## Terahertz difference-frequency generation by tilted amplitude front excitation |

Optics Express, Vol. 20, Issue 27, pp. 28573-28585 (2012)

http://dx.doi.org/10.1364/OE.20.028573

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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 LiNbO_{3} pumped at 1.3 *μ*m and GaAs pumped at 1.55 *μ*m shows its efficiency.

© 2012 OSA

## 1. Introduction

*ω*

_{1}and

*ω*

_{2}in a quadratic nonlinear medium that results in the generation of a third wave with a difference (terahertz) frequency Ω =

*ω*

_{1}−

*ω*

_{2}. 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*and terahertz phase velocity

_{g}*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

1. T. Yajima and K. Inoue, “Submillimeter-wave generation by optical difference-frequency mixing of ruby R_{1} and R_{2} 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]

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]

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 *μ*m for variety of potential applications,” Appl. Phys. Lett. **84**, 1635–11637 (2004). [CrossRef]

_{3}(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

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]

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]

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]

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]

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]

*v*in the direction of the beam, i.e., at the first order diffraction angle

_{g}*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*cos

_{g}*α*. 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*cos

_{g}*α*=

*v*

_{THz}(Ω). Thus, one can achieve phase matching with the quasiplane terahertz wave propagating normally to the boundary.

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]

*μ*J [21

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

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 LiNbO_{3},” Appl. Phys. Lett. **98**, 091106 (2011). [CrossRef]

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]

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]

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]

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]

*μ*m wavelength, like in Refs. [18

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]

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]

*μ*m.

## 2. Generation scheme and model

*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

**E**

_{0}= (0,0,

*E*

_{0}) but with different frequencies

*ω*

_{1,2}: The incident optical intensity is thus

*I*(0−,

*y*,

*t*) ∝ exp(−

*y*

^{2}/

*a*

^{2}) and the standard full width at half-maximum (FWHM) is

*a*is assumed to be large enough so that we can neglect the diffraction divergence of the laser beam on the crystal length

*L*.

*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*= 2

*π*/

*d*is the grating groove frequency (with

*d*the grating period), and

*x*-component of the wave vector [with

*n*=

_{m}*n*(

*ω*) the optical refractive index of the crystal at the frequency

_{m}*ω*and

_{m}*c*the speed of light]. The diffraction angle

*α*(Fig. 1) is defined as sin

*α*=

*λ*̄/(

*nd*) where

*λ*̄ = 2

*πc*/

*ω*̄ is the central (vacuum) wavelength,

*n*=

*n*(

*ω*̄), and

*ω*̄= (

*ω*

_{1}+

*ω*

_{2})/2 is the central frequency. Expanding

*h*in power series up to the first order of Ω (Ω =

_{m}*ω*

_{1}−

*ω*

_{2}, Ω ≪

*ω*̄) and

*γ*, the integral in Eq. (2) can be evaluated as where

*ξ*=

*t*−

*n*/(

_{g}x*c*cos

*α*),

*η*

_{±}=

*y*∓

*x*tan

*α*, and

*n*is the optical group refractive index. The next terms in the power series expansion are insignificant: the ∝ Ω

_{g}^{2}term gives only negligible corrections to the direction and absolute value of the diffracted beam phase velocity; the ∝

*γ*

_{2}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*) cos

_{g}*α*(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*) cos

_{g}*α*=

*v*

_{THz}(Ω). Thus, one can achieve phase matching with the quasiplane terahertz wave propagating along the

*x*-axis.

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]

*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

*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}= (2

*n*cos

*α*)

^{−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*≠ 0. In LN, such configuration occurs if the optical axis of the crystal is along the

_{z}*z*-axis, and

*d*

_{eff}=

*d*

_{33}= 168 pm/V [29

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]

*p*is achieved for a 〈110〉-cut crystal with the [001] axis oriented at ≈ 55° to the

_{z}*z*-axis [30

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]

*p*is defined by

_{z}*d*

_{eff}= (4/3)

^{1/2}

*d*

_{14}[30

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]

*d*

_{eff}= 65.6 pm/V [26

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]

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]

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}/(2

*π*) = 7.44 THz,

*ε*

_{∞}= 10,

*ε*

_{0}= 24.4, and

*ν*/(2

*π*) = 1.3 THz. For GaAs, we will use

*ω*

_{TO}/(2

*π*) = 8.2 THz,

*ε*

_{0}= 12.9,

*ε*

_{∞}= 11, and

*ν*/(2

*π*) = 0.25 THz [34

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 *μ*m fiber laser pulses,” Appl. Phys. Lett. **85**, 3974–3976 (2004). [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]

