Terahertz difference-frequency generation by tilted amplitude front excitation

doi:10.1364/OE.20.028573

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Terahertz difference-frequency generation by tilted amplitude front excitation

M. I. Bakunov, M. V. Tsarev, and E. A. Mashkovich »View Author Affiliations

Optics Express, Vol. 20, Issue 27, pp. 28573-28585 (2012)
http://dx.doi.org/10.1364/OE.20.028573

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▸ Article Outline
– Abstract
1. Introduction
2. Generation scheme and model
3. Theoretical formalism
4. Results and discussion
5. Conclusion
– References and links

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Abstract

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.

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 v_g and terahertz phase velocity v_THz(Ω): v_g = v_THz(Ω). Collinear phase-matched terahertz DFG was achieved in a few isotropic semiconductor compounds, such as ZnTe [1

T. Yajima and K. Inoue, “Submillimeter-wave generation by optical difference-frequency mixing of ruby R₁ and R₂ laser lines,” Phys. Lett. A 26, 281–282 (1968). [CrossRef]

, 2

T. Yajima and K. Inoue, “Submillimeter-wave generation by difference-frequency mixing of ruby laser lines in ZnTe,” IEEE J. Quantum Electron. 5, 140–146 (1969). [CrossRef]

] and GaP [3

T. Taniuchi and H. Nakanishi, “Collinear phase-matched terahertz-wave generation in GaP crystal using a dual-wavelength optical parametric oscillator,” J. Appl. Phys. 95, 7588–7591 (2004). [CrossRef]

]. A widely tunable terahertz generation using collinear DFG was implemented in GaSe due to its large birefringence [4

W. Shi, Y. J. Ding, N. Fernelius, and K. Vodopyanov, “Efficient, tunable, and coherent 0.18–5.27-THz source based on GaSe crystal,” Opt. Lett. 27, 1454–1456 (2002). [CrossRef]

, 5

W. Shi and Y. J. Ding, “A monochromatic and high-power terahertz source tunable in the ranges of 2.7–38.4 and 58.2–3540 μm for variety of potential applications,” Appl. Phys. Lett. 84, 1635–11637 (2004). [CrossRef]

]. 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

Y. J. Ding and J. B. Khurgin, “A new scheme for efficient generation of coherent and incoherent submillimeter to THz waves in periodically-poled lithium niobate,” Opt. Commun. 148, 105–109 (1998). [CrossRef]

] and non-collinear [7

Y. Avetisyan, Y. Sasaki, and H. Ito, “Analysis of THz-wave surface-emitted difference-frequency generation in periodically poled lithium niobate waveguide,” Appl. Phys. B 73, 511–514 (2001). [CrossRef]

Y. Sasaki, Y. Avetisyan, K. Kawase, and H. Ito, “Terahertz-wave surface-emitted difference frequency generation in slant-stripe-type periodically poled LiNbO₃ crystal,” Appl. Phys. Lett. 81, 3323–3325 (2002). [CrossRef]

–9

Y. Sasaki, Y. Avetisyan, H. Yokoyama, and H. Ito, “Surface-emitted terahertz-wave difference-frequency generation in two-dimensional periodically poled lithium niobate,” Opt. Lett. 30, 2927–2929 (2005). [CrossRef] [PubMed]

] 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

D. E. Thompson and P. D. Coleman, “Step-tunable far infrared radiation by phase matched mixing in planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 22, 995–1000 (1974). [CrossRef]

W. Shi and Y. J. Ding, “Designs of terahertz waveguides for efficient parametric terahertz generation,” Appl. Phys. Lett. 82, 4435–4437 (2003). [CrossRef]

H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett. 29, 1751–1753 (2004). [CrossRef] [PubMed]

V. Berger and C. Sirtori, “Nonlinear phase matching in THz semiconductor waveguides,” Semicond. Sci. Technol. 19, 964–970 (2004). [CrossRef]

