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We demonstrate experimentally the efficiency of a recently proposed scheme of terahertz generation based on Cherenkov emission from ultrashort laser pulses in a sandwich structure. The structure has a thin nonlinear core covered with a prism of low terahertz absorption. Using an 8 mm long Si-LiNbO3-BK7 structure with a 50 μm thick LiNbO3 core, we converted 40 μJ, 50 fs Ti:sapphire laser pulses into terahertz pulses of ~ 3 THz bandwidth with a record efficiency of over 0.1%.
©2009 Optical Society of America
1. Introduction
Amplified ultrashort laser systems provides a variety of ways for tabletop generation of high-energy terahertz pulses [1]. The recently proposed [2] method of terahertz generation via mixing of the optical fundamental and the second harmonic in a laser produced plasma is still under development [1], although terahertz pulses of as high energy as ~ 30 nJ with 0.3 – 7 THz spectrum were already generated with 0.5 mJ Ti:sapphire laser pulses in ionized air [3]. Commonly used are biased photoconductive switches gated by femtosecond laser pulses. Such devices can emit up to 400 nJ (at 1 kHz repetition rate) [4] and 800 nJ (at 10 Hz rate) [5] terahertz pulses at bias voltages up to 45 and 11 kV, respectively. However, the spectral peak of the pulses generated by photoconductive switches typically lies well below 1 THz (in the 0.3-0.5 THz range, in Refs. [4, 5]). Alternatively, optical rectification of femtosecond laser pulses in electro-optic crystals is an established way of generating broadband terahertz radiation.
A widely used scheme of optical rectification is illumination of a slab of ZnTe by a large (as compared to the terahertz wavelength) aperture beam of Ti:sapphire laser pulses (~ 800 nm wavelength). This results in the phase-matched excitation of ~ 2.5 THz quasiplane wave propagating collinearly to the laser beam [6, 7, 8]. However, ZnTe as an emitter has several drawbacks, such as a moderate (as compared to some other materials [9]) nonlinear coefficient, a relatively strong terahertz absorption (~10 cm-1) at room temperature [10, 11], and strong two-photon absorption of the Ti:sapphire laser radiation at high laser intensities. The latter factor leads to saturation of terahertz yield at high laser intensities [7] and, thus, the optical-to-terahertz conversion efficiency in ZnTe is typically ≤ 10-6 – 10-5. For example, the highest terahertz pulse energy of 1.5 μJ measured from a ZnTe emitter was obtained with 48 mJ Ti:sapphire laser pulses. This corresponds to the conversion efficiency of 3 × 10-5 [8].
A promising material for optical-to-terahertz conversion is LiNbO3. Its nonlinear coefficient is ~ 2.5 times larger than that of ZnTe [9]. LiNbO3 has a wider bandgap and, thus, does not suffer from two-photon absorption, when illuminated by a Ti:sapphire laser. However, collinear velocity matching is impossible in this material because its terahertz refractive index is more than two times larger than the optical group refractive index. Additionally, LiNbO3 has a strong terahertz absorption at room temperature ~ 16 – 170 cm -1 at 1 – 2.5 THz [9, 12].
To compensate the velocity mismatch, quasi-phase-matching schemes based on periodically poled lithium niobate (PPLN) structures are used [13, 14]. They indeed provide an increase in the terahertz yield but only via generation of multi-cycle (narrow-band) terahertz wavepackets, not increasing the peak terahertz power. A promising way to achieve phase matching in LiNbO3 is to use pump pulses with tilted intensity front [15]. This approach has yielded single-cycle terahertz pulses with 3.3 μJ energy at 1 kHz repetition rate [16] and 30 μJ energy at 100 Hz repetition rate [17]. The conversion efficiency reached record values of 7 × 10-4 [16] and 10-3 [17]. A drawback of using tilted-front pulses is the short propagation distance of such pulses due to significant diffractive distortions. This restricts the thickness of the nonlinear crystal by a few millimeters. A strong terahertz absorption in LiNbO3 still remains an essential limitation.