*n*= 2.15,

*n*= 2.19 for LN at 1.3

_{g}*μ*m [37

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 LiNbO_{3} crystal using mid-infrared difference-frequency generation,” Opt. Commun. **268**, 110–114 (2006). [CrossRef]

*n*= 3.37,

*n*= 3.53 for GaAs at 1.55

_{g}*μ*m [34

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 *μ*m fiber laser pulses,” Appl. Phys. Lett. **85**, 3974–3976 (2004). [CrossRef]

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

**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

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]

*κ*

_{2}= (Ω/

*c*)

^{2}

*ε*−

*g*

^{2}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): and in the case of a holographic grating, we obtain from Eq. (5):

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]

*y*domain by taking inverse transform in the following form

*x*=

*L*), one can use the usual Fresnel transmission coefficient in the near-phase-matched regime (

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]

*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

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]

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]

*z*-axis for the optical and terahertz pulses.

## 4. Results and discussion

*α*. Using the general solution of Sec. 3, we analyze further both schemes for two materials – LN pumped at 1.3

*μ*m and GaAs pumped at 1.55

*μ*m.

### 4.1. Tilt angles and characteristic lengths

*a*→ ∞. In this limit, the function

*a*exp(−

*g*

^{2}

*a*

^{2}/4)/(2

*π*

^{1/2}) 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

*n*/ cos

_{g}*α*. 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

*x*, according to Eq. (12).

*α*on the phase-matched frequency Ω for LN pumped at 1.3

*μ*m and GaAs pumped at 1.55

*μ*m. If the phase matching condition (13) is fulfilled and the losses are included, the terahertz field is given by with

*x*all the way to the exit boundary of the crystal. Due to the losses, the growth saturates at

*x*∼

*L*with

_{ℓ}*π*) = 1 THz, we obtain

*L*≈ 1.4 mm for LN and

_{ℓ}*L*≈ 4.7 cm for GaAs.

_{ℓ}*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

**28**, 1724–1734 (2011). [CrossRef]

**28**, 1724–1734 (2011). [CrossRef]

*L*

_{tw}≈ 0.5

*a*

_{FWHM}for LN and

*L*

_{tw}≈ 5.3

*a*

_{FWHM}for GaAs. If

*L*

_{tw}>

*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

_{ℓ}*x*→ ∞. However, if

*L*

_{tw}<

*L*, the amplitude saturates at a smaller value, which is defined by the parameter

_{ℓ}*a*

_{FWHM}.

**28**, 1724–1734 (2011). [CrossRef]

*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}*ω*

_{1}−

*ω*

_{2}but also the central wavelength

*λ*̄ of the laser beam to preserve the phase matching condition (13). Assuming that a 10% variation of

*λ*̄ is acceptable not to affect the diffraction efficiency noticeably, we estimate the corresponding variation of the tilt angle

*α*as Δ

*α*∼ 10

^{−1}tan

*α*, i.e., Δ

*α*∼ 11.5° for LN and Δ

*α*∼ 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

*μ*m wavelength and 4 ×10

^{6}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/(2

*π*) = 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 ×10

^{6}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

**16**, 7493–7498 (2008). [CrossRef] [PubMed]

**17**, 6676–6681 (2009). [CrossRef] [PubMed]

*z*-axis, as in Ref. [18

**16**, 7493–7498 (2008). [CrossRef] [PubMed]

*a*

_{FWHM}= 5 mm [Fig. 3(a)], the transverse walkoff length

*L*

_{tw}≈ 2.5 mm is larger than

*L*≈ 1.4 mm and, therefore, the amplitude of the phase matched wave saturates at

_{ℓ}*x*≈

*L*≈ 1.4 mm to the maximum value defined by Eq. (14). For

_{ℓ}*a*

_{FWHM}= 0.5 mm [Fig. 3(b)], the transverse walkoff effect is more pronounced:

*L*

_{tw}≈ 0.3 mm is smaller than

*L*, and the saturation occurs at the distance

_{ℓ}*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).

*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

*μ*m. 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*and

_{ℓ}*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*≈ 1.4 mm. For

_{ℓ}*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*, the amplitude of the phase matched wave reaches its maximum at

_{ℓ}*x*≈

*L*

_{tw}; however, the width of the terahertz beam (and therefore the terahertz power) continues to grow with

*x*until

*x*≈

*L*(Fig. 3). For

_{ℓ}*L*

_{tw}>

*L*, on the contrary, the width of the terahertz beam reaches its steady state value at

_{ℓ}*x*≈

*L*whereas the amplitude of the terahertz wave continues to grow until

_{ℓ}*x*≈

*L*

_{tw}(Fig. 3).

