Enhancing the temporal contrast and peak power of femtosecond laser pulses

Dmitry Silin; Efim Khazanov

doi:10.1364/OE.447635

1. Introduction

The temporal contrast and peak power of ultrashort laser pulses are of great importance for studies of the behavior of matter in extreme light fields. The temporal contrast is, actually, the ratio of the intensity at the peak of a pulse to the intensity at its wings. The temporal contrast may be enhanced using plasma mirrors [1], second harmonic generation [2], cross-polarized wave generation [3], filtered self-phase-modulation-broadened spectra [4,5], and a nonlinear Mach-Zehnder interferometer [6]. In recent works laser pulse peak power has been increased using the method based on self-phase modulation of a pulse in a nonlinear medium with its subsequent compression by means of a chirped mirror introducing negative dispersion [7–15] (Fig. 1 а). The method is called TFC (Thin Film Compression) [8], CafCA (Compression after Compressor Approach) [9,11], or post-compression [12]. Hereinafter we will call it CafCA.

Fig. 1. Nonlinear compression without contrast enhancement (Reference case) (a), nonlinear polarization interferometer (b), modified nonlinear polarization interferometer with three crystals (c).

Download Full Size | PDF

A nonlinear polarization interferometer (NPI), that enables simultaneous enhancement of contrast and peak power, was recently proposed in the work [16]. The idea of NPI (Fig. 1(b)) is to install between two crossed polarizers two birefringent plates, which work as a λ/1 plate of minus first order (introducing zero phase incursion) for low-intensity radiation, and as a λ/2 plate due to the incursion of the nonlinear phase π for maximum intensity. As a result, for low intensity the transmission of the entire system is equal to zero, and for maximum intensity to unity. The birefringent plates are made of the same uniaxial crystal, have the same thickness L, the same angle θ (the angle between the optical axis and the wave vector) but, in a general case, different angles φ (the angle of rotation around the optical axis). The ordinary and extraordinary waves in the plates have polarizations oriented along the x and y axes and are turned by 90° relative to each other around the z axis, providing a zero phase difference in the linear mode. The incursion of nonlinear phase difference π is attained either due to cubic nonlinearity anisotropy (n₂ depends on φ) or due to different phase intensities in cross-polarizations, which is achieved by rotating the polarizers by angles π/4±δ to the y axis. The nonlinear phase incursion results in self-phase modulation of the pulse at the NPI output, which allows the pulse to be compressed after reflection from the chirped mirror. An important merit of the NPI is its in-line geometry that does not require spatial beam separation unlike the Mach-Zehnder interferometer.

In the work [16] the NPI was investigated using two significant simplifications. First, it was assumed that pulses with o- and e-polarizations propagate almost all the path in both the first and second crystals without overlapping in space, i.e. cross-interaction of the pulses was not taken into account. This assumption holds true only for short pulses having a duration τ_in, such that the length l of their separation in space l=сτ_in/|n₀-n_e| is much less than the crystal length L. The second approximation was neglect of dispersion (both linear and nonlinear) in crystals, which, on the contrary, holds true for long pulses. Here, we present a detailed study of the NPI on the basis of a numerical solution of equations for the propagation of a laser pulse of arbitrary polarization in a uniaxial crystal with cubic nonlinearity without the simplifications mentioned above. The performed analysis demonstrated that for long pulses cross-interaction significantly disrupts NPI operation. For such pulses, we propose to modify the NPI by adding one more crystal (see Fig. 1(c)).

2. Modified NPI with three crystals

Clearly, the cross-interaction of pulses with o- and e-polarizations negatively impacts on NPI operation, as one pulse is always ahead of the other (Fig. 1(b)). Consequently, the first pulse always acts on the leading edge of the second pulse, while the second pulse always influences the trailing edge of the first pulse. As a result, after crystal II, the pulses have nonlinear phase maxima that do not coincide in time. When the pulses are combined in the second polarizer, the intensity of the pulse at the NPI output is modulated and, as a consequence, its power decreases after compression. The modified NPI with three crystals, the scheme of which is shown in Fig. 1(c), is free from this drawback. In this scheme, crystal II coincides with crystal II in the original NPI scheme (Fig. 1(b)), and crystal I in the original NPI scheme is split into two identical L/2-thick parts located ahead of and behind crystal II (Fig. 1(c)). In the modified NPI, pulses with o- and e-polarizations interact with each other for a longer time, thus increasing the influence of cross-interaction but this influence becomes symmetric: the time during which the pulse propagates ahead of and behind the other pulse is the same. Below we will show that at large values of τ_in this leads to a pronounced power enhancement of the output pulse.

