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Abstract
A nonlinear optical converter of femtosecond laser pulses to terahertz radiation, which combines the tilted-pulse-front pumping and prism coupling techniques, is proposed and experimentally tested. In contrast to the conventional tilted-pulse-front scheme with a prism-shaped LiNbO3 crystal, the converter consists of a plane-parallel LiNbO3 plate sandwiched between two dielectric prisms. One prism is used to couple the pump beam into the LiNbO3 plate, another prism couples the generated terahertz radiation out of the plate. The proposed scheme enables scaling to large-diameter LiNbO3 wafers and large-aperture high-energy pump laser beams resulting in generation of strong terahertz fields. In a proof-of-principle experiment with a 1-mm thick, small size (1 × 2 cm2) LiNbO3 plate pumped by a mJ-class laser, the conversion efficiency comparable to that of the conventional scheme (∼0.1%) was demonstrated.
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1. Introduction
Optical rectification of tilted-pulse-front femtosecond Ti:sapphire laser pulses in LiNbO$_3$ (LN) is a convenient and reliable method to generate strong (up to MV/cm and Tesla level) near-single-cycle electromagnetic fields in the low-terahertz ($\sim$0.1-2 THz) frequency range. Strong terahertz fields in this frequency range are required for a variety of applications, including terahertz-driven electron acceleration [1], ultrafast manipulation of magnetization in materials [2], compression and characterization of ultrafast electron pulses [3], nonlinear terahertz spectroscopy [4], and terahertz-field-induced nonlinear optical microscopy [5]. By using tilted-pulse-front pumping, terahertz pulse energies as high as 0.125 mJ [6], 0.2 mJ [7], and 0.4 mJ [8] were achieved for an optical pump energy of 45 mJ, 70 mJ, and 58 mJ, respectively. Recently, 1.4 mJ terahertz pulses were generated through cryogenically cooling the crystal, tailoring the pump laser spectrum, chirping the pump pulse, and increasing the laser energy to 214 mJ [9].
In the conventional tilted-pulse-front setup [10–12], a prism-shaped LN crystal is used to secure emission of the generated terahertz radiation from the crystal through its tilted (at $\approx 63^\circ$) output face. The pump pulse-front tilt is achieved by diffracting the laser beam off an optical grating, which is imaged into the LN prism by a lens or telescope to reproduce the diffracted pulse at the grating image. The grating image plane and pulse front are adjusted parallel to the output face of the prism.
To generate high terahertz energies, high pump energies are required. Scaling up the pump laser energy requires, however, simultaneously enlarging the LN prism size to avoid the crystal damage and reduce the detrimental effects of high-order nonlinearities [13–16]. For example, two stacked LN prisms with sides as large as $\approx 6$-7 cm and a total 8-cm height were used in the recent record experiment [9]. To further increase the pump energy (up to a J-level), even larger LN prisms should be used, and even so, only a limited increase in the terahertz energy can be achieved [17,18]. Stacking more prisms of a moderate size and pumping by a strongly elliptical laser beam will reduce the focusability of the generated terahertz beam.
The usage of LN prisms in the conventional tilted-pulse-front scheme is rather inefficient. Indeed, due to strong angular dispersion, the pump laser pulse experiences rapid broadening outside the vicinity of the grating image plane. Therefore, only a thin ($\leq$1-mm thick for a $\leq$200-fs pump pulse [19]) layer of the LN crystal in the vicinity of the grating image plane, where the optical intensity is maximal (pulse duration is close to Fourier-limited value), mainly contributes to the terahertz generation [19,20]. To avoid strong terahertz absorption in LN, this working region of the crystal should be close to the output face of the crystal. Thus, considerable part of the crystal volume preceding the grating image plane is not used for terahertz generation; in fact, it plays the role of a matching prism to introduce the pump beam into the working region of the crystal.
If longer laser pulses, with the angular dispersion length comparable to the LN prism size, are used for pumping, the problem of strongly nonuniform terahertz generation across the pump beam comes forward. The terahertz waves generated at the opposite edges of the pump beam (one near the prism apex and the other near the base) have different optical-terahertz interaction lengths. Additionally, the edges of the pump beam experience different nonlinear distortions caused by the back effect of the generated terahertz radiation and self-phase modulation [14–16]. All this leads to a bad quality of the generated terahertz beam with nonuniform intensity and even time dependence across the beam. To solve the problem, the hybrid schemes based on using a plane-parallel LN slab with an echelon structure on its input surface were proposed and demonstrated [21–23]. A scheme with a reflection grating on the back surface of a plane-parallel LN slab was also considered [24].
In this paper, we propose and experimentally test a simpler scheme, where a plane-parallel LN plate is used instead of a LN prism or echelon slab. The LN plate is sandwiched between two prisms – one is dielectric, another is made of high-resistivity Si. The dielectric prism is used to couple the pump beam into the LN plate, instead of the frontal part of the LN prism in the conventional scheme. In particular, a prism-shaped liquid-filled cuvette can be used as the dielectric prism. Alternatively, the dielectric prism can be omitted to allow the pump beam to impinge obliquely on the coated LN plate. We study this case as well. The Si prism couples the generated terahertz radiation out of the LN plate. This prism is required due to oblique incidence of the terahertz radiation onto the output face of the LN plate in the proposed scheme [19]. Commercially available large-diameter LN wafers can be used in the scheme thus enabling scaling up the generated terahertz energy by pumping with large-aperture terawatt-power laser beams.
2. Conversion scheme
The proposed conversion scheme is shown in Fig. 1. A plane-parallel plate of LN of thickness $L$ is sandwiched between two prisms – the input dielectric prism with the optical refractive index $n_d$ and apex angle $\theta$ and the output Si prism with the terahertz refractive index $n_{\rm Si}=3.42$ and apex angle $\psi$. The optical axis of the LN crystal ($z$ axis) is orthogonal to the drawing plane. The pump laser beam is diffracted from a diffraction grating and the pump beam spot on the grating is imaged into the LN plate by a lens or telescope (not shown in Fig. 1) through the entrance face of the input prism. The phase fronts of the pump pulse are parallel to the prism entrance face (normal incidence). The pulse is polarized along the optical axis of the LN crystal. In the input prism, the pulse propagates in the incidence direction and then is refracted into the LN plate, with the optical refractive index $n_{\rm opt}=2.16$, at angle $\varphi$ with respect to the plate normal ($x$ axis). In the plate, the pulse front is tilted at angle $\alpha$, which is close to the noncollinear phase-matching angle $63^\circ$ [11], with respect to the phase fronts. The pulse propagates with group velocity $V_g=c/n_g$ ($n_g=2.23$ is the group refractive index of LN, $c$ is the speed of light) in the direction normal to the phase fronts. The plane of the diffraction grating image is assumed to be coplanar with the $y,z$ plane and situated at a coordinate $x_0$. In further calculations, $x_0$ is set to maximize the generated terahertz energy. Due to strong angular dispersion, the pump laser pulse experiences significant dispersive broadening outside the vicinity of the grating image plane. Therefore, only a thin region in the vicinity of the plane mainly contributes to the terahertz generation [19,20]. The terahertz radiation is emitted perpendicularly to the pump pulse front and incident on the exit surface of the LN plate at angle $\alpha -\varphi$. After refraction from LN with the terahertz refractive index $n_{\rm THz}\approx 5$ [25] into the Si prism at angle $\psi$, the terahertz radiation is transmitted to free space through the prism’s output face in the regime of normal incidence. For $\sin \theta >n_d^{-1}$, the prism should be separated from LN by a 1-2-$\mu$m-thick air gap to ensure total reflection of the optical pump at the LN-air interface and thereby to prevent free-carrier photogeneration in Si. For $\sin \theta <n_d^{-1}$, an optically reflective or absorbing coating of the exit LN surface should be used.
