We investigate the scaling properties of mode-locked all-normal dispersion fiber oscillators in terms of output pulse energy and compressed pulse duration. Experimental results are achieved by stepwise variation of the resonator dispersion, total fiber length, and the spectral filter bandwidth. Adjustment of these parameters enables pulse duration scaling down to 31 fs and increase of output pulse energy up to 84 nJ.
©2012 Optical Society of America
Femtosecond lasers are versatile tools in science and industry addressing a growing number of applications in material processing, microscopy, and medicine. The demands of a large number of these applications can already be satisfied by diode-pumped fiber lasers, which achieve output pulse energies of up to 100 µJ  and compressed pulse durations down to 8 fs . However, an increasing number of applications, especially in material processing and microscopy, takes advantage from sub-50 fs pulses with energies of several tens of nJ or even µJ [3, 4]. No fiber lasers or any other diode-pumped lasers operating in this parameter range have been demonstrated so far, and more expensive Ti:sapphire lasers must be used. These applications would benefit from a cheap, user-friendly and compact fiber laser alternative.
Shorter pulse durations from master-oscillator power-amplifier (MOPA) fiber laser systems can be obtained by pulse duration and energy scaling of the seed femtosecond oscillator. In a MOPA scheme the output pulse duration and energy depend mainly on the corresponding parameters of the seed oscillator. Shorter pulse durations from the seed oscillator result directly in shorter pulse durations after amplification. Whereas, higher output pulse energies reduce the required amount of amplification and the accompanied increase in pulse duration due to gain-narrowing of the pulses .
A large step in terms of pulse energy scaling was achieved by introducing of all-normal dispersion fiber oscillators [6, 7], whose scaling properties have been an active research topic during the last years [7–14]. Papers [9, 10] present the currently demonstrated pulse duration and energy limitations of all-normal dispersion fiber oscillators which can be realistically integrated into compact and adjustment-free fiber setups. However, recently a new type of mode-locked fiber oscillators has been demonstrated [11–14]. These oscillators combine a free-space resonator with an active fiber section and can be considered as solid-state oscillators in which the laser crystal is replaced by the active fiber. The free-space resonator allows the generation of pulse energies comparable to that of bulk Yb-crystal oscillators, whereas strong gain and nonlinearity in the fiber section enable dissipative soliton dynamics supporting sub-100 fs pulse durations. Thus, by sacrificing the compactness and adjustment freedom of all-fiber setups, a new range of output pulse parameters from diode-pumped oscillators has been accessed, providing sub-100 fs pulses with energies of several hundreds of nJ [13, 14]. In this publication, we analyze the scaling properties of this type of fiber oscillators and demonstrate pulses with a duration of 31 fs and energies of 84 nJ from an all-normal dispersion step-index fiber oscillator.
To derive the scaling properties of the above described all-normal dispersion fiber oscillators which contain only active fibers, the well-known method of dimensional analysis  can be used. The dimensional analysis is based on the principal that every physically relevant relation must have a representation which is independent of the choice of units of the physical quantities. It has been already applied in ultrafast fiber optics to derive the asymptotic parabolic-pulse solutions of normal dispersion fiber amplifiers . A major benefit of the dimensional analysis is that an exact mathematical formulation of a physical problem is not necessary required. To obtain useful information or even a complete solution, it is often sufficient to just determine the governing physical parameters.
The pulse dynamics of all-normal dispersion fiber oscillators, mode-locked by use of nonlinear-polarization-evolution (NPE) in combination with a birefringent spectral filter, which consist of only the active fiber and a free-space resonator are fully determined by the pulse propagation equation inside the active fiber, the transmission function of the spectral filter, and the nonlinear transmission of the saturable absorber. The pulse propagation inside the active fiber is described by the generalized nonlinear Schrödinger equation :7] and spectral filtering inside the resonator. The transmission function of the NPE saturable absorber, which is also used as laser output, is determined by three dimensionless parameters (i = 1,2,3) which are the rotation angles of the three waveplates used for the adjustment of laser polarization . The sinusoidal transmission of the spectral filter is characterized by its full width at half maximum (FWHM) . Thus, the pulse dynamics of this type of all-normal dispersion oscillators depend on a set of eight parameters: , , , , , and .