3. Equations for simulation

The propagation of a laser pulse with a plane wave front in a uniaxial crystal with cubic nonlinearity is described by the following equations [17]:

(1)$$\frac{{{\partial ^2}{E_j}}}{{\partial {z^2}}} - \frac{1}{{{c^2}}}\frac{{{\partial ^2}{E_j}}}{{\partial {t^2}}} - \frac{{4\pi }}{{{c^2}}}\frac{{{\partial ^2}P_j^{(L )}}}{{\partial {t^2}}} = \frac{{4\pi }}{{{c^2}}}\frac{{{\partial ^2}P_j^{(3 )}}}{{\partial {t^2}}},\; \; j = o,e$$

Here, E_o(z,t) and E_e(z,t) are the electric field strengths of the ordinary (o) and extraordinary (e) waves, respectively, P_o^(L)(z,t) and P_e^(L)(z,t) are the o- and e-components of the linear part of the polarization, and P_o⁽³⁾(z,t) and P_e⁽³⁾(z,t) are the o- and e-components of the nonlinear part of the polarization. The components of the electric field strength are sought for by the method of slowly varying envelope approximation (SVEA):

(2)$${E_j}({z,t} )= \frac{1}{2}{A_j}({z,t} ){e^{i{\omega _0}t - i{k_j}({{\omega_0}} )z}} + c.c.,\; \; j = o,e, $$

where ${k_j} = \frac{\omega }{c}{n_j}$, n_o and n_e are the indices of refraction for o- and e-waves, respectively, and с.с. is the complex-conjugate term. Following [17], in the third order of the dispersion theory we obtain

(3)$$\begin{array}{l} \left( {\frac{\partial }{{\partial z}} + \frac{1}{{{u_j}}}\frac{\partial }{{\partial t}} - \frac{i}{2}{{\left( {\frac{{{\partial^2}{k_j}}}{{\partial {\omega^2}}}} \right)}_{{\omega_0}}}\frac{{{\partial^2}}}{{\partial {t^2}}} - \frac{1}{{12{k_j}({{\omega_0}} )}}{{\left( {\frac{{{\partial^3}k_j^2}}{{\partial {\omega^3}}}} \right)}_{{\omega_0}}}\frac{{{\partial^3}}}{{\partial {t^3}}}} \right.\\ \left. { + \frac{i}{{2{k_j}({{\omega_0}} )}}\left[ {\frac{{{\partial^2}}}{{\partial {z^2}}} - \frac{1}{{u_j^2}}\frac{{{\partial^2}}}{{\partial {t^2}}}} \right]} \right){A_j}({z,t} )= \frac{{4\pi }}{{{c^2}}}\frac{i}{{{k_j}({{\omega_0}} )}}\frac{{{\partial ^2}P_{0j}^{(3 )}}}{{\partial {t^2}}},\; \; j = o,e \end{array}$$

Here, ${u_j} = {[{{{({{{\partial {k_j}} / {\partial \omega }}} )}_{{\omega_0}}}} ]^{ - 1}}$ are the group velocities of the o- and e-waves and $P_{0j}^{(3)}$ are the complex amplitudes of the nonlinear polarization:

(4)$$P_j^{(3)}({z,t} )= \frac{1}{2}P_{0j}^{(3)}({z,t} ){e^{i{\omega _0}t - i{k_j}({{\omega_0}} )z}} + c.c.,\; \; j = o,e$$

Passing over to the moving reference frame (z = z, t’=t–z/u_j), we can show that, in the left-hand side of Eq. (3), the sum of the term with the third time derivative and the term with a square bracket is $- \frac{1}{6}{\left( {\frac{{{\partial^3}{k_j}}}{{\partial {\omega^3}}}} \right)_{{\omega _0}}}\frac{{{\partial ^3}}}{{\partial {t^3}}}{A_j}({z,t} )$ to an accuracy of the third order of smallness.

In a crystal with cubic nonlinearity we have

(5)$$P_j^{(3 )}({z,t} )= \mathop \sum \limits_{l,m,n = o,e} \chi _{jlmn}^{(3 )}{E_l}{E_m}{E_n},\; \,j = o,e$$

where $\chi _{jlmn}^{(3 )}$ are the components of the tensor of nonlinear susceptibility. We did not take into account cascade quadratic nonlinearity, since its influence is quite small for the NPI parameters discussed in the following sections. In Eq. (5) we neglected nonstationarity of the cubic nonlinearity, as for the femtosecond pulses with an intensity on the order of a few TW/cm² propagating in transparent dielectrics, the time of the medium nonlinear response is much shorter than the pulse duration. By substituting Eq. (2,4,5) into Eq. (3) and taking into consideration the Kleinmann symmetry [18], we obtain a system of equations for A_o(z,t) and A_e(z,t):

(6)$$\begin{array}{l} \frac{{\partial {A_o}}}{{\partial z}} + \frac{1}{{{u_o}}}\frac{{\partial {A_o}}}{{\partial t}} - \frac{i}{2}{\left( {\frac{{{\partial^2}{k_o}}}{{\partial {\omega^2}}}} \right)_{{\omega _0}}}\frac{{{\partial ^2}{A_o}}}{{\partial {t^2}}} - \frac{1}{6}{\left( {\frac{{{\partial^3}{k_o}}}{{\partial {\omega^3}}}} \right)_{{\omega _0}}}\frac{{{\partial ^3}{A_0}}}{{\partial {t^3}}}\\ + \frac{{3\pi }}{{2{n_o}c}}({\chi_{oooo}^{(3 )}{F_1} + \chi_{oooe}^{(3 )}{F_2} + \chi_{ooee}^{(3 )}{F_3}{e^{ - i\Delta kz}} + \chi_{oeee}^{(3 )}{F_4}{e^{ - i\Delta kz}}} )= 0 \end{array}$$