A. C. Chiang, T. D. Wang, Y. Y. Lin, S. T. Lin, H. H. Lee, Y. C. Huang, and Y. H. Chen, “Enhanced terahertz wave parametric generation and oscillation in lithium niobate waveguides at terahertz frequencies,” Opt. Lett. 30, 3392–3394 (2005). [CrossRef]

C. Staus, T. Kuech, and L. McCaughan, “Continuously phase-matched terahertz difference frequency generation in an embedded waveguide structure supporting only fundamental modes,” Opt. Express 16, 13296–13303 (2008). [CrossRef] [PubMed]

–16

Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express 17, 13502–13515 (2009). [CrossRef] [PubMed]

]. Cherenkov radiation mechanism is also used to synchronize optical and terahertz waves [17

D. A. Bagdasaryan, A. D. Makaryan, and P. S. Pogosyan, “Cerenkov radiation from a propagating nonlinear polarization wave,” JETP Lett. 37, 594–596 (1983).

, 18

K. Suizu, T. Shibuya, T. Akiba, T. Tutui, C. Otani, and K. Kawase, “Cherenkov phase-matched monochromatic THz wave generation using difference frequency generation with a lithium niobate crystal,” Opt. Express 16, 7493–7498 (2008). [CrossRef] [PubMed]

]. 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

K. Suizu, K. Koketsu, T. Shibuya, T. Tsutsui, T. Akiba, and K. Kawase, “Extremely frequency-widened terahertz wave generation using Cherenkov-type radiation,” Opt. Express 17, 6676–6681 (2009). [CrossRef] [PubMed]

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 v_g 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 v_g 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: v_g cosα = v_THz(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

J. Hebling, G. Almási, I. Z. Kozma, and J. Kuhl, “Velocity matching by pulse front tilting for large-area THz-pulse generation,” Opt. Express 10, 1161–1166 (2002). [CrossRef] [PubMed]

] 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

K. L. Yeh, M. C. Hoffman, J. Hebling, and K. A. Nelson, “Generation of 10 μJ ultrashort terahertz pulses by optical rectification,” Appl. Phys. Lett. 90, 171121 (2007). [CrossRef]

A. G. Stepanov, L. Bonacina, S. V. Chekalin, and J.-P. Wolf, “Generation of 30 μJ single-cycle terahertz pulses at 100 Hz repetition rate by optical rectification,” Opt. Lett. 33, 2497–2499 (2008). [CrossRef] [PubMed]

A. G. Stepanov, S. Henin, Y. Petit, L. Bonacina, J. Kasparian, and J.-P. Wolf, “Mobile source of high-energy single-cycle terahertz pulses,” Appl. Phys. B 101, 11–14 (2010). [CrossRef]

–24

J. A. Fülöp, L. Pálfalvi, S. Klingebiel, G. Almási, F. Krausz, S. Karsch, and J. Hebling, “Generation of sub-mJ terahertz pulses by optical rectification,” Opt. Lett. 37, 557–559 (2012). [CrossRef] [PubMed]

] and a peak terahertz field exceeding 1.2 MV/cm [25

H. Hirori, A. Doi, F. Blanchard, and K. Tanaka, “Single-cycle terahertz pulses with amplitudes exceeding 1 MV/cm generated by optical rectification in LiNbO₃,” Appl. Phys. Lett. 98, 091106 (2011). [CrossRef]

]. 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

J. A. Fülöp, L. Pálfalvi, G. Almási, and J. Hebling, “Design of high-energy terahertz sources based on optical rectification,” Opt. Express 18, 12311–12327 (2010). [CrossRef] [PubMed]

, 27

M. I. Bakunov, S. B. Bodrov, and E. A. Mashkovich, “Terahertz generation with tilted-front laser pulses: dynamic theory for low-absorbibg crystals,” J. Opt. Soc. Am. B 28, 1724–1734 (2011). [CrossRef]

]. 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

J. A. Fülöp, L. Pálfalvi, G. Almási, and J. Hebling, “Design of high-energy terahertz sources based on optical rectification,” Opt. Express 18, 12311–12327 (2010). [CrossRef] [PubMed]

]. We propose to use the contact-grating scheme [28

L. Pálfalvi, J. A. Fülöp, G. Almási, and J. Hebling, “Novel setups for extremely high power single-cycle terahertz pulse generation by optical rectification,” Appl. Phys. Lett. 92, 171107 (2008). [CrossRef]

] 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 ω_1,2:

E (0 -, y, t) = \sum_{m = 1}^{2} E_{0} exp (i ω_{m} t - \frac{y^{2}}{2 a^{2}}) .

