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To circumvent a velocity mismatch between optical pump and terahertz waves in electro-optic crystals, we propose to use dual-wavelength optical beams tilted with respect to their planes of equal amplitude. The tilt is achieved by transmission of a dual-wavelength laser beam through a diffraction grating placed on the crystal boundary. The proposed technique extends optical rectification of tilted-front femtosecond laser pulses to difference-frequency generation with longer (nanosecond) pulses. Our analysis of the technique for LiNbO3 pumped at 1.3 μm and GaAs pumped at 1.55 μm shows its efficiency.
© 2012 Optical Society of America
1. Introduction
Difference-frequency generation (DFG) is an established way of producing tunable narrow-linewidth terahertz radiation. This technique is based on the parametric interaction of two optical waves with close frequencies ω1 and ω2 in a quadratic nonlinear medium that results in the generation of a third wave with a difference (terahertz) frequency Ω = ω1 − ω2. The efficiency of this nonlinear process depends on the fulfillment of the phase-matching condition, which, in the case of collinear interaction, turns into the condition of the equality of the optical group velocity vg and terahertz phase velocity vTHz(Ω): vg = vTHz(Ω). Collinear phase-matched terahertz DFG was achieved in a few isotropic semiconductor compounds, such as ZnTe [1, 2] and GaP [3]. A widely tunable terahertz generation using collinear DFG was implemented in GaSe due to its large birefringence [4, 5]. In the dielectric materials, like LiNbO3 (LN), with large second-order nonlinearity, a high damage threshold, and the absence of two-photon absorption for typical pump wavelengths, the optical group velocity is more than twice as large as the terahertz phase velocity. To overcome the large velocity mismatch, the concept of quasi-phase-matching in periodically poled lithium niobate (PPLN) structures was developed for both collinear [6] and non-collinear [7–9] geometry. Another approach for achieving phase matching, and reducing simultaneously the diffraction broadening of the optical and terahertz beams, is to use waveguide structures supporting overlapped optical and terahertz modes [10–16]. Cherenkov radiation mechanism is also used to synchronize optical and terahertz waves [17, 18]. The state-of-the-art Cherenkov-type DFG scheme includes a thin LN slab waveguide, which confines the optical pump, and a Si-prism coupler to output the generated terahertz radiation [19].
In this paper, we propose another method to achieve phase-matched DFG. We propose pumping the crystal by a dual-wavelength optical beam tilted with respect to its planes of equal amplitude. The tilt can be achieved by transmitting a non-tilted dual-wavelength laser beam through a diffraction grating placed on the crystal boundary. Amplitude beating is inherent in the dual-wavelength beam. In the diffracted beam, the planes of equal amplitude are parallel to the crystal boundary (in the simple case of normal incidence of the laser beam on the boundary) and propagate with the group velocity vg in the direction of the beam, i.e., at the first order diffraction angle α to the boundary normal. The projection of this velocity on the direction perpendicular to the planes of equal amplitude (and the crystal boundary) is vg cosα. By the proper choice of the angle α, this projection can be made equal to the phase velocity of a terahertz wave with a difference frequency Ω: vg cosα = vTHz(Ω). Thus, one can achieve phase matching with the quasiplane terahertz wave propagating normally to the boundary.
The proposed approach extends the concept of broadband terahertz generation via optical rectification of tilted-front femtosecond laser pulses [20] to the case of terahertz DFG with longer (nanosecond) pulses. The tilted-pulse-front pumping of Mg-doped stoichiometric LN is the most efficient technique nowadays in generating near-single-cycle terahertz pulses, it provides terahertz pulses with energies as high as 10–125 μJ [21–24] and a peak terahertz field exceeding 1.2 MV/cm [25]. As we demonstrate below, extending this technique to quasi-cw tilted-amplitude-front pumping can result in efficient generation of narrow-band terahertz radiation. One should emphasize that, in contrast to femtosecond tilted-front pulses, the quasi-cw tilted-front beams do not suffer from the strong broadening caused by angular and material dispersions, the main factor restricting the efficiency of the tilted-pulse-front pumping technique [26, 27]. This allows one to use thicker crystals for terahertz DFG and, therefore, to increase the interaction length between the optical pump and terahertz wave.