By use of dimensional analysis, as described in , one can derive the following scaling laws for the electric field envelope :
The amplitude of the electric field is proportional to the inverse square root of the nonlinear parameter and the active fiber length . All other parameters are related to the electric field through the dimensionless function of the normalized time and length . The dimensionless parameters of shown in square brackets are the total gain per roundtrip, the product of the spectral filter bandwidth and the square root of the total resonator dispersion, and the saturable absorber parameters . The dimensionless function itself is unknown and can only be determined by numerical simulations. However, Eq. (2) shows that fixing of the parameters inside the square brackets results in self-similar scaling of both, the electric field envelope and the pulse dynamics, with respect to the total resonator dispersion and the active fiber length. The normalized time in illustrates a linear relationship between the square root of the resonator dispersion and the time-scale of the electric field envelope. Thus, all temporal characteristics e.g. the values of compressed, Fourier-transform-limited (FTL), and chirped pulse durations are proportional to the square root of the resonator dispersion. In agreement with Eq. (2) the dimensional analysis predicts the following dependencies of the output pulse energy , and the compressed pulse duration on the oscillator parameters:
According to Eq. (2-4), down-scaling of the active fiber length should result in increase of the output pulse energy and decrease in the pulse duration as long as the arguments of the functions inside the square brackets are constant. Since the total gain per roundtrip and the saturable absorber parameters are continuously tunable, this can be achieved by simultaneous scaling of the spectral filter bandwidth with the square root of the resonator dispersion.
3. Experimental setup
In order to investigate the scaling properties predicted above, the experimental setup illustrated in Fig. 1 has been used. The fiber section consists only of the ytterbium-doped double-clad fiber (YDF: Yb1200-10/125DC from Liekki) with a core-diameter of 10 µm, a numerical aperture of 0.08, and an estimated dispersion of 20 ps2/km. This fiber has been chosen due to its strong pump light absorption of 6.5 dB/m, which enables the use of fiber lengths as short as 20 cm. The free-space section of the oscillator contains four waveplates in combination with a polarizing beam splitter (PBS) to achieve mode-locking by use of NPE inside the fiber segment. The spectral filter is realized by a birefringent quartz plate (BP) in front of an additional PBS. A free-space isolator ensures uni-directional laser operation and the NPE-rejection port at the first PBS is used as the oscillator output.
The pump-light at 975 nm is provided by a fiber-coupled multimode diode and is coupled through a dichroic mirror (DM) into the active fiber. An aperture with a diameter of approximately 3 mm is placed in front of the output port to block the unabsorbed pump light. Additional apertures inside the output laser beam further reduce the amount of co-propagating unabsorbed pump light.
At short active fiber lengths only a fraction of the total pump light can be absorbed, which results in low laser-efficiencies and output powers. To avoid pump-limitation of the output pulse energy, the repetition rate of the oscillator is reduced using a Herriott-cell as optical delay line inside the free-space section. It is important to note that the free-space propagation inside the Herriott-cell does not affect the pulse dynamics inside the fiber section and has no influence on the pulse energy limitations. Furthermore, by use of active fibers with stronger absorptions , the Heriott-cell can be removed to achieve the same pulse parameters at higher repetition rates.
Finally, the positively chirped output pulses are compressed using a 300 groove/mm grating compressor with a transmission of 53%, and the temporal profiles of the compressed pulses have been measured by frequency-resolved optical gating (FROG).
According to the scaling laws (2-4), stepwise reduction of the YDF length and increase of the spectral filter bandwidth have been performed in order to simultaneously increase the output pulse energy and to reduce the compressed pulse durations. The lasing performances of the different realized oscillator setups are listed in Table 1 . Due to the low output powers at short fiber lengths no self-starting mode-locking could be obtained and mode-locking had to be initiated by rotation of the half-waveplate in front of the NPE-port. Furthermore, at low output powers no mode-locking could be obtained with the spectral filter bandwidths required by dimensional analysis. Therefore, narrower spectral filters with a bandwidth of 33 nm were used at the YDF lengths of 0.2 m and 0.32 m. However, according to reference  application of narrower spectral filters should have only a weak influence on the obtained output pulse parameters.
At each YDF length we observed a number of similar operation states with slightly varying pulse parameters. For every oscillator the waveplates were adjusted to achieve the shortest compressed pulse duration and the state with the highest pulse energy was recorded. At all fiber lengths the pulse durations and energies were limited by pulse break-up and onset of multi-pulsing, except for the YDF length of 0.2 m, where the pulse energy was pump-limited. The recorded output pulse energies, the compressed and the FTL pulse durations, the peak powers of the compressed pulses, and the pulse chirps of the output pulses for all oscillators are plotted in Fig. 2 to demonstrate the scaling of these parameters with the active fiber length. Additionally, Fig. 2 shows the scaling of each parameter predicted by the dimensional analysis, where the dimensionless parameters of the functions shown in square brackets have been assumed constant for all fiber lengths. The function values themselves have been fitted to the measured pulse parameters. As can be seen, the measured values are in good agreement with the theoretical predictions and verify the scaling properties derived from the dimensional analysis. The observed deviations, especially at the YDF length of 1.95 m, can be attributed to slight fluctuations in the adjustment of the oscillator and saturable absorber parameters. These fluctuations result mainly from the variation of fiber parameters such as mode-field area and birefringence which affect both the nonlinear pulse propagation and the saturable absorber.
