1. Introduction

Ytterbium-doped material crystals have received a great interest in recent years, mainly for its broadband fluorescence spectrum, simple energy structure, and favorable absorption wavelength for diode pumping. However, its quasi-three-level operating scheme has a rigorous requirement on the pumping rate, and the thermal populating in the ground state laser level would cause strong re-absorption around the emission wavelengths. These defects have restricted their application in mode locked laser systems.

Recently, several newly developed Yb-doped material crystals such as Yb:GSO, Yb:GYSO and Yb:LYSO have been demonstrated to be able to overcome these problems [1–4]. Yb:GSO crystal has a quasi-four-level system, high absorption cross-sections and particularly broad emission bandwidth (about 72 nm) [1]. Yb:GYSO not only retains the high laser performance of Yb:GSO, but also contains the good mechanical properties of Yb:Y₂SiO₅ (Yb:YSO) [3]. In the continuous-wave laser operation of Yb:LYSO, a maximal slope efficiency of 96% and output power of 7.8 W have been achieved [4]. All these properties with their ultra-broad fluorescence bands indicate that they are excellent media for the femtosecond lasers. Pulses as short as 343 fs have already been achieved in the diode-pumped Yb:GSO femtosecond laser [2]. However, there have not been reliable dispersion data available for these three new crystals. For further shortening the pulse width, accurate dispersion data are required. Therefore, measurement for the GDD of these crystals becomes necessary.

In this paper, we report the GDD of these new crystals (10%-doped Yb:GSO, 5%-doped Yb:GSO, 5%-doped Yb:GYSO and 5%-doped Yb:LYSO) measured by a home-made white-light interferometer. The measured GDD data are highly precise for the wavelengths of 1000nm to 1200nm, covering the tuning bands of those lasers. The GDD data should be useful in developing diode pumped femtosecond lasers using those new promising crystals.

2. White-light Interferometer

The GDD measurement was performed with a white-light interferometer which was proposed by Naganuma et al. [5], and we made some modifications. The schematic of the interferometer is shown in Fig. 1. The interferometer was basically a Michelson. The white-light source was a fiber-coupled metal-halogen lamp and was polarized by a polarization beam splitter cube. The scanner was composed of a solenoid and an iron rod, and was driven by a waveform generator and a power amplifier. To make precise calibration, another Michelson interferometer was built on the back of the scanner and the light source was a helium-neon (He-Ne) laser. Two silver mirrors were glued on both ends of the rod. A triangle current wave was used to drive the scanner and to generate the interferograms. The repetition frequency of the scanner was about 40Hz and the scan range was adjusted to best cover the full fringe. The interferograms were recorded with two InGaAs photodiodes on both sides of the system.

 

Fig. 1. Schematic of the white-light interferometer. SM: single mode fiber; BS: cubic beam splitter; PBS: polarization beam splitter; M1~M4: Silver mirrors in the Michelson interferometers; M5~M9: Silver mirrors for optical axes alignment. The GDD of the apparatus on a blank sample condition would first be measured and would be subtracted from the subsequent measurement on samples.

Download Full Size | PDF

The key problem was that the scanner could not translate precisely and linearly with respect to the scanning time. Although two parallel Michelsons had already been used to correct the nonlinearity of the scanner, the translation of the scanner should be as linear as possible, so that the interference fringe could be sampled uniformly in the temporal axis. Both piezo translator and the solenoid were tried as the scanner and the linearity of the piezo translator was found much worse than that of the solenoid. Thus the solenoid was employed in this experiment as the scanner. Slight adjustment of the repetition frequency and the driving current was still necessary to ensure the He-Ne interference fringes to look equally spaced in time.

3. Data Processing

Yb:GSO, Yb:GYSO and Yb:LYSO are all biaxial monoclinic crystals. The direction that the light propagates through our sample is determined as the b-axis. With unpolarized light when the PBS in the white-light interfermometer was removed, we could identify two white-light interferograms due to the different group delay of orthogonal polarizations. The interferogram at the early delay corresponds to the polarization that sees a smaller refractive index.

