Terahertz generation by means of optical lasers

Laser Phys. Lett. 5, No. 8, 559–576 (2008) / DOI 10.1002/lapl.200810039 559

Abstract: Generation of high-power radiation in terahertz fre-quency range 0.3 – 10 THz is a fast developing research area at-tracting considerable interest over the past years due to its im-portant applications in spectroscopy, communication, biomedicalimaging and tomography. A brief review of laser-based genera-tion methods is presented, starting from the schemes of resonancelaser excitation of photoconductive materials, organic moleculesand ionized gases, and proceeding with more detailed descriptionof non-resonance schemes of optical rectification and differencefrequency generation in materials with high optical nonlinearity.Recent progress achieved in both groups of methods is compared.

Opticalpump Crystal

Terahertzoutput

x

θ

θ

k1

k2 kTHz

Difference-frequency generation in a bulk non-linear crystal.Schematic diagram and orientations of the wave-vectors insidethe crystal

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Terahertz generation by means of optical lasersG.Kh. Kitaeva ∗

Faculty of Physics, M.V. Lomonosov Moscow State University, Moscow 119991, Russia

Received: 24 March 2008, Revised: 13 April 2008, Accepted: 15 April 2008Published online: 7 May 2008

Key words: terahertz range; difference frequency generation; optical rectification; resonance laser excitation; phase matching; period-ically poled crystal

PACS: 07.57.Hm, 42.72.Ai, 42.79.Nv

1. Introduction

The problem of efficient generation of radiation in the ter-ahertz frequency range 0.3 – 10 THz (cited sometimes as“terahertz gap”) has attracted considerable interest overthe past ten years due to a lot of important promising ap-plications of terahertz (THz) waves. The long-wave edgeof optical phonons, molecular vibrations, intrinsic excita-tions of subnanometer-scale objects in the most of materi-als falls near, or just into the THz range. Therefore, on theone hand, attenuation of THz waves in atmosphere gases,crystals, organic materials and so on, is not too large andthe waves can propagate to certain objects inside the me-dia. And, on the other hand, a vast number of eigenfre-quencies which are typical for vibrations or rotations oflarge molecules, molecular chains, electron transitions innano-structured materials, and so on, keep in resonancewith THz-range waves and can be used for the investi-

gation and diagnostics of relevant objects. Among the at-tractive applications of the THz radiation there are THzspectroscopy for chemical identification of large complexmolecules, sensing of chemical processes, industrial or en-vironmental diagnostics, single-shot THz imaging, nonin-vasive biomedical imaging and medical tomography, ma-nipulation of charged and magnetic particles, as well asof nanoparticles [1–5], monitoring of electronic intersub-band processes and carrier dynamics in semiconductors,gases, dielectrics, liquids and nanostructured materials [6–12]. Another group of applications is concerned with THzradio-vision [5]: communication, radioastronomy, aeron-omy.

Destructive influence of Rayleigh scattering, whichscales as λ−4, is less for THz circuits than for near-infraredor visible ones. Nevertheless, although a variety of real-lifeimportant applications have already been demonstrated onthe lab scale, their further advancement into a mass utiliza-

∗ Corresponding author: e-mail: [email protected]

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560 G.Kh. Kitaeva: Terahertz generation by means of optical lasers

tion still requires considerable progress in the developmentof effective THz sources.

Quite different physical approaches are considerednow to solve this problem. The primary classification ofthe various THz generation methods can be started withthe general Maxwell wave equation:

∆E(r, t) − 1c2

∂2E(r, t)∂t2

= (1)

=4π

c

∂2P (r, t)∂t2

+4π

c

∂j(r, t)∂t

.

The source functions are on the right-hand side of theequation, so that the electric field E can be generated viathe time variation of the local current density j, or via themodulation of the local polarization P . Moreover, in thelatter case there are two different opportunities, the onewhere the THz response is somehow delayed with respectto external excitation, and the other where the response isalmost instantaneous, depending on the resonance or non-resonance character of the excitation. According to thisdivision, the basic principles for THz generation can besorted into three vast groups: the methods where a variablecurrent density j appears as a source, the ones where mod-ulation of the medium polarization P has a resonance char-acter, and the ones based on instantaneous non-resonantmodulation of P . For the methods using the movement offree carriers in conductive media or ionized gases, distinc-tion between the first and second groups is not strict andconventionally they will be attributed to the second group.

The first-group methods are realized with free elec-trons in such different devices as vacuum electron tubesand charged particle accelerators. Remarkably, namelythese devices occupy the bottom (vacuum tubes) and thetop (relativistic photonics) of the power-scaled list of THzsources available today. The highest possible THz pow-ers are now achievable using accelerated relativistic elec-trons [13], in free-electron lasers (FEL, with pulse ener-gies in the µJ range) [14,15] and synchrotron radiationsources [16]. FEL devices are characterized by high di-rectivity, tunable wavelength, narrow frequency band, highdegree of coherence. However, the cost effective accessto these large-scale facilities is limited. For mass appli-cations, much desirable are compact table-top sources,which can operate at room temperature and, at the sametime, produce powerful and coherent THz radiation. Thereare well-developed compact sources in the adjacent gi-gahertz range, such as vacuum-electronic submillimetertraveling-wave or backward-wave oscillators, solid-statetunnel-injection transit-time diodes, Gunn devices and oth-ers [17–20]. Their output power decreases rapidly whileapproaching the THz range [3], and the energy per spec-tral and spatial mode (radiance) is still too low, so specialattempts are required for applications in THz spectroscopyand THz communication [18].

This paper concerns the methods of laser-induced THzgeneration via polarization effects, belonging to the resid-ual two groups. To describe all the variety of THz gen-eration methods one has to add also the available THz

lasers (masers) – sources with the inversion of popula-tion at THz-shifted energy levels. Among the possibleTHz lasers there are CO2-pumped or Raman gas lasers[21,22], and several types of semiconductor lasers [23–26]. Gas lasers provide generation at a number of nar-row lines. More perspective are frequency-tunable com-pact laser sources such as semiconductor strained p-Gelaser [23], quantum cascade THz lasers on the basis ofSiGe/Si, AlGaAs/GaAs, InAlAs/InGaAs heterostructures[24–26]. Necessity of low-temperature operation is one ofthe main problems within this field of research.

The subsequent part of the paper is organized as fol-lows. A brief review of various methods relating to thesecond group, based on the resonance non-linear optical-to-terahertz susceptibilities of different materials, is pre-sented in Sec. 2, including the methods based on thecurrent surge in photoconductive materials (Subsec. 2.1),semiconductor surface emission (Subsec. 2.2), intramolec-ular photoinduced-charge transfer (Subsec. 2.3), andphoto-ionization of gases (Subsec. 2.4). Sec. 3. presentsmore detailed information on the generation methods ofthe third group based on the polarization excitation vianon-resonance non-linear susceptibilities. The schemes ofTHz parametric oscillation and difference-frequency gen-eration under two-laser pumping are considered in Sub-sec. 3.1, and the optical rectification schemes are describedin Subsec. 3.2, including those based on Cherenkov-typegeometry for bulk crystals (3.2.1), the tilted pulse front(3.2.2), and quasi-phase matching (3.2.3). Summarizingthe reviewed information, conclusions will be drawn inSec. 4 on the current progress and perspectives of the lasermethods for the THz generation.

2. Resonance polarization effects

Within the scope of this group, the THz pulses of sub-µJ-scale energy are generated under resonant excitation of asource medium by high-power laser pulses. In the frame-work of the general phenomenological approach, all pro-cesses can be presented in terms of the 2nd or 3rd - orderoptical susceptibilities of the medium, in which the imag-inary parts are substantial [27]. Non-coherently generatedTHz pulses have high directional divergence and occupy awide spectral range. The spectral range of maximal gen-eration efficiency is limited by the relaxation time of thecharged particles involved in the resonance process.

2.1. Photoconductive emitters and photo-mixers

Biased photoconductive (PC) emitters (antennas,switches) are considered to be among the most well-proven and commonly used THz emitters [28]. Conven-tionally, a PC emitter is a semiconductor material with avoltage applied. Under illumination of its surface by vis-ible or near-infrared ultrafast laser pulse (of 100 – 200 fs

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Laser Phys. Lett. 5, No. 8 (2008) 561

Generation Optical pump THz output Energy Ref.scheme efficiency

Pulse energy/ Repetition Wavelength Pulse Mean FrