Period doubling and deterministic chaos in continuously pumped regenerative amplifiers

doi:10.1364/OPEX.12.001759

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Period doubling and deterministic chaos in continuously pumped regenerative amplifiers

Jochen Dörring, Alexander Killi, Uwe Morgner, Alexander Lang, Max Lederer, and Daniel Kopf »View Author Affiliations

Optics Express, Vol. 12, Issue 8, pp. 1759-1768 (2004)
http://dx.doi.org/10.1364/OPEX.12.001759

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– Abstract
1. Introduction
2. Theoretical description
4. Experimental results
5. Conclusion
– References and links

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Abstract

Multi-energy and chaotic pulse energy output from a continuously pumped regenerative amplifier is observed for dumping rates around the inverse upper state lifetime of the gain medium. The relevant regimes of operation are analyzed numerically and experimentally in a diode-pumped Yb:glass regenerative amplifier. The boundaries between stable and unstable pulsing are identified and stability criteria in dependence on the amplifier gate length and pump power are discussed.

1. Introduction

In the past several techniques for the generation of ultrashort optical pulses with high energies have been proposed and developed. For pulse energies in the µJ- or mJ-range the large majority of the laser systems employ chirped pulse amplification [1

H. Yoshida, E. Ishii, R. Kodama, H. Fujita, Y. Kitagawa, Y. Izawa, and T. Yamanaka, “High-power and high-contrast optical parametric chirped pulse amplification in beta -BaB₂O₄ crystal,” Opt. Lett. 28, 257–259.(2003). [CrossRef] [PubMed]

–3

C. Horvath, A. Braun, H. Liu, T. Juhasz, and G. Mourou, “Compact directly diode-pumped femtosecond Nd:glass chirped-pulse-amplification laser system,” Opt. Lett. 22, 1790–1792 (1997) [CrossRef]

]. For gain media with a short upper state lifetime multipass amplification systems may be a viable solution to achieve high pulse energies [4

W. H. Lowdermilk and J. E. Murray, “The multipass amplifier: Theory and numerical analysis,” J. Appl. Phys. 51, 2436–2444 (1980). [CrossRef]

], but gain media with a long upper state lifetime can accumulate a large population inversion and are capable of storing higher energies. Unfortunately, the corresponding low emission cross section prevents the use of long-lifetime media in a multipass scheme. Nevertheless, these media have been successfully used in regenerative amplifiers (RA). In the last years ytterbium-based regenerative amplifiers have become a focus of research and continue to be of interest for the amplification of ultrashort pulses to high energies. The gain medium advantageously combines a small quantum defect, a broad absorption and emission bandwidth, and an absorption band in a spectral region where high brightness laser diodes are available [5

C. Hönninger, R. Paschotta, M. Graf, F. Morier-Genoud, G. Zhang, M. Moser, S. Biswal, J. Nees, A. Braun, G. A. Mourou, I. Johannsen, A. Giesen, W. Seeber, and U. Keller, “Ultrafast ytterbium-doped bulk lasers and laser amplifiers,” Appl. Phys. B 69, 3–17 (1999). [CrossRef]

]. The long fluorescence lifetime makes ytterbium the ideal candidate for RAs. As a matter of fact, ytterbium based RAs have been demonstrated in the sub-picosecond regime up to the millijoule-energy level [6

C. Hönninger, I. Johannsen, M. Moser, G. Zhang, A. Giesen, and U. Keller, “Diode-pumped thin-disk Yb:YAG regenerative amplifier,” Appl. Phys. B 65, 423–426 (1997). [CrossRef]

H. Liu, S. Biswal, J. Paye, J. Nees, G. Mourou, C. Hönninger, and U. Keller, “Directly diode-pumped millijoule subpicosecond Yb:glass regenerative amplifier,” Opt. Lett. 24, 917–919 (1999) [CrossRef]

Many applications like micromachining or refractive eye surgery, for instance, would benefit from a high pulse repetition rate. However, as we will show in this paper, a high repetition rate may lead to irregular pulsing and even chaotic behaviour in the energy sequence of the extracted pulses. This paper is organized as follows: After a brief description of the underlying rate equations we present some numerical solutions, followed by the experimental results. In particular, the physical limit for the single pulse energy output of an Yb:glass-based RA is discussed, as well as the irregular behaviour once the limit is surpassed.