(7)$$\begin{array}{l} \frac{{\partial {A_e}}}{{\partial z}} + \frac{1}{{{u_e}}}\frac{{\partial {A_e}}}{{\partial t}} - \frac{i}{2}{\left( {\frac{{{\partial^2}{k_e}}}{{\partial {\omega^2}}}} \right)_{{\omega _0}}}\frac{{{\partial ^2}{A_e}}}{{\partial {t^2}}} - \frac{1}{6}{\left( {\frac{{{\partial^3}{k_e}}}{{\partial {\omega^3}}}} \right)_{{\omega _0}}}\frac{{{\partial ^3}{A_e}}}{{\partial {t^3}}}\\ + \frac{{3\pi }}{{2{n_e}c}}({\chi_{eeee}^{(3 )}{F_4} + \chi_{oeee}^{(3 )}{F_3} + \chi_{ooee}^{(3 )}{F_2}{e^{i\Delta kz}} + \chi_{oooe}^{(3 )}{F_1}{e^{i\Delta kz}}} )= 0 \end{array}$$

In this system we introduced the following notation

(8)$${F_1} = i{\omega _0}{|{{A_o}} |^2}{A_o} + 2{A_o}\frac{{\partial {{|{{A_o}} |}^2}}}{{\partial t}} + 2{|{{A_o}} |^2}\frac{{\partial {A_o}}}{{\partial t}}$$

(9)$$\begin{aligned} {F_2} &= i{\omega _0}A_o^2A_e^\ast {e^{i\Delta kz}} + 2i{\omega _0}{|{{A_o}} |^2}{A_e}{e^{ - i\Delta kz}} + 2A_e^\ast \frac{{\partial A_o^2}}{{\partial t}}{e^{i\Delta kz}} + 2A_o^2\frac{{\partial A_e^\ast }}{{\partial t}}{e^{i\Delta kz}}\\ & + 4{A_e}\frac{{\partial {{|{{A_o}} |}^2}}}{{\partial t}}{e^{ - i\Delta kz}} + 4{|{{A_o}} |^2}\frac{{\partial {A_e}}}{{\partial t}}{e^{ - i\Delta kz}} \end{aligned}$$

(10)$$\begin{aligned} {F_3} &= i{\omega _0}A_e^2A_o^\ast {e^{ - i\Delta kz}} + 2i{\omega _0}{|{{A_e}} |^2}{A_o}{e^{i\Delta kz}} + 2A_o^\ast \frac{{\partial A_e^2}}{{\partial t}}{e^{ - i\Delta kz}} + 2A_e^2\frac{{\partial A_o^\ast }}{{\partial t}}{e^{ - i\Delta kz}}\\ & + 4{A_o}\frac{{\partial {{|{{A_e}} |}^2}}}{{\partial t}}{e^{i\Delta kz}} + 4{|{{A_e}} |^2}\frac{{\partial {A_o}}}{{\partial t}}{e^{i\Delta kz}} \end{aligned}$$

(11)$${F_4} = i{\omega _0}{|{{A_e}} |^2}{A_e} + 2{A_e}\frac{{\partial {{|{{A_e}} |}^2}}}{{\partial t}} + 2{|{{A_e}} |^2}\frac{{\partial {A_e}}}{{\partial t}}$$

(12)$$\Delta k = ({{n_e} - {n_o}} )\frac{{{\omega _0}}}{c}$$

Terms with the second time derivatives are omitted in Eq. (8–11) because these terms are of the second order of smallness in the parameter 1/ω₀τ, where τ is a pulse duration. This neglect is consistent with SVEA.

The system of equations (6,7) contains the components of the tensor χ⁽³⁾ in laboratory reference frame. The expressions for determining these components through the tensor components in the intrinsic coordinate system of the crystal (x,y,z) for uniaxial crystals of arbitrary symmetry were presented, for example, in [19]. In particular, for KDP and DKDP crystals belonging to the tetragonal symmetry class $42\bar{m}$, these expressions have the form

(13)$$\chi _{oooo}^{(3 )} = \frac{1}{4}({\chi_{xxxx}^{(3 )}({3 + \cos 4\varphi } )+ 6\chi_{xxyy}^{(3 )}{{\sin }^2}2\varphi } )$$

(14)$$\chi _{oooe}^{(3 )} = \frac{1}{4}\cos \theta \sin 4\varphi ({\chi_{xxxx}^{(3 )} - 3\chi_{xxyy}^{(3 )}} )$$

(15)$$\chi _{ooee}^{(3 )} = \frac{1}{2}{\cos ^2}\theta ({\chi_{xxxx}^{(3 )}{{\sin }^2}2\varphi + \chi_{xxyy}^{(3 )}({3{{\cos }^2}2\varphi - 1} )} )+ \chi _{xxzz}^{(3 )}{\sin ^2}\theta $$

