1. Introduction
The chirped pulse amplification (CPA) technique[Reference Strickland and Mourou1, Reference Mourou, Tajima and Bulanov2] is commonly used to build high-power laser systems for various experiments and applications, such as laser-driven plasma accelerators and fast ignition in inertial confinement fusion. Peak laser powers of up to 1 PW 
$(1\ \text{PW}=10^{15}\ \text{W})$ are now achievable, and are expected to be boosted to the EW level 
$(1\ \text{EW}=10^{18}\ \text{W})$ in the near future[3].
In CPA ultrashort laser systems, the dispersion elements include stretcher, compressor and amplifier material[Reference Cheriaux, Rousseau, Salin and Chambaret4–Reference Yang, Liu, Wang, Xie, Sun, Xu, Gao, Guo and Lin10]. The compressor can completely compensate for group delay dispersion (GDD) and partly compensate for third-order dispersion (TOD), which are generated by the stretcher and amplifier material[Reference Treacy6, Reference Yang, Guo, Xie, Zhang, Sun, Gao, Li and Lin7]. However, compensating for the residual high-order dispersion is extremely important in the entire system to obtain the shortest pulse duration and highest pulse contrast, particularly for ultrashort pulses (
${<}50\ \text{fs}$)[3, Reference Backus, Durfee III, Murnane and Kapteyn5]. Thus, setting up an independent dispersion compensation element is necessary to compensate for the residual high-order dispersion of the system.
Ray-tracing sketch for a pair of identical isosceles prisms in a parallel face configuration.
The prism pair is an important element in dispersion compensation. In 1984, Fork et al. indicated that a pair of prisms could provide negative GDD and TOD, which are suitable for the dispersion compensation element[Reference Fork, Martinez and Gordon11–Reference Cojocaru14]. However, the equations used by Fork to evaluate the GDD and TOD of a pair of prisms are only for tip-to-tip propagation in the prisms and do not consider the effect of the spectral bandwidth. Another approach to evaluate the dispersion of a pair of prisms was presented by Arissian et al. [Reference Arissian and Diels15, Reference Corral, Aguilar and Mejia16]; the technique is based on accurate calculation of the optical path of the pair of prisms. However, the equations employed by Arissian et al. to evaluate GDD and TOD of the pair of prisms do not consider limited spectral bandwidth and continuous TOD compensation, which are very important for dispersion compensation. For a particular ultrashort laser system, dispersion compensation can be considered to be quantitatively meaningful only under the premise of a certain spectral bandwidth; otherwise, this method is qualitative. For example, for a 30 fs ultrashort laser system, the spectral bandwidth of dispersion compensation should remain greater than 140 nm in order to obtain a short compressed pulse and a high pulse contrast ratio. But, in practical applications, the possible spectral bandwidth is only 30 nm when the spectral bandwidth is not considered in the dispersion compensation. In that time, a short compressed pulse (e.g., 30 fs) cannot be obtained. Therefore, it is very important to consider the spectral bandwidth in dispersion compensation, particularly in practical applications.
In this study, we focus on independent and continuous TOD compensation using a pair of prisms. The dispersion of the pair of prisms is analyzed in detail using a ray-tracing method operating at other than tip-to-tip propagation of the prisms. The variations of GDD and TOD for the pair of prisms are calculated with respect to the incident position and the separation between the prisms. A pair of prisms can provide a wide range of independent and continuous TOD compensation. The effect of residual TOD (RTOD) on the pulse contrast ratio and pulse duration is calculated. The RTOD not only worsens the pulse contrast ratio, but also greatly increases the pulse duration to the hundreds of femtosecond range for a tens of femtosecond pulse, even when small RTOD is employed. These phenomena are helpful in compensating for residual high-order dispersion and in understanding its effect on the pulse contrast ratio in ultrashort laser systems.
2. Model
Figure 1 shows the ray-tracing sketch for a pair of identical isosceles prisms in a parallel face configuration. The blue curve represents the shortest wavelength trajectory and the black curve represents any single wavelength trajectory after dispersion through the prism pairs.
