1. Introduction

Extremely high (exceeding 10 MV/cm) terahertz (THz) electric fields are of great interest for electron acceleration [1–3], charged particle beam manipulation [4] and post acceleration of protons/ions [5] and THz-assisted attosecond generation [6]. Enhancement of the energy and peak electric field strength of THz pulses requires development of THz sources.

Optical rectification of ultrashort laser pulses in nonlinear crystals is a promising way for THz pulse generation; THz pulse energy of 0.9 mJ, and electric field strength of 42 MV/cm was achieved in the few THz range in DSTMS [7]. LiNbO₃ (LN) is also an excellent material for THz generation due to its advantageous properties, especially to the high value of its d₃₃ second order nonlinear optical tensor element [8]. In order to achieve the necessary matching between the group velocity of the pump pulses and the phase velocity of the THz the tilted-pulse-front (TPF) geometry is the exclusive solution because of the significant refractive index difference in the visible and the THz range [9]. The noncollinear velocity matching reads as

In a conventional TPF setup the pump beam is diffracted by an optical grating, and the illuminated spot on the grating is imaged into the nonlinear crystal. Optimization of the TPF setup was a subject of several works [12–14]. As a drawback of the setup imaging errors lead to the broadening of the pump pulses resulting in a significant decrease of the conversion efficiency especially in the case of high pump energies, when wide beams are used [12, 15].

In order to avoid this effect the contact grating (CG) scheme was proposed [15]. In this scheme the tilted-pulse-front is introduced by creating a grating directly on the surface of the nonlinear crystal (typically by etching), hence imaging optics can be omitted.

Several publications deal with the theoretical design and the practical realization of the CG based THz source (especially LN and ZnTe). Maximizing the diffraction efficiency at the CG has key importance for efficient THz generation. Nagashima and associates performed theoretical optimization with respect to the grating constant, and the fine structure of binary grating profile. The merely 20% maximal diffraction efficiency of the air/LN CG could be increased to 90% by using an air/fused silica/LN structure [16, 17]. The drawback of this structure is the problematic technical implementation, since filling a submicron structure on the surface of LN by fused silica is not yet solved. This technical problem can be solved by using RIML prism at the air/crystal boundary for coupling the pump beam into the LN. 98% diffraction efficiency was predicted by using RIML having index of refraction equivalent to that of BK7 [18]. The necessity of the prism limits the use of wide beams.

Making the coupling of the pump beam into the LN more efficient, a Fabry-Perot resonator was proposed by applying a semi-reflecting layer between the grating and the LN [19]. By this solution 71% efficiency was reached [20], which is ~10% lower than the theoretically predicted value probably because of the difficulties with the technical implementation.

For typical pumping wavelengths the value of the groove density falls in the 2500 – 3000 1/mm range in all the above mentioned cases [17–20]. According to Eq. (2) of [21], Clausnitzer et al. in the case of air/LN boundary at 1030 nm wavelength the groove density has to be in the range of 1398 – 1942 1/mm in order to achieve high diffraction efficiency in the −1st diffraction order. Furthermore, results of our earlier investigations show, that the diffraction efficiency decreases drastically with the deviation from the ideal rectangular grating profile. It was demonstrated, that slight (only a few degrees) deviation from perpendicular side walls can cause significant decrease in the efficiency [22]. Experimental results show that fabrication of good quality grating with large groove density (exceeding 2000/mm for LN) causes difficulties since the grating profile becomes slurred. This fact does not allow efficient THz generation with LN based simple CG, the diffraction efficiency lags behind the theoretically predicted.

The unsolved fabrication technological problems concerning the LN CG, the efficiency considerations given above and the drawback of the conventional TPF scheme coming from the effects of imaging errors motivated our research developing new setup for THz generation at high pump energy level.