(16)$$\chi _{oeee}^{(3 )} = \frac{1}{4}({3\chi_{xxyy}^{(3 )} - \chi_{xxxx}^{(3 )}} ){\cos ^3}\theta \sin 4\varphi $$

(17)$$\chi _{eeee}^{(3 )} = \frac{1}{4}\chi _{xxxx}^{(3 )}{\cos ^4}\theta ({3 + \cos 4\varphi } )+ \frac{3}{2}\chi _{xxzz}^{(3 )}{\sin ^2}2\theta + \frac{3}{2}\chi _{xxyy}^{(3 )}{\cos ^4}\theta {\sin ^2}2\varphi + \chi _{zzzz}^{(3 )}{\sin ^4}\theta $$

In what will follow, we will limit ourselves to considering only KDP and DKDP crystals. This choice is explained, firstly, by the fact that these crystals can be grown up to a large aperture and, secondly, that all components of the nonlinear susceptibility tensor have been measured for them [20]. Note that, similarly to [20], we assumed that the tensor χ⁽³⁾ is identical for these two crystals. Therefore, in the calculations presented below, these crystals differed only by dispersion properties. We used the Sellmeier relations from the works [21,22]. The components of the tensor χ⁽³⁾ used in numerical simulation are given in Table 1.

Table 1. The Parameters of KDP and DKDP Crystals Used in Numerical Simulation (λ=910 nm, Deuteration Level of DKDP is 0.96)

View Table

After passing through the NPI, the spectral phase of the pulse, due to the nonlinear phase incursion, becomes a complex function of frequency with a quadratic component. This component may be compensated for (subtracted) by the reflection of the pulse from a chirped mirror, which allows enhancing the peak power of the pulse (CafCA). This was also taken into account in numerical simulation: the chirped mirror introduced the spectral phase $\psi ={-} \alpha \frac{{{{({\omega - {\omega_0}} )}^2}}}{2}$. The magnitude of α was chosen so that the peak intensity of the output pulse should be maximal.

It is useful to compare the peak intensities of the output pulse in the NPI and in the traditional CafCA method (one isotropic nonlinear plate and a chirped mirror, Fig. 1(а)). In the work [23] it was shown that, in the absence of dispersion, the increase in the peak intensity of the pulse at the CafCA output P_out compared to the input intensity P_in can be estimated from the following formula

(18)$$\frac{{{P_{out}}}}{{{P_{in}}}} = 1 + {B_0}/2, $$

where B₀= kLn₂I_in is the B-integral, k is the wave number in vacuum, n₂ is the nonlinear index of refraction (n = n₀+ n₂I), and I_in is the peak intensity of the input pulse. The B-integral cannot be arbitrarily increased for enhancing peak intensity of P_out because of the effect of small-scale self-focusing [24]. For pulses with an intensity of about 1 TW/cm² it was demonstrated that small-scale self-focusing may be avoided up to B₀= 20–25 [25] for a beam of a small diameter and up to B₀= 19 for a beam 18 cm in diameter of a subpetawatt laser [15,26]. It can be expected that, for equal values of B₀, the NPI with a chirped mirror at the output will give a smaller value of P_out/P_in than the traditional CafCA method. For comparison we also modeled CafCA (Fig. 1 а): one crystal (KDP or DKDP, o-wave) and a chirped mirror with optimal dispersion. Hereinafter we will call it a Reference case. The crystal thickness in the Reference case was chosen such that the nonlinear phase incursion was equal to the arithmetic mean of the nonlinear phase incursion of the o- and e-pulses at the NPI output < B_out>=(B_o+B_e)/2.

4. Nonlinear polarization interferometer with equal intensities of o- and e-pulses

In this section we present results of calculations of the laser pulse passing through the NPI, in which δ=0 and the incursion of the nonlinear phase difference of o- and e-pulses is reached due to cubic nonlinearity anisotropy. In line with [16], we set optimal values of the angles for crystals of $42\bar{m}$ symmetry class: θ=π/2, φ_I= 0, and φ_II=π/4. The results of NPI (Fig. 1(b)) simulation for an input Gaussian pulse having duration τ_in= 50 fs (by the level 1/e), wavelength 910 nm, and peak intensity of 2 TW/cm² are presented in Fig. 2. The KDP crystal thickness L was 3.1 mm and was optimized to the maximum peak intensity of the pulse past the chirped mirror. Figure 3 is analogous to Fig. 2 but corresponds to the pulse with a duration of 300 fs and the modified NPI with three crystals (Fig. 1(c)), L = 4.1 mm. The input pulse contrast was 10⁷ in both cases.

Fig. 2. Results of calculations of Gaussian pulse with 50 fs duration and 2 TW/cm² peak intensity propagating through NPI (KDP, δ=0, θ=π/2, φ_I= 0, and φ_II=π/4) and chirped mirror. < B_out>=10.9, L = 3.1 mm, α_opt=−170 fs².

Download Full Size | PDF

Fig. 3. Results of calculations of Gaussian pulse with 300 fs duration and 2 TW/cm² peak intensity propagating through modified NPI with three crystals (KDP, δ=0, θ=π/2, φ_I= 0, and φ_II=π/4) and chirped mirror. < B_out>=25.4, L = 4.1 mm, α_opt=−919 fs².

Download Full Size | PDF

Figures 2(a-b) and 3(a-b) show, in addition to the input (green curves) and output (black curves) pulses, two cross-polarized pulses in front of the output polarizer: the pulse that propagated as an e-wave in crystal I and as an o-wave in crystal II (red curves) and vice versa (blue curves). The waveforms of these two pulses differ rather strongly due to the different dispersions of the o- and e-waves in the KDP crystal. Despite the fact that each of the two pulses propagates through the NPI in turn as an o- and an e-wave, the impact of dispersion on the final waveform of the pulse strongly depends on the order in which the pulse was the o- or the e-wave. This effect is weaker in the DKDP crystal, as the difference in the dispersions of o- and e-waves is smaller. However, the results of calculations for the NPI with a DKDP crystal turned out to be very close to the NPI with KDP, so in what will follow we will present results only for KDP.

The displacement of the pulse to the right relative to t = 0 reflects its time delay compared to the linear case due to nonlinear dispersion (the last term in Eq. (8–11)). In addition, the thin curves in Figs. 2(a) and 3(a) show the time dependence of the pulse phase on time, and the purple curves the phase difference of cross-polarized pulses in front of the second polarizer. Note that the phase difference of the pulses with o- and e-polarizations in Fig. 2(a) in the area of intensity peaks is lower than the expected value of π, while in Fig. 3(a) this difference is more than π. This fact is explained by the optimization of the crystal thickness L by the parameter P_out/P_in: for short pulses, it is more beneficial to slightly reduce L (and, consequently, the phase difference) for reducing the negative impact of dispersion, while for long pulses, the dispersion is not so important; therefore, it is better to increase L for increasing the B-integral, see Eq. (18).

From Figs. 2(b) and 3(b) it is clear that the pulse spectrum after the NPI has become much wider and the phase has acquired a parabolic component. After the chirped mirror (α=−170 fs²), the pulse has an almost five times higher peak intensity and a six times shorter duration than at the NPI input (Fig. 2(c)). For the modified NPI with three crystals (α=−919 fs²), the intensity of a pulse having a duration of 300 fs increases by more than 8 times (Fig. 3(c)). In both interferometers these values are approximately equal to 80% of the Reference case. This is a consequence of the inevitable energy loss in the NPI in contrast to the lossless Reference case. The NPI energy efficiency η (the ratio of the pulse energy at the NPI input to the pulse energy at the output) for these cases is 80% and 75.6%.

We will now analyze contrast enhancement for the NPI (Fig. 1(b)) in three temporal ranges: i) |t|<τ_in; ii) τ_in<|t|<τ_oe, where τ_oe= L·|1/u_o–1/u_e| is the delay time of the second pulse relative to the first pulse after passing crystal I; and iii) |t|>τ_oe. For |t|<τ_in, the NPI provides a much better contrast than that in the Reference case (Fig. 2(с)), which can be explained as follows. Even if at t = 0 the nonlinear phase incursion is equal to π, at the fronts of the input pulse (i.е. at |t| on the order of τ_in) the nonlinear phase incursion is less than π and, hence, the NPI introduces losses. However, this improves the quality of compression, as the chirp introduced by the self-phase modulation at such times strongly differs from the linear one and is poorly compensated by the chirp mirror. At larger times (τ_in<|t|<τ_oe; τ_oe= 528 fs in Fig. 2(d)), contrast enhancement is almost time independent. This is due to the following. When propagating in crystal I, the o- and e-pulses are separated in time τ_oe, as a result of which the rear peripheral part of the first pulse, despite the low intensity, acquires a nonlinear phase due to cross-interaction with the maximum of the second pulse. During propagation in crystal II, the process occurs in the opposite direction and this phase doubles. At the same time, the rear peripheral part of the second pulse never interacts with the first pulse and has a zero nonlinear phase. An analogous situation occurs in the front peripheral part. So, at |t|<τ_oe the nonlinear phase difference is nonzero, which limits contrast enhancement. Finally, at |t|>τ_oe both pulses propagate only in a linear mode, and the contrast within this model tends to infinity. In practice, the contrast will be determined by the quality of the surfaces of nonlinear crystals, which is discussed in detail in [16]. For the modified NPI with three crystals (Fig. 1(c)), the pulse duration τ_in is usually comparable to τ_oe= L/2·|1/u_o–1/u_e| (in Fig. 3(d), τ_in= 300 fs and τ_oe= 349 fs). For |t|>τ_in,τ_oe, within this model the contrast also tends to infinity. Note that for |t|>τ_in the contrast in the Reference case persists to be equal to the contrast of the input pulse (see Fig. 2,3(d)), since the peripheral part of the pulse does not participate in the nonlinear interaction because of its low intensity.