Fig. 1. (a) Schematic set-up of the RA, and (b) an illustration of the relevant time constants: the gating time T_G, the round trip time T_R, and the dumping rate is defined by

{T_{D}}^{- 1}

2. Theoretical description

The modelling of the RA is based on the schematic set-up as depicted in Fig. 1(a). The RA consists of a laser cavity with an actively controlled output coupling. All involved mirrors are highly reflective and the switch inside the cavity controls the energy ejection out of the cavity. At the same time the switch modulates the Q-value of the cavity drastically. Two phases can be distinguished: The high-Q phase and the low-Q phase. During the low-Q phase the gain medium is pumped optically to steadily build up the population inversion in the gain medium. Due to the high losses no laser action is possible. In the general case of an RA a low-energy pulse from a laser oscillator is seeded into the RA cavity at the beginning of the following high-Q phase. Once it is injected the pulse passes typically tens to hundreds of times through the gain medium to collect the stored energy. As depicted in Fig. 1(b) the dumping frequency of the high energy pulses is defined by the time period TD corresponding to the duration of one high-Q cycle and one low-Q cycle. The gating time TG defines the duration of the high-Q phase and represents an integer multiple of the round trip time TR of the RA cavity. The high energy pulses can be dumped at the desired energy level after the corresponding number of round trips.

Since the relevant time constant of the gain and the cavity are orders of magnitude higher than the pulse duration, energy and gain do not bear a physical dependence on the pulse width. Therefore, the processes in the RA can be conveniently described by rate equations for the pulse energy and the gain. Let us consider the governing rate equations for both phases:

2.1 Low-Q phase

During the time period T_D - T_G the build up of the gain g takes place when no lasing is possible. The gain is governed by the equation:

\frac{\partial g}{\partial t} = \frac{g_{0} - g}{τ_{L}}

(1)

The crucial constants are the lifetime τ_L of the upper laser level and the small signal gain g0, which is proportional to the absorbed pump power. As a result, the low-Q phase can be treated analytically. Given the gain g(0)=g₁ at the beginning of the low-Q phase the integration of Eq. (1) leads to the gain g as a function of time. The gain at the beginning of the following high-Q phase g₂=g(T_D-T_G) is then given by

g_{2} = g_{0} + (g_{1} - g_{0}) e^{(- \frac{T_{D} - T_{G}}{τ_{L}})} .

(2)

2.2 High-Q phase

During the High-Q phase the gain medium is also continuously pumped and the first term is the same as in Eq.(1). However, in this phase the gain is depleted by stimulated emission, represented by the negative term in Eq. (3), which is also the dominant term during the high-Q phase. It couples the differential equation of the gain to the one of the pulse energy:

\frac{\partial g}{\partial t} = \frac{g_{0} - g}{τ_{L}} - \frac{g E}{E_{sat} T_{R}}

(3)

\frac{\partial E}{\partial t} = \frac{E}{T_{R}} (g - l)

(4)

The boundary conditions for gain and energy are given by g(0)=g₂ and E(0)=E_seed. In the High-Q phase no analytical solution is known, and Eqs.(3) and (4) must be solved numerically. The model treats the amplification of an injected pulse and the amplification of a noise fluctuation originated inside the RA cavity in the same way. The only difference when modelling both cases is the different pulse energy E_seed at the beginning of the regenerative process. As a result a different number of round trips are required to reach the desired output energy.

3. Numerical simulations

For the numerical solution a fourth order Runge-Kutta method was employed using the parameters of Table 1, which were either taken from literature or were measured during the experiment. The only exception is the seed energy E_seed, which had to be obtained by matching the numerical to the experimental data.

In order to reach a steady state, the equations have been solved for 500 subsequent low-and high-Q phases. If we assume a linear dependence between pump power and initial gain g0, we find that pump powers between 3.5W and 5.4W correspond approximately to a power gain of 1.22>G₀=exp(g₀)>1.46 or to the value 0.2>g₀>0.38.