In a conventional pulse-front-tilting setup, tilting a femtosecond pulse front is introduced by diffracting the pulse off an optical grating. A lens (or a two-lens telescope) is used to relay-image the pump spot on the grating into the LN crystal. In such a scheme, however, aberrations caused by the imaging optics can introduce strong asymmetry into the transverse profile of the generated terahertz beam and significant curvature of the terahertz wavefronts leading to a strong divergence of the beam [26]. We propose to use the contact-grating scheme [28] in which the imaging optics is omitted and the transmission grating is placed directly on the entrance boundary of the crystal.
We develop a detailed analysis of terahertz generation with quasi-cw tilted-amplitude-front optical beams in the contact-grating scheme. Our theory accounts for a finite transverse size of the pump optical beam. This allows us to study the important effects of Cherenkov radiation and the transverse walkoff of terahertz waves [27]. Estimates are made for LN pumped at ≈ 1.3 μm wavelength, like in Refs. [18, 19], and for GaAs pumped at ≈ 1.55 μm.
2. Generation scheme and model
We assume that a collimated dual-wavelength laser beam propagating along the x-axis impinges normally on the transmission diffraction grating placed at the entrance boundary x = 0 of an electro-optic crystal (Fig. 1). The electric field E(x,y,t) of the beam consists of two Gaussian beams with the same transverse size a (along the y-axis) and amplitude vector E0 = (0,0,E0) but with different frequencies ω1,2:
The incident optical intensity is thus I(0−,y,t) ∝ exp(−y2/a2) and the standard full width at half-maximum (FWHM) is . The size a is assumed to be large enough so that we can neglect the diffraction divergence of the laser beam on the crystal length L.Applying Fourier transform with respect to y (γ is the corresponding Fourier variable) to Eq. (1), optical beams diffracted to the ±1st orders can be described as
where T±1 are the ±1st order transmission coefficients, G = 2π/d is the grating groove frequency (with d the grating period), and is the x-component of the wave vector [with nm = n(ωm) the optical refractive index of the crystal at the frequency ωm and c the speed of light]. The diffraction angle α (Fig. 1) is defined as sinα = λ̄/(nd) where λ̄ = 2πc/ω̄ is the central (vacuum) wavelength, n = n(ω̄), and ω̄= (ω1 + ω2)/2 is the central frequency. Expanding hm in power series up to the first order of Ω (Ω = ω1 − ω2, Ω ≪ ω̄) and γ, the integral in Eq. (2) can be evaluated as where ξ = t − ngx/(c cosα), η± = y ∓ x tanα, and ng is the optical group refractive index. The next terms in the power series expansion are insignificant: the ∝ Ω2 term gives only negligible corrections to the direction and absolute value of the diffracted beam phase velocity; the ∝ γ2 term describes the diffraction divergence of the beam (assumed negligible due to a large a) and its phase modulation, which will not affect the nonlinear polarization created by the beam in the crystal. According to Eq. (3), the diffracted beams have a tilted-amplitude-front structure: the beam and its phase fronts propagate at the angle α to the x-axis whereas the planes of constant amplitude ξ = const propagate along the x-axis with the reduced group velocity (c/ng) cosα (Fig. 1). By choosing the tilt angle α, this velocity can be made equal to the phase velocity of a terahertz wave with the frequency Ω: (c/ng) cosα = vTHz(Ω). Thus, one can achieve phase matching with the quasiplane terahertz wave propagating along the x-axis.In our analysis, we will consider two experimental configurations of interest [28]: (i) a blazed grating which diffracts the incident laser beam into only one of the ±1st orders (T+1 ≠ 0, T−1 = 0) and (ii) a grating with equal efficiencies in the ±1st orders (T+1 = T−1), such as holographic gratings. In the case of a blazed grating, the nonlinear polarization created by the optical field E+1 [see Eq. (3)] via DFG becomes