The simplest way to achieve phase-matching in such materials as LiNbO3 is the Cherenkov radiation mechanism [18]. To produce a Cherenkov cone of terahertz waves, the optical pulse should be focused to a size of the order of (or smaller than) the terahertz wavelength. Focusing into a line allows one to scale up the generated terahertz energy and produce more convenient for applications Cherenkov wedge, rather than cone [19, 20]. A strong terahertz absorption in LiNbO3 can be minimized by aligning the laser beam parallel with and near the lateral surface of the crystal and putting, additionally, a Si-prism coupler on the surface to output the Cherenkov radiation from the crystal [21]. As a further improvement of the Cherenkov scheme, it was proposed recently [22] to sandwich a thin layer of LiNbO3 (or another nonlinear material) between two prisms (or a prism and substrate) made of a material (or materials) with low terahertz absorption (for example, high-resistivity Si). This sandwich structure constrains the laser pulse within the core and provides its guiding for a long distance. Therefore, not only the structure reduces the terahertz losses but also increases the length on which terahertz radiation is emitted.
In this paper, we demonstrate experimentally that the proposed in Ref. [22] scheme is, indeed, very efficient. Using a Si-LiNbO3 -BK7 structure, that is more simple for fabrication than the Si-LiNbO3-Si structure considered in Ref. [22], we obtained a record conversion efficiency of over 0.1% with a μJ-level pump. Our theoretical results agree well with the experiment and show ways to further improving the efficiency.
2. Sandwich preparation and experimental setup
Schematics of the Si-LiNbO3-BK7 sandwich structure and experimental setup is shown in Fig. 1. To prepare the sandwich, a slab of congruent LiNbO3 was glued to a BK7 glass substrate. After its front and back facets were polished, the slab was ground down to a desirable thickness and its upper surface was also polished. Using this procedure, we prepared two structures with LiNbO3 layers of 30 and 50 μm thickness. The length of both structures was 8 mm. This length is approximately equal to the characteristic length of filamentation instability at the typical pump intensity of ~ 100 GW/cm2 [22]. Using longer structures at such optical intensities could cause breakdown damage of the LiNbO3 layer. To complete the sandwich, a prism with base angle of 45° made of a high-resistivity Si was pressed against the upper surface of the LiNBO3 layer in orientation shown in Fig. 1.
As an optical pump we used amplified Ti:Sapphire laser system Tsunami, Spitfire Pro (Spectra-Physics) which delivered 50 fs long pulses with 780 nm central wavelength at 1 kHz repetition rate. The laser pulse energy was varied in the range of 0 – 130 μJ by use of a polarization attenuator. To study the effect of laser pulse duration on the terahertz generation, we used 50 and 200 fs pulses, the latter were obtained by dispersive broadening of the 50 fs pulses in a 6.5 cm thick glass plate. The laser beam had a diameter of 5.3 mm (FWHM) and was focused onto the entrance facet of the LiNbO3 core by a cylindrical lens (Fig. 1). The focal length of the lens was chosen, according to calculations of Ref. [22], to optimize the excitation of the fundamental mode of the dielectric slab waveguide. The width of the focal line was about 13 and 21 μm, respectively. The accuracy of the laser beam focusing was monitored by translating (with ~ 10 times amplification) the optical intensity distribution from the exit facet of the core to a CCD camera with use of a short-focus lens. The optical spectrum after propagation through the structure was measured by a spectrometer S150 (Solar Laser Systems).
According to Ref. [22], terahertz radiation is generated most efficiently by the y-component of the nonlinear polarization produced in the LiNbO3 core by the optical pulse (Fig. 1). To maximize this component, the optical axis of the crystal and the laser beam polarization were directed along the y-axis.