Based on Figure 1, the optical path of any one wavelength is given by
$$\begin{eqnarray}\displaystyle p & = & \displaystyle 2[n(BC+DE)+CD+EF]\nonumber\\ \displaystyle & = & \displaystyle 2\left\{n\left[\frac{h_{1}\sin {\it\alpha}}{\cos {\it\gamma}_{2}({\it\lambda})}+\frac{h_{2}\sin {\it\alpha}}{\cos {\it\gamma}_{1}({\it\lambda})}\right]+\frac{d}{\cos i_{2}({\it\lambda})}\right.\nonumber\\ \displaystyle & & \displaystyle +\,\left.\left[L-\frac{h_{2}\cos {\it\gamma}_{2}({\it\lambda})}{\cos {\it\gamma}_{1}({\it\lambda})}\right]\sin i_{1}\right\}\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle h_{2}=h_{s}\!+\!d\tan i_{2}({\it\lambda}_{s})\!-\!d\tan i_{2}({\it\lambda})\!+\!\frac{h_{1}\sin {\it\alpha}}{\cos {\it\gamma}_{2}({\it\lambda}_{s})}\!-\!\frac{h_{1}\sin {\it\alpha}}{\cos {\it\gamma}_{2}({\it\lambda})} & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$ where 
${\it\lambda}$ is the wavelength under consideration, 
${\it\lambda}_{s}$ is the shortest wavelength, 
$n({\it\lambda})$ is the refractive index of the prism, 
$h_{1}$ is the distance of the beam input trajectory to the apex of the prism, 
$h_{2}$ is the distance of the trajectory at a particular wavelength to the apex of the second prism, 
$h_{s}$ is the distance of the trajectory of the shortest wavelength to the apex of the second prism, 
$L$ is the waist length of the isosceles prisms, 
${\it\alpha}$ is the apex angle of the prism, 
$i_{1}$ is the incidence angle, 
$i_{2}({\it\lambda})$ is the exit angle of the prism, 
${\it\gamma}_{1}({\it\lambda})$ is the refraction angle in the incidence plane, and 
${\it\gamma}_{2}({\it\lambda})$ is the incidence angle in the exit plane of the first prism at a particular wavelength.
At the central wavelength, GDD and TOD change with 
$h_{1}$. Material: 
$\text{CaF}_{2}$; simulation parameters: 
$i_{1}=69.9^{\circ }$, 
${\it\lambda}_{0}=808\ \text{nm}$, 
${\rm\Delta}{\it\lambda}=140\ \text{nm}$, 
${\it\lambda}_{s}=738\ \text{nm}$, 
$d=800\ \text{mm}$, and 
$h_{s}=1\ \text{mm}$.
To ensure high transmission efficiency of the prism pairs, Brewster angle incidence is adopted. The values of these angles can be calculated using 
$i_{1}=\arctan [n({\it\lambda}_{0})]$, 
${\it\gamma}_{1}({\it\lambda})=\arcsin [\frac{\sin i_{1}}{n({\it\lambda})}]$, 
${\it\gamma}_{2}({\it\lambda})={\it\alpha}-{\it\gamma}_{1}({\it\lambda})$, 
$i_{2}({\it\lambda})=\arcsin$ 
$[n({\it\lambda})\sin {\it\gamma}_{2}({\it\lambda})]$, where 
${\it\lambda}_{0}$ is the central wavelength.
The total phase for the pair of prisms yields
$$\begin{eqnarray}{\it\phi}({\it\omega})=\frac{{\it\omega}p}{c}\end{eqnarray}$$ where 
${\it\omega}$ is the angular frequency and 
$c$ is the speed of light in vacuum.