(1)

The incident optical intensity is thus I(0−,y,t) ∝ exp(−y²/a²) and the standard full width at half-maximum (FWHM) is

a_{FWHM} = 2 \sqrt{ln 2} a

. 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.

Fig. 1 Generation scheme.

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

E_{\pm 1} (x, y, t) = \sum_{m = 1}^{2} \frac{a}{{(2 π)}^{1 / 2}} T_{\pm 1} E_{0} e^{i ω_{m} t \mp i G y} \int_{- \infty}^{\infty} d γ e^{- i γ y - i h_{m} x - {(γ a)}^{2} / 2},

(2)

where T_±1 are the ±1st order transmission coefficients, G = 2π/d is the grating groove frequency (with d the grating period), and

h_{m} = {[ω_{m}^{2} n_{m}^{2} c^{- 2} - {(γ \pm G)}^{2}]}^{1 / 2}

is the x-component of the wave vector [with n_m = n(ω_m) the optical refractive index of the crystal at the frequency ω_m 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 h_m 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

E_{\pm 1} (x, y, t) \approx 2 T_{\pm 1} E_{0} cos (Ω ξ / 2) e^{- η_{\pm}^{2} / (2 a^{2}) + i \bar{ω} t \mp i G y - i G x cot α},

(3)

where ξ = t − n_gx/(c cosα), η_± = y ∓ x tanα, and n_g 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/n_g) 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/n_g) cosα = v_THz(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

P^{NL} = p e^{i Ω ξ - η_{+}^{2} / a^{2}},

(4)

whereas in the case of a holographic grating the nonlinear polarization created by the field E₊₁ + E₋₁ is

P^{NL} = p e^{i Ω ξ} [e^{- η_{+}^{2} / a^{2}} + e^{- η_{-}^{2} / a^{2}} + 2 cos (2 G y) e^{- (y^{2} + x^{2} {tan}^{2} α) / a^{2}}] .

(5)

In both cases, the absolute value of the amplitude vector p is

p = d_{eff} T_{+ 1}^{2} E_{0}^{2}

, where d_eff 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 p_x,y = 0 and p_z ≠ 0. In LN, such configuration occurs if the optical axis of the crystal is along the z-axis, and d_eff = d₃₃ = 168 pm/V [29

J. Hebling, K.-L. Yeh, M. C. Hoffmann, B. Bartal, and K. A. Nelson, “Generation of high-power terahertz pulses by tilted-pulse-front excitation and their application possibilities,” J. Opt. Soc. Am. B 25, B6–B19 (2008). [CrossRef]

]. In GaAs, a maximum of p_z is achieved for a 〈110〉-cut crystal with the [001] axis oriented at ≈ 55° to the z-axis [30

Q. Chen, M. Tani, Z. Jiang, and X.-C. Zhang, “Electro-optic transceivers for terahertz-wave applications,” J. Opt. Soc. Am. B 18, 823–831 (2001). [CrossRef]

]. The maximum value of p_z is defined by d_eff = (4/3)^1/2d₁₄[30

Q. Chen, M. Tani, Z. Jiang, and X.-C. Zhang, “Electro-optic transceivers for terahertz-wave applications,” J. Opt. Soc. Am. B 18, 823–831 (2001). [CrossRef]

]. For GaAs, we will use d_eff = 65.6 pm/V [26

J. A. Fülöp, L. Pálfalvi, G. Almási, and J. Hebling, “Design of high-energy terahertz sources based on optical rectification,” Opt. Express 18, 12311–12327 (2010). [CrossRef] [PubMed]

To characterize the crystal at the terahertz frequencies, we use a one-phonon-resonance dielectric function

ε = ε_{\infty} + \frac{(ε_{0} - ε_{\infty}) ω_{TO}^{2}}{ω_{TO}^{2} - Ω^{2} + i ν Ω} .