Table 1. Defining parameters of the Yb:glass RA:

Parameter [unit]	Value
Wavelength λ [nm]	1040
emission cross section σ_L [10^-24m²]	0.2
upper state lifetime τ_L [ms]	1.3
focus beam waist radius w₀ [µm]	60
saturation energy E_sat [mJ]	5.4
seed pulse energy E_seed [pJ]	0.1
loss l [%]	2

Fig. 2. Calculated bifurcation diagram (main graph) in dependence on the gate length at a dumping rate of

{T_{D}}^{- 1}

=10kHz, and a small signal gain of g₀=0.3. For any gate length eleven subsequent pulse energies are plotted. The inlay illustrates the actual sequence of the dumped pulse energies at an arbitrarily chosen gate length in the P2 regime.

For a dumping rate of

{T_{D}}^{- 1}

=10kHz and a small signal gain of g₀=0.3, the pulse energies of eleven subsequent pulses have been plotted in the main graph of Fig. 2 as a function of the duration of the high-Q phase, the gate length T_G. The result is a very clear bifurcation route to deterministic chaos. At small gate lengths single-energy pulsing (P1) was followed by the first bifurcation point around T_G=3µs, multiple energy regimes (Pn), and chaotic regimes (P∞).

Clearly, the dynamics is governed by the main control parameter: the gate length. For a small gate length the circulating pulse cannot extract the whole stored energy from the amplifier medium, and a large fraction of the initial gain remains at the end of the high-Q phase, giving the RA the opportunity to recover the gain during the following low-Q phase. Then the system is in stable P1 state. However, with increasing gate length the energy of the amplified pulse increases while the remaining gain g₁ at the end of the high-Q phase decreases. For a very small g₁ the gain cannot recover the previous value during the low-Q phase, and the system becomes unstable: High pulse energy E leads to a small g₁ and consequently a small g₂, which makes the next E small. This keeps a large g₁, g₂ and leads again to a large E in the next phase. An example for an energy and gain sequence in the P2 state is given in Fig.3. This irregular behaviour does not occur at arbitrary dumping frequencies, which is confirmed by the numerically obtained curve of the maximum pulse energy difference between subsequent pulses shown in Fig. 4. The Pn regimes only appear at dumping frequencies from 0.5kHz up to 40.0kHz. Interestingly, the dumping frequency corresponding to the inverse upper state lifetime f_L=1/τ_L≅0.8kHz is located very close to the peak of the multi-energy region. This behaviour can be compared to a resonance phenomenon, where the maximum amplitude is located at the eigenfrequency f_L of the system. If the RA is operated below 0.5kHz the gain medium is given enough time to recover and also if the RA had been operated above 40.0kHz the dumping cycles would have come so frequently not to affect the gain sufficiently enough for causing a memory effect.

Fig. 3. Illustration of the interplay of gain and pulse energy during the low-Q and high-Q cycles in the case of a P2 state. The duration of the low-Q phase is not drawn to scale.

Fig. 4. Maximum pulse energy difference in dependence on the dumping frequency at g₀=0.3. Zero energy difference means stable P1 state.

Fig. 5. Numerically obtained normalized intra-cavity energy during a high-Q phase in the single-energy (regular) regime at

{T_{D}}^{- 1}

=0.4kHz and g₀=0.3.

For maximum single-energy output the dumping process should be ideally initiated close to the energy maximum while operating the RA at repetition rates below 0.5kHz. As shown in Fig. 5, the gain medium was given sufficient time to completely recover from the high-Q phase and only one energy curve was observed. The curve reveals a typical shape: During the time before the maximum the oscillating laser light collects increasingly more energy from the gain medium at each round trip. At the maximum the remaining gain exactly compensates for the intra-cavity losses and after that the losses play the major role and the curve begins to decay exponentially towards the intra-cavity energy corresponding to continuous-wave operation.

As mentioned in the first section, it is not relevant for the dynamics to occur whether the RA is seeded with a picosecond or sub-picosecond seed pulse from outside the cavity or it is initiated by energy fluctuations inside the cavity, which leads to a pulse duration basically determined by the length of the cavity. Consequently, to avoid any complications due to nonlinearity, dispersion, and possible damage, the active Q-switching regime of the RA is the preferable set-up to observe clean multi-energy pulsing and deterministic chaos.

Fig. 6. Experimental setup of the Yb:glass RA; TFP: thin film polarizer; λ/4: quarter wave plate; EOM: electro-optical Pockels-cell.

