1. Introduction

For Chirped Pulse Amplification (CPA) laser systems, optical pumping in multipass or regenerative amplifiers induces a thermal load that becomes a critical issue as the repetition rate of the laser increases. This thermal load, originating mainly from the difference between the pump and the amplified beam photon energies, is responsible for a transverse-varying index gradient that leads to thermal lensing in the amplifier medium and often alters the quality of the wavefront [1–6]. To counteract this thermal lensing, which causes a reduction in the amplified beam mode size, possibly resulting in damage to the optical components of the laser system, several solutions have been proposed. One can compensate its presence by a passive optical system (generally a negative lens), use this thermal lensing as an advantage with a thermal eigenmode multipass amplifier design [7], or remove its existence with appropriate cooling of the amplifier medium. Cryogenic cooling results in favorable changes in thermal and mechanical properties of the amplifying medium and is most commonly used to remove or reduce the thermal lensing effect [5,8–10].

In this article, we perform what is, to our knowledge, the first time-resolved measurement of thermal lensing in cryogenically cooled Ti:sapphire at 1 kHz. While the cryogenic cooling method has been shown to decrease the lensing effect to an acceptable level [5,9], to our knowledge no direct measurement of the thermal lensing value has been reported when cooling a Ti:sapphire amplifier down to liquid Nitrogen (LN) temperature (77K). Furthermore, a recent analytical calculation of thermal effect in laser amplifiers [11] has predicted that the cooling rate for sapphire at LN temperature should be sufficiently rapid to allow considerable relaxation of the thermal profile between shots at 1 kHz repetition rate. In this case one would expect that a steady-state measurement of thermal lensing, like all measurements that have been done previously in Ti:sapphire [5,6], would tend to result in an underestimate of the lensing effect.

2. Characterization of thermal effects

Thermal effects in optical amplifiers have been characterized with various methods, all examining changes in a probe beam going through the gain medium after the optical pump excitation occurred. Measuring the modification in the probe beam size [5] or its divergence [6,12,13] are the most straightforward methods to obtain the thermal lensing value, but they are not sensitive enough for measuring residual thermal lensing that could still be present when cooling at cryogenic temperatures. The change in spatial phase of the probe beam experiencing the non-uniform radial temperature profile has also been measured with spatial interferometry in Ruby and Nd:glass [14–17] or with a Shack-Hartmann wavefront sensor in Ti:sapphire [5,6]. Measuring the spatial phase rather than just an effective focal length also brings the ability to quantify higher-order spatial phase aberrations. For Ti:sapphire, the only time-resolved measurements of the refractive index pump-induced changes were made with beam-deflection spectroscopy [18,19] and time-domain interferometry [20], which do not provide a value of the thermal lensing. Here we choose to use a version of spatially-resolved spectral interferometry, 2D Fourier Transform Spectral Interferometry [21–23] for our time-resolved measurement of the pump-induced spatial phase changes. The probe beam is the same infrared beam that is amplified in our 4-pass cryogenically cooled amplifier, and not a probe beam at a different wavelength, as is common practice [5,6,12–20]. This allows us to synchronize the probe beam with respect with the pump beam to easily perform time-resolved measurements.

In 1D spectral interferometry, the beam is split into two arms, which are then combined collinearly and sent to a spectrometer, with a delay on the order of picosecond between the arms to produce spectral fringes. In spatially-resolved spectral interferometry, the spatial information is preserved by using an imaging spectrometer incorporating a bidimensional CCD camera. As a result, a 2D interferogram S(r,λ) contains the difference in spatiospectral phase between both arms of the interferometer. While 2D spectral interferometry limits our study to one dimension of space, along the slit of the imaging spectrometer, it brings several advantages as compared to other methods. It is less sensitive to pointing fluctuations than spatial interferometry as the uncertainty in the spectral fringes is mainly caused by the time jitter of probe pulse. In our study, where we are only interested in the spatial information, 2D spectral interferometry offers the potential of higher sensitivity since there are many fringes of data for each spatial point, allowing an average over several wavelengths to improve the signal-to-noise ratio.

3. Experimental setup and data acquisition

The experimental measurement scheme is depicted in Fig. 1. The cylindrical Ti:sapphire crystal (16 mm in diameter and 15 mm long) is held in a copper cold finger with a collet mount that allows for rotation of the normal-incidence crystal for proper alignment of the crystal axes. The crystal collet is mounted in a cold finger below a reservoir of LN. This cryogenic chamber was designed in collaboration with Abbess Instruments and tested with a thermal load of 100 W, which gave an increase in temperature of 1.5 K from a base temperature of 77 K. This cryogenic cooled crystal is used inside a 4-pass amplifier pumped by four Nd:YLF lasers, each of them providing 20 W at 532nm in 200 ns. The amplifier layout is not changed to perform the thermal lensing measurement. A R=50 % @ 800 nm beamsplitter is inserted on the incoming beam before the multipass amplifier, creating an adjustable reference arm. The probe beam double passes the thermal lensing medium and is recombined with the reference beam at the beamsplitter.

