2. Theory

For a high-energy CPA system, a large aperture of the laser crystal is necessary for saturable pumping or damage flux existing. Unfortunately, PL will be obtained more readily with the enlarging of the aperture, thereby reducing the amplification efficiency of the system. In this section, the principal condition of parasitic lasing is discussed, and it is shown that splicing technology can suppress the parasitic lasing theoretically. The possible issues resulting from the splicing Ti:sapphire crystal technology, and the influences of these issues, such as spectral modulation, the temporal profile of the compressed pulse, and the spatial distribution, are also discussed.

The threshold of parasitic lasing depends on the gain of the crystal. The gain of the crystal based on the amplifier (which in this case is Ti:sapphire) can be calculated with inverted population density, which can be obtained by using the Frantz–Nodvik (F–N) equation^{[Reference Frantz and Nodvik7]}:

(1) $$\begin{eqnarray} \label {eq1} G &=& \exp \left \{\frac {W_{0}}{2} [\exp (-\alpha _{p} \cdot z) + \exp (-\alpha _{p} \cdot (L_{z} -z))] \right .\nonumber \\ && \left .\times \left (\frac {\alpha _{p} }{\hslash \cdot \omega _{p}}\right ) \cdot 2D\cdot 3e^{-19}\right \}, \end{eqnarray}$$

where $G$ is the gain of the crystal, $W_{0}$ is the density of the pump energy, $\alpha _{p}$ is the absorption coefficient, $L_{z}$ is the thickness of the Ti:sapphire, $\hslash ,\omega _{p}$ are the Planck constant and the angular frequency, $D$ is the clear aperture, and $z$ isthe distance from the surface to the inner of the crystal. The gain of the transverse ASE is the highest on the surface, and the parasitic lasing is also the strongest; thus the situation of parasitic lasing on the surface is used to represent the crystal^{[Reference Ertel, Hooker, Hawkes, Parry and Collier8]}; in this model, $z=0$ . Equation (1) indicates that a stronger pump generates higher $G$ , and the threshold of the parasitic lasing can be more easily reached. The equation $R_{1}\cdot R_{2}\cdot G_{\mathrm{ASE}}=1$ can be used as the criterion for parasitic lasing, where $R_{1}$ and $R_{2}$ are the reflectivities of the crystal side surfaces. $G_{\mathrm{ASE}}$ is the gain at the crystal surface, which also indicates the threshold of the parasitic lasing previously mentioned. Then Equation (1) can be expressed as follows:

(2) $$\begin{eqnarray} \label {eq2} && R_{1}\cdot R_{2} \cdot \exp \left \{\frac {W_{0}}{2}[\exp (-\alpha _{p} \cdot z) + \exp (-\alpha _{p} \cdot (L_{z} -z))] \right .\nonumber \\ && \left .{\quad \times } \left (\frac {\alpha _{p} }{\hslash \cdot \omega _{p}}\right )\cdot 2\cdot (2\cdot r)\cdot 3e^{-19}\right \} = 1, \end{eqnarray}$$

where $2r=D$ . Equation (2) indicates that there should be a threshold of pump energy with a special diameter.

The result is shown by the dashed curve in the Figure 1.

Figure 1. Theoretic relationship between the radius and the threshold of pump energy with a single crystal (dashed line), $2\times 2$ splicing crystals (dot line), and $4\times 4$ splicing crystals (solid line). The absorption coefficient is 0.94.

The dashed curve from Figure 1 indicates that the density of the pump energy should be decreased to avoid the occurrence of parasitic lasing, even if the aperture is increased. A common method is cladding with refractive-index-matched liquid or direct absorber coating on the side surfaces^{[Reference Liang, Leng, Wang, Li, Lin, Zhao, Jiang, Lu, Hu, Zhang, Lu, Yin, Jiang, Lu, Wei, Zhu, Li and Xu1]} of the crystals to decrease the reflectivity. This method has been used widely in PW-level systems. However, in a system with higher peak power output, the aperture of the crystal is enlarged to avoid damage. Equation (2) shows that the threshold of PL would be decreased^{[Reference Chu, Liang, Yu, Xu, Xu, Ma, Lu, Liu, Leng, Li and Xu4]}.