According to Equation (3), the dispersions can be divided into various orders, such as the second-, third- and fourth-order dispersions (i.e., GDD, TOD and FOD, respectively).
$$\begin{eqnarray}\displaystyle \text{GDD}|_{{\it\omega}_{0}} & = & \displaystyle \left.\frac{{\it\lambda}^{3}}{2{\it\pi}c^{2}}\frac{d^{2}P}{d{\it\lambda}^{2}}\right|_{{\it\lambda}_{0}}\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \text{TOD}|_{{\it\omega}_{0}} & = & \displaystyle \left.-\frac{{\it\lambda}^{4}}{4{\it\pi}^{2}c^{3}}\left(3\frac{d^{2}P}{d{\it\lambda}^{2}}+{\it\lambda}\frac{d^{3}P}{d{\it\lambda}^{3}}\right)\right|_{{\it\lambda}_{0}}\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \text{FOD}|_{{\it\omega}_{0}} & = & \displaystyle \left.\frac{{\it\lambda}^{5}}{8{\it\pi}^{3}c^{4}}\left(12\frac{d^{2}P}{d{\it\lambda}^{2}}+8{\it\lambda}\frac{d^{3}P}{d{\it\lambda}^{3}}+{\it\lambda}^{2}\frac{d^{4}P}{d{\it\lambda}^{4}}\right)\right|_{{\it\lambda}_{0}}\end{eqnarray}$$
${\it\omega}_{0}$ is the central angular frequency.3. Numerical results
According to the model constructed in Section 2, we can calculate the GDD and TOD of the prisms. The influences of the distances 
$h_{1}$, 
$h_{s}$, and 
$d$ on GDD and TOD are determined. We then discuss independent and continuous TOD compensation by changing the distances 
$h_{1}$ and 
$d$ (or 
$h_{2}$ and 
$d$) simultaneously. The effect of RTOD on the pulse contrast ratio and pulse duration is also discussed.
During the simulation, we adopted a typical Sellmeier series equation to describe glass material dispersion[Reference Mourou, Tajima and Bulanov2]. The Sellmeier series equation is written as
$$\begin{eqnarray}n^{2}=1+\frac{B_{1}{\it\lambda}^{2}}{{\it\lambda}^{2}-C_{1}}+\frac{B_{2}{\it\lambda}^{2}}{{\it\lambda}^{2}-C_{2}}+\frac{B_{3}{\it\lambda}^{2}}{{\it\lambda}^{2}-C_{3}}\end{eqnarray}$$ where the wavelength 
${\it\lambda}$ is expressed in 
${\rm\mu}\text{m}$. The Sellmeier coefficient values for 
$B_{1},B_{2},B_{3},C_{1},C_{2},C_{3}$ are shown in Table 1.
3.1. Influence of distances 
$h_{1}$ and $h_{s}$ on dispersion
At the central wavelength, GDD and TOD change with distance 
$h_{1}$ or 
$h_{s}$ (Figures 2 and 3, respectively). GDD changes from negative to positive and TOD reduces rapidly when 
$h_{1}$ or 
$h_{s}$ increases. For instance, when 
$h_{1}<9.84\ \text{mm}$, the value of GDD is reduced rapidly (Figure 2). However, when 
$h_{1}>9.84\ \text{mm}$, the value of GDD increases rapidly. Thus, 
$h_{1}=9.84\ \text{mm}$ is the critical value; at 
$h_{1}=9.84\ \text{mm}$, the value of GDD is zero and the value of TOD is 
$-1135\ \text{fs}^{3}$. When 
$h_{s}<11.7\ \text{mm}$, the value of GDD reduces rapidly. However, when 
$h_{s}>11.7\ \text{mm}$, the value of GDD increases rapidly. Thus, 
$h_{s}=11.7\ \text{mm}$ is the critical value; at 
$h_{s}=11.7\ \text{mm}$, the value of GDD is zero and the value of TOD is 
$-1134\ \text{fs}^{3}$.
Figures 2 and 3 show that the simultaneous occurrence of GDD and TOD with the same or different signs is possible when 
$h_{1}$ or 
$h_{s}$ changes. Therefore, the prisms can compensate for GDD and TOD with the same signs, as well as GDD and TOD with different signs, by changing 
$h_{1}$ or 
$h_{s}$.