Fig. 1. Optical-to-terahertz converter of tilted-pulse-front laser pulses comprising a LN plate sandwiched between two prisms.
By applying Snell’s law to the dielectric-LN and LN-Si interfaces, we calculated the dependences of the output prism apex angle $\psi$ on the input prism apex angle $\theta$ for different refractive indices $n_d$ of the input prism material [Fig. 2(a)]. In particular, the following values were taken: $n_d = 1$ (no input prism), $n_d = 1.33$ (the prism is a prism-shaped water-filled cuvette), $n_d = 1.5$ (a glass prism or a cuvette filled with organic liquid), $n_d = 2.16$ (the prism material is a linear counterpart of LN), and $n_d = 2.5$ (a chalcogenide glass prism). In Fig. 2(a), three characteristic angles are shown for the curve with $n_d=2.5$. The minimal angle $\theta _{\rm min}$, given by
Fig. 2. (a) The output prism apex angle $\psi$ as a function of the input prism apex angle $\theta$ for different $n_d$ (labeled next to the corresponding curves). (b) The characteristic angles $\theta _{\rm min}$, $\theta _0$, and $\theta _{\rm max}$ as functions of $n_d$.
3. Theoretical analysis
To calculate the generated terahertz radiation, we use the approach based on an analytical solution of the Maxwell equations with a nonlinear source determined by the intensity envelope of the pump pulse [19,26–28]. We neglect the pump pulse distortion due to nonlinear effects, such as self-phase modulation, self-focusing, multiphoton absorption, and the inverse effect of the terahertz radiation on the optical pump. We also neglect material dispersion of LN, which is much weaker than the angular dispersion associated with the tilt of the pump pulse front. Thus, we write the pump pulse duration in the vicinity of the grating image plane as [20]
where $\tau _0$ is the minimal (Fourier-limited) pulse duration, which is achieved at the grating image plane $x=x_0$, and $L_d$ is the dispersion length, which can be conveniently expressed as [19] $L_d[{\rm mm}]\approx 2.3\times 10^{-5}\times \cos \varphi \tau ^2_{0{\rm FWHM}}[{\rm fs}]$ with $\tau _{0{\rm FWHM}}=\tau _02\sqrt {\ln 2}\approx 1.7\tau _0$ the standard full-width-at-half-maximum (the factor $\cos \varphi$ appears as we introduce $L_d$ along the $x$ axis, rather than the propagation direction). Then, the optical intensity envelope in LN is written as [19] with peak value I0, local time $\xi =t-[x\cos (\alpha -\varphi ) + y\sin (\alpha -\varphi )]/(V_g\cos \alpha )$, the transverse (with respect to the propagation direction) profile of the pump beam $G(x,y)=\exp [-(y+x\tan \varphi )^2\cos ^2\theta /a^2]$, $a_{\rm FWHM}=a2\sqrt {\ln 2}$, and the power transmission coefficient of the laser beam from free space through the input prism into the LN plate $T_{\rm entr}$. If antireflection coatings are used to suppress the reflection of the laser beam at the interfaces (as we assume below), then $T_{\rm entr}=\cos \theta /\cos \varphi$. Otherwise, $T_{\rm entr}$ is determined by the Fresnel equations.The nonlinear polarization induced in LN by the pump pulse via optical rectification is directed along the $z$ axis and equals $P^{\rm NL}(x,y,t)=8\pi d_{\rm NL}I(x,y,t)/(cn_{\rm opt})$, where $d_{\rm NL}$ is the nonlinear optical coefficient of LN, $d_{\rm NL}=168$ pm/V [25,29]. The terahertz electric field $E_z(x,y,t)$ generated by the nonlinear polarization inside the crystal can be represented as [19,26–28]