4. Experimental results

The experimental set-up is depicted in Fig. 6. The pump diode had a maximum output power of 5.4W emitting at 976.0nm, which corresponds to an effective absorbed pump power of 3W. The RA operated at 1040nm. The pump beam was focussed through a dichroic mirror into the 3.3mm-long Yb:glass. Apart from the gain medium the cavity consisted of a thin film polarizer (TFP), a quarter wave plate, an electro-optic modulator (EOM) and several highly reflective, dielectric folding mirrors. When high voltage was applied to the EOM the RA was in high-Q state and otherwise in low-Q state. In low-Q state the wave plate (in double pass) rotated the polarization of the beam by 90° and the light was dumped out of the cavity at the TFP, which was installed for horizontal polarization transmission. During this state laser action was suppressed and the gain medium was accumulating gain for the next high-Q phase. The high-Q phase was established once the voltage was applied to the EOM shifting the polarization another 90° (in double pass). Then the total polarization shift was 180° and the beam remained inside the cavity. At the beginning of the next low-Q phase, the intra-cavity pulse was dumped. The pulse duration of 14.5ns (68.8 MHz) was determined by the cavity length.

In order to monitor the intra-cavity energy transient during the high-Q phase, some light leaking through one of the mirrors was detected by a fast photo diode. The normalized intra-cavity energy curves during 200 consecutive high-Q phases in the chaotic regime are shown in Fig. 7. The origin at t=0 was set to the beginning of the high-Q phase. The dumping occurs after 5.6µs. Each curve reflects another initial gain parameter, which is the consequence of the coupling of subsequent high-Q phases. In contrast, the regular regime produces only one pulse energy and therefore only one energy curve is existent as shown in Fig. 5. For P2, P4 and the chaotic regime two, four and infinite number of such energy curves can be found, respectively. However, all of the multi-energy regimes represent a major drawback for most applications.

Fig. 7. Experimentally obtained intra-cavity energy curves in the chaotic regime at

{T_{D}}^{- 1}

=10kHz and P_P=5.4W. (200 curves of consecutive dumping cycles are displayed).

Fig. 8. Bifurcation diagrams for the pump power of (a) P_p=5.4W, (b) P_p=5.1W, and (c) P_p=4.7W at the dumping rate of

{T_{D}}^{- 1}

=10kHz.

Single and multi-energy regimes were also observed in dependence on the gate length. In Fig. 8(a), for a pump power of P_p=5.4W and a repetition rate of

{T_{D}}^{- 1}

=10kHz, the absolute pulse energy is plotted in dependence on the gate length. Also here 200 energy values of consecutive output pulses for each gate length were recorded. As predicted by the numerical calculations the multi-energy regime starts in the vicinity of the gate length where the maximum pulse energy for single-energy output is expected. For high power applications this might be a serious problem since it is desirable to remain in single-energy regime. Several operation regimes could be realized as the gate length was increased. Below T_G=2.65µs the system was in P1 and above it changed to P2 state. Afterwards it turned into P4 and entered the P∞ regime. Interestingly – and typical for the deterministic chaos [8

J. Briggs and F. D. Peat, Turbulent Mirror (Harper & Row, Publishers Inc.,1989).

] – the RA switched to the multi-energy regime (P3 & P4) in a small window of 0.4µs to return to P∞ once again.

From Fig. 8(a)–(c) we learn that the sequence of the Pn regimes remained qualitatively the same when changing the pump power. The bifurcation diagram is only stretched in time for decreasing pump powers. As the gain (pump power) was decreased more round trips were needed to reach comparable pulse energy levels and therefore the first bifurcation point is reached at higher gate lengths. We observed a total shift of the first bifurcation of ~3µs varying the pump power from 3.2W up to 5.4W (see Fig. 9(a)).

Fig. 9. Bifurcation gate length (a) in dependence on the pump power and (b) in dependence on the dumping frequency.

The P1 gate length – which is the maximum gate length before the first bifurcation occurs -depends on the pump power as well as on the dumping frequency. Changing the dumping frequency is equivalent to manipulating the low-Q phase duration, which governs the coupling of the boundary conditions of every high-Q phase to the preceding high-Q phase. The more the high-Q phases are temporally separated the weaker is their coupling. The experiments confirm this dependence. The P1 gate length increases almost linearly with the dumping frequency as plotted in Fig. 9(b). It could be shifted more than 1µs over the dumping frequency range of 1 to 10kHz.