In the simulation shown in Figure 2, the following parameters are employed: 
$i_{1}=69.9^{\circ }$, 
${\it\lambda}_{0}=808\ \text{nm}$, 
${\rm\Delta}{\it\lambda}=140\ \text{nm}$, 
${\it\lambda}_{s}=738\ \text{nm}$, 
$d=800\ \text{mm}$, and 
$h_{s}=1\ \text{mm}$; material: 
$\text{CaF}_{2}$.
In the simulation shown in Figure 3, the following parameters are employed: 
$i_{1}=69.9^{\circ }$, 
${\it\lambda}_{0}=808\ \text{nm}$, 
${\rm\Delta}{\it\lambda}=140\ \text{nm}$, 
${\it\lambda}_{s}=738\ \text{nm}$, 
$d=800\ \text{mm}$, and 
$h_{1}=1\ \text{mm}$; material: 
$\text{CaF}_{2}$.
At the central wavelength, GDD and TOD change with 
$h_{s}$. Material: 
$\text{CaF}_{2}$; simulation parameters: 
$i_{1}=69.9^{\circ }$, 
${\it\lambda}_{0}=808\ \text{nm}$, 
${\rm\Delta}{\it\lambda}=140\ \text{nm}$, 
${\it\lambda}_{s}=738\ \text{nm}$, 
$d=800\ \text{mm}$, and 
$h_{1}=1\ \text{mm}$.
At the central wavelength, GDD and TOD change with 
$d$. Material: 
$\text{CaF}_{2}$; simulation parameters: 
$i_{1}=69.9^{\circ }$, 
${\it\lambda}_{0}=808\ \text{nm}$, 
${\rm\Delta}{\it\lambda}=140\ \text{nm}$, 
${\it\lambda}_{s}=738\ \text{nm}$, 
$h_{1}=6\ \text{mm}$, and 
$h_{s}=1\ \text{mm}$.
3.2. Influence of distance $d$ on dispersion
At the central wavelength, GDD and TOD change with 
$d$ (Figure 4). GDD and TOD change simultaneously from positive to negative with increasing 
$d$. For instance, when 
$d<498.2\ \text{mm}$, the value of GDD reduces rapidly. However, 
$d>498.2\ \text{mm}$, the value of GDD increases rapidly. Thus 
$d=498.2\ \text{mm}$ is the critical value; at 
$d=498.2\ \text{mm}$, GDD is zero and TOD is 
$-753\ \text{fs}^{3}$.
During the simulation, the following parameters are employed: 
$i_{1}=69.9^{\circ }$, 
${\it\lambda}_{0}=808\ \text{nm}$, 
${\rm\Delta}{\it\lambda}=140\ \text{nm}$, 
${\it\lambda}_{s}=738\ \text{nm}$, 
$h_{1}=6\ \text{mm}$, and 
$h_{s}=1\ \text{mm}$; material: 
$\text{CaF}_{2}$.
At the central wavelength, TOD changes with 
$h_{1}$ and 
$d$ when 
$\text{GDD}=0$. Simulation parameters: 
$i_{1}=69.9^{\circ }$, 
${\it\lambda}_{0}=808\ \text{nm}$, 
${\rm\Delta}{\it\lambda}=140\ \text{nm}$, 
${\it\lambda}_{s}=738\ \text{nm}$, and 
$h_{s}=1\ \text{mm}$; material: 
$\text{CaF}_{2}$.
At the central wavelength, TOD changes with 
$h_{1}$ and 
$d$ when 
$\text{GDD}=0$. Simulation parameters: 
$i_{1}=60.6^{\circ }$, 
${\it\lambda}_{0}=808\ \text{nm}$, 
${\rm\Delta}{\it\lambda}=140\ \text{nm}$, 
${\it\lambda}_{s}=738\ \text{nm}$, and 
$h_{s}=1\ \text{mm}$; material: SF10.
Effect of RTOD on the pulse contrast ratio. (b) is a magnified section of (a).