whereas in the case of a holographic grating the nonlinear polarization created by the field E+1 + E−1 is In both cases, the absolute value of the amplitude vector p is , where deff is the effective nonlinear coefficient of the crystal. In our further calculations, we assume 100% diffraction efficiency, thus putting T+1 = (n cos α)−1/2 for a blazed grating and T+1 = T−1 = (2n cosα)−1/2 for a holographic grating. If a more accurate evaluation is required, a correction factor can be easily applied to our final results. The orientation of p is determined by the crystallographic orientation of the crystal. We assume px,y = 0 and pz ≠ 0. In LN, such configuration occurs if the optical axis of the crystal is along the z-axis, and deff = d33 = 168 pm/V [29]. In GaAs, a maximum of pz is achieved for a 〈110〉-cut crystal with the [001] axis oriented at ≈ 55° to the z-axis [30]. The maximum value of pz is defined by deff = (4/3)1/2d14[30]. For GaAs, we will use deff = 65.6 pm/V [26].To characterize the crystal at the terahertz frequencies, we use a one-phonon-resonance dielectric function
For LN, we will use the parameters of 0.68 mol% stoichiometric LN [31–33]: ωTO/(2π) = 7.44 THz, ε∞ = 10, ε0 = 24.4, and ν/(2π) = 1.3 THz. For GaAs, we will use ωTO/(2π) = 8.2 THz, ε0 = 12.9, ε∞ = 11, and ν/(2π) = 0.25 THz [34–36]. In the optical range, we will use n = 2.15, ng = 2.19 for LN at 1.3 μm [37] and n = 3.37, ng = 3.53 for GaAs at 1.55 μm [34, 38].3. Theoretical formalism
To find the terahertz radiation generated by the moving nonlinear polarization PNL(x,y,t) [Eqs. (4) or (5)], we apply Fourier transform with respect to y (g is the corresponding Fourier variable; ˜ will denote quantities in the Fourier domain) to Maxwell’s equations with PNL included as a source. Eliminating the magnetic field, we obtain an equation for the terahertz electric field transform Ẽz(x,g,t) [27]:
where κ2 = (Ω/c)2ε − g2 and P̃NL is the Fourier transform of PNL. In the case of a blazed grating, we obtain for P̃NL from Eq. (4): and in the case of a holographic grating, we obtain from Eq. (5):Using the slowly varying envelope approximation (see, for example, [39]) we obtain the solution of Eq. (7) as
Solution (10) in the Fourier domain can be transformed to the y domain by taking inverse transform in the following formTo describe the transmission of the terahertz field through the exit boundary of the crystal (x = L), one can use the usual Fresnel transmission coefficient in the near-phase-matched regime ( ) under consideration [40]. We assume, however, that an antireflection coating is deposited onto the exit boundary and the intensity transmission coefficient equals unity. To calculate the terahertz power emitted from the crystal (per unit length along the z-axis), we integrate the x-component of the time-averaged Poynting vector in vacuum (at x = L+) over the infinite interval −∞ < y < ∞. To compare conveniently with experimental data, in particular, in Refs. [18, 19], we define the optical-to-terahertz conversion efficiency as a ratio of the terahertz and optical pulse energies assuming Gaussian temporal envelopes and spatial profiles along the z-axis for the optical and terahertz pulses.
4. Results and discussion
In terms of the highest terahertz energy, the blazed grating scheme is apparently advantageous over the scheme with a holographic grating. Indeed, for the same optical intensity at the entrance boundary of the crystal the ±1st order optical beams after their separation in the holographic grating scheme have two times smaller intensities than the intensity of the single (for example, +1st order) beam in the blazed grating scheme. Since DFG is a second-order nonlinear process, the total terahertz power generated by the two separate optical beams in the holographic grating scheme will be two times smaller than in the blazed grating scheme. Nevertheless, the easier availability of holographic gratings makes them attractive for practical use. Additionally, in the holographic grating scheme with a semiconductor as an electro-optic crystal the ±1st order pump beams can remain overlapped even for a large crystal thickness due to a small required tilt angle α. Using the general solution of Sec. 3, we analyze further both schemes for two materials – LN pumped at 1.3 μm and GaAs pumped at 1.55 μm.