In the present paper the design of a hybrid tilted-pulse-front (HTPF) scheme is given. This solution overcomes the problems concerning the large groove density and also reduces the optical imaging error. In this setup the pulse front tilt needed for velocity matching is created in two steps. First an optical grating following with imaging introduces a smaller than needed initial tilt, and the remaining (additionally needed) tilt is created by diffraction on the CG. The concept of the pulse front tilt division was to obtain near equal values for the tangent of the pulse front tilt angle change in both steps. On the one hand imaging errors can be significantly reduced by the HTPF scheme as compared to the conventional one, since the angular dispersion in the beam arriving to the lens is smaller in the former case. On the other hand due to the initial pulse front tilt only a lower value of the groove density is necessary, making possible the precise nanofabrication of the CG structure. Furthermore advantageously the application of RIML can be avoided making the structure simpler. The contact grating structure of the proposed setup is also significantly simpler than the one based on Fabry-Perot resonator [19, 20].

The paper is organized as follows. In Sec. 2 the description of the hybrid TPF scheme is given. In Sec. 3 the optimization procedure concerning the diffraction efficiency of the CG of the proposed scheme is described, considering practical aspects as well. In Sec. 4 instructions are given how to design the optimal geometrical configuration in order to minimize the effect of imaging errors.

2. The hybrid tilted-pulse-front scheme

The schematic drawing of the proposed HTPF setup is shown in Fig. 1. The notations and acronyms used in Fig. 1 together with other symbols used in the paper are explained in Table 1.

 

Fig. 1 The scheme of the HTPF THz source. The four short green lines on the ray symbolize the pulse front.

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Table 1. Notations and acronyms used in the text and in the figures.

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The pump beam represented by the ray shown in Fig. 1 can be considered also as the optical axis of the system. The pump beam arrives on G1 at an angle of incidence of${\theta }_{\text{i}}$. The diffraction on G1 results an angular dispersion of the pump beam and hence initial pulse front tilt. Figure 1 shows a transmission grating as an example, but a reflection one (used typically in earlier experiments) can also be adequate. The advantage of the former one is the typically higher diffraction efficiency compared to the latter one, and the good separation of the beam diffracted into the −1st order used for the THz generation from the incident one. The diffracted beam is directed to the surface of the CG at an angle of ${\theta }_{\text{i2}}$ through a lens. The beam enters the nonlinear crystal (NLC) in the −1st diffraction order. In the crystal the pump beam propagates in a direction making a diffraction angle of ${\theta }_{\text{d2}}$ determined by the grating equation

In order to get high THz generation efficiency under given pump conditions (wavelength, transform limited pulse length, pulse energy, beam width) the pump pulse broadening caused by imaging errors has to be low and the diffraction efficiency on the CG has to be large.

On the one hand larger initial tilt of the pump pulse front allows the application of lower groove density for the CG which is advantageous from practical point of view. Furthermore, as it will be shown in Section 3, enhancement of the diffraction efficiency can be reached for larger initial tilt. On the other hand larger initial pulse front tilt results in larger imaging errors. The goal is to find a trade-off in geometry with the following features: (i) CG with groove density lower than ~2000/mm, (ii) high diffraction efficiency at the CG, (iii) reduced imaging errors.

In the following the optimization procedure concerning the LN based HTPF will be given. At first, the maximization of the diffraction efficiency at the CG will be discussed followed by the optimization of the imaging system. The examined situations correspond to 1030 nm pump wavelength and 200 fs transform limited pump pulse length. The THz generator crystal was chosen to be 0.6 mol% Mg doped LN in order to avoid photorefraction for the pump beam [24] and minimize THz absorption [25]. The temperature was supposed to be 100 K [26].

3. Maximizing the diffraction efficiency at the CG

Beside the grating Eq. (2) the incidence and diffraction angles at the CG (see Fig. 1) have to satisfy Eq. (3) as well which corresponds to the relation between the initial and the final pulse front tilt. The deduction is found in Appendix A.

 

Fig. 2 Angle of incidence on the CG and the wedge angle versus the grating period for initial tilt angle of${\gamma }_0=66°$.

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As seen in Fig. 2 the wedge angles are significantly lower than for the conventional TPF. The squares belong to the geometry predicting the highest efficiency during the optimization method (see below).

The diffraction efficiency has been calculated by using the GSolver software (Grating Solver Development Company). The software uses the rigorous coupled-wave analysis (RCWA) method [18, 27].