3.3. TOD independent and continuous compensation
Based on the analyses in Sections 3.1 and 3.2, we can conclude that the prisms can independently compensate for the TOD of the system by selecting the appropriate values of 
$h_{1}$ and 
$d$ or 
$h_{s}$ and 
$d$. At the central wavelength, TOD changes with 
$h_{1}$ and 
$d$ when 
$\text{GDD}=0$ (Figures 5 and 6). The figures indicate group-specific values 
$(h_{1},d)$ that correspond to a particular TOD compensation value. Thus, the RTOD of the laser system is compensated by adjusting distances 
$h_{1}$ and 
$d$ simultaneously.
In the simulation shown in Figure 5, the following parameters are employed: 
$i_{1}=69.9^{\circ }$, 
${\it\lambda}_{0}=808\ \text{nm}$, 
${\rm\Delta}{\it\lambda}=140\ \text{nm}$, 
${\it\lambda}_{s}=738\ \text{nm}$, and 
$h_{s}=1\ \text{mm}$; material: 
$\text{CaF}_{2}$. In the simulation shown in Figure 6, the following parameters are employed: 
$i_{1}=60.6^{\circ }$, 
${\it\lambda}_{0}=808\ \text{nm}$, 
${\rm\Delta}{\it\lambda}=140\ \text{nm}$, 
${\it\lambda}_{s}=738\ \text{nm}$, and 
$h_{s}=1\ \text{mm}$; material: SF10. By comparison, we find that the 
$\text{CaF}_{2}$ prism pair can provide a smaller TOD, which is suitable for small RTOD compensation (Figure 5). By contrast, the SF10 prism pair can provide a larger TOD, which is suitable for large RTOD compensation (Figure 6) in the CPA laser system. The 
$\text{CaF}_{2}$ prisms are suitable for compensating RTOD under 
$10^{3}\ \text{fs}^{3}$, whereas the SF10 prisms are suitable for compensating RTOD under 
$10^{4}\ \text{fs}^{3}$.
Effect of RTOD on pulse duration. The figure on the right is a magnified section of the figure on the left.
3.4. Effect of RTOD on pulse contrast ratio and pulse duration
To demonstrate the effect of RTOD on the pulse contrast in CPA lasers, we employed a 30 fs compressed pulse laser system as a example. The central wavelength is 
${\it\lambda}_{0}=808\ \text{nm}$ and the spectral width is 140 nm. The spectral functions of the pulse exiting the compressor are assumed to have a Gaussian shape. The final output pulse contrast is calculated using a model in Ref. [Reference Mourou, Tajima and Bulanov2].
Figure 7 shows the effect of RTOD on the pulse contrast ratio. Even when RTOD is up to 
$10^{4}\ \text{fs}^{3}$, the output pulse contrast ratio in the 10 ps point is almost equal to the case when RTOD is zero (Figure 7(a)). However, the output pulse contrast ratios change in the range 
$-300$ to 300 fs (Figure 7(b)). For an ultrashort pulse (
${<}50\ \text{fs}$), the RTOD not only affects the output contrast ratio, but also the pulse duration from 
$-300$ to 300 fs. When RTOD is 
$1\times 10^{4}\ \text{fs}^{3}$, the output pulse duration is increased from 30 to 36 fs (Figure 8).
4. Conclusion
In summary, a ray-tracing model is presented to calculate the dispersion of a pair of prisms operating at other than tip-to-tip propagation of the prisms. The pair of prisms can provide a wide range of independent and continuous TOD compensation by employing appropriate values of 
$h_{1}$ and 
$d$ or 
$h_{s}$ and 
$d$ simultaneously, which is helpful in compensating the residual high-order dispersion of the CPA laser system. RTOD not only worsens the pulse contrast ratio, but also increases the pulse duration to the hundreds of femtosecond range for a tens of femtosecond pulse, even at small RTOD. These phenomena are helpful in understanding the effect of residual high-order dispersion on the pulse contrast ratio in ultrashort pulse laser systems.
Acknowledgements
This work is supported by the National High-Tech Committee of China and the National Nature Science Foundation of China.
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