{circ}*t*) is then identical to the incident pulse power

*P*(

_{inc}*t*) multiplied by

*F/π*, where

*F*is the resonator finesse. Only when

*δϕ*(

*ω*)=0 can the net round trip phase shift be zero for each comb component, allowing the incoming and circulating light to add coherently at the resonator input. Any dispersion in the cavity will result in an additional, frequency-dependent round-trip phase shift, lowering the efficiency of the superposition of the input and circulating light. We define the power coupling efficiency

*β*(Ω) to be the ratio between the actual circulating power at a particular frequency, and the ideal (

*δϕ*(

*ω*)=0) case for which

*β*=1:

*R*results in multiplication of the complex amplitude by the complex reflectivity [7], which we denote

*h*(Ω):

ℰ ˜

_{inc}(Ω), then in a lossless and symmetric mirror system, the initial complex amplitude spectrum inside the cavity is

ℰ ˜

_{circ}(Ω) gives the steady-state relative phase

*ϕ*(Ω) of the circulating comb. The power coupling efficiency can be determined from the cirulating power spectrum |

_{circ}ℰ ˜

_{circ}(Ω)|

^{2}. For an unchirped incident OFC,

*δϕ*(Ω) for these mirrors, obtained by integrating the published GVD (

*ϕ*″(

*ω*)) data, is given in Fig. 1(a). Henceforth in this paper we will assume the use of mirrors with these dispersion characteristics. The resulting circulating phase and coupling efficiency are given in Figs. 1(b) and 1(c) as calculated for

*R*=99.8% (currently available as a stock component) and

*R*=99.99%.

*F*is increased, because increasing the number of passes increases the circulating phase shift, or equivalently, because the resonator modes are reduced in linewidth and thus the incoupled power is more sensitive to dispersive detuning of the cavity resonances. The limited circulating power bandwidth corresponds to a lower bound on the length of pulses which can be efficiently coupled into the cavity. Shortening the incident pulses will thus eventually result in relative broadening of the circulating pulses and reduced coupling efficiency.

## 4. Circulating pulse characteristics

*ℰ*

_{circ}(

*t*) of the pulse in the time domain can be obtained via the discrete Fourier transform, giving in turn the power

*τ*/

_{circ}*τ*of the circulating pulse depends on the input pulse length

_{inc}*τ*. The form of this dependence can be examined by extracting

_{inc}*τ*from the calculated circulating pulse power envelopes (by finding the time-between half-power points). The total power coupling efficiency,

_{circ}*τ*, are plotted in Fig. 3(a). For sufficiently long incident pulses, the circulating pulses are negligibly broadened, and the total power coupling efficiency approaches unity. For shorter input pulses, more of the comb spectrum lies outside of the efficient-coupling bandwidth, so the total power coupling efficiency decreases and the circulating pulse is relatively broadened. These effects become more pronounced as the finesse is increased because of the associated reduction in coupling bandwidth.

_{inc}*F/π*times the energy per incident pulse. Using a thin-disk gain element to facilitate the dissipation of heat from the pump beam, Brunner

*et al*[9

9. F. Brunner*et al*, “240-fs pulses with 22-W average power from a mode-locked thin-disk Yb:KY(WO_{4})_{2} laser,” Opt. Lett. **27**, 1162–1164 (2002). [CrossRef]

*µ*J, 240-fs pulses at a repetition rate of 25 MHz; locking such an oscillator to a resonator built from CVI LGVD mirrors (reflectance 99.8%) would result in a circulating pulse energy of 1.8 mJ. If 99.99%-reflectance cavity mirrors with the same dispersion characteristics were used, a 36-mJ circulating pulse could be produced.

*P*varies inversely with the circulating pulse length, so at first, shortening the incident pulses increases

_{max}*P*. However, this can only continue until the comb bandwidth approaches the coupling bandwidth of the cavity, at which point the circulating pulse will begin to relatively broaden and the total power coupling

_{max}*β*will decrease. Thus there exists an optimum incident pulse length which maximises

_{net}*P*for a given finesse. The maximum instantaneous power

_{max}*P*, obtained from the calculated circulating pulse power envelope, is plotted in Fig. 3(b) as a function of the incident pulse length

_{max}*τ*. The highest peak power that can be reached with

_{inc}*R*=99.8% is 8.3 GW per

*µ*J of incident pulse energy, with 30 fs incident pulses. When

*R*=99.99%, the optimal incident pulse length is 101 fs, producing a peak power in the cavity of 54 GW per

*µ*J of incident pulse energy. This is only about 6.5 times higher than for the lower reflectance, despite a factor of 20 increase in finesse, because the effect of intracavity dispersion becomes more pronounced as the finesse is increased.