(6)

For LN, we will use the parameters of 0.68 mol% stoichiometric LN [31

J. Hebling, A. G. Stepanov, G. Almási, B. Bartal, and J. Kuhl, “Tunable THz pulse generation by optical rectification of ultrasort laser pulses with tilted pulse fronts,” Appl. Phys. B 78, 593–599 (2004). [CrossRef]

L. Pálfalvi, J. Hebling, J. Kuhl, Á. Péter, and K. Polgár, “Temperature dependence of the absorption and refraction of Mg-doped congruent and stoichiometric LiNbO₃ in the THz range,” J. Appl. Phys. 97, 123505 (2005). [CrossRef]

–33

N. S. Stoyanov, T. Feurer, D. W. Ward, E. R. Statz, and K. A. Nelson, “Direct visualization of a polariton resonator in the THz regime,” Opt. Express 12, 2387–2396 (2004). [CrossRef] [PubMed]

]: ω_TO/(2π) = 7.44 THz, ε_∞ = 10, ε₀ = 24.4, and ν/(2π) = 1.3 THz. For GaAs, we will use ω_TO/(2π) = 8.2 THz, ε₀ = 12.9, ε_∞ = 11, and ν/(2π) = 0.25 THz [34

M. Nagai, K. Tanaka, H. Ohtake, T. Bessho, T. Sugiura, T. Hirosumi, and M. Yoshida, “Generation and detection of terahertz radiation by electro-optical process in GaAs using 1.56 μm fiber laser pulses,” Appl. Phys. Lett. 85, 3974–3976 (2004). [CrossRef]

R. H. Stolen, “Far-infrared absorption in high resistivity GaAs,” Appl. Phys. Lett. 15, 74–75 (1969). [CrossRef]

–36

D. Grischkowsky, S. Keiding, M. van Exter, and Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990). [CrossRef]

]. In the optical range, we will use n = 2.15, n_g = 2.19 for LN at 1.3 μm [37

L. H. Deng, X. M. Gao, Z. S. Cao, W. D. Chen, Y. Q. Yuan, W. J. Zhang, and Z. B. Gong, “Improvement to Sellmeier equation for periodically poled LiNbO₃ crystal using mid-infrared difference-frequency generation,” Opt. Commun. 268, 110–114 (2006). [CrossRef]

] and n = 3.37, n_g = 3.53 for GaAs at 1.55 μm [34

, 38

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94, 6447–6455 (2003). [CrossRef]

3. Theoretical formalism

To find the terahertz radiation generated by the moving nonlinear polarization P^NL(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 P^NL included as a source. Eliminating the magnetic field, we obtain an equation for the terahertz electric field transform Ẽ_z(x,g,t) [27

\frac{\partial^{2} {\tilde{E}}_{z}}{\partial x^{2}} + κ^{2} {\tilde{E}}_{z} = - \frac{4 π Ω^{2}}{c^{2}} {\tilde{P}}^{NL},

(7)

where κ₂ = (Ω/c)²ε − g² and P̃^NL is the Fourier transform of P^NL. In the case of a blazed grating, we obtain for P̃^NL from Eq. (4):

{\tilde{P}}^{NL} (x, g, t) = \frac{p a}{2 π^{1 / 2}} e^{i Ω ξ - g^{2} a^{2} / 4 + i g x tan α}