Electro-optic sampling with a 3 mm thick ZnTe crystal was used to characterize the temporal profile of the terahertz pulses emitted from the Si-prism. A liquid-He-cooled InSb hot electron bolometer (QMS Instruments Ltd.) was used to measure the energy of the terahertz pulses. Terahertz radiation was collected into the entrance window of the bolometer by a teflon lens. Since the bolometer was calibrated by the manufacturer only for cw radiation, we calibrated it for short terahertz pulses (of ~ 1 ps duration). For this purpose, we used ≈ 30 ns pulses at ~ 100 GHz that were produced by gating a cw source of mm radiation with a pulse voltage generator. Since the pulse duration (≈ 30 ns) is an order of magnitude smaller than the response time of the bolometer (~ 300 ns), the pulses can serve as a reference signal for short pulse bolometer calibration. Using two reference signals at 91.6 and 118.1 GHz (to average a possible frequency dependence of the bolometer responsivity) with known repetition rates and average powers we derived the bolometer pulse responsivity of 3 V/nJ. To verify the calibration, we measured terahertz energy generated in a 2 mm thick ⟨110⟩-cut ZnTe crystal by 50 fs Ti:sapphire laser pulses and compared the result with Ref. [8]. In Ref. [8], a 0.5 mm thick ⟨110⟩-cut ZnTe crystal and 30 fs Ti:sapphire laser pulses were used, and terahertz energy was measured by calibrated pyroelectric detector (Coherent-Molectron). For the same optical fluence of 100 μJ/cm2, the measured conversion efficiency was ~1.5× 10-5 in Ref. [8] and ~2× 10-5 in our experiment. The minor discrepancy (by a factor of 1.3) between these two values can be attributed to different thickness of the crystals. Indeed, according to Ref [23], the conversion efficiency for a 2 mm ZnTe crystal should be ~ 2 times as high as for a 0.5 mm one.
3. Theoretical model and calculation
We adapted the theory, developed in Ref. [22] for a symmetric Si-LiNbO3-Si structure, to an asymmetric Si-LiNbO3-BK7 structure using the same approximations as in Ref. [22]. The structure, optical beam, and terahertz fields are treated as two-dimensional [independent of y (Fig. 1)]. The optical pulse propagates in the core (∣x∣ < a/2) without distortions as a fundamental mode of the dielectric slab waveguide with a transverse intensity profile ∝ cos2(πx/a) and Gaussian temporal envelope. Transient effects at the entrance (z = 0) and exit (z = L) boundaries of the sandwich are neglected.
The nonlinear polarization produced by the optical pulse in the core P NL (x, t - ngz/c), with ng = 2.23 the optical (at 800 nm wavelength) group refractive index of LiNbO3 [9], is written in terms of the optical intensity and the nonlinear coefficient of LiNbO3 d 33 = 168 pm/V [9]. To find the terahertz radiation emitted by the moving polarization P NL (x, t - ngz/c), we apply the Fourier transform with respect to t - ngz/c to Maxwell’s equations, solve the resultant equations in the homogeneous regions ∣x∣ < a/2 and ∣x∣ > a/2, match the solutions by the boundary conditions at x = ±a/2, and then take the inverse Fourier transform (see details in Ref. [22]). The optical-to-terahertz conversion efficiency is calculated as a ratio of the terahertz energy emitted through the output surface of the Si-prism to the energy of the pump laser pulse. The terahertz reflection at the output surface of the Si-prism (30%) and optical reflection at the entrance facet of the LiNbO3 core (14%) are included. This reduces the efficiency by a factor of ≈ 0.53. When calculating the efficiency we take into account that the laser beam has a finite width in the y-direction Since the terahertz energy is proportional to the square of the optical intensity, its y-profile is more narrow (by a factor of 1/√2 ≈ 0.7 for a Gaussian profile) than the optical intensity profile [7, 24]. This reduces the efficiency by a factor of ~ 0.7.
For the refractive indices (n LN, n Si, and n BK7) and amplitude absorption coefficients (α LN, α BK7, and α Si) of the materials (congruent LiNbO3, Si, and BK7, respectively) in the terahertz range we use following formulas (v is in THz): n LN = 5.12+ 0.027v 2+ 0.0063 v 4 [12], α LN[cm-1] = 24.8 +2.15v 2 + 5v 4 (fitting of the data of Ref. [12]), n Si = 3.418 [25], α Si[cm-1] = 6 × 10-3 v 2 (fitting of the data of Ref. [25]), n BK7 = 2.62 + 0.017v + 0.03v 2 (fitting of the data of Ref. [26]), α BK7[cm-1] = 24v 2 (fitting of the data of Ref. [26]). The optical refractive index of LiNbO3 is 2.16 [9].