If the grating constant and the incident angle is given the diffraction efficiency belonging to a given diffraction order depends on the two parameters what determine the fine structure of the grating profile namely on the $F=p^{\ast }/p$ filling factor and the h groove depth (Fig. 3(a)).

 

Fig. 3 A schematic figure showing the binary grating profile (a). Diffraction efficiency of the CG versus the filling factor and the groove depth for${\gamma }_0=66°$, ${\theta }_{\text{i2}}=28°$, and $p_2=0.63\text{μm}$(b).

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We performed the detailed efficiency optimization as the function of the grating constant under the constraint of velocity matching condition. For various grating constant values the efficiency was maximized by numerical simulations varying both F and h. From the aspect of fabrication technology F close to 0.5 is advantageous.

Figure 3(b) shows the diffraction efficiency as a function of the groove depth h and the filling factor F in a contour plot for a very promising example of${\gamma }_0=66°$, ${\theta }_{\text{i2}}=28°$, and$p_2=0.63\text{μm}$. As it is seen 94% efficiency can be reached, and advantageously the extension of the high efficiency region is rather large. Furthermore the lower than 0.5 µm groove depth and F ≈0.4 filling factor belonging to the high efficiency region is also advantageous from the point of view of fabrication technology.

The detailed optimization of the diffraction efficiency by varying F and h at fixed $p_2$ (and hence fixed${\theta }_{\text{i2}}$) was performed in the ${\gamma }_0=60-70°$ range. Figure 4(a) shows the highest diffraction efficiency versus ${\gamma }_0$ and ${\theta }_{\text{i2}}$ and Fig. 4(b) shows the corresponding grating period and the necessary wedge angle. The typical F and h values and their tolerance belonging to the high efficiency points of Fig. 4 are very similar to the case shown in Fig. 3(b). The efficiency peaking point of Fig. 3(b) corresponds to the “+” sign and to the encircled points in Figs. 4(a) and 4(b), respectively. As it is seen in Fig. 4(a) the peak efficiency increases monotonously with the initial pulse front tilt. (For larger initial tilt the additional tilt introduced by the CG can be smaller, and as a consequence, the grating constant, too.) For ${\gamma }_0=60°$ peak efficiency of ~70% can be reached at angle of incidence of${\theta }_{\text{i2}}=36°$. For ${\gamma }_0=66°$ the efficiency reaches nearly 95% at incident angle of 28°. For ${\gamma }_0=70°$ efficiency higher than 90% can be reached in the angle of incidence interval of 35 − 50°, and the efficiency approaches 100% for ${\theta }_{\text{i2}}=48°$. As Fig. 4(b) shows there is an abrupt jump in the grating period between ${\gamma }_0=65°$ and 66°. This jump is connected to the jump between the two branches corresponding to the smaller and larger angle of incidence of the efficiency plot in Fig. 4(a). The geometrical parameters corresponding to the high efficiency points are given in Table 2 for a few${\gamma }_0$. This table also contains the parameters belonging to an optimized conventional TPF setup, and to the one consisting of only a simple CG containing RIML as well. Please notice again that the grating constant values belonging to the optimized HTPF schemes are significantly smaller than in the case of simple CG making possible the fabrication of good quality grating structure.

 

Fig. 4 Diffraction efficiency of the CG versus the angle of incidence and initial pulse-front-tilt angle (a). The grating period and wedge angle versus the initial pulse front tilt angle (b). The geometrical parameters belonging to a few different ${\gamma }_0$ are shown in Table 2.

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Table 2. Geometrical parameters belonging to the optimized HTPF, conventional TPF, and the simple CG setups. For HTPF setups parameter values belonging to efficiency maximum are given for initial tilt angles of${\gamma }_0=60,63,66\text{and}\text{70°}$. The groove density of G1 (HTPF setups) and of the optical grating in the conventional setup is 1400/mm. The focal length of the lens is 200 mm in the setups containing imaging. The notation can be clarified by Table 1. Note that the simple CG setup (last line in Table 2) contains RIML.

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The δ wedge angles are advantageously significantly smaller than in the case of the conventional TPF setup (Table 2). This is also an important fact from the point of view of THz beam quality.