*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

**16**, 7493–7498 (2008). [CrossRef] [PubMed]

**17**, 6676–6681 (2009). [CrossRef] [PubMed]

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

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 LiNbO_{3},” Appl. Phys. Lett. **98**, 091106 (2011). [CrossRef]

*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.

### 4.3. GaAs pumped at 1.55 μm

*L*

_{tw}(see Sec. 4.1). Additionally, due to a low terahertz absorption in GaAs [see Eq. (6)] the attenuation length

*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

_{ℓ}*x*in comparison to LN. This means that terahertz generation can be very efficient in thick GaAs crystals.

*μ*m 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 ×10

^{6}W/cm. The difference frequency is again Ω/(2

*π*) = 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*≈ 4.7 cm. Correspondingly, the conversion efficiency, shown in Fig. 7 for several values of

_{ℓ}*a*

_{FWHM}, reaches saturation at

*x*≈

*L*≈ 4.7 cm. Comparing Figs. 7 and 4 one can conclude that GaAs provides an efficiency an order of magnitude higher than LN.

_{ℓ}*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}.

## 5. Conclusion

^{−6}for LN pumped by 1.3-

*μ*m wavelength, 2-mJ energy, 15-ns duration laser pulses, and even higher, i.e., ∼ 5 ×10

^{−5}, for GaAs pumped by 1.55-

*μ*m wavelength laser pulses of the same energy and duration.

## Acknowledgments

## References and links

1. | T. Yajima and K. Inoue, “Submillimeter-wave generation by optical difference-frequency mixing of ruby R |

2. | T. Yajima and K. Inoue, “Submillimeter-wave generation by difference-frequency mixing of ruby laser lines in ZnTe,” IEEE J. Quantum Electron. |

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. |

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. |

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 |

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. |

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 |

8. | Y. Sasaki, Y. Avetisyan, K. Kawase, and H. Ito, “Terahertz-wave surface-emitted difference frequency generation in slant-stripe-type periodically poled LiNbO |

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. |

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. |

11. | W. Shi and Y. J. Ding, “Designs of terahertz waveguides for efficient parametric terahertz generation,” Appl. Phys. Lett. |

12. | H. Cao, R. A. Linke, and A. Nahata, “Broadband generation of terahertz radiation in a waveguide,” Opt. Lett. |

13. | V. Berger and C. Sirtori, “Nonlinear phase matching in THz semiconductor waveguides,” Semicond. Sci. Technol. |

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. |

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. | 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. | D. A. Bagdasaryan, A. D. Makaryan, and P. S. Pogosyan, “Cerenkov radiation from a propagating nonlinear polarization wave,” JETP Lett. |

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 |

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 |

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 |

21. | K. L. Yeh, M. C. Hoffman, J. Hebling, and K. A. Nelson, “Generation of 10 |

22. | A. G. Stepanov, L. Bonacina, S. V. Chekalin, and J.-P. Wolf, “Generation of 30 |

23. | 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 |

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. |

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 LiNbO |

26. | J. A. Fülöp, L. Pálfalvi, G. Almási, and J. Hebling, “Design of high-energy terahertz sources based on optical rectification,” Opt. Express |

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. | 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. |

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 |

30. | Q. Chen, M. Tani, Z. Jiang, and X.-C. Zhang, “Electro-optic transceivers for terahertz-wave applications,” J. Opt. Soc. Am. B |

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 |

32. | L. Pálfalvi, J. Hebling, J. Kuhl, Á. Péter, and K. Polgár, “Temperature dependence of the absorption and refraction of Mg-doped congruent and stoichiometric LiNbO |

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 |

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 |

35. | R. H. Stolen, “Far-infrared absorption in high resistivity GaAs,” Appl. Phys. Lett. |

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 |

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 LiNbO |

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. |

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 |

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. |

**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

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- 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]
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- 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]
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- 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]
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- 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]
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- J. A. Fülöp, L. Pálfalvi, G. Almási, and J. Hebling, “Design of high-energy terahertz sources based on optical rectification,” Opt. Express18, 12311–12327 (2010). [CrossRef] [PubMed]
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- 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. B25, B6–B19 (2008). [CrossRef]
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- 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. B78, 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]
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- 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]
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- 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]

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