The interferograms from both Michelson were recorded simultaneously. The He-Ne laser interference fringe should be uniformly spaced in the temporal axis, with which we could calibrate the delay axis and apply to the white-light fringe. However, by checking the time interval between the neighboring sampled points in the delay axis, we found that the spacing was not exactly the same, because of the residual nonlinearity of the scanner translation. Therefore, based on the original sampled data, we used piecewise cubic Hermite interpolation [6] to rebuilt the data sets and made the delay axis uniform. The procedure was, for every half-period of the fringes of He-Ne laser, the horizontal axis was calibrated by taking arccosine of the normalized amplitude and the phase angle was projected on to the horizontal axis in from 0 to π. Then the horizontal axis was re-arranged in equal space by cubic Hermite interpolation. This new horizontal axis was then applied to calibrate the delay axis of the white-light interference fringes. In this way, the delay axis of the white-light fringe became uniform in time. Because our original data were densely sampled with respect to time (about 30 points per period), we had quite enough data points for ensuring the accurate interpolation in the new time frame. Then the Fourier transform of the calibrated fringes in cross correlation yields:

where Ẽ_s(ω) was the Fourier transform of the electric field transmitting through the sample arm and Ẽ_r ^*(ω) was the complex conjugate of the signal in the reference arm. The phase φ(ω) was expressed as φ(ω) = φ_s(ω) + φ_system(ω) - ωτ ₀, where φ_s (ω) was the phase shift due to the crystal and φ_system(ω) was caused by the apparatus inherent bias in group delay, while τ ₀ was the delay representing the path length offset between the two arms in the interferometer. GDD of the crystal was the second derivative of φ_s(ω) with respect to ω where τ ₀ would be vanished:

In practice, φ(ω) was fitted with a 4^th order polynomial from ω = 1.5708fs^-1 to 1.8850fs^-1, correspondent to wavelength from 1200nm to 1000nm. Then the second derivative was taken, and d ² φ_system(ω)/dω ² would be subtracted through another measurement on a blank sample condition. Thirty data sets were taken and the phase was averaged before the curve fitting. To estimate the accuracy, we calculated the root mean squared error (RMSE) for the fitted curve of the phase, where the phase was normalized in the arithmetic mean sense. The RMSE was very small, ranging from 1.5326×10^-5 to 6.3445×10^-5 for these four crystals. We also calculated the standard deviation of these thirty data sets for each point. The average standard deviation of these four crystals was ranged from 0.66264fs²/mm to 3.4308fs²/mm, indicating the consistency of our measurement.

The fitted GDD formulae valid in the wavelength range of 1.0μm~1.2μm are summarized in Table 1, and plotted in Fig. 2 – Fig. 5.

Table 1. Fitted single pass GDD in the wavelength range of 1.0μm~1.2μm

View Table

 

Fig. 2. Fitted GDD of 10%-doped Yb:GSO crystal.

Download Full Size | PDF

 

Fig. 3. Fitted GDD of 5%-doped Yb:GSO crystal.

Download Full Size | PDF

 

Fig. 4. Fitted GDD of 5%-doped Yb:GYSO crystal.

Download Full Size | PDF

 

Fig. 5. Fitted GDD of 5%-doped Yb:LYSO crystal.

Download Full Size | PDF

In Fig. 2–Fig. 5, axis x ₁ and axis x ₂ are the two principal axes in Fresnel’s ellipsoid. They are orthogonal to the biaxial monoclinic crystal’s b-axis and are also perpendicular to each other. Axis x ₁ corresponds to the polarization parallel to the axis with smaller refractive index, and its GDD is also smaller. All the GDD decreases with increasing wavelength, and would reach zero at a longer wavelength, indicating these crystals have high refractive indices.

The GDD of 5%-doped Yb:GSO is a bit smaller than that of 10%-doped Yb:GSO. We then measured the refractive index of Yb:GSO by optical transmission measurements in the 1000~1200nm region according to methodology described by J. C. Manifacier et al. [7]. The result showed that 5%-doped Yb:GSO had a smaller refractive index than the 10%-doped Yb:GSO. This is consistent with our common sense that the higher refractive index material has a higher GDD. We concluded that the higher concentration of the Yb³⁺ ion in the substrate would lead to a higher refractive index and also a higher GDD in Yb:GSO.

It can also be seen that the GDD of these crystals is quite large and drops rapidly (from 100 to 75 fs²/ mm for Yb:GSO, 95 to 70 fs²/mm for Yb:GYSO and 95 to 60 fs²/mm for Yb:LYSO, respectively). The rapid drop of GDD implies a large third order dispersion (TOD). For example, by taking derivative of the formulae listed in Table 1, we obtained the TOD for axis x ₁ of Yb:GSO which was approximately 101fs³/mm at the wavelength of 1064nm, and the TOD for axis x ₂ of Yb:LYSO was even larger, approximately 170fs³/mm at 1064nm. Therefore, the dispersion compensation should include TOD for femtosecond pulse generation.

4. Conclusions

We have measured the GDD of Yb:GSO, Yb:GYSO and Yb:LYSO crystal along the two orthogonal axes with respect to b-axis, over the wavelengths from 1000nm to 1200nm, which is positive and decreases with increasing wavelength. The measured GDD provided here is convincing and reliable, and would be useful for the dispersion compensation and for further short pulse generation in femtosecond lasers where these new crystals are used as laser media.