Concluding from the above observed tendency of the diffraction efficiency and the abrupt change of the grating constant with the initial pulse–front-tilt angle the optimal initial tilt angle for a practically realizable LN based setup is${\gamma }_0=66°$.

Beside the diffraction efficiency in the CG the resultant THz generation efficiency is affected by the imaging errors as well, hence the examination of this effect is also important.

4. Optimization of imaging optics

Beside the high diffraction efficiency of the CG, efficient THz generation with the HTPF setup requires minimal deviation of the pump pulse length from the transformation limited value along the TPF. This is satisfied, if inside the nonlinear crystal the image of the illuminated area on G1 is tangential to the pulse front [12].

Based on the above considerations a recipe was deduced for the construction of the optimized geometry (details are given in Appendix B). The incident angle on G1, and the distances between the elements of the optimized setup is given by the following formulae:

The $p_2,{\theta }_{\text{i2}},\text{and}{\theta }_{\text{d2}}$parameters are supposed to be known from the diffraction efficiency optimization procedure discussed in the previous section. The corresponding grating periods and the diffraction geometry at the gratings are given in Table 2 for the case of the examined examples. $\gamma$ is determined by the velocity matching condition (1). The $s$ depth inside the crystal, where the most intensive THz generation is expected is also considered as fixed parameter with a value of a few mm.

Exemplarily for LN at ${\lambda }_0=1030\text{nm}$ with parameter values of $n=2.16$, $n_g=2.22$, $1/p_1=1400/\text{mm}$ (the same as that of the reflection grating used in earlier experiments [11]), $f=200\text{mm}$, ${\gamma }_0=66°$, ${\theta }_{\text{i2}}=28°$, ${\theta }_{\text{d2}}=32.8°$ (results of optimization, see Section 3), and $s=3\text{mm}$ one can obtain $k=2.04$, $a=1.17$ from Eqs. (7) and (8), ${\theta }_{\text{i}}=46.3°$from Eq. (4), $s_1=417\text{mm}$from Eq. (5) and $s_2=383\text{mm}$ from Eq. (6).

For the characterization of the effect of imaging errors the local pump pulse length was determined by ray tracing analyses along the TPF in the image plane of G1 inside LN supposing transform limited pulse length of 200 fs. This pulse duration was chosen since recently this value was predicted to be the optimal one. In Fig. 5 the local pump pulse length is shown versus the (x) transversal coordinate (see inset) in the plane of the angular dispersion. The curves unambiguously show the advantage of the HTPF setup to the conventional TPF due to the significantly lower pulse lengthening across the whole beam diameter. The pulse lengthening at the edges of the pump beam is increasing with${\gamma }_0$. This effect has to be taken into consideration together with the results of diffraction efficiency optimization and fabrication technological aspects (Section 3). In the preferred case of ${\gamma }_0=66°$ the pulse length remains below 500 fs for ~10 mm beam diameter, and around 250 fs inside a 5 mm region. Supposing 200 GW/cm² pump beam intensity similarly as used in earlier experiments [11] pumping at the 100 mJ level is available with pump beam cross section of 10 × 30 mm, where the 10 mm measure corresponds to the plane of angular dispersion. Expectedly THz energy exceeding the 1 mJ can be generated by such source. We note that the limitation of the efficiency of TPF schemes originating from cascading effect in conjunction with angular dispersion [28–30] cannot be avoided either by the HTPF method.

 

Fig. 5 Comparison of the conventional TPF and the HTPF setups based on the pump pulse length changes along the diameter (x) of the pump beam in the plane of angular dispersion.

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5. Conclusions

A hybrid TPF THz source was developed, which is a combination of the conventional TPF and the simple contact grating setup. The solution was proposed for the efficient application of THz generator nonlinear materials which requires large pulse-front-tilt on the one hand, and cannot be used as simple CG due to the required large groove density leading to problematic fabrication on the other hand. The tilt of the pulse front is realized in two steps hence the effect of the imaging errors on the pulse length can be significantly reduced. Furthermore the moderated value of the necessary groove density of the contact grating results in unproblematic fabrication.

