Detailed analysis of the tilted-pulse-front pumping scheme used for ultrashort THz pulse generation by optical rectification of femtosecond laser pulses is presented. It is shown that imaging errors in a pulse-front-tilting setup consisting of a grating and a lens can lead to a THz beam with strongly asymmetric intensity profile and strong divergence, thereby limiting applications. Optimized setup parameters are given to reduce such distortions. We also show that semiconductors can offer a promising alternative to LiNbO3 in high-energy THz pulse generation when pumped at longer wavelengths. This requires tilted-pulse-front pumping, however the small tilt angles allow semiconductors to be easily used in such schemes. Semiconductors can be advantageous for generating THz pulses with high spectral intensity at higher THz frequencies, while LiNbO3 is better suited to generate THz pulses with very large relative spectral width. By using optimized schemes the upscaling of the energy of ultrashort THz pulses is foreseen.
©2010 Optical Society of America
CorrectionsJózsef András Fülöp, László Pálfalvi, Gábor Almási, and János Hebling, "Design of high-energy terahertz sources based on optical rectification: erratum," Opt. Express 19, 22950-22950 (2011)
Tilting the pump pulse front has been proposed for efficient phase-matched THz generation by optical rectification (OR) of femtosecond laser pulses in LiNbO3 (LN) . By using amplified Ti:sapphire laser systems for pumping, this technique has recently resulted in generation of near-single-cycle THz pulses with energies on the 10-µJ scale [2, 3]. Such high-energy THz pulses have opened up the field of sub-picosecond THz nonlinear optics and spectroscopy [4, 5, 6, 7, 8]. Besides this emerging research field also many other promising applications, such as single-shot THz imaging, THz radar, THz electron spin resonance, etc. require THz pulses with even higher energies. THz generation by OR is becoming a versatile and widely used technique, where further upscaling of the THz energy can be expected . Furthermore, triggered by the development of compact femtosecond laser sources operating in the infrared spectral range, a trend towards using such longer wavelengths for pumping OR-based THz sources can be recognized [10, 11, 12, 13, 14].
With increasing wavelength the group index decreases, and for most of the suitable materials the use of the tilted-pulse-front pumping (TPFP) scheme is required. However, a detailed analysis of this scheme and its design details are not yet given in the literature. Therefore, the goal of the present work is to give a guideline in designing high-energy ultrafast THz sources based on OR of femtosecond pulses, and to assess the possibilities of further upscaling the THz energy. The key issues are selection of the nonlinear material, optimal pumping conditions (wavelength, pulse duration, intensity), and optimization of the pulse-front-tilting setup.
The paper is organized as follows. In Sec. 2 the theoretical model used for describing THz generation is given. In Sec. 3 we address the effects of the longitudinal spatial variation of pump pulse duration on the THz generation process in the TPFP scheme. In Sec. 4 the performance of TPFP THz sources will be analyzed in terms of the output THz beam characteristics. In Sec. 5 optimization of the pulse-front-tilting setup consisting of a grating and a lens will be presented. In Sec. 6 the performance of selected semiconductor nonlinear materials suitable for OR will be compared to that of LN.
2. Theoretical model
Efficient THz generation by OR requires phase matching, i.e. matching the group velocity of the optical pump pulse to the phase velocity of the THz radiation [1, 15]. Collinear velocity matching is possible only for specific wavelengths and materials (see Sec. 6). For most of the commonly used laser wavelengths and materials the group velocity v g(ω) of the pump pulses differs from the phase velocity v(Ω) of the THz field, where ω and Ω are the optical pump and THz frequencies, respectively. If n g(ω) < n(Ω), noncollinear velocity matching can be achieved by tilting the pump pulse front . Here, n is the refractive index and n g = c/v g is the group index, c is the speed of light in vacuum. In this case, velocity matching reads as
where ω 0 and Ω0 are the phase-matching pump and THz frequencies, respectively, and γ is the required pulse front tilt (PFT) angle.
In order to assess the suitability of different nonlinear materials for THz generation in the TPFP scheme and to compare them to each other in a quantitative way we have performed numerical calculations. In the calculations we have taken into account
- the noncollinear propagation direction of pump and THz beams,
- the absorption at THz frequencies due to the complex dielectric function (determined by phonon resonances) and to free carriers generated by pump absorption.