The curves for the parameters P_out /P_in, τ_in/τ_out (the ratio of the durations of the input and compressed pulses), and < B_out> as a function of τ_in are plotted in Fig. 4. The graphs for the efficiency η, crystal thickness L, and optimal dispersion α_opt of the chirped mirror are presented in Fig. 5. One can see in Fig. 4 that for short pulses (for an intensity of 2 TW/cm² at τ_in< 150 fs and for an intensity of 5 TW/cm² at τ_in< 60 fs) an increase in the duration of the input pulse in the original NPI scheme leads, on the average, to an increase of the parameters P_out /P_in, τ_in /τ_out, and < B_out>, which is associated with a decrease in dispersion for long pulses, as well as with an increase of the influence of the cross-interaction of the pulses with o- and e-polarizations.

Fig. 4. Pulse parameters P_out/P_in, τ_in/τ_out, and < B_out> versus τ_in (δ=0, θ=π/2, φ_I= 0, and φ_II=π/4) for 2 TW/cm² (а) and 5 TW/cm² (b) peak intensities.

Download Full Size | PDF

Fig. 5. Efficiency η, L, and optimal dispersion α_opt of chirped mirror versus τ_in (δ=0, θ=π/2, φ_I= 0, and φ_II=π/4).

Download Full Size | PDF

Maximum efficiency (Fig. 5(a)) is attained at τ_in∼100 fs for the 2 TW/сm² pulses and at τ_in∼50–70 fs for the 5 TW/cm² pulse. However, at larger values of τ_in, the parameter P_out /P_in starts to reduce. The modified NPI scheme with three crystals has higher values of P_out /P_in and efficiency for longer pulses (with a duration of more than ∼100 fs), whereas for shorter pulses the original NPI with two crystals outperforms the NPI with three crystals.

The comparison of the NPI and the Reference case in terms of the parameter P_out /P_in (Fig. 4) shows that, for the input pulse with a peak intensity of 2 TW/cm², the Reference case is a little better as a rule. However, for an intensity of 5 TW/cm² in the range of the τ_in values from ∼45 to ∼65 fs, the NPI is even better than the Reference case. Bearing in mind that the NPI contributes to a significant contrast enhancement, its advantage over the traditional CafCA (Fig. 1(а)) is obvious.

5. Nonlinear polarization interferometer with different intensities of the o- and e-pulses

Here we present the results of calculations for the NPI, in which δ>0 and the nonlinear phase incursion of the o- and e-pulses is attained due to different intensities. Figures 6 and 7 are analogous to Figs. 4 and 5. In all these figures P_out /P_in, τ_in/τ_out, < B_out>, η, L, and α_opt are plotted as a function of the input pulse duration τ_in for the original and modified NPI at δ=4°, θ=π/2, and φ_I=φ_II= 0. The curves in Figs. 6 and 7 are similar to those for the NPI with identical intensities of the o- and e-pulses (Figs. 4 and 5), but upon the whole the ratio P_out/P_in in Figs. 6 and 7 is a little smaller than in Figs. 4 and 5. Besides, the efficiency η is markedly lower for long pulses in the modified NPI scheme, while < B_out> for this scheme is larger.

Fig. 6. Pulse parameters P_out/P_in, τ_in/τ_out, and < B_out> versus τ_in (δ=4°, θ=π/2, φ_I=φ_II= 0) for 2 TW/cm² (а) and 5 TW/cm² (b) peak intensities.

Download Full Size | PDF

Fig. 7. Efficiency η, L, and optimal dispersion α_opt of chirped mirror versus τ_in (δ=4°, θ=π/2, φ_I=φ_II= 0).

Download Full Size | PDF

The dependence of P_out/P_in and < B_out> on the angle δ is presented in Fig. 8 for the input pulse durations τ_in= 25, 50, 100, and 200 fs with the input pulse peak intensity 2 TW/cm² and 5 TW/cm²; θ=π/2, φ_I=φ_II= 0. For the pulses with τ_in from 25 to 100 fs, the calculations were made for the original NPI scheme, and for pulses with τ_in= 200 fs for the modified NPI scheme with three crystals. The graphs presented in Fig. 8 make it possible to determine the optimal value of the angle δ for different durations of the input pulse at a given restriction on the maximal value of < B_out> .

Fig. 8. P_out/P_in and < B_out> versus angle δ for different τ_in and input pulse intensities (θ=π/2, φ_I=φ_II= 0).

Download Full Size | PDF

Note that NPIs with different intensities of the o- and e-pulses have practical advantages. Their parameters can be readily matched by varying δ; besides, a wider scope of nonlinear crystals can be used.