4.1. Tilt angles and characteristic lengths
To define the required tilt angles for LN and GaAs, we consider first the limiting case of an infinitely wide incident optical beam: a → ∞. In this limit, the function a exp(−g2a2/4)/(2π1/2) in Eq. (8) transforms to the delta function δ(g). Substitution of δ(g) into Eq. (11) gives (at ν → 0)
Equation (12) is similar to that describing DFG with ordinary (non-tilted) optical beams but in a virtual medium with the optical group refractive index ng/ cosα. Varying the tilt angle α is equivalent to changing the optical group refractive index of the virtual medium and, thus, allows one to satisfy the phase matching condition . If this condition is fulfilled, the amplitude of the generated terahertz field grows linearly with distance x, according to Eq. (12).With the losses included, the tilt angle is given by
Figure 2 shows the dependence of α on the phase-matched frequency Ω for LN pumped at 1.3 μm and GaAs pumped at 1.55 μm. If the phase matching condition (13) is fulfilled and the losses are included, the terahertz field is given by with . According to Eq. (14), even in the phase matched regime (13) the amplitude of the terahertz field does not grow linearly with x all the way to the exit boundary of the crystal. Due to the losses, the growth saturates at x ∼ Lℓ with . Taking, for example, Ω/(2π) = 1 THz, we obtain Lℓ ≈ 1.4 mm for LN and Lℓ ≈ 4.7 cm for GaAs.
Fig. 2 Tilt angle α vs. phase-matched frequency Ω for LN pumped at 1.3 μm and GaAs pumped at 1.55 μm.
For a finite width a of the laser beam, the growth of the amplitude of the phase matched terahertz wave is limited not only by the losses but also by the transverse (in the −y-direction in Fig. 1) walkoff of the wave from the region where the laser beam propagates [27]. For the material parameters we use (Sec. 2), the transverse walkoff length [27] can be estimated as Ltw ≈ 0.5aFWHM for LN and Ltw ≈ 5.3aFWHM for GaAs. If Ltw > Lℓ, the effect of the transverse walkoff is insignificant and the steady-state amplitude of the phase matched wave equals the maximum value given by Eq. (14) at x → ∞. However, if Ltw < Lℓ, the amplitude saturates at a smaller value, which is defined by the parameter aFWHM .
Besides the phase matched wave, the laser beam of a finite width also generates Cherenkov radiation. By analogy with the theory of the tilted-pulse-front pumping [27], we can conclude that, if aFWHM is larger than the terahertz wavelength in the crystal, the Cherenkov radiation should be strongly asymmetric: it should exist mainly on one side of the laser beam, namely, in the gray shaded area in Fig. 1. Since the Cherenkov radiation propagates in the same (+x) direction with the phase matched wave, these two components of the terahertz field cannot be distinguished in the area pointed out above. The attenuation length of both components along the x-axis equals, obviously, Lℓ.
Using Fig. 2 we can estimate the tunability of the method. The tuning of the generation frequency Ω requires one to vary not only the frequency difference ω1 − ω2 but also the central wavelength λ̄ of the laser beam to preserve the phase matching condition (13). Assuming that a 10% variation of λ̄ is acceptable not to affect the diffraction efficiency noticeably, we estimate the corresponding variation of the tilt angle α as Δα ∼ 10−1 tanα, i.e., Δα ∼ 11.5° for LN and Δα ∼ 1.3° for GaAs. According to Fig. 2, this provides tunability in the frequency range of 0–5 THz for LN and 0–2 THz for GaAs.
4.2. LN pumped at 1.3 μm
Let us now consider the blazed grating scheme. Figure 3 shows the radiation patterns, calculated numerically on the basis of Eqs. (8), (10), and (11), for LN excited by a dual-wavelength laser beam with ≈1.3 μm wavelength and 4 ×106 W/cm total power (per unit length along the z-axis) for two beam widths aFWHM = 5 and 0.5 mm. The difference frequency was set equal to Ω/(2π) = 1 THz, and the corresponding optimal value of the tilt angle α = 63.8° (Fig. 2) was used in the calculations. From the experimental point of view, the pump power of 4 ×106 W/cm corresponds, for example, to the laser of a 2 mJ pulse energy and 15 ns pulse duration (similar to that in Refs. [18, 19]) and a 0.3 mm beam width (FWHM) along the z-axis, as in Ref. [18]. For aFWHM = 5 mm [Fig. 3(a)], the transverse walkoff length Ltw ≈ 2.5 mm is larger than Lℓ ≈ 1.4 mm and, therefore, the amplitude of the phase matched wave saturates at x ≈ Lℓ ≈ 1.4 mm to the maximum value defined by Eq. (14). For aFWHM = 0.5 mm [Fig. 3(b)], the transverse walkoff effect is more pronounced: Ltw ≈ 0.3 mm is smaller than Lℓ, and the saturation occurs at the distance x ≈ Ltw ≈ 0.3 mm. This means that the amplitude of the phase matched wave does not reach the maximum value defined by Eq. (14). However, since the optical intensity for aFWHM = 0.5 mm is 10 times higher than for aFWHM = 5 mm, the steady-state amplitude of the phase matched wave is still larger in Fig. 3(b) than in Fig. 3(a).