These effects were included into the one-dimensional equation for the Fourier component E(Ω, z) of the THz field [15, 18] at the angular frequency Ω, which follows from Maxwell’s equations in the slowly varying envelope approximation:
where the Fourier component of nonlinear polarization, P NL, can be expressed through the material nonlinear susceptibility χ(2) as
Here ε 0 and µ 0 are the permittivity and permeability of free space, respectively, α is the intensity absorption coefficient as discussed below, z is the THz propagation coordinate, z/cos γ is the coordinate in the pump propagation direction (see Fig. 1). We note that for a more accurate simulation of high-power THz generation, also cascaded χ(2) processes should be taken into account (see e.g. ). However, such effects were neglected here for simplicity.
In case of collinear THz generation, the wave-vector mismatch Δk is given by the following relation :
Here, the approximation was made under the condition Ω ≪ ω 0. For noncollinear geometry this should be modified as
since the pump pulse propagation distance is 1/cos γ times larger than the THz propagation distance.
Absorption at THz frequencies has two main contributions:
The first one, αε, is determined by the complex dielectric function of the material while the second term, α fc, is due to free carriers generated in the medium by the pump pulse (therefore its position dependence). The density of free carriers, N fc, generated by the pump pulse can be calculated as
Here, τ = τ(z/cos γ) is the pump pulse duration, I = I(z/cos γ) is the time-averaged pump intensity, λ 0 =2πc/ω 0 is the pump phase-matching (central) wavelength, α 0 = αε(ω 0), β 2, and β 3 are the linear, two-, and three-photon absorption coefficients, respectively. The free-carrier absorption (FCA) coefficient α fc was calculated using the Drude-model :
where ε ∞ is the high-frequency dielectric constant, τ sc is the electron scattering time, ω p = eN 1/2 fc(ε 0 ε ∞ m eff)−1/2 is the plasma frequency, e and m eff are the electron charge and effective mass, respectively. Note that α fc depends on the pump propagation coordinate z/cos γ, since N fc depends on it. The material parameters used in the calculations are listed in Table 1.
The model described here is a simplified one-dimensional model of noncollinear THz generation. Effects due to finite pump beam cross section, such as the decrease of interaction length due to spatial walk-off, or variation of pump pulse duration and intensity across the beam, are not included in this model. The latter effect is discussed in Sec. 4.
3. Effective THz generation length
In order to analyze the effects of the choice of pump pulse duration and its spatial variation with the propagation coordinate on the THz generation process in the TPFP scheme, we have carried out numerical calculations based on the model described in Sec. 2. Here, and in the following two Sections stoichiometric LN was chosen as the nonlinear material pumped at 800 nm wavelength, which is used in many experiments.
Figure 2(a) shows the variation of the pump pulse duration with the pump propagation distance in the nonlinear material. Each curve corresponds to different Fourier transform-limited (TL) pulse durations (τ 0). An appropriate amount of initial chirp has been assumed in the calculation which ensures that the pulse duration reaches its Fourier-limit at the center of the medium. Experimentally, in a chirped-pulse amplification laser system this can be achieved e.g. by tuning the grating separation in the pulse compressor. Since in case of LN a PFT as large as 62.7° is required for phase matching the dominant pulse-stretching effect is group-velocity dispersion (GVD) originating from angular dispersion , which is proportional to tan2(γ) . Note that the dispersion length L d , over which the pump pulse duration increases from τ 0 to √2τ 0 is proportional to τ 2 0.
Figure 2(b) shows the corresponding build-up of the THz field in terms of pump-to-THz energy conversion efficiency at a constant pump fluence level of 5.1 mJ/cm2. At this moderate pumping level saturation of THz generation (e.g. due to three-photon absorption) can be neglected (see e.g. ). As can be seen from the curves the THz field experiences a significant gain only in the region where the pump pulse is shortest (i.e. close to its respective TL value). Outside this region the pump intensity drops and THz absorption overcomes the gain leading to decreasing THz intensity. We define an effective THz generation length L eff as the distance measured in the THz propagation direction over which the THz intensity grows from 5% to its peak value. For a particular pump pulse duration it is useful to choose the crystal dimensions according to this effective length (see Fig. 1). The crystal should be positioned along the pump beam such that the maximum of the THz efficiency is reached at the crystal output surface.
For many experiments it is more important to reach the maximum THz field strength rather than the maximum conversion efficiency. An effective length can be defined for the THz peak electric field strength in a similar manner. This length is usually shorter than that defined above for the efficiency since with increasing propagation length dispersion tends to broaden the THz pulse thereby decreasing its field strength.