6. Conclusion

The presented detailed analysis of two variants of NPI based on KDP/DKDP crystals with a chirped mirror (Fig. 1(b) and (c)) allows us to make the following conclusions:

1. Contrast enhancement at times longer than the delay time τ_oe of the second pulse relative to the first one after passage of crystal I is not limited by the nonlinear effects and is determined only by the crystal manufacturing quality and polarizer contrast, as was predicted in the work [16]. At shorter times (up to the input pulse duration τ_in), contrast enhancement is also pronounced (see Figs. 2, 3(d)).
2. For pulses with a duration of tens of femtoseconds, the original scheme presented in [16] and Fig. 1(b) is more preferable, and for pulses with a duration of hundreds of femtoseconds, the modified scheme in Fig. 1(с). In both cases power enhancement is, as a rule, only a little lower (and sometimes higher) than in the traditionally used scheme of nonlinear compression without interferometer (Fig. 1(а)).
3. The NPI with different intensities of o- and e-pulses (δ≠0) allows obtaining parameters close to the NPI with equal intensities, but has practical advantages, such as readily matched parameters using the variation of δ and a wider scope of nonlinear crystals.
4. The NPI and the chirped mirror can find a wide application thanks to the in-line geometry, simple adjustment, and a possibility to be used in any laser with a power varying from TWs to PWs without changing the optical scheme.

Funding

Ministry of Science and Higher Education of the Russian Federation (Agreement No. 075-15-2020-906, Center of Excellence “Center of Photonics”).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. A. Lévy, T. Ceccotti, P. D’Oliveira, F. Réau, M. Perdrix, F. Quéré, P. Monot, M. Bougeard, H. Lagadec, P. Martin, J.-P. Geindre, and P. Audebert, “Double plasma mirror for ultrahigh temporal contrast ultraintense laser pulses,” Opt. Lett. 32(3), 310–312 (2007). [CrossRef]

2. S. Y. Mironov, V. V. Lozhkarev, V. N. Ginzburg, and E. A. Khazanov, “High-efficiency second-harmonic generation of superintense ultrashort laser pulses,” Appl. Opt. 48(11), 2051–2057 (2009). [CrossRef]

3. A. Jullien, O. Albert, F. Burgy, G. Hamoniaux, J.-P. Rousseau, J.-P. Chambaret, F. Augé-Rochereau, G. Chériaux, J. Etchepare, N. Minkovski, and S. M. Saltiel, “10¹⁰ temporal contrast for femtosecond ultraintense lasers by cross-polarized wave generation,” Opt. Lett. 30(8), 920–922 (2005). [CrossRef]

4. J. Buldt, M. Müller, R. Klas, T. Eidam, J. Limpert, and A. Tünnermann, “Temporal contrast enhancement of energetic laser pulses by filtered self-phase-modulation-broadened spectra,” Opt. Lett. 42(19), 3761–3764 (2017). [CrossRef]

5. S. Y. Mironov, M. V. Starodubtsev, and E. A. Khazanov, “Temporal contrast enhancement and compression of output pulses of ultra-high power lasers,” Opt. Lett. 46(7), 1620–1623 (2021). [CrossRef]

6. E. A. Khazanov and S. Y. Mironov, “Nonlinear interferometer for increasing the contrast ratio of intense laser pulses,” Quantum Electron. 49(4), 337–343 (2019). [CrossRef]

7. V. Chvykov, C. Radier, G. Cheriaux, G. Kalinchenko, V. Yanovsky, and G. Mourou, “Compression of Ultra-high Power Laser Pulses,” in Conference on Lasers and Electro-Optics (CLEO)/Quantum Electronics and Laser Science Conference (QELS), (IEEE, San Jose, CA, 2010).

8. G. Mourou, S. Mironov, E. Khazanov, and A. Sergeev, “Single cycle thin film compressor opening the door to Zeptosecond-Exawatt physics,” Eur. Phys. J. Spec. Top. 223(6), 1181–1188 (2014). [CrossRef]

9. V. N. Ginzburg, I. V. Yakovlev, A. S. Zuev, A. A. Korobeinikova, A. A. Kochetkov, A. A. Kuz’min, S. Y. Mironov, A. A. Shaykin, I. A. Shaykin, and A. E. Khazanov, “Compression after compressor: threefold shortening of 200-TW laser pulses,” Quantum Electron. 49(4), 299–301 (2019). [CrossRef]

10. S. K. Lee, J. Y. Yoo, J. I. Kim, R. Bhushan, Y. G. Kim, J. W. Yoon, H. W. Lee, J. H. Sung, and C. H. Nam, “High energy pulse compression by a solid medium,” in Conference on Lasers and Electro-Optics (CLEO), (IEEE, 345 E 47TH st, New York, NY 10017 USA, 2018).

11. Е. А. Khazanov, S. Y. Mironov, and G. Mourou, “Nonlinear compression of high-power laser pulses: compression after compressor approach,” UFN 62, 1096–1124 (2019). [CrossRef]

12. P. Balla, A. B. Wahid, I. Sytcevich, C. Guo, A.-L. Viotti, L. Silletti, A. Cartella, S. Alisauskas, H. Tavakol, U. Grosse-Wortmann, A. Schönberg, M. Seidel, A. Trabattoni, B. Manschwetus, T. Lang, F. Calegari, A. Couairon, A. L’Huillier, C. L. Arnold, I. Hartl, and C. M. Heyl, “Postcompression of picosecond pulses into the few-cycle regime,” Opt. Lett. 45(9), 2572–2575 (2020). [CrossRef]