Fig. 3 Snapshots of |Ez(x,y,t)| in LN for (a) aFWHM = 5 mm and (b) aFWHM = 0.5 mm, the blazed grating scheme. Optical beams are shown by dashed lines. The pump power is 4 ×106 W/cm, Ω/(2π) = 1 THz, and α = 63.8°.
Figure 4 shows the conversion efficiency as a function of the LN crystal thickness L for several values of aFWHM. The dashed segment of the curve for aFWHM = 0.1 mm is unrealistic. It corresponds to the thicknesses L where the mutual shift (in the y-direction) of the two frequency components of the pump optical beam due to the angular dispersion inserted by the diffraction grating exceeds aFWHM/2 = 50 μm. Unlike the amplitude of the phase-matched wave (Fig. 3), the efficiency reaches saturation at the maximum (rather than minimum) of the two lengths Lℓ and Ltw. Indeed, for the curves with aFWHM = 1, 0.5, and 0.1 mm (Ltw ≈ 0.5, 0.25, and 0.05 mm, respectively), the efficiency reaches saturation at L ≈ Lℓ ≈ 1.4 mm. For aFWHM = 5 mm (Ltw ≈ 2.5 mm), the efficiency reaches saturation at L ≈ Ltw ≈ 2.5 mm. Physically, this can be explained as follows. For Ltw < Lℓ, the amplitude of the phase matched wave reaches its maximum at x ≈ Ltw; however, the width of the terahertz beam (and therefore the terahertz power) continues to grow with x until x ≈ Lℓ (Fig. 3). For Ltw > Lℓ, on the contrary, the width of the terahertz beam reaches its steady state value at x ≈ Lℓ whereas the amplitude of the terahertz wave continues to grow until x ≈ Ltw (Fig. 3).
Fig. 4 Efficiency as a function of the LN crystal thickness L for several values of aFWHM (labeled next to the corresponding curves). The pump parameters are the same as in Fig. 3.
One can draw a practical conclusion from Fig. 4: the optimal thickness of the LN crystal is ≈ 2 mm. Indeed, a decrease in the thickness reduces the efficiency (Fig. 4), whereas an increase causes additional absorption of the pump laser beam and spatial separation of its two frequency components due to the angular dispersion. Thus, these two effects (not accounted for in our analysis) also reduce the efficiency. Another practical conclusion is that focusing the pump laser beam from aFWHM = 5 mm to 0.5 mm increases the efficiency by a factor of ≈ 2; however, further focusing adds little to the conversion efficiency. The maximum efficiency for aFWHM ∼ 0.1 −1 mm (∼ 8 × 10−6) is about 200 times higher in comparison to Refs. [18, 19]. Naturally, this efficiency is smaller than for femtosecond pulses with much higher optical intensity [21–25].
Now let us discuss briefly the case of holographic grating. Figure 5 shows the radiation patterns, calculated numerically on the basis of Eqs. (9), (10), and (11), for a LN crystal with a 2 mm thickness. For aFWHM = 5 mm, the terahertz fields generated by the ±1st order optical beams overlap and form a terahertz beam with a flat central part. The conversion efficiency is 2.2 ×10−6. For aFWHM = 0.5 mm, two separate terahertz beams are generated with the efficiency 3.6×10−6. Reducing the crystal thickness to 0.5 mm allows one to combine the terahertz beams into a single beam with an almost flat central part [a dotted line in Fig. 5(b)]. The conversion efficiency becomes 1.9 ×10−6. Thus, even in the case of holographic grating a terahertz beam of good quality can be generated using relatively wide optical beams or relatively thin crystals but at the cost of somewhat decreased efficiency.
Fig. 5 Snapshots of |Ez(x,y,t)| in LN for (a) aFWHM = 5 mm and (b) aFWHM = 0.5 mm, the holographic grating scheme. Optical beams are shown by dashed lines. Profiles Ez(y) are shown in the air at x = 2 mm (solid) or x = 0.5 mm (dotted). The pump parameters are the same as in Fig. 3.