9. F. Brunner*et al*, “240-fs pulses with 22-W average power from a mode-locked thin-disk Yb:KY(WO_{4})_{2} laser,” Opt. Lett. **27**, 1162–1164 (2002). [CrossRef]

*et al*[10

10. T. Beddard*et al*, “High-average-power, 1-MW peak-power self-mode-locked Ti:sapphire oscillator,” Opt. Lett. **24**, 163–165 (1999). [CrossRef]

9. F. Brunner*et al*, “240-fs pulses with 22-W average power from a mode-locked thin-disk Yb:KY(WO_{4})_{2} laser,” Opt. Lett. **27**, 1162–1164 (2002). [CrossRef]

## 5. Conclusion

## Acknowledgments

## Footnotes

1 | In the time domain, this corresponds to the requirement that the cavity round trip time 2L must exactly equal the delay 1/_{opt}/cf between pulses, so that the circulating pulse is reinforced by the next incident pulse on each pass._{rep} |

## References and links

1. | 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 |

2. | R.J. Jones and J.-C. Diels, “Stabilization of femtosecond lasers for optical frequency metrology and direct optical to radio frequency synthesis,” Phys. Rev. Lett. |

3. | S. Link, H.A. Dürr, and W. Eberhardt, “Femtosecond spectroscopy,” J. Phys.: Condens. Matter |

4. | J.C. Petersen and A.N. Luiten, “Resonant Polarization Interferometry with Ultrashort Laser Pulses,” Phys. Rev. A (in preparation). |

5. | R.J. Jones and J. Ye, “Femtosecond pulse amplification by coherent addition in a passive optical cavity,” Opt. Lett. |

6. | T. Udem, J. Reichert, R. Holzwarth, M. Niering, M. Weitz, and T.W. Hänsch, “Measuring the Frequency of Light with Mode-Locked Lasers,” in A.N. Luiten (Ed.), |

7. | A.E. Siegman, |

8. | CVI Laser Corporation (www.cvilaser.com). |

9. | F. Brunner |

10. | T. Beddard |

**OCIS Codes**

(230.5750) Optical devices : Resonators

(320.0320) Ultrafast optics : Ultrafast optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 13, 2003

Revised Manuscript: October 27, 2003

Published: November 3, 2003

**Citation**

J. Petersen and A. Luiten, "Short pulses in optical resonators," Opt. Express **11**, 2975-2981 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-22-2975

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### References

- 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 31, 97 (1983). [CrossRef]
- R.J. Jones and J.-C. Diels, �??Stabilization of femtosecond lasers for optical frequency metrology and direct optical to radio frequency synthesis,�?? Phys. Rev. Lett. 86, 3288 (2001). [CrossRef] [PubMed]
- S. Link, H.A. D¨urr and W. Eberhardt, �??Femtosecond spectroscopy,�?? J. Phys.: Condens. Matter 13 7873-7884 (2001). [CrossRef]
- J.C. Petersen and A.N. Luiten, �??Resonant Polarization Interferometry with Ultrashort Laser Pulses,�?? Phys. Rev. A (in preparation).
- R.J. Jones and J. Ye, �??Femtosecond pulse amplification by coherent addition in a passive optical cavity,�?? Opt. Lett. 27, 1848 (2002) [CrossRef]
- T. Udem, J. Reichert, R. Holzwarth, M. Niering, M. Weitz and T.W. H¨ansch, �??Measuring the Frequency of Light with Mode-Locked Lasers,�?? in A.N. Luiten (Ed.), Frequency Measurement and Control, Topics Appl. Phys. 79 275 (Springer-Verlag, Berlin, 2001). [CrossRef]
- A.E. Siegman, Lasers (University Science Books, Sausalito, 1986).
- CVI Laser Corporation (www.cvilaser.com).
- F. Brunner et al, �??240-fs pulses with 22-W average power from a mode-locked thin-disk Yb:KY(WO4)2 laser,�?? Opt. Lett. 27, 1162-1164 (2002). [CrossRef]
- T. Beddard et al, �??High-average-power, 1-MW peak-power self-mode-locked Ti:sapphire oscillator,�?? Opt. Lett. 24, 163-165 (1999). [CrossRef]

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