The terahertz energy $W$ emitted to free space (per unit length along the $z$ axis) is found by integration of the $x$ component of the Poynting vector in the Si prism (at $x=L+$) over $-\infty <y<\infty$ and $-\infty <t<\infty$ and multiplying the result by the Si-air Fresnel power transmission coefficient $T_{\rm Si/air}=4n_{\rm Si}(1+n_{\rm Si})^{-2}$. By using Eqs. (4) and (5), this gives $W=\int _{0}^{\infty }w(\omega )d\omega$ with the spectral density
Figure 3(a) shows how the efficiency changes with $\theta$ for different $n_d$. For $n_d\leq 2.16$, the efficiency is zero at the ends of the interval $\theta _{\rm min}\leq \theta \leq 90^\circ$ and has a maximum inside the interval. The maximum is achieved at $\theta \approx \theta _{\rm min}+10^\circ$ for $n_d=1.33$ and $n_d=1$, and at $\theta \approx 63^\circ$ for $n_d=2.16$. Importantly, the maximal efficiency is only $\sim$2-3 times smaller for $n_d=1.33$ and 1 than for $n_d=2.16$. For $n_d > 2.16$, the efficiency increases monotonically with $\theta$ in the interval $\theta _{\rm min}\leq \theta \leq \theta _{\rm max}$.
Fig. 3. (a) The efficiency as a function of $\theta$ for different $n_d$ (labeled next to the corresponding curves). The dashed curve is for the case of uncoated LN entrance surface. (b) The efficiency as a function of the crystal thickness $L$ for different $\theta$ and $n_d$. Other parameters are $\tau _{0 {\rm FWHM}}=100$ fs, $a_{\rm FWHM}=5$ mm, and $I_0=10$ GW/cm$^2$.
The behavior of the curves in Fig. 3(a) results from three $\theta$-dependent factors: (i) Fresnel losses of terahertz radiation at the LN-Si boundary, (ii) pump beam expansion (contraction) by refraction at the input prism-LN boundary, and (iii) variation of the angle between the pump pulse front and grating image plane in LN. The first factor leads to the efficiency drop at $\theta \rightarrow \theta _{\rm min}$ for any $n_d$ as the incidence angle $\alpha -\phi$ of the terahertz radiation on the LN-Si boundary approaches the critical angle of total internal reflection $\approx 44^\circ$. Due to the second factor, the pump beam transmission into the LN plate leads to the beam expansion for $n_d<2.16$ or contraction for $n_d>2.16$. As a result, the optical intensity decreases or increases thus providing a decrease or increase of the efficiency for $\theta \rightarrow \pi /2$ and $\theta \rightarrow \theta _{\rm max}$, respectively. The third factor is only substantial for $\theta >\alpha$ providing a decrease of the efficiency with $\theta$ due to shortening of the interaction length between the pump pulse and generated terahertz wave [19]. In all cases, an antireflection coating increases the conversion efficiency [shown only for $n_d=1$ in Fig. 3(a)].
Figure 3(b) shows that the efficiency increases with the LN crystal thickness reaching saturation more rapidly for larger $n_d$. The latter can be explained by a decrease of the dispersion length $L_d$ with $n_d$ due to the dependence $L_d\propto \cos \varphi$. The saturated values are higher for larger $n_d$. A practical conclusion from Fig. 3(b) is that using crystals thicker than 3-4 mm adds little to the efficiency (using thicker crystals may be beneficial for longer pump pulses).
To determine the parameters of the optical scheme (diffraction grating and telescope) for tilting the pump laser pulse front and imaging it to the LN plate through the input dielectric prism, we modified the formulas [19] as follows
Fig. 4. (a) The grating groove density and (b) telescope demagnification as functions of $\theta$ for different $n_d$ (labeled next to the corresponding curves).