 

Fig. 1. Experimental 2D spectral interferometry setup for measuring the thermal effect in the cryogenically-cooled kHz multipass amplifier. BS: 50 % beamsplitter. M: Flat mirror. DM: 45° dichroic mirror HR 532 nm HT 800nm. BW: Brewster windows.

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A Mach-Zehnder type interferometer would allow single pass measurements in the crystal but the double-pass design increases the amount of thermal focusing the infrared (probe) beam experiences thereby improving the sensitivity of the measurement. When the amplifier is used to amplify the beam in the 4-pass configuration, the pump beam and the infrared beam to amplify are both Gaussian in shape with a FWHM of 1.9 mm. To perform the thermal lensing measurement, the pump beam size is kept the same, but the size of the probe beam is increased with a magnifying telescope before the interferometer to a value of FWHM=5.2 mm, in order to sample a crystal area larger than the pump size. The pump-induced phase changes are measured for both polarizations of the probe beam by using a half-wave plate before the interferometer. Because of the birefringence of sapphire, the delay between the reference and crystal arm is slightly adjusted to get similar fringe spacing in both cases.

A 2D spectral interferogram S(r,λ) contains the difference in spatiospectral phase between the reference arm and the arm containing the crystal, including the static spatial phase aberrations as a result of crystal imperfections and the cryogenic chamber windows. To measure the deformations resulting only from the thermal effects, we first perform a reference measurement by saving a 2D spectral interferogram of the unpumped crystal. The spatiospectral phase obtained from this interferogram is then subtracted from subsequent measurements when the crystal is pumped. For each 2D spectral interferogram S(r,λ), the spatiospectral phase is deduced using a bi-dimensional Fourier transform extraction method [21–24]. As we are interested only in the spatial phase information (1D wavefront), the useful information is present in lineouts of the spatiospectral phase at each wavelength. In order to decrease the noise in the measurement, we use the available spectral dimension to perform an average of the spatial phase lineouts for several wavelengths in the spectrum. From the wavefront curvature measured at the spectrometer, we calculate the thermal lensing value with an ABCD matrix propagation to account for the crystal double-pass and the propagation distances between each pass and up to the spectrometer. To experimentally check the validity of this calculation, we place a f=10 m singlet lens in place of the crystal and obtain the 2D spectral interferogram and the spatiospectral phase that are shown in Fig. 2(a) and 2(b). Spatial phase lineouts are depicted in Fig. 2(c) for three different wavelengths (with an arbitrary constant phase shift of 1 radian at r=0 for better visibility) and the average spatial phase on twenty wavelengths is shown in Fig. 2(d). The extraction of the focal length from the quadratic fit of this measured spatial phase gives a value of f=10.4 m, validating the way we extract the thermal lensing value. Figures 2(c) and 2(d) also illustrate the averaging process of the spatial phase as a function of wavelength that is performed on each interferogram. In this case, the slight change in the spatial phase curvature for different wavelengths is due to the chromatic aberration (i.e. different focal length for each wavelength) of the singlet lens.

 

Fig. 2. (a) Spectral interferogram for a 10 m singlet lens in place of crystal. (b) Corresponding spatiospectral phase. (c) Lineouts of spatiospectral phase showing the spatial phase at different wavelengths. (d) Average of all spatial phase lineouts (solid line) and quadratic fit (dashed line).

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4. Results and discussion

The time-resolved study of the thermal lensing is accomplished by electronically varying the delay of the pump beam with respect to the probe beam, which then probes a different thermal state of the crystal for each pump-probe delay. When changing the delay of the pump, the probe beam itself experiences different level of amplification, hence reducing the contrast of the fringes in the pump region at maximum amplification. The contrast of the fringes always stayed between 0.5 to 0.8 for the whole range of delay. Figure 3 depicts the measured spatial phase as a function of the pump-probe delay τ for the σ-polarization (laser polarization perpendicular to the c-axis of the crystal), the zero delay corresponding to the probe beam being synchronized with the maximum of the pump fluorescence signal. The phase shift is the smallest for negative pump-probe delays τ, which corresponds to the thermal state of the crystal right before the pump pulse arrives and roughly one millisecond after the previous pump pulse. There is an important increase in the phase shift when the probe pulse arrives in the crystal at the same time and after the pump pulse (τ≥0). To obtain the lensing values from the phase measurements, we perform a fourth-order polynomial fit of the spatial phase on the pump region (from -2mm to +2mm), and deduce the focal lens from the quadratic term in r, as was done for our test lens previously. We obtain f=23 m in the σ-polarization for the strongest lensing, which corresponds to the pump-probe delay where the energy extraction is the best (τ=0.5 µs). When the probe arrives slightly before the pump, the lensing value drops to f=92 m (τ=-2 µs). Clearly a time-average measurement of the thermal lensing would give, in this case, an underestimate of the value at maximum extraction.