The output pulse spectrum, the intensity autocorrelation of the compressed pulses, and the pulse shape retrieved from the measured FROG trace are shown in Fig. 3 for the YDF lengths of 1.95 m, 1.00 m, and 0.32 m. As expected from the dimensional analysis, the optical spectra and temporal pulse profiles of all setups exhibit similar features on different wavelength and time scales. All spectra exhibit the characteristics of dissipative solitons with maxima at the wings of the spectrum and steep spectral side-edges [6, 7]. They are asymmetric in shape with a total maximum at long wavelengths and a smaller local maximum at short wavelengths. However, with decreasing fiber lengths more energy is contained in the wings of the pulse spectrum, the energy content of the long wavelength maximum increases, and the center of the pulse spectrum shifts to shorter wavelengths. These effects can be explained by an increasing impact of self-steepening , which depends on the spectral bandwidth and has been neglected to allow for the dimensional analysis. All temporal pulse profiles exhibit a main pulse surrounded by a pedestal containing from 12% up to 53% of the total pulse energy. These pedestals result from uncompensated higher-order phase terms accumulated due to nonlinearities in the active fiber. However, we observed no correlation between the energy content inside the pedestal, the deviation from the FTL limited pulse duration, and the YDF length. Therefore, the variation of these parameters can be also attributed to small deviations in oscillator alignments.
In agreement with the dimensional analysis, the shortest pulse durations and the highest output pulse energies have been obtained simultaneously at the shortest YDF length of 0.2 m. At a pump power of 22 W stable single-pulse mode-locking is achieved by tuning of the half-waveplate in front of the output port. The oscillator operates at a repetition rate of 5.7 MHz with an average output power of 0.48 W, which corresponds to the output pulse energy of 84 nJ. Figure 4(a) shows the optical spectrum with a FWHM of 86 nm and a FTL pulse duration of 28 fs. The peak at 975 nm appears due to co-propagating unabsorbed pump light. The output pulses have a positive chirp of 0.007 ps2 and are compressed to the pulse duration of 31 fs. The temporal pulse profile of the compressed pulses and intensity autocorrelation retrieved by FROG are shown in Fig. 4(b) together with the measured intensity autocorrelation. The retrieved and measured intensity autocorrelations are in good agreement. The compressed pulses have a peak power of 0.8 MW and the pedestal surrounding the main pulse contains 26% of the total pulse energy. The FROG-trace of the compressed pulses has been measured with a commercially available FROG-device from the APE-Berlin with a spectral resolution of 10 nm and a temporal resolution of 5.8 fs. The measured and retrieved FROG-traces are shown in Fig. 4(c) and (d). Both traces are in good agreement with an error of 0.8%. The fringes in the retrieved FROG-trace, performed with the spectral resolution of 2.9 nm, cannot be resolved in the measured FROG-trace.
To verify stable single-pulse operation of the oscillator, the output pulse train has been recorded with a 10 GHz photodiode in combination with a 6 GHz oscilloscope and an rf-spectrum-analyzer. The oscilloscope traces and the radio-frequency spectra of the photodiode signal are shown in Fig. 4(e) and (f). No satellite pulses are visible in the oscilloscope trace and in the 150 ps span of the intensity autocorrelator which confirms single-pulse operation.
The radio-frequency spectrum at the fundamental repetition rate of 5.7 MHz has been recorded with a resolution of 1 Hz. The constant heights of the radio-frequency peaks, as well as the noise-suppression of 80 dB, confirm stable mode-locked operation without Q-switching, period-doubling, or higher-harmonic mode-locking .
In summary, scaling properties of mode-locked all-normal dispersion ytterbium fiber oscillators have been studied in terms of pulse duration and energy. We focused on all-normal dispersion oscillators consisting only of an active fiber and a free-space section without passive fibers. The dimensional analysis has been applied to obtain analytic expressions for the scaling of the laser pulse characteristics. By stepwise variation of the YDF length from 1.95 m to 0.2 m we reduced the compressed pulse duration from 108 fs down to 31 fs and increased the output pulse energy from 31 nJ up to 84 nJ, thereby confirming the theoretical predictions. These parameters correspond to the shortest pulse duration and the highest peak power per mode-field area from an all-normal dispersion fiber oscillator. The only limitation on pulse duration and energy scaling has been set by the low pump light absorption at short YDF lengths. By use of YDF fibers with stronger cladding absorptions  and smaller dispersion values  further pulse duration and energy scaling should be feasible.
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