4. Proof-of-principle experiment
As a proof-of-principle experiment, we used a prism-shaped ($\theta =70^\circ$) water-filled cuvette with one of its walls made of a 1-mm thick LN plate and others made of silica (Fig. 5). Due to antireflection coating on the entrance surface of the LN plate the reflection loss at the water-LN interface was reduced to a few percents. A Si prism with $\psi =43^\circ$ was pressed to the exit surface of the LN plate. Tilting the pump pulse front was accomplished by means of a diffraction grating ($\Lambda ^{-1}=2000$ mm$^{-1}$) and telescope ($M=1.8$) consisting of a 36-cm focal-length spherical mirror and 20-cm focal-length achromatic lens. To compare results with the conventional scheme, we changed the cuvette to the standard $63^\circ$ prism-cut LN crystal (right-angle prism with $7.6\times 7.6$ mm$^2$ entrance face) and used another grating with $\Lambda ^{-1}=1700$ mm$^{-1}$. As a light source, two Ti:sapphire laser systems (800-nm wavelength) were used: a commercial 1-kHz amplified laser (Astrella, Coherent) delivering 3-mJ, 60-fs pulses, and a home-made 10-Hz, 10-mJ, 150-fs laser. The laser beam width was varied by means of a telescope placed before the diffraction grating. To increase the 1-kHz laser pulse duration from 60 to 120 fs, a narrowband (10-nm bandwidth) optical filter (FB800-10, Thorlabs) was used. The terahertz power was measured by a Golay cell (GC-1P, Tydex) positioned next to the output face of the Si prism. To measure the terahertz power spectrum, terahertz band-pass filters were inserted between the Si prism and Golay cell.
Fig. 5. Experimental setup. In the LN-plate converter, a water-filled cuvette is used as an input prism.
Figure 6 shows the efficiency of the LN-plate converter as a function of the pump pulse energy for different widths of the laser beam before the grating $d_{\rm FWHM}$ and different pulse durations $\tau _{\rm FWHM}$ (in comparison with the LN prism). For pumping by the 1-kHz laser with $d_{\rm FWHM}=5$ mm and $\tau _{\rm FWHM}=60$ fs [Fig. 6(a)], the converter is $\sim$5 times less efficient than the LN prism. However, unlike the LN prism, the converter efficiency does not saturate at the energies up to 1 mJ (this was the largest available energy due to energy losses in the optical scheme). The 1 mJ energy corresponds to the optical intensity of 90 GW/cm$^2$ just after the diffraction grating and 240 GW/cm$^2$ in the LN prism. In the LN prism, the saturation is caused by the distortion of the pump pulse due to nonlinear effects of self-phase modulation and sum- and difference-frequency generation between the optical and terahertz fields [14–16]. The absence of the saturation in the LN-plate converter can be explained by a reduction of the optical intensity due to a pump beam expansion after refraction at the water-LN boundary (it equals 200 GW/cm$^2$ at the 1 mJ pump energy and the intensity of 160 GW/cm2 just after the grating).
Fig. 6. The efficiency as a function of the pump pulse energy for the LN-plate converter (labeled "Plate") and the conventional LN prism (labeled "Prism"). The parameters are (a) $d_{\rm FWHM}=5$ mm and $\tau _{\rm FWHM}=60$ fs, (b) $d_{\rm FWHM}=3.7$ mm, $\tau _{\rm FWHM}=60$ fs (hollow circles and squares) and 120 fs (filled circles and squares), (c) $\tau _{\rm FWHM}=150$ fs, $d_{\rm FWHM}=6$ mm (hollow triangles and stars) and 4 mm (filled stars).
Reducing the pump beam width to $d_{\rm FWHM}=3.7$ mm leads to a higher slope of the initial linear segment of the curves, lower saturation energy for the LN prism (with practically the same saturation intensity of 220 GW/cm$^2$ in LN), and appearance of saturation for the LN-plate converter [Fig. 6(b)]. The maximum efficiency for $d_{\rm FWHM}=3.7$ mm is practically the same as for $d_{\rm FWHM}=5$ mm [Fig. 6(a)].