Meanwhile, the transverse gain can be decreased and the threshold can be increased significantly when the optical path length of the ASE can be cut off, which can be achieved with splicing technology. Figure 2 shows that the larger crystal can be composed of $2\times 2$ , $4\times 4$ , or more crystals. The gap can be filled with an index-matched material, liquid or solid; and the material hashigh absorption coefficient about ASE avoiding the reflection of ASE at the side face of Ti:sapphire crystal. In this way, the threshold can be increased, and the result is shown in Figure 1, which is calculated from Equation (2). The $2\times 2$ splicing crystals (the dot curve) and $4\times 4$ splicing crystals (the dashed curve) have a higher threshold than the single crystal (the solid curve). For example, when the radius is 5 cm, the threshold is $6.68\ \mathrm{J\ cm}^{-2}$ for the $4\times 4$ splicing crystals; the threshold is $4.72\ \mathrm{J\ cm}^{-2}$ for the $2\times 2$ splicing crystals, and the threshold is $2.36\ \mathrm{J\ cm}^{-2}$ for the single crystal.

Figure 2. Scheme of splicing crystals.

Figure 1 also indicates that this technology has the advantage of short growing period and low cost. For example, in a 10 PW CPA system, when the pulse width of the extraction laser is 30 fs, the output energy would be 300 J. Considering the efficiency of the compressor and the conversion efficiency from pump to extraction, the pump energy of the final amplifier will be more than 850 J. With the ordinary index-matched material having a reflectivity index of 0.048%, the aperture is much more than for the largest Ti:sapphire that has been reported. Even when better index-matched materials are used, the aperture would still be more than 300 mm. However, using the splicing technology of the Ti:sapphire crystal, only four 120 mm $\times $ 120 mm crystals are needed to compose a large-aperture splicing crystal with clear aperture of 240 mm.

In the period of the CPA system running, the accurate threshold of the PL, in our experience, is always lower than the results previously shown, and the calculation process is also more complicated because of the influence of some uncertain factors, such as the beam homogeneity, and the delay jitter between the pump laser and the extraction laser. However, these factors can hardly affect the conclusion of this section, because the splicing crystals have similar effects to those of the single crystal. For example, the threshold of PL with the single crystal decreased from $2.36\ \mathrm{J\ cm}^{-2}$ to $1.18\ \mathrm{J\ cm}^{-2}$ , while the threshold of the $2\times 2$ splicing crystals decreased from $4.72\ \mathrm{J\ cm}^{-2}$ to $2.36\ \mathrm{J\ cm}^{-2}$ under the same conditions.

According to the analysis above, crystal splicing technology has the potential to increase the threshold of PL in a CPA system. However, as previously mentioned, possible issues should be considered, including spectral modulation, stretching or splitting of temporal profile, and the sidelobe in spatial domain which may worsen the consistency of the focusing spot in the spatial domain or time domain compared with the single crystal amplifier. Furthermore, the solving of these issues is limited to inevitable errors in the process of crystal splicing, such as the different thicknesses of the crystals, angle errors of the optical axis, and the gap of the splicing crystal. Thus, details of these issues will be discussed theoretically in what follows.

(i) Spectral modulation and the temporal profile of the compressed pulse.

Unlike the single crystal amplifier, the errors mentioned before can be hardly corrected by manual adjustment during the period of the system running. For example, as the $2\times 2$ splicing crystals have four crystals and four optical axes correspondingly, if one of the crystals is adjusted and its optical axis is corrected, the errors of the other crystals may be worsened, which would generate spectral modulation and a sidelobe pulse in the time domain as well as thickness differences. For these reasons, these kinds of influence should be calculated.

The Ti:sapphire crystal is a uniaxial crystal, and the grating in the compressor system is sensitive to the laser pulse polarization^{[Reference Kong, Jin, Liu, Chen, Guan, He, Du and He9]}. Therefore, a Ti:sapphire crystal and grating are equivalent to a birefringent plate and a polarizer, and the angle errors (Figure 3) would modulate the spectra of the laser^{[Reference Lu, Li, Leng, Wang, Zhang, Liang, Li and Xu10, Reference Leng, Lin, Wang, Jiang, Tang and Xu11]}. The equation for the spectral modulation is as follows:

(3) $$\begin{eqnarray} \label {eq3} T(\lambda ) &=& 1 - \sin ^{2} (2\phi ) \frac {n_{o}^{4} -n_{o}^{2} \cos ^{2}(\theta )} {(n_{o}^{2} -\cos ^{2}\phi \cos ^{2}\theta )^{2}} \nonumber \\ && \times \, \sin ^{2} \left \{\frac {\pi tn_{e} [1+\cos ^{2}\theta \cos ^{2}\phi /n_{e}^{2} -\cos ^{2}\theta \cos ^{2}\phi /n_{o}^{2} ]} {\lambda [1-\cos ^{2}\theta \sin ^{2}\phi /n_{e}^{2} -\cos ^{2}\theta \cos ^{2}\phi /n_{o}^{2} ]^{1/2}} \right .\nonumber \\ && - \,\left .\frac {\pi tn_{o}}{\lambda [1-\cos ^{2}\theta /n_{o}^{2} ]^{ {1/2}}}\right \}, \end{eqnarray}$$

where $T(\lambda )$ is the transmittance related to the wavelength $(\lambda )$ ; $n_{o}$ , $n_{e}$ are the principal refractive indices; and $\theta $ , $\varphi $ are the angle errors.

Figure 3. Schematic of $\theta $ and $\varphi $ .

Equation (3) shows that the spectral modulation is influenced by three factors: the thickness of the crystal, and the angle errors $\theta $ and $\varphi $ ; different values of these factors would generate different situations of spectral modulation. Figure 4a shows the situation of spectral modulation at $\theta =10^{\circ }$ , $\varphi =18.5^{\circ }$ ; the thickness of the crystal is 10 mm, and the beam single-passes the crystal.

Figure 4. (a) Situation of spectral modulation (single-pass) at $\theta =10$ , $\varphi =18.5$ ; the thickness of the crystal is 10 mm. (b) Sidelobes generated by spectral modulation.

As we know, the spectral modulation changes the temporal profile of the compressed pulse. Thus, using Fourier transformation, the temporal profile from the spectral modulation can be calculated, as shown in Figure 4a shows. Figure 4b is then obtained, indicating that two sidelobes are generated and the intensity is 10% of the main pulse^{[Reference Preuuss and Gole12]}.

Another inevitable error that can influence the temporal profile of the compressed pulse is the thickness difference, show in Figure 5. Different thicknesses would stretch or even split the pulse duration. For example, in the splicing crystal CPA system with a duration of 30 fs, only about 5–6 mm of thickness differences (the influence of the GDD are ignored) can stretch the pulse duration about 60 fs, or even cause it to be split. This phenomenon is a very serious problem for ultra-short lasers, in which the pulse duration is of fs order. Thus, this error should be given special attention in the process of splicing the crystals.

Figure 5. Schematic of the influence of thickness difference on the temporal profile.

(ii) The spatial distribution of the focused point.

The existence of a gap for the compression of parasitic lasing is inevitable, which may affect the spatial distribution. This phenomenon can be explained easily by diffraction theory^{[Reference Born and Wolf13]}. In fact, the result can be easily obtained through FFT (fast Fourier transformation) using Matlab. Figure 6 shows the simulated result, which indicates that the gap would generate a series of sidelobes in the frequency domain. In this case, the second intensity sidelobe is about 10% of the main peak; the width of the gap is 10% of the beam diameter.

3. Experiment

As we discussed before, some inevitable errors in the processing of crystal splicing generate possible issues, and the issues are solved in two aspects [(i) and (ii)]. However, this study focuses on the influence of these possible issues, that is, whether these inevitable errors can be limited to be small enough to be ignored using existing crystal processing technology. Thus, in this study, some experiments have been carried out to test the influence of these issues with the two aspects in theory, and the parameters used below are processed with ordinary accuracy. Notably, the experiment is limited to the aperture of the splicing crystals (10 mm); an experiment about suppressing PL has not been carried out.

Figure 6. Theoretical relationship between the spot in the near field and the spot in the far field.