The developed setup makes possible THz generation with high pump energy i.e. with extended beam size. Instructions were given for the construction of the optimized HTPF setup geometry with minimal imaging errors, and numerical analyses were performed for LN as the nonlinear media. It was shown, that for LN the necessary groove density of the contact grating can be advantageously reduced to the 1500 – 1800/mm range. By detailed numerical analyses it was shown that with practically realizable setup the diffraction efficiency can approach 100%. Further advantage of the proposed scheme is that only a moderate wedge angle is needed, which is advantageous from the point of view of the THz beam quality. Considering the relevant effects (imaging errors, diffraction efficiency) together with aspects of technical implementation a trade-off solution can be find which allows pumping at the 100 mJ level expectedly leading to the generation of single- cycle THz pulses with energy above one mJ.

Appendix A

Figure 6 illustrates the formation of the pulse front tilt in course of the diffraction at the CG. Let the incident beam have initial pulse front tilt of${\gamma }_0$. At time, when point B of the TPF arrives to B’ the point A arrives to A’. This means, that

(A.1)$$\frac{BC+CD}{c}+\frac{DB\text{'}}{c}{n}_g=\frac{AA\text{'}}{c}{n}_g$$

with

(A.2)$$BC=AD\cos \left({\theta }_{\text{i2}}\right)\tan {\gamma }_0,$$

(A.3)$$CD=AD\sin \left({\theta }_{\text{i2}}\right),$$

(A.4)$$DB\text{'}=AD\sin \left({\theta }_{\text{d2}}\right),$$

(A.5)$$AA\text{'}=AD\cos \left({\theta }_{\text{d2}}\right)\tan \gamma .$$

From Eqs. (A.1)-(A.5) one can obtain

(A.6)$$\tan {\gamma }_0=\left(\tan \left(\gamma \right)-\tan \left({\theta }_{\text{d2}}\right)-\frac{1}{n_g}\frac{\sin \left({\theta }_{\text{i2}}\right)}{\cos \left({\theta }_{\text{d2}}\right)}\right)\frac{\cos \left({\theta }_{\text{d2}}\right)}{\cos \left({\theta }_{\text{i2}}\right)}{n}_g$$

for the initial pulse front tilt.

 

Fig. 6 Formation of the pulse front tilt in course of the diffraction at the CG.

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Appendix B

From Eq. (2) follows:

(B.1)$$\Delta {\theta }_{\text{d2}}=-\frac{\cos \left({\theta }_{\text{i2}}\right)}{n\cos \left({\theta }_{\text{d2}}\right)}\Delta {\theta }_{\text{i2}}.$$

The difference between incident angles at the CG belonging to wavelength of ${\lambda }_0$ and ${\lambda }_0+\Delta \lambda$ is (see Fig. 7):

(B.2)$$\Delta {\theta }_{\text{i2}}=\Delta {\epsilon }_2,$$

were due to the imaging

(B.3)$$\Delta {\epsilon }_2=-\frac{s_1-f}{f}\Delta {\epsilon }_1.$$

By using Eqs. (B.1)–(B.3) the $M_A=-\Delta {\theta }_{\text{d2}}/\Delta {\epsilon }_1$ angular magnification (ratio of the transversal and longitudinal magnification) of this imaging can be expressed as

(B.4)$$M_A=-\frac{s_1-f}{nf}\frac{\cos \left({\theta }_{\text{i2}}\right)}{\cos \left({\theta }_{\text{d2}}\right)}.$$

The relation between angles, ${\theta }_{\text{d}}$ and $\theta$ (illustrated in Fig. 7) expressed with the angular magnification:

(B.5)$$\tan \left(\theta \right)=\frac{1}{M_A}\tan \left({\theta }_{\text{d}}\right).$$

Combining Eq. (B.4) and (B.5) one can obtain

(B.6)$$\tan \left(\theta \right)=-\frac{nf}{s_1-f}\frac{\cos \left({\theta }_{\text{d2}}\right)}{\cos \left({\theta }_{\text{i2}}\right)}\tan \left({\theta }_{\text{d}}\right).$$

According to the geometry shown in Fig. 7 $\left(\Delta {\epsilon }_1\text{>0,}\Delta {\epsilon }_2<0,\Delta {\theta }_{\text{d2}}\text{>}0\right)$ :