Increasing the pump pulse duration to $\tau _{\rm FWHM}=120$ fs leads to a higher efficiency for both schemes [Fig. 6(b)]. Since the effect is more pronounced for the LN-plate converter, the ratio of the schemes’ efficiencies reduces to only $\sim$3 at 0.3 mJ, with the efficiency of the LN-plate converter as high as $\sim$0.03$\%$.
To study the regime of pumping by longer and higher energy optical pulses, the 10-Hz laser system was used. For $d_{\rm FWHM}=6$ mm, both schemes have approximately the same maximum efficiency $\sim$0.07$\%$ [Fig. 6(c)]. It is, however, achieved at different pump pulse energies, i.e., $\sim$0.7 mJ for the LN prism and $\sim$(2.5-3) mJ for the LN-plate converter (the corresponding peak optical intensities in LN are $\sim$50 and $\sim$150 GW/cm$^2$). Thus, the maximum terahertz energy, achieved at the pump pulse energy of 3 mJ, was even larger for the LN-plate converter. By reducing the pump beam width to 4 mm, we achieved a higher slope of the efficiency growth for the LN-plate converter at the pump pulse energies smaller than 1.5 mJ but with saturation on the level of 0.06$\%$ at 1.5 mJ.
The efficiency of generating four spectral components of terahertz radiation (around 0.3, 0.5, 1, and 2 THz) are shown in Fig. 7(a) and (b). For the pump energies $\geq$0.5 mJ, the generation efficiency of high frequency components by the LN prism saturates and even drops [Fig. 7(b)], whereas the LN-plate converter continues to generate these components with a growing efficiency [Fig. 7(a)]. For the pump energies below the saturation, the power spectra are practcially coincide for both schemes [Fig. 7(c)].
Fig. 7. The efficiency at 0.3 THz (black circles), 0.5 THz (blue diamonds), 1 THz (green squares), and 2 THz (red crosses) as a function of the pump pulse energy for the LN-plate converter (a) and LN prism (b). (c) The normalized power spectrum for the LN-plate converter (red circles) and LN prism (blue diamonds) at the pump energy below the saturation. The parameters are $d_{\rm FWHM}=5$ mm and $\tau _{\rm FWHM}=60$ fs (1-kHz laser).
To estimate the quality of the output terahertz beam, we measured its width in the plane of drawing in Figs. 1 and 5 near the output face of the Si prism and at the distance of 17 cm from the face. The width was $D_{0 {\rm FWHM}}\approx 2$ mm near the prism face and $D_{\rm FWHM}\approx 2.5$ cm at the 17-cm long distance. This agrees well with the theoretical estimate $D_{\rm FWHM}=D_{0{\rm FWHM}}[1+(z/z_R)^2]^{1/2}\approx 2$ cm, where the Rayleigh length $z_R=2\pi (D_{0{\rm FWHM}}/1.7)^2/\lambda _{\rm THz}\approx 1.7$ cm is taken at the terahertz wavelength $\lambda _{\rm THz}=0.5$ mm. Thus, one can conclude that the terahertz beam was close to collimated.
5. Conclusion
To summarize, the proposed LN-plate converter provides the efficiency and spectral characteristics comparable to those of the conventional tilted-pulse-front scheme and high quality of the generated terahertz beam. Additionally, the converter has the potential of scaling up the generated terahertz energy by using large-aperture terawatt-power pump laser beams and commercially available large-diameter LN wafers, instead of expensive large-size LN prisms necessary for the conventional tilted-pulse-front scheme. An interesting possibility, which should be verified in future experiments, is omitting the input dielectric prism and introducing the pump beam directly to a LN wafer through its antireflectively coated front face. The rear face of the wafer needs to be reflectively coated for the pump beam to prevent free-carrier generation in the output Si prism.
Funding
Ministry of Science and Higher Education of the Russian Federation (0729-2020-0035); Russian Foundation for Basic Research (20-32-90080).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.
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