The experiment was carried out in a typical CPA system based on Ti:sapphire. The scheme is shown in Figure 7a–c. The experiment shows (a) measurement of the energy, spot in the near and the far fields, and the spectra; (b) measurement of the autocorrelator trace; and (c) a photo of $2\times 2$ spliced Ti:sapphire $7.5\times 7.5\ \mathrm{mm}^{2}$ in size, 10 mm in thickness, and an absorptivity of 80% for 532 nm. The extraction beam passed through the splicing crystals twice when the crystal was pumped. The wavelength of the pump beam is 532 nm, and that of the extraction beam is from 760 nm to 830 nm. A spectrometer was used to measure the situation of the spectral modulation; a calorimeter was used to measure the energy of the amplified beam, and a CCD was used to observe the spatial distribution. Finally, a Glan prism was used to replace the gating to measure the temporal profile of the compressed pulse with autocorrelation and CCD.

Figure 7. Scheme of experiment: (a) measurement of the energy, spot in the near and the far fields, and spectra; (b) measurement of the autocorrelator trace; (c) photo of the $2\times 2$ splicing crystals.

The diameters of the extraction and pump beams are 10 mm; the extraction beam is 0.5 mJ and the pump is 1.7 J; we obtained 1.1 mJ output for a single pass and 3.1 mJ for a double pass. Considering that the absorptivity of the crystal without coating is 80% for the pump, this result is in accordance with the theoretically expected result.

Figure 8. Spectrum of the seed: the dashed curve is the original spectrum; the dotted curve is the spectrum amplified by a single pass; the solid curve is the spectrum amplified by a single pass.

(i) Experiment of the spectral modulation and the temporal profile.

The calculation result shows that the angle errors generate spectral modulation. However, when the angle errors are less than $1^{\circ }$ (ordinary accuracy), this kind of spectral modulation can be ignored. Figure 8 shows the spectra of the whole beam measured by the spectrometer, and Figure 9 shows the spectra of each beam passing through the four smaller crystals of the splicing crystals.

Figure 9. Spectrum of the beam passed through the smaller crystals of the $2\times 2$ splicing crystals.

In Figure 8, the dashed curve indicates the original spectrum of the extraction beam, the dotted curve indicates the extraction beam single passed through the crystal and the solid curve is that for a double pass. The spectra did not have any obvious change, and the spectra of each beam that passed through the four smaller crystals of the splicing crystals in Figure 8 also support this point. The solid line in Figure 9 indicates a single pass, and the dotted line indicates a double pass.

The temporal profile of the compressed pulse can also be measured in the experiment, and the result is shown in Figure 10.

The result in Figure 10 was measured by autocorrelation and CCD. The width of the curve respects the pulse duration^{[Reference Salin, Georges, Roger and brun14]}: (a) is the autocorrelation trace of the beam that did not pass through splicing crystals, and (b) is the autocorrelation trace of the beam that passed through the splicing crystals. The pulse duration is 63 fs in (a) and 62 fs in (b), respectively. The results show that the pulse duration did not stretch.

Figure 10. Autocorrelation trace in the experiment: (a) original trace; (b) the trace of the splicing crystals.

(ii) Experiment of the spatial distribution of the focused point.

There are two problems about the spatial distribution: the diffraction effect in the near field and the spatial sidelobes in the far field. Figure 11a shows the diffraction of the extraction beam that passed through the splicing crystals in the near field, and the width of the gap is about 10% of the aperture. The diffraction effect is intense, which may damage the optical elements. However, a soft-edge aperture can solve this problem. Figure 11b shows the spot with the soft-edge aperture, and the shape is $+$ , as Figure 11c shows. This figure indicates that the diffraction effect significantly decreased.

Figure 11. Spot of the extraction beam in the near field: (a) with soft-edge aperture; (b) without edge aperture, and (c) schematic of the soft-edge aperture.

Figure 12. Spot of the extraction beam in the far field: (a) did not pass through the crystal, (b) passed through the crystal without the soft-edge aperture, and (c) passed through the crystal with the soft-edge aperture.

The spatial distribution in the far field is also measured. Figure 12a shows the original spot in the far field in which the beam did not pass through the crystal, (b) shows the spot single-passed through the splicing crystal without the aperture, and (c) shows the spot single-passed through the splicing crystal with a soft-edge aperture. These results show that, in the far field, the spot that passed through the crystal (Figure 12b) has similar spot distribution to the original spot distribution (Figure 12a), where the gap is about the 10% of the width of the clear aperture. Figure 12c shows that the sidelobes are generated because the soft-edge aperture is too wide (about 20% of the clear aperture).