(B.7)$$b\approx \frac{s_1\Delta {\epsilon }_1+s_2\Delta {\epsilon }_2}{\cos \left({\theta }_{\text{i2}}\right)},$$

and

(B.8)$$b\cos \left({\theta }_{\text{d2}}\right)\approx s\Delta {\theta }_{\text{d2}}.$$

From Eqs. (B.1)–(B.3), (B.7) and (B.8) one can obtain

(B.9)$$s_2=\frac{f{s}_1}{s_1-f}-\frac{s}{n}\frac{{\cos }^2\left({\theta }_{\text{i2}}\right)}{{\cos }^2\left({\theta }_{\text{d2}}\right)},$$

for the lens − CG distance. The grating equation for G1is:

(B.10)$$\sin \left({\theta }_{\text{i}}\right)+\sin \left({\theta }_d\right)=\frac{{\lambda }_0}{p_1},$$

and the beam experiences an angular dispersion of:

(B.11)$$\frac{\text{d}{\epsilon }_1}{\text{d}\lambda }=\frac{1}{p_1\cos \left({\theta }_d\right)},$$

behind G1. This is modified by the angular magnification of the lens to

(B.12)$$\frac{\text{d}{\epsilon }_2}{\text{d}\lambda }=-\frac{1}{p_1\cos \left({\theta }_d\right)}\frac{s_1-f}{f}.$$

The relation between the pulse front tilt and the angular dispersion [31] can be expressed as:

(B.13)$$\tan {\gamma }_0=-{\lambda }_0\frac{\text{d}{\epsilon }_2}{\text{d}\lambda }.$$

Combining Eqs. (B.12) and (B.13) one can obtain

(B.14)$$\tan {\gamma }_0=\frac{{\lambda }_0}{p_1\cos \left({\theta }_d\right)}\frac{s_1-f}{f},$$

In order to fulfill the requirement declared in the first paragraph of Section 4

(B.15)$$\theta =-\gamma$$

has to be satisfied. From Eqs. (B.6), (B.14) and (B.15) the geometry of the optimized setup $\left({\theta }_{\text{i}},s_1,s_2\right)$ can be given. The diffraction and incident angle at G1 can be given as

(B.16)$$\sin \left({\theta }_{\text{d}}\right)=a\frac{{\lambda }_0}{p_{1}n{n}_g}k,$$

and according to Eq. (B.10)

(B.17)$$\sin \left({\theta }_{\text{i}}\right)=\frac{{\lambda }_0}{p_1}\left(1-\frac{ak}{n{n}_g}\right),$$

where

(B.18)$$a=\frac{p_1{n}^2{n}_g^2}{2{\lambda }_0{k}^2}\sqrt{\frac{{\lambda }_0^2}{p_1^2{\tan }^4{\gamma }_0}+\frac{4k^2}{n^2{n}_g^2}}-\frac{n^2{n}_g^2}{2{\tan }^2{\gamma }_0{k}^2},$$

and

(B.19)$$k=\frac{\cos \left({\theta }_{\text{i2}}\right)}{\cos \left({\theta }_{\text{d2}}\right)}\frac{\tan \gamma }{\tan {\gamma }_0}{n}_g$$

coming also from the combination of Eqs. (B.6), (B.14) and (B.15).

 

Fig. 7 Imaging with the lens into the crystal.

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The G1 − lens distance can be given as

(B.20)$$s_1=f\left(\sqrt{a}+1\right),$$

and the lens – CG distance can be determined from Eq. (B.9).

Please notice, that in the case of the “conventional setup” limit, with ${\theta }_{\text{i2}}=0$, $p_2\to \infty$ and $\tan {\gamma }_0=n_g\tan \gamma$ (i.e. $k=1$) the value of a parameter expressed in Eq. (B.19) becomes identical to the $a$ reported in [32], Fülöp et al.

Acknowledgment

Financial support from Hungarian Scientific Research Fund (OTKA) grant number 113083 is acknowledged. The present scientific contribution is dedicated to the 650th anniversary of the foundation of the University of Pécs, Hungary.

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