The bifurcation diagrams for different dumping rates (Fig. 10(a)–(c)) show clearly that the RA behaves more chaotic at high dumping rates. At the dumping rate of

{T_{D}}^{- 1}

=1kHz only P1 and P2 regimes were present. In order to suppress also the P2 state the dumping rate needs to be well below 0.5kHz (compare Fig. 4), or much higher than 50 kHz which was not feasible with the current EOM driving electronics.

First calculations reveal that a possible solution to overcome the instabilities might be the employment of an active feedback control, which modulates the pump diode current as a function of the detected output pulse energy. Therefore a constant population inversion may be reached independently of the gain parameter g₂ at the end of each high-Q cycle. This active decoupling of the high-Q cycles by controlling the gain could be realized by already established feedback stabilization techniques, which were applied for Q-Switching suppression in passively mode-locked lasers and is currently under investigation [9

T.R. Schibli, K.E. Robinson, U. Morgner, S. Mohr, D. Kopf, and F.X. Kärtner, “Control of Q-switching instabilities in passively mode-locked lasers,” Trends in Optics and Photonics 68, 498–504, Springer 2002

Fig. 10. Bifurcation diagrams for the dumping rates (a)

{T_{D}}^{- 1}

=1kHz, (b)

{T_{D}}^{- 1}

=4kHz, and (c)

{T_{D}}^{- 1}

=7kHz at the pump power P_p=5.4W.

5. Conclusion

In summary, we presented in the pulse trains from a diode-pumped Yb:glass-based regenerative amplifier (RA), for the first time to our knowledge, unstable pulsing with bifurcation routes to chaos. In the (regular) single-energy regime the RA generated pulse energies of up to 620µJ and 39µJ at dumping rates of 1kHz and 10kHz, respectively. The limiting factor for the dumping rate selection maintaining the single-energy regime is the upper state lifetime of the gain medium. The route to multi-energy regimes depended on pump power, dumping rate and gate length, and the instability occurred coincidentally in the parameter range where the highest output pulse energy was expected. The typical bifurcation diagram contained up to four distinguishable pulse energies and sections with chaotic pulse energy behaviour. The comparison of the numerical and experimental results shows that the multi-energy pulsing is very well described by a simple rate equation model. This gives rise to the expectation that active stabilization techniques may be developed in the future to suppress the instability and reach the output energy maximum.

Acknowledgments

The authors gratefully acknowledge the funding by the EU-CRAFT project within the contract G1ST-CT-2002-50266 (DACO).

References and Links

1.	H. Yoshida, E. Ishii, R. Kodama, H. Fujita, Y. Kitagawa, Y. Izawa, and T. Yamanaka, “High-power and high-contrast optical parametric chirped pulse amplification in beta -BaB₂O₄ crystal,” Opt. Lett. 28, 257–259.(2003). [CrossRef] [PubMed]
2.	V. Petrov, F. Noack, F. Rotermund, V. Pasiskevicius, A. Fragemann, F. Laurell, H. Hundertmark, P. Adel, and C. Fallnich, “Efficient all-diode-pumped double stage femtosecond optical parametric chirped pulse amplification at 1-kHz with periodically poled KTiOPO₄ ,” Jpn. J. Appl. Phys. 42, L1327–1329 (2003). [CrossRef]
3.	C. Horvath, A. Braun, H. Liu, T. Juhasz, and G. Mourou, “Compact directly diode-pumped femtosecond Nd:glass chirped-pulse-amplification laser system,” Opt. Lett. 22, 1790–1792 (1997) [CrossRef]
4.	W. H. Lowdermilk and J. E. Murray, “The multipass amplifier: Theory and numerical analysis,” J. Appl. Phys. 51, 2436–2444 (1980). [CrossRef]
5.	C. Hönninger, R. Paschotta, M. Graf, F. Morier-Genoud, G. Zhang, M. Moser, S. Biswal, J. Nees, A. Braun, G. A. Mourou, I. Johannsen, A. Giesen, W. Seeber, and U. Keller, “Ultrafast ytterbium-doped bulk lasers and laser amplifiers,” Appl. Phys. B 69, 3–17 (1999). [CrossRef]
6.	C. Hönninger, I. Johannsen, M. Moser, G. Zhang, A. Giesen, and U. Keller, “Diode-pumped thin-disk Yb:YAG regenerative amplifier,” Appl. Phys. B 65, 423–426 (1997). [CrossRef]
7.	H. Liu, S. Biswal, J. Paye, J. Nees, G. Mourou, C. Hönninger, and U. Keller, “Directly diode-pumped millijoule subpicosecond Yb:glass regenerative amplifier,” Opt. Lett. 24, 917–919 (1999) [CrossRef]
8.	J. Briggs and F. D. Peat, Turbulent Mirror (Harper & Row, Publishers Inc.,1989).
9.	T.R. Schibli, K.E. Robinson, U. Morgner, S. Mohr, D. Kopf, and F.X. Kärtner, “Control of Q-switching instabilities in passively mode-locked lasers,” Trends in Optics and Photonics 68, 498–504, Springer 2002