(8)

and in the case of a holographic grating, we obtain from Eq. (5):

\begin{array}{l} {\tilde{P}}^{NL} (x, g, t) & = & \frac{p a}{2 π^{1 / 2}} e^{i Ω ξ - g^{2} a^{2} / 4} \\ \times & [e^{i g x tan α} + e^{- i g x tan α} + e^{- {(x / a)}^{2} {tan}^{2} α - G^{2} a^{2}} (e^{g G a^{2}} + e^{- g G a^{2}})] . \end{array}

(9)

Using the slowly varying envelope approximation (see, for example, [39

K. L. Vodopyanov, “Optical generation of narrow-band terahertz packets in periodically-inverted electro-optic crystals: conversion efficiency and optimal laser pulse format,” Opt. Express 14, 2263–2276 (2006). [CrossRef] [PubMed]

]) we obtain the solution of Eq. (7) as

{\tilde{E}}_{z} (x, g, t) = \frac{2 π Ω^{2}}{i κ c^{2}} \int_{0}^{x} d x^{'} {\tilde{P}}^{NL} (x^{'}, g, t) e^{i κ (x^{'} - x)} .

(10)

Solution (10) in the Fourier domain can be transformed to the y domain by taking inverse transform in the following form

E_{z} (x, y, t) = \int_{- \infty}^{\infty} d g {\tilde{E}}_{t} (x, g, y) e^{- i g y} .

(11)

To 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 (

n_{g} / cos α \approx \sqrt{ε}

) under consideration [40

M. I. Bakunov, A. V. Maslov, and S. B. Bodrov, “Fresnel formulas for the forced electromagnetic pulses and their application for optical-to-terahertz conversion in nonlinear crystals,” Phys. Rev. Lett. 99, 203904 (2007). [CrossRef]

]. 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)/(2π^1/2) in Eq. (8) transforms to the delta function δ(g). Substitution of δ(g) into Eq. (11) gives (at ν → 0)

E_{z} (x, t) = \frac{2 π p Ω x}{c \sqrt{ε}} sinc [\frac{Ω x}{2 c} (\frac{n_{g}}{cos α} - \sqrt{ε})] sin [Ω t - \frac{Ω x}{2 c} (\frac{n_{g}}{cos α} + \sqrt{ε})] .

(12)

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 n_g/ 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

n_{g} / cos α - \sqrt{ε} = 0

. 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

cos α = n_{g} / Re \sqrt{ε (Ω)} .

(13)

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

E_{z} (x, t) \approx \frac{2 π p}{\sqrt{| ε |} Im \sqrt{ε}} (1 - e^{\frac{Ω x}{c} Im \sqrt{ε}}) sin (Ω t - \frac{Ω x}{2 c} Re \sqrt{ε})

(14)

with

Im \sqrt{ε} < 0

. 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 ∼ L_ℓ with

L_{ℓ} = c / (Ω | Im \sqrt{ε} |)

. Taking, for example, Ω/(2π) = 1 THz, we obtain L_ℓ ≈ 1.4 mm for LN and L_ℓ ≈ 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

L_{tw} = n_{g} a_{FWHM} {(ε_{0} - n_{g}^{2})}^{- 1 / 2}

[27

] can be estimated as L_tw ≈ 0.5a_FWHM for LN and L_tw ≈ 5.3a_FWHM for GaAs. If L_tw > L_ℓ, 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 L_tw < L_ℓ, the amplitude saturates at a smaller value, which is defined by the parameter a_FWHM .

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 a_FWHM 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, L_ℓ.

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 a_FWHM = 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 a_FWHM = 5 mm [Fig. 3(a)], the transverse walkoff length L_tw ≈ 2.5 mm is larger than L_ℓ ≈ 1.4 mm and, therefore, the amplitude of the phase matched wave saturates at x ≈ L_ℓ ≈ 1.4 mm to the maximum value defined by Eq. (14). For a_FWHM = 0.5 mm [Fig. 3(b)], the transverse walkoff effect is more pronounced: L_tw ≈ 0.3 mm is smaller than L_ℓ, and the saturation occurs at the distance x ≈ L_tw ≈ 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 a_FWHM = 0.5 mm is 10 times higher than for a_FWHM = 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 |E_z(x,y,t)| in LN for (a) a_FWHM = 5 mm and (b) a_FWHM = 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 a_FWHM. The dashed segment of the curve for a_FWHM = 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 a_FWHM/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 L_ℓ and L_tw. Indeed, for the curves with a_FWHM = 1, 0.5, and 0.1 mm (L_tw ≈ 0.5, 0.25, and 0.05 mm, respectively), the efficiency reaches saturation at L ≈ L_ℓ ≈ 1.4 mm. For a_FWHM = 5 mm (L_tw ≈ 2.5 mm), the efficiency reaches saturation at L ≈ L_tw ≈ 2.5 mm. Physically, this can be explained as follows. For L_tw < L_ℓ, the amplitude of the phase matched wave reaches its maximum at x ≈ L_tw; however, the width of the terahertz beam (and therefore the terahertz power) continues to grow with x until x ≈ L_ℓ (Fig. 3). For L_tw > L_ℓ, on the contrary, the width of the terahertz beam reaches its steady state value at x ≈ L_ℓ whereas the amplitude of the terahertz wave continues to grow until x ≈ L_tw (Fig. 3).

Fig. 4 Efficiency as a function of the LN crystal thickness L for several values of a_FWHM (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 a_FWHM = 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 a_FWHM ∼ 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

K. L. Yeh, M. C. Hoffman, J. Hebling, and K. A. Nelson, “Generation of 10 μJ ultrashort terahertz pulses by optical rectification,” Appl. Phys. Lett. 90, 171121 (2007). [CrossRef]

A. G. Stepanov, S. Henin, Y. Petit, L. Bonacina, J. Kasparian, and J.-P. Wolf, “Mobile source of high-energy single-cycle terahertz pulses,” Appl. Phys. B 101, 11–14 (2010). [CrossRef]

–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 a_FWHM = 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 a_FWHM = 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 |E_z(x,y,t)| in LN for (a) a_FWHM = 5 mm and (b) a_FWHM = 0.5 mm, the holographic grating scheme. Optical beams are shown by dashed lines. Profiles E_z(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 L_tw (see Sec. 4.1). Additionally, due to a low terahertz absorption in GaAs [see Eq. (6)] the attenuation length L_ℓ ≈ 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 a_FWHM = 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 ≈ L_tw ≈ 2.7 cm for a_FWHM = 5 mm [Fig. 6(a)] and at x ≈ L_tw ≈ 2.7 mm for a_FWHM = 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 a_FWHM in Fig. 6, the steady-state width of the terahertz beam is reached at x ≈ L_ℓ ≈ 4.7 cm. Correspondingly, the conversion efficiency, shown in Fig. 7 for several values of a_FWHM, reaches saturation at x ≈ L_ℓ ≈ 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 |E_z(x,y,t)| in GaAs for (a) a_FWHM = 5 mm and (b) a_FWHM = 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 a_FWHM (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 a_FWHM = 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 a_FWHM = 0.5 mm and L = 8 mm [Fig. 8(b)], the conversion efficiency is 6.6 ×10⁻⁶.

Fig. 8 Snapshots of |E_z(x,y,t)| in GaAs for (a) a_FWHM = 5 mm and (b) a_FWHM = 0.5 mm, the holographic grating scheme. Optical beams are shown by dashed lines. Profiles E_z(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.

References and links

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2.	T. Yajima and K. Inoue, “Submillimeter-wave generation by difference-frequency mixing of ruby laser lines in ZnTe,” IEEE J. Quantum Electron. 5, 140–146 (1969). [CrossRef]
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6.	Y. J. Ding and J. B. Khurgin, “A new scheme for efficient generation of coherent and incoherent submillimeter to THz waves in periodically-poled lithium niobate,” Opt. Commun. 148, 105–109 (1998). [CrossRef]
7.	Y. Avetisyan, Y. Sasaki, and H. Ito, “Analysis of THz-wave surface-emitted difference-frequency generation in periodically poled lithium niobate waveguide,” Appl. Phys. B 73, 511–514 (2001). [CrossRef]
8.	Y. Sasaki, Y. Avetisyan, K. Kawase, and H. Ito, “Terahertz-wave surface-emitted difference frequency generation in slant-stripe-type periodically poled LiNbO₃ crystal,” Appl. Phys. Lett. 81, 3323–3325 (2002). [CrossRef]
9.	Y. Sasaki, Y. Avetisyan, H. Yokoyama, and H. Ito, “Surface-emitted terahertz-wave difference-frequency generation in two-dimensional periodically poled lithium niobate,” Opt. Lett. 30, 2927–2929 (2005). [CrossRef] [PubMed]
10.	D. E. Thompson and P. D. Coleman, “Step-tunable far infrared radiation by phase matched mixing in planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 22, 995–1000 (1974). [CrossRef]
11.	W. Shi and Y. J. Ding, “Designs of terahertz waveguides for efficient parametric terahertz generation,” Appl. Phys. Lett. 82, 4435–4437 (2003). [CrossRef]
12.	H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett. 29, 1751–1753 (2004). [CrossRef] [PubMed]
13.	V. Berger and C. Sirtori, “Nonlinear phase matching in THz semiconductor waveguides,” Semicond. Sci. Technol. 19, 964–970 (2004). [CrossRef]
14.	A. C. Chiang, T. D. Wang, Y. Y. Lin, S. T. Lin, H. H. Lee, Y. C. Huang, and Y. H. Chen, “Enhanced terahertz wave parametric generation and oscillation in lithium niobate waveguides at terahertz frequencies,” Opt. Lett. 30, 3392–3394 (2005). [CrossRef]
15.	C. Staus, T. Kuech, and L. McCaughan, “Continuously phase-matched terahertz difference frequency generation in an embedded waveguide structure supporting only fundamental modes,” Opt. Express 16, 13296–13303 (2008). [CrossRef] [PubMed]
16.	Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express 17, 13502–13515 (2009). [CrossRef] [PubMed]
17.	D. A. Bagdasaryan, A. D. Makaryan, and P. S. Pogosyan, “Cerenkov radiation from a propagating nonlinear polarization wave,” JETP Lett. 37, 594–596 (1983).
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OCIS Codes
(260.3090) Physical optics : Infrared, far
(190.4223) Nonlinear optics : Nonlinear wave mixing

ToC Category:
Nonlinear Optics

History
Original Manuscript: October 31, 2012
Revised Manuscript: November 26, 2012
Manuscript Accepted: November 26, 2012
Published: December 10, 2012

Citation
M. I. Bakunov, M. V. Tsarev, and E. A. Mashkovich, "Terahertz difference-frequency generation by tilted amplitude front excitation," Opt. Express 20, 28573-28585 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-27-28573