Figure 3 shows the dependence of the maximal THz yield on the pump pulse duration at a constant pump fluence level of 5.1 mJ/cm2 together with the effective THz generation length L eff. The useful crystal thickness in the THz propagation direction increases monotonically with the pump pulse duration. In contrast to this, the corresponding THz generation efficiency has a maximum at τ 0 = 350 fs, which indicates the optimal choice of pump pulse duration for high-energy THz pulse generation. For shorter TL pulse duration, the L eff effective length is small, which limits the THz yield. For longer TL pulse duration, the lower pump intensity results in larger L eff, increased THz absorption, and reduced THz yield.
For many applications, such as THz time-domain spectroscopy, not only the THz pulse energy is important but also its spectrum, i.e. bandwidth and position of spectral peak. Figure 4 shows the normalized THz spectra for different pump pulse durations corresponding to the respective optimal crystal dimensions (see Fig. 3). In all cases phase matching was set for 1 THz. In case of τ 0 = 50 fs the spectral peak is located at 1.0 THz and the FWHM bandwidth is 1.7 THz. Pumping with τ 0 = 350 fs pulses the THz generation efficiency can be increased by more than a factor of five (Fig. 3), however the spectrum shifts to lower frequencies with a peak at 0.3 THz and bandwidth of 0.7 THz. Further increasing the pump pulse duration leads to even lower THz frequencies, as shown by the spectrum corresponding to τ 0 = 600 fs. The dependence of the THz peak frequency on the TL pump pulse duration is shown in the inset figure. Hence, in designing a THz source for a specific application a suitable trade-off has to be found between optimizing the THz energy, on the one hand, and providing the required spectral range, on the other hand.
4. THz beam properties
The pulse-front-tilting setup used in recent experiments [2, 3, 5] consists of a grating and a lens (Fig. 5). However, even if an achromatic lens is used the remaining geometric aberrations together with the relatively large angular dispersion and an extended pump beam size can lead to distorted THz beam profile and phase fronts. Such effects can limit experimental applications and high-energy THz pulse generation, which requires an extended pump spot due to pump intensity limitations (see Sec. 6).
In order to explore the performance and limitations of a TPFP setup in detail, we carried out numerical calculations with an extended pump beam. In the calculations LN was chosen as the nonlinear material pumped at 800 nm wavelength, for the TL pump pulse duration the typical value of τ 0 = 100 fs (FWHM) was chosen, the pump beam had Gaussian spatial and temporal intensity profile. For the detailed analysis of the setup it was essential to take into account the spatial variation of the pump pulse duration inside the nonlinear crystal both in the direction of pump propagation as well as across the pump beam. For simplicity, only the plane of grating dispersion (x-z plane) was considered. The pump pulse duration at a given (x, z) position inside the crystal, as well as the shape of the pump pulse front were determined from the group delays of the long- and short-wavelength components of the pump spectrum by using the ray tracing method described in Ref. . For calculating the spatial distribution of the THz generation efficiency, η(x, z), the nonlinear crystal was decomposed into thin slabs of equal thicknesses parallel to the pump pulse fronts. Eq. (2) was solved numerically for each slab with varying pump pulse duration across the pump spot. The output THz beam profile was obtained by summing up the THz amplitudes from each slab in the THz propagation direction z, thereby accounting for the non-collinearity of the THz generation process. For simplicity, absorption in the THz range was neglected here.
The calculated THz beam profiles can be seen in Fig. 6(d) for three different TPFP setups shown in Figs. 6(a–c). The pump beam diameter was 5 mm, measured at 1/e 2-value of intensity maximum. In case of setups (a) and (b) the pulse front tilt is introduced by a grating-lens system as shown in Fig. 5 (but not shown in Fig. 6). The groove density of the grating was 1800 mm−1 (a), and 1600 mm−1 (b), and the lens focal length was f = 75 mm for both setups. Setup (a) is an optimized one (for details see Sec. 5), where the tilted pump pulse front (red/solid line inside the LN prism in Fig. 6(a)) coincides with the image of the grating (black/dashed line) at the pump beam center. The color plot inside the crystal prism shows the spatial distribution of THz generation efficiency η(x, z). Please note that it is not the intensity distribution of the generated THz pulse, since it is not a snapshot of it. As can be seen in Fig. 6(a), the loci of most intense THz growth, given by the peak values of η(x, z), coincide well with the image of the grating. We note that the image of the grating gives, to a good approximation, the loci of shortest pump pulse duration (where the group delays of short- and long-wavelength spectral components are equal). The image of the grating has stronger curvature than that of the pump pulse front, which introduces asymmetry with respect to the THz wavefronts. This leads to a moderately asymmetric THz beam profile with 9.3 mm 1/e 2-width, as shown in Fig. 6(d).