Figure 2(a) shows the spatial distribution of the electric field Ey calculated for a Si-LiNbO3-BK7 structure with a = 50 μm. The radiation pattern is strongly asymmetric. The opening angle of the Cherenkov wedge is θ Si = arcsin(ng/n Si) ≈ 41° in Si and θ BK7 = arcsin(ng/n BK7) ≈ 58° in BK7. In the substrate, the terahertz field attenuates rapidly with distance from the core due to strong terahertz absorption in BK7, while in the Si-prism the field forms a long Cherenkov wedge. The gradually fading oscillations in the field distribution across the Cherenkov wedge can be attributed to multiple reflections of the generated in the core terahertz waves at the core-prism and core-substrate interfaces.
Figure 2(b) shows the calculated spectra of the terahertz field in the prism (at x = a/2 + 0) and substrate (at x = -a/2 - 0). The spectra have the same maximum frequency of 1.2 THz and the same bandwidth of ~ 0.1 – 2.5 THz. There is only a minor divergence between the spectra near 0.6 THz. The spectra are similar to the spectrum for a symmetric Si-LiNbO3-Si structure (Fig. 2(b) and Ref. [22]). This similarity can be explained by the proximity of the terahertz refractive indices of BK7 and Si. We will exploit the similarity of terahertz generation in Si-LiNbO3-BK7 and Si-LiNbO3-Si structures referring to some results of Ref. [22]. In particular, we will refer to a prediction that the maximum frequency of the terahertz spectrum scales as ∝= a -1 for short laser pulses with duration τ FWHM ≪ a(n 2 LN -n 2 g)1/2/(5c) [or τ FWHM(fs) ≪ 2.8a, with a in μm].
Fig. 2. (a) Snapshot of the electric field Ey produced in a Si-LiNbO3 -BK7 structure with a = 50 μm by a Ti:sapphire laser pulse with duration of 50 fs and peak intensity of 100 GW/cm2. (b) Corresponding terahertz field spectra in Si (solid) and BK7 (dotted). The spectrum for a Si-LiNbO3-Si structure (dashed) is shown for reference. The spectrum generated in the Si-LiNbO3-BK7 structure by a 10 μm wide laser pulse (dashed-dotted) models the effect of self-focusing instability (the curve is reduced by a factor of 2). All spectra have the same normalization.
To model the effect of self-focusing instability, that can develop at high optical intensities, on the terahertz generation, we plotted in Fig. 2(b) the terahertz spectrum generated in the structure with a = 50 μm by a laser pulse with the width (FWHM) of 10 μm, that is 2 times smaller than the width of the fundamental mode of the dielectric slab waveguide (21 μm, Sec. 2). It is seen from Fig. 2(b) that a decrease in the transverse size of the optical pulse leads to an enhanced generation of higher frequency components in the terahertz spectrum. The dips in the spectrum at 0.65 and 1.9 THz can be attributed to the destructive interference of the terahertz waves reflected from the core boundaries. The position of the dips (interference minimums) is given by the formula vk = c(2an LNcosθ LN)-1(2k- 1), where θ LN ≈ 26° is the opening angle of the Cherenkov cone in LiNbO3 and k= 1,2,…
4. Experimental results and discussion
Figure 3(a) shows experimental high-resolution terahertz spectra generated by a 50 fs Ti:sapphire laser pulse in the structures with a = 50 and 30 μm. The corresponding terahertz waveform (for a = 50 μm) is shown in the inset. The numerous narrow dips in the spectra are identified as absorption lines of water vapor. For a = 50 μm, the position of the main maximum of the spectrum at 0.8 – 1.2 THz agrees well with our theory [cf. Fig. 2(b)]. For a = 30 μm, the spectrum maximum is shifted to ~ 1.5 THz in accord with the theoretical prediction of Ref. [22] for short (with τFWHM ≪ 84 – 140 fs, for a = 30 – 50 μm) laser pulses (Sec. 3). What is not predicted by the quasilinear theory of Ref. [22] is the second spectrum maximum at 2 – 2.5 THz [Fig. 3(a)], the similar maximum exists in the spectra for 200 fs laser pulses [Fig. 3(b)]. The increase of the maximum with increasing pump energy [Fig. 3(b)] allows us to attribute this effect to self-focusing instability of the optical pulse. Indeed, such an instability, developing at high optical intensities, causes a decrease in the transverse size of the optical pulse. According to Fig. 2(b), this should enhance the generation of higher frequency components in the terahertz spectrum (up to 3 THz at decreasing the transverse size to 10 μm). The position of the interference dip in Fig. 3 agrees well with Fig. 2(b) and the analytical formula for vk (Sec. 3).