OCIS Codes
(140.1540) Lasers and laser optics : Chaos
(140.3280) Lasers and laser optics : Laser amplifiers
(140.3480) Lasers and laser optics : Lasers, diode-pumped

ToC Category:
Research Papers

History
Original Manuscript: March 2, 2004
Revised Manuscript: April 8, 2004
Published: April 19, 2004

Citation
Jochen Dörring, Alexander Killi, Uwe Morgner, Alexander Lang, Max Lederer, and Daniel Kopf, "Period doubling and deterministic chaos in continuously pumped regenerative amplifiers," Opt. Express 12, 1759-1768 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-8-1759

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References

H. Yoshida, E. Ishii, R. Kodama, H. Fujita, Y. Kitagawa, Y. Izawa, and T. Yamanaka,??? High-power and high-contrast optical parametric chirped pulse amplification in beta ???BaB2O4 crystal,??? Opt. Lett. 28, 257-259.(2003). [CrossRef] [PubMed]
V. Petrov, F. Noack, F. Rotermund, V. Pasiskevicius, A. Fragemann, F. Laurell, H. Hundertmark, P. Adel, and C. Fallnich, ???Efficient all-diode-pumped double stage femtosecond optical parametric chirped pulse amplification at 1-kHz with periodically poled KTiOPO4,??? Jpn. J. Appl. Phys. 42, L1327-1329 (2003). [CrossRef]
C. Horvath, A. Braun, H. Liu, T. Juhasz, and G. Mourou, ???Compact directly diode-pumped femtosecond Nd:glass chirped-pulse-amplification laser system,??? Opt. Lett. 22, 1790-1792 (1997). [CrossRef]
W. H. Lowdermilk, and J. E. Murray, ???The multipass amplifier: Theory and numerical analysis,??? J. Appl. Phys. 51, 2436-2444 (1980). [CrossRef]
C. Hönninger, R. Paschotta, M. Graf, F. Morier-Genoud, G. Zhang, M. Moser, S. Biswal, J. Nees, A. Braun, G. A. Mourou, I. Johannsen, A. Giesen, W. Seeber, U. Keller, ???Ultrafast ytterbium-doped bulk lasers and laser amplifiers,??? Appl. Phys. B 69, 3-17 (1999). [CrossRef]
C. Hönninger, I. Johannsen, M. Moser, G. Zhang, A. Giesen, U. Keller, ???Diode-pumped thin-disk Yb:YAG regenerative amplifier, ??? Appl. Phys. B 65, 423-426 (1997). [CrossRef]
H. Liu, S. Biswal, J. Paye, J. Nees, G. Mourou, C. Hönninger, and U. Keller, ???Directly diode-pumped millijoule subpicosecond Yb:glass regenerative amplifier,??? Opt. Lett. 24, 917-919 (1999) [CrossRef]
J. Briggs, and F. D. Peat, Turbulent Mirror (Harper & Row, Publishers Inc.,1989).
T.R. Schibli, K.E. Robinson, U. Morgner, S. Mohr, D. Kopf, F.X. Kärtner, ???Control of Q-switching instabilities in passively mode-locked lasers,??? Trends in Optics and Photonics 68, 498-504, Springer 2002

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