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References

T. Yajima and K. Inoue, “Submillimeter-wave generation by optical difference-frequency mixing of ruby R1 and R2 laser lines,” Phys. Lett. A26, 281–282 (1968). [CrossRef]
T. Yajima and K. Inoue, “Submillimeter-wave generation by difference-frequency mixing of ruby laser lines in ZnTe,” IEEE J. Quantum Electron.5, 140–146 (1969). [CrossRef]
T. Taniuchi and H. Nakanishi, “Collinear phase-matched terahertz-wave generation in GaP crystal using a dual-wavelength optical parametric oscillator,” J. Appl. Phys.95, 7588–7591 (2004). [CrossRef]
W. Shi, Y. J. Ding, N. Fernelius, and K. Vodopyanov, “Efficient, tunable, and coherent 0.18–5.27-THz source based on GaSe crystal,” Opt. Lett.27, 1454–1456 (2002). [CrossRef]
W. Shi and Y. J. Ding, “A monochromatic and high-power terahertz source tunable in the ranges of 2.7–38.4 and 58.2–3540 μm for variety of potential applications,” Appl. Phys. Lett.84, 1635–11637 (2004). [CrossRef]
Y. J. Ding and J. B. Khurgin, “A new scheme for efficient generation of coherent and incoherent submillimeter to THz waves in periodically-poled lithium niobate,” Opt. Commun.148, 105–109 (1998). [CrossRef]
Y. Avetisyan, Y. Sasaki, and H. Ito, “Analysis of THz-wave surface-emitted difference-frequency generation in periodically poled lithium niobate waveguide,” Appl. Phys. B73, 511–514 (2001). [CrossRef]
Y. Sasaki, Y. Avetisyan, K. Kawase, and H. Ito, “Terahertz-wave surface-emitted difference frequency generation in slant-stripe-type periodically poled LiNbO3 crystal,” Appl. Phys. Lett.81, 3323–3325 (2002). [CrossRef]
Y. Sasaki, Y. Avetisyan, H. Yokoyama, and H. Ito, “Surface-emitted terahertz-wave difference-frequency generation in two-dimensional periodically poled lithium niobate,” Opt. Lett.30, 2927–2929 (2005). [CrossRef] [PubMed]
D. E. Thompson and P. D. Coleman, “Step-tunable far infrared radiation by phase matched mixing in planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech.22, 995–1000 (1974). [CrossRef]
W. Shi and Y. J. Ding, “Designs of terahertz waveguides for efficient parametric terahertz generation,” Appl. Phys. Lett.82, 4435–4437 (2003). [CrossRef]
H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett.29, 1751–1753 (2004). [CrossRef] [PubMed]
V. Berger and C. Sirtori, “Nonlinear phase matching in THz semiconductor waveguides,” Semicond. Sci. Technol.19, 964–970 (2004). [CrossRef]
A. C. Chiang, T. D. Wang, Y. Y. Lin, S. T. Lin, H. H. Lee, Y. C. Huang, and Y. H. Chen, “Enhanced terahertz wave parametric generation and oscillation in lithium niobate waveguides at terahertz frequencies,” Opt. Lett.30, 3392–3394 (2005). [CrossRef]
C. Staus, T. Kuech, and L. McCaughan, “Continuously phase-matched terahertz difference frequency generation in an embedded waveguide structure supporting only fundamental modes,” Opt. Express16, 13296–13303 (2008). [CrossRef] [PubMed]
Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express17, 13502–13515 (2009). [CrossRef] [PubMed]
D. A. Bagdasaryan, A. D. Makaryan, and P. S. Pogosyan, “Cerenkov radiation from a propagating nonlinear polarization wave,” JETP Lett.37, 594–596 (1983).
K. Suizu, T. Shibuya, T. Akiba, T. Tutui, C. Otani, and K. Kawase, “Cherenkov phase-matched monochromatic THz wave generation using difference frequency generation with a lithium niobate crystal,” Opt. Express16, 7493–7498 (2008). [CrossRef] [PubMed]
K. Suizu, K. Koketsu, T. Shibuya, T. Tsutsui, T. Akiba, and K. Kawase, “Extremely frequency-widened terahertz wave generation using Cherenkov-type radiation,” Opt. Express17, 6676–6681 (2009). [CrossRef] [PubMed]
J. Hebling, G. Almási, I. Z. Kozma, and J. Kuhl, “Velocity matching by pulse front tilting for large-area THz-pulse generation,” Opt. Express10, 1161–1166 (2002). [CrossRef] [PubMed]
K. L. Yeh, M. C. Hoffman, J. Hebling, and K. A. Nelson, “Generation of 10 μJ ultrashort terahertz pulses by optical rectification,” Appl. Phys. Lett.90, 171121 (2007). [CrossRef]
A. G. Stepanov, L. Bonacina, S. V. Chekalin, and J.-P. Wolf, “Generation of 30 μJ single-cycle terahertz pulses at 100 Hz repetition rate by optical rectification,” Opt. Lett.33, 2497–2499 (2008). [CrossRef] [PubMed]
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