Fig. 3. (a) Normalized terahertz spectra generated in the structures with a = 50 (solid) and 30 μm (dotted) by a 50 fs laser pulse with energy of 17 and 10 μJ, respectively. Inset: corresponding terahertz waveform for a = 50 μm. (b) Terahertz spectra generated in the structure with a = 50 μm by a 200 fs laser pulse with energy of 9 (solid), 30 (dashed), and 130 μJ (dotted).
We verified that, in accord with our theory, the terahertz radiation is linearly polarized -along the y-axis for the optimal, in the same direction, laser polarization. When we rotated the direction of the laser polarization, a monotonic decrease in the terahertz signal was observed. For example, for the laser polarization in the x-direction, the peak terahertz signal was 30 times smaller than for the optimal polarization.
Figure 4(a) shows the conversion efficiency as a function of the pump pulse energy. To take into account a limited spectral range of the bolometer (> 1.5 THz), we corrected the measured terahertz energy using a shape of the spectrum (in a.u.) obtained with electro-optic sampling (the bolometer data were multiplied by the ratio of the area under the whole spectrum to its part below 1.5 THz). For low pump energies, the conversion efficiency grows linearly with the energy, the saturation starts at lower energies for smaller a and τ FWHM [Fig. 4(a)]. Maximum conversion efficiency ~ 0.12% is achieved for three combinations of a and τ FWHM [Fig. 4(a)]. For a = 30 μm and τ FWHM = 50 fs, the efficiency saturates at a lower level ~ 0.075%. In terms of the optical intensity, the saturation occurs at a 300–500 GW/cm2 level for all curves in Fig. 4(a). (The saturation was observed also in measuring the terahertz pulses with electro-optic sampling). Similarly to Ref. [20], the physical nature of the saturation can be attributed to three-photon absorption of the pump optical pulses in LiNbO3. In the experiment, three-photon absorption manifested itself also in a decrease of the transmitted through the structure optical energy with increasing the pump energy.
The discrepancy between the experimental and theoretical dependencies in Fig. 4(a) can be explained by the following reasons. First, in the experiment, only 30% of the incident optical energy was transmitted through the sandwich at low pump energies. This transmitted energy is 2.5 times smaller than 75% expected from the optical reflection at the entrance and exit facets of the core. Additional optical losses, that can be attributed to imperfect laser focusing onto the entrance facet and scattering of the light on the core defects, reduce, evidently, the conversion efficiency. Second, there was, probably, an air gap between the core and the Si-prism, that increased the reflection of the generated in the core terahertz waves from the LiNbO3-Si interface and their further absorption in the BK7 substrate.
It is interesting to estimate a gain in efficiency provided by the sandwich structure as compared to terahertz generation in bulk LiNbO3. Due to high terahertz absorption in congruent LiNbO3 (~ 30 cm-1) only a ~ 0.25 mm thick layer near the exit boundary of the bulk crystal can contribute to the terahertz emission. In the sandwich structure, terahertz radiation is emitted from the whole length (8 mm) of the structure. Thus, the total gain in efficiency can be estimated as ~ 8/0.25 = 32. The relative contributions to the overall gain from the two factors - lower terahertz attenuation and longer propagation length of the laser pulse in the structure (Sec. 1) - depend on the core thickness. For the structures used in the experiment with a = 30 and 50 μm the laser beam width was about 13 and 21 μm, respectively (Sec. 2). For such beam widths, the double Rayleigh length, i.e., the propagation length of the laser pulse in the bulk crystal, equals 2 and 5 mm, respectively. Therefore, the efficiency gain due to guiding is 8/2=4 (8/5=1.6) for a = 30 (50) μm. The rest of the efficiency gain is related to the reduced terahertz attenuation.
Fig. 4. (a) Conversion efficiency as a function of the laser pulse energy for a = 50 μm and τFWHM = 200 fs (filled red circles), a = 30 μm and τFWHM = 200 fs (empty blue circles), a = 50 μm and τFWHM = 50 fs (filled green squares), and a = 30 μm and τFWHM = 50 fs (empty black squares). (b) Relative red shift of the optical pump spectrum versus the laser pulse energy for a = 30 μm and τFWHM = 200 fs. Solid line is a linear fit to the experimental points for the energies < 30 μJ. Inset: an output optical spectrum S(λ) for the pump energy of 29 (solid) and 0.5 μJ (dotted). The dashed straight lines in (a) and (b) show the corresponding theoretical dependencies for a = 30 μm and τFWHM = 200 fs.
Efficient optical-to-terahertz conversion should result in a red shift of the pump optical spectrum (a photon from the high-frequency wing of the spectrum decays into a terahertz photon plus a red-shifted photon corresponding to the low-frequency wing of the spectrum) [27]. Such a shift, indeed, observed in our experiments (for the scheme with tilted-front pulses the similar effect was reported in Refs. [28, 29]). Figure 4(b) shows the relative shift Δλ̄/λ̄ of the central optical wavelength λ̄ measured for a = 30 μm and τ FWHM = 200 fs as a function of the pump pulse energy. The central wavelength λ (the center of weight of the spectrum) was calculated using the formula [29]
with S(λ) the measured optical spectrum after the sandwich. The spectrum S(λ) is shown in the inset to Fig. 4(b) for two pump energies - 0.5 and 29 μJ. For the lower pump energy (0.5 μJ), the spectrum is slightly distorted with respect to the spectrum of the incident laser pulse (not shown), its central wavelength is practically not shifted. For the higher pump energy (29 μJ), the distortion is more pronounced and a substantial boost of spectral components on the red side of the spectrum is observed. This leads to a red shift of the central wavelength λ̄. According to Fig. 4(b), the relative red shift depends linearly on the pump pulse energy for the energies < 30 μJ. This dependence agrees with the second-order nature of the optical-to-terahertz conversion. At ~ 30 μJ the linear growth of the red shift stops. This effect may be attributed to three-photon absorption of the optical pump and corresponding photogeneration of free carriers in LiNbO3. Indeed, a laser pulse experiences a blue shift when it propagates through a plasma with growing in time density (see, for example, [30]). According to our estimation of the photogenerated carrier density, the blue shifting process can compensate the linear growth of the red shift at pump energies > 30 μJ.
In Fig. 4(b), a theoretical dependence of the red shift (dashed straight line) was calculated as a ratio of the emitted from the core (to the prism and substrate) terahertz energy to the optical energy inside the core. Indeed, the emitted terahertz energy equals a decrease in the optical pulse energy and, therefore, defines (at a constant number of optical photons [27]) a decrease in the average photon energy (frequency). The discrepancy between the theoretical and experimental dependencies in Fig. 4(b) can be partially explained by the above mentioned additional optical losses.
5. Conclusion
We have demonstrated experimentally the high efficiency of the optical-to-terahertz conversion scheme proposed in Ref. [22]. Using a Si-LiNbO3-BK7 structure, we converted 40 μJ, 50 fs Ti:sapphire laser pulses into broadband (of ~ 3 THz bandwidth) terahertz pulses with a record efficiency of over 0.1%. This efficiency is higher than that for the tilted-front laser pulses [16, 17] at a μJ-level. The insensitivity of the scheme with the sandwich structure to the optical wavelength allows one to use a fiber laser as a pump. To preserve a high optical intensity at a lower pump energy, focusing of the laser beam in the plane of the sandwich structure (along the y-axis in Fig. 1) may be used.
A further increase in the conversion efficiency can be achieved by changing congruent LiNbO3 with Mg-doped stoichiometric composition of the crystal, by using quartz substrate (that will totaly reflect the generated in the core terahertz radiation) instead of BK7 glass, and by putting an antireflective coating onto the entrance facet of the sandwich structure.
Acknowledgement
This work was supported in part by RF President Grant No. MK-3749.2008.2 and RFBR Grant Nos. 08-02-99052, 08-02-00988, and 08-02-92216. S.B.B. also acknowledges partial support from the fund “Dynasty”.
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