By using a non-optimized setup (Fig. 6(b)), where the grating’s image has a much stronger curvature, and a different slope at beam center than the pump pulse front, the asymmetry becomes more pronounced. In addition, a very strong narrowing-down of the THz beam can be observed, caused by the narrower projection of the THz generating region, extending along the grating’s image, onto the THz wavefronts. A THz beam with such narrow-peaked (2.5 mm 1/e 2-width, 1.0 mm FWHM) asymmetric beam profile, and the corresponding large divergence is disadvantageous for many applications even if the THz energy is similar to that in the optimized case.
We note that Stepanov et al.  reported on a measured THz beam divergence of 133 mrad in a TPFP setup, which significantly exceeds the diffraction limit of 48 mrad estimated from the spot size. Our model calculations predict a divergence of 150 mrad for their setup, which is close to their measured value. It can be attributed to the curvature of the pump pulse front as introduced by the pulse-front-tilting setup, leading to curved THz wavefronts. The measured THz beam profile also shows asymmetry.
Recently, we have proposed to omit the imaging optics and to bring the grating in contact with the crystal surface . This geometry eliminates imaging errors and allows to use a large pump cross section. In such a setup, the loci of shortest pump pulse duration lie exactly along the plane pump pulse fronts (Fig. 6(c)). This results in plane THz wavefronts and symmetric THz beam profile, as shown in Fig. 6(d).
5. Optimization of the pulse-front-tilting setup
In order to obtain optimal pump-to-THz conversion and optimal THz beam characteristics in a TPFP setup the following conditions have to be fulfilled:
- Velocity matching of pump and THz requires a certain tilt angle g of the pump pulse front inside the crystal given by Eq. (1).
- The pump pulse duration has to be minimal across the tilted pulse front.
- The pump pulse front has to be plane in the crystal.
All conditions can be fulfilled to very good accuracy in the contact-grating setup. In case of setups with imaging optics condition (ii) implies that the image of the grating should coincide with the tilted pulse front over the whole pump beam cross section. Parameters of optimized setups satisfying conditions (i) and (approximately) (ii), such as in case of Fig. 6(a), can be found by simple analytical calculations based on paraxial ray optics, which will be outlined below. Details of the calculations are given in the Appendix.
The pulse-front-tilting setup and its parameters are shown in Fig. 5. The incidence and diffraction angles at the grating are θ i and θ d, respectively. Please note that θ d also gives the tilt angle of the grating (Fig. 5). Let us assume that the grating with a grating period length of p and the lens with a focal length of f are given. Furthermore, we assume that the distance s between the image of the grating and the crystal input surface along the optical axis is also given. s is arbitrarily chosen such that the region of effective THz generation (gray-shaded area in Fig. 1) is not truncated. In case of given p, f, and s parameters, and given nonlinear material, one can obtain (see Appendix) for the geometrical parameters of the setup (Fig. 5):
where a is given by:
Please note that for a given material and wavelength θ d depends on the grating period only. Therefore, according to Eq. (10), the angle of incidence, θ i, also depends on p only.
The solid line in Fig. 7 shows the required angle of incidence (θ i) vs. the groove density of the grating (p −1) calculated from Eqs. (9), (10), and (13) for LN, λ 0 = 800 nm, Ω0/2π = 1 THz, and f = 75 mm. It is advantageous to use an incidence angle as close to the Littrow-angle, θ Littrow = sin−1(λ 0/2p), as practically possible, since the diffraction efficiency of a grating is usually maximal in Littrow-geometry (dashed-dotted line in Fig. 7). Hence, the grating period p can be optimally chosen according to θ i = θ Littrow +Δθ. Here, Δθ is a small (positive or negative) deviation from the Littrow-angle necessary to separate the incident and the diffracted beams (dashed and dotted lines in Fig. 7). The typical value of Δθ = −10° (+10°) gives an optimal groove density of 1800 mm−1 (2200 mm−1). In recent experiments 2000 mm−1 groove density and θ i ≈ 60° were used [2, 5], which are not far from optimum (Fig. 7).