 

Fig. 3. Time evolution of the pump-induced spatial phase shift for σ-polarization

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The shape of the pump-induced spatial phase is evolving with time, and is also dependant on the polarization. As an illustration, Fig. 4 depicts some measured spatial phase that provide different ratio between the orders of the polynomial fit (see Figure caption for details on the value of the fit). Notably, the spherical aberration term (proportional to r⁴) is changing sign between these two cases and is adding to the focus term at maximum amplification (Fig. 4(b)). To evaluate the time evolution of the lensing effects, we choose to calculate for each pump-probe delay the peak-to-valley value of the spatial phase Δφ (difference between maximum and minimum values of the pump-induced spatial phase). By using this value, rather than the second order term of the fit, we quantify the amount of pump-induced phase shift independently of the shape of the spatial phase.

 

Fig. 4. Pump-induced spatial phase. (Blue crosses) Measurement. (Red solid line) Fourth-order polynomial fit: φ(ρ)=φ₂ ρ² + φ₃ ρ³ + φ₄ ρ⁴ with ρ=r/r_max (r_max=2mm). (a) σ-polarization at τ=-2 µs. φ₂=0.62 φ₃=0.03 φ₄=-0.19 (b) π-polarization at τ=0.5 µs φ₂=0.4 φ₃=0.11 φ₄=0.5

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Figure 5 shows the time-dependence of this phase difference for both polarizations (solid and dotted lines). As Fig. 3 indicated, in the case of the σ-polarization, there is a strong increase of the phase shift followed by a relaxation with a time constant of 5 µs (dashed line in Fig. 5). Following the analysis of Lausten [11], the thermal time constant is approximately τ _th=r ²/κα ² ₀₁, where r is the pump radius and α₀₁ is the first zero of the zeroth-order Bessel function. The material parameter that controls the cooling rate is the thermal diffusivity coefficient κ=K/c_p, where K is the thermal conductivity and c _p is the specific heat. The changes in K and c _p with cooling lead to an increase in κ by almost 300x from 300K to 77K. At a crystal temperature of 77K, τ _th ~45µs, whereas at 300K, τ _th ~12ms. The cooling timescale at 77K is much less than the pump cycle time, but much longer than our measurement range. The observed fast relaxation has a timescale more commensurate with the fluorescence decay of the Ti³⁺ ions [20,22], which is 3.85 µs at 77K [25]. A similar signal has been observed in photothermal spectroscopy deflection experiments, and has been attributed to a photoelastic effect in the TiO₆ complex in its excited state [20–22]. Our measured timescale of 5 µs indicates that the relaxation is primarily non-thermal; consequently a better cooling rate would not decrease drastically the pump-induced lensing strength.

 

Fig. 5. Phase difference as a function of pump-probe delay. Dashed line : π-polarization. Solid line : σ-polarization. Dotted line : Exponential fit for σ-polarization.

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The non-thermal mechanism of the pump-induced phase shift is known to exhibit a strong anisotropic behavior [20,22]. This is consistent with our observations of the π-polarization signal (dotted line in Fig. 5); this is the polarization of interest when amplifying, as it supplies the most gain. While the faster transient signal is not noticeable for this polarization, an increase in thermal lensing strength is nevertheless visible around the zero delay (f=83m at τ=-2 µs and f=62 m at τ=0.5 µs). In that case the relaxation occurs on a much longer timescale than the 8 µs measurement duration, but still faster than the intra-pulse time of 1 ms. The existence of an increase in thermal lensing at the pump arrival, indicates that there is some cooling between laser shots, as predicted by our estimates. Concerning the geometrical stability of our multipass ring amplifier, a 62 m thermal lensing value (the focal length measured at maximum energy extraction) is low enough so that it does not negatively influence the amplification process.

In conclusion, we measured the thermal lensing effect in Ti:sapphire cooled at liquid Nitrogen temperature. By performing a time-resolved characterization of the thermal effects, we have demonstrated that an average measurement of the thermal lensing would give a false evaluation of the thermal lensing strength, which is found to be higher when the amplified beam is timed for maximum amplification. These measurements also demonstrate the sensitivity of 2D spectral interferometry as a characterization method for transient phase shifts such as the pump-induced spatial phase. In our system, the measured thermal lensing value shows efficient removal of detrimental focusing effects due to the thermal load. The non-thermal (photoelastic) processes at these low temperatures dominate the time evolution of the phase shift induced by the optical pumping, indicating that no further cooling is required.