13. V. N. Ginzburg, I. V. Yakovlev, A. S. Zuev, A. P. Korobeynikova, A. A. Kochetkov, A. A. Kuzmin, S. Y. Mironov, A. A. Shaykin, I. A. Shaikin, and E. A. Khazanov, “Two-stage nonlinear compression of high-power femtosecond laser pulses,” Quantum Electron. 50(4), 331–334 (2020). [CrossRef]

14. V. Ginzburg, I. Yakovlev, A. Zuev, A. Korobeynikova, A. Kochetkov, A. Kuzmin, S. Mironov, A. Shaykin, I. Shaikin, E. Khazanov, and G. Mourou, “Fivefold compression of 250-TW laser pulses,” Phys. Rev. A 101(1), 013829 (2020). [CrossRef]

15. V. Ginzburg, I. Yakovlev, A. Kochetkov, A. Kuzmin, S. Mironov, I. Shaikin, A. Shaykin, and E. Khazanov, “11 fs, 1.5 PW laser with nonlinear pulse compression,” Opt. Express 29(18), 28297–28306 (2021). [CrossRef]

16. E. Khazanov, “Nonlinear polarization interferometer for enhancement of laser pulse contrast and power,” Opt. Express 29(11), 17277–17285 (2021). [CrossRef]

17. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses 1992nd Edition (American Institute of Physics, 1992).

18. R.L. Sutherland, Handbook of Nonlinear Optics 2nd Edition (CRC Press, 2003).

19. P. S. Banks, M. D. Feit, and M. D. Perry, “High-intensity third-harmonic generation,” J. Opt. Soc. Am. B 19(1), 102–118 (2002). [CrossRef]

20. I. A. Kulagin, R. A. Ganeev, R. I. Tugushev, A. I. Ryasnyansky, and T. Usmanov, “Analysis of third-order nonlinear susceptibilities of quadratic nonlinear-optical crystals,” J. Opt. Soc. Am. B 23(1), 75–80 (2006). [CrossRef]

21. F. Zernike, “Refractive Indices of Ammonium Dihydrogen Phosphate and Potassium Dihydrogen Phosphate between 2000 Å and 1.5 µ,” J. Opt. Soc. Am. 54(10), 1215–1220 (1964). [CrossRef]

22. V. V. Lozhkarev, G. I. Freidman, V. N. Ginzburg, E. A. Khazanov, О. V. Palashov, A. M. Sergeev, and I. V. Yakovlev, “Study of broadband optical parametric chirped pulse amplification in a DKDP crystal pumped by the second harmonic of a Nd:YLF laser,” Laser Phys. 15, 1319–1333 (2005).

23. V. N. Ginzburg, A. A. Kochetkov, I. V. Yakovlev, S. Y. Mironov, A. A. Shaykin, and E. A. Khazanov, “Influence of the cubic spectral phase of high-power laser pulses on their self-phase modulation,” Quantum Electron. 46, 106–108 (2016). [CrossRef]

24. V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3(12), 307–310 (1966).

25. S. Mironov, V. Lozhkarev, G. Luchinin, A. Shaykin, and E. Khazanov, “Suppression of small-scale self-focusing of high-intensity femtosecond radiation,” Appl. Phys. B: Lasers Opt. 113(1), 147–151 (2013). [CrossRef]

26. A. Shaykin, V. Ginzburg, I. Yakovlev, A. Kochetkov, A. Kuzmin, S. Mironov, I. Shaikin, S. Stukachev, V. Lozhkarev, A. Prokhorov, and E. Khazanov, “Use of KDP crystal as a Kerr nonlinear medium for compressing PW laser pulses down to 10 fs,” High Pow Laser Sci Eng 9, e54 (2021). [CrossRef]

Parameter	KDP	DKDP
$χ_{x x x x}^{(3)}$ , m²/V²	3.7·10⁻²²
$χ_{z z z z}^{(3)}$ , m²/V²	2.3·10⁻²²
$χ_{x x y y}^{(3)}$ , m²/V²	5.18·10⁻²³
$χ_{x x z z}^{(3)}$ , m²/V²	1.11·10⁻²²
n_o(ω₀)	1.49822	1.49602
n_e(ω₀)	1.46170	1.45987
u_o, cm/s	1.966·10¹⁰	1.980·10¹⁰
u_e, cm/s	2.034·10¹⁰	2.038·10¹⁰
$(\partial^{2} k_{o} / \partial ω^{2})_{ω_{0}}$ , fs²/mm	11.36	23.46
$(\partial^{2} k_{e} / \partial ω^{2})_{ω_{0}}$ , fs²/mm	28.64	29.83
$(\partial^{3} k_{o} / \partial ω^{3})_{ω_{0}}$ , fs³/mm	65.78	46.43
$(\partial^{3} k_{e} / \partial ω^{3})_{ω_{0}}$ , fs³/mm	30.06	27.97

Enhancing the temporal contrast and peak power of femtosecond laser pulses

Abstract

1. Introduction

2. Modified NPI with three crystals

3. Equations for simulation

4. Nonlinear polarization interferometer with equal intensities of o- and e-pulses

5. Nonlinear polarization interferometer with different intensities of the o- and e-pulses

6. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Tables (1)

Equations (18)

Optics Express