The effect of the lens focal length f was also investigated in case of the optimized setup, i.e. where the grating’s image coincides with the tilted pump pulse front at the beam center. Pump pulse fronts and images of the grating are compared in Fig. 8 for f = 75 mm, corresponding to the previous case in Fig. 6(a), and f = 150 mm when using the same grating and angle of incidence. In case of the longer focal length the curvatures of both the pump pulse front and the grating’s image are reduced. Thus, using longer focus is advantageous for reducing THz divergence and THz beam asymmetry. We note that in order to further reduce the image-field curvature a corrected flat-field lens can be used.
6. Selection of the nonlinear material
In the design of a TPFP THz source the selection of the nonlinear material is a key step. The effective nonlinear coefficient, THz absorption, and phase matching between pump and THz fields are the most important issues to consider. The effective nonlinear coefficient of LN is one of the highest among all materials suitable for OR. However, the large pulse front tilt required for phase matching and the large absorption coefficient in the THz range can limit high-energy THz pulse generation in LN. Therefore, it is of great practical importance to consider alternative materials.
Some of the materials most suitable for OR are listed in Table 2. The PFT necessary for velocity matching at 1 THz and the possible lowest order of linear or multiphoton pump absorption are given for three practically important pump wavelengths (800 nm, 1064 nm and 1560 nm). In all materials but LN a PFT of γ ≈ 30° or smaller is sufficient for velocity matching, in contrast to γ ≈ 63° in LN. The much smaller angular dispersion, and hence, the reduced variation of pump pulse duration can allow for longer effective THz generation lengths than in LN (GVD ∝ tan2(γ)). In some cases this can compensate for the smaller nonlinear coefficient of the semiconductor materials (see below). Please note that collinearly phase-matched THz generation is possible only in ZnTe near 800 nm pumping. Collinear pumping can eventually be used in cases where the PFT is smaller than about 10°, such as GaP at ~ 1064 nm pumping . However, for high-energy THz pulse generation with large pump beam diameters exact phase matching provided by TPFP is optimal. Table 2 also shows that in the considered materials, only higher-order multiphoton absorption will be effective if longer pump wavelength is used. This allows for higher pump intensities and more efficient THz generation, since free carriers causing strong THz absorption will be generated by the pump only at higher intensities.
In order to compare the performance of various materials, we carried out numerical calculations based on the model described in Sec. 2. Figure 9 shows the maximal achievable pump-to-THz energy conversion efficiencies as a function of pump intensity (temporal average at beam center) for LN and some of the semiconductors. The TL pump pulse duration τ 0 was chosen such that the highest THz energy is obtained for a given pump fluence. The material length was equal to the effective THz generation length L eff. In Fig. 9(a) phase matching was set for 1 THz, while in Fig. 9(b) it was 3 THz and 5 THz. The saturation of THz generation appears for the semiconductor materials due to FCA caused by multiphoton absorption at the pump wavelength. In lack of reliable data we have not included multiphoton absorption for LN in the calculations. Experimentally, Hoffmann et al.  observed the saturation of THz generation above 85 GW/cm2 pump intensity (shaded areas in Fig. 9), which can be attributed to FCA due to four-photon absorption at 1064 nm pump wavelength. On the basis of this one can expect that below this intensity level FCA has negligible influence on the THz yield and the numerical results can be compared to those for semiconductors. The results at 1 THz (Fig. 9(a)) show that semiconductors such as GaP or GaSe can compete with LN at low pump intensities. Using longer pump wavelength can suppress, for example, 2PA in GaSe, which allows for more intense pumping and higher THz yield. However, LN can still be pumped at higher intensities giving higher THz efficiencies.
We note that due to its relatively strong THz absorption  and 2PA ZnTe is less efficient when pumped at 1064 nm, despite its relatively large d eff. However, with 1560-nm pumping at low temperatures (80 K in Fig. 9) its THz absorption and 2PA can be suppressed and efficient THz generation becomes possible. Since no data were available for β 3 in ZnTe we used the values of GaP and GaSe.
If phase matching is set at higher THz frequencies, as shown in Fig. 9(b) for Ω0 = 3 THz and 5 THz, semiconductors such as GaAs or GaP can compete with LN in the entire pump intensity range. They can provide higher THz efficiency than LN even at a much weaker pump intensity level. LN has high THz absorption in this spectral range, which reduces its efficiency. Semiconductors such as GaAs or GaP have smaller THz absorption. This, together with the significantly larger maximal effective THz generation length resulting from the much smaller PFT can allow for higher THz efficiency than in LN. Hence, semiconductors can be promising candidates in high-energy THz generation setups. The corresponding optimal TL pump pulse duration τ 0 and the effective THz generation length L eff are plotted in Fig. 10. These plots indeed show larger possible L eff for the semiconductors than in LN for pump intensity levels below the onset of multiphoton pump absorption.