4.3. GaAs pumped at 1.55 μm
In GaAs, the tilt angle required for phase matching at a given frequency Ω is significantly smaller as compared to LN (Fig. 2). Using smaller tilt angles reduces the transverse walkoff effect, i.e., increases the transverse walkoff length Ltw (see Sec. 4.1). Additionally, due to a low terahertz absorption in GaAs [see Eq. (6)] the attenuation length Lℓ ≈ 4.7 cm is ∼ 30 times as large as in LN. Thus, the steady-state amplitude of the terahertz wave and saturation of the conversion efficiency are reached in GaAs at much longer distances x in comparison to LN. This means that terahertz generation can be very efficient in thick GaAs crystals.
Figure 6 shows the radiation patterns, calculated numerically on the basis of Eqs. (8), (10), and (11), for GaAs excited by a dual-wavelength laser beam of an ≈1.55 μm wavelength for two beam widths aFWHM = 5 and 0.5 mm in the blazed grating scheme. The total pump power (per unit length along the z-axis) is the same as in Sec. 4.2, i.e., 4 ×106 W/cm. The difference frequency is again Ω/(2π) = 1 THz, and the tilt angle is optimal for this frequency: α = 12.2° (Fig. 2). The steady-state amplitude of the terahertz wave is reached at x ≈ Ltw ≈ 2.7 cm for aFWHM = 5 mm [Fig. 6(a)] and at x ≈ Ltw ≈ 2.7 mm for aFWHM = 0.5 mm [Fig. 6(b)]. The values of the amplitude are about twice as large as the corresponding values for LN (Fig. 3). For both widths aFWHM in Fig. 6, the steady-state width of the terahertz beam is reached at x ≈ Lℓ ≈ 4.7 cm. Correspondingly, the conversion efficiency, shown in Fig. 7 for several values of aFWHM, reaches saturation at x ≈ Lℓ ≈ 4.7 cm. Comparing Figs. 7 and 4 one can conclude that GaAs provides an efficiency an order of magnitude higher than LN.
Fig. 6 Snapshots of |Ez(x,y,t)| in GaAs for (a) aFWHM = 5 mm and (b) aFWHM = 0.5 mm, the blazed grating scheme. Optical beams are shown by dashed lines. The pump power is 4 ×106 W/cm, Ω/(2π) = 1 THz, and α = 12.2°.
Fig. 7 Efficiency as a function of the GaAs crystal thickness L for several values of aFWHM (labeled next to the corresponding curves). The pump parameters are the same as in Fig. 6.
In the case of holographic grating, Fig. 8 demonstrates two examples of generating a flat terahertz beam. For aFWHM = 5 mm and L = 2.4 cm [Fig. 8(a)], the conversion efficiency is 1.2 ×10−5, i.e., more than half of the maximum efficiency 2 ×10−5 achieved at L ≥ 4.7 cm. For aFWHM = 0.5 mm and L = 8 mm [Fig. 8(b)], the conversion efficiency is 6.6 ×10−6.
Fig. 8 Snapshots of |Ez(x,y,t)| in GaAs for (a) aFWHM = 5 mm and (b) aFWHM = 0.5 mm, the holographic grating scheme. Optical beams are shown by dashed lines. Profiles Ez(y) are shown in the air at x = 2.4 cm (a) and x = 0.8 cm (b). The pump parameters are the same as in Fig. 6.
5. Conclusion
Using dual-wavelength optical beams with tilted planes of equal amplitude allows one to achieve phase matching in terahertz DFG in different electro-optic crystals for a wide range of optical wavelengths and terahertz frequencies. Due to phase matching, the internal optical-to-terahertz conversion efficiency of the tilted-amplitude-front optical beams can be as high as ∼ 8 ×10−6 for LN pumped by 1.3-μm wavelength, 2-mJ energy, 15-ns duration laser pulses, and even higher, i.e., ∼ 5 ×10−5, for GaAs pumped by 1.55-μm wavelength laser pulses of the same energy and duration.
Acknowledgments
This work was supported in part by the Ministry of Education and Science of the Russian Federation through Agreement Nos. 11.G34.31.0011 and 14.B37.21.0770 and RFBR grant Nos. 11-02-92107 and 12-02-31395.
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