It is also important to compare the corresponding THz spectra. As can be seen in Fig. 9(c), at Ω0 = 1 THz phase matching frequency, the semiconductor spectra are peaked at this frequency. In contrast to this, the spectrum from LN is peaked at only 0.4 THz, which can be attributed to its rapidly increasing absorption with increasing THz frequency. However, the relative width of the THz spectrum from LN is larger than that of semiconductors. This behavior is even more obvious at the higher THz phase matching frequencies (Ω0 = 3 THz and 5 THz), as can be seen in Fig. 9(d). Thus, when large relative bandwidth is needed, LN might still be the material of choice, despite the eventually higher efficiency of some semiconductors. Semiconductors can be advantageous for generating THz pulses with higher spectral intensity at higher frequencies.
Figure 11 shows a comparison of the results of our calculations to experimental results published in the literature for ZnTe pumped by 150-fs , and 30-fs  pulses at 800 nm wavelength. Both experimental and theoretical data show the saturation of THz generation with increasing pump fluence. While the calculation predicts slowly decreasing THz generation efficiencies at higher pump levels due to FCA caused by 2PA, in the experiments approximately constant THz yield was observed. Thus, our simple model, which is based on the Drude model, overestimates the influence of FCA. This can be attributed to the saturation of FCA, which also has been observed in recent experiments . Saturation of FCA causes the saturation of THz generation efficiency to appear at even higher intensities for semiconductors in Figs. 9(a) and (b), so that they can be pumped at even higher intensities, making them more efficient. (We note that the variation of the difference between calculated and experimental efficiencies with pump pulse duration in Fig. 11 can be attributed to the transient behavior of FCA , which also depends on pump pulse duration, and was not included in the model.)
In summary, a detailed analysis of the TPFP THz generation scheme was given. It was shown that there is an optimum pump pulse duration, and a corresponding effective THz generation length, which give the highest THz generation efficiency. We showed that imaging errors of the lens in a pulse-front-tilting setup can lead to a THz beam with strongly asymmetric intensity profile and strong divergence, thereby limiting applications. Optimized setup parameters were given to reduce such distortions.
By comparing semiconductor materials to LN we showed that semiconductors can offer a promising alternative to LN for high-energy THz pulse generation. Semiconductors have to be pumped at longer wavelengths in order to avoid multiphoton pump absorption, which results in strong FCA in the THz range. Longer pump wavelengths require to use TPFP, but the necessary PFT angles are much smaller than in LN. Besides allowing for large effective THz generation lengths, small PFT angles relax the requirements on the pulse-front-tilting setup, and are better suited for realizing the contact-grating scheme. Semiconductors can be advantageous for generating THz pulses with high spectral intensity at higher THz frequencies, while LN is better suited to generate THz pulses with very large relative spectral width. Such optimized schemes with large pumped areas will expectedly allow for high-energy ultrashort THz pulse generation driven by medium- and large-scale laser facilities with pulse energies on the joule scale.
The starting point in the derivation of Eqs. (9), and (11)–(13) is to determine the connection between the tilt angle of the grating, θ d, and that of its image, θ, with respect to the x-axis (Fig. 12). This can be found by calculating the image P(x, z) inside the crystal of an arbitrary point P1(x 1, z 1) on the grating surface. The correspondence between the coordinates of P1 and P can be found by using the rays shown in Fig. 12:
By using tan θ d = z 1/x 1 and eliminating x 1 and z 1 one obtains
which shows that a line is imaged onto a line. It follows that the tilt angle θ of the grating’s image is
The other important point in the derivation is to find the connection between the angular dispersion introduced by the grating and that inside the nonlinear crystal. The incident beam experiences an angular dispersion of dε 1/dλ = −(p cos θ d)−1 when diffracted from the grating. This is modified by the angular magnification of the lens to dε 2/dλ = −(dε 1/dλ)(s 1 − f)/f and, subsequently, by refraction at the crystal input surface to dε/dλ = n −1dε 2/dλ. The pulse front tilt γ inside the nonlinear material is related to the angular dispersion inside the medium by :
Financial support from Hungarian Scientific Research Fund (OTKA), grant numbers 76101 and 78262, and from Science, Please! Research Team on Innovation (SROP-4.2.2/08/1/2008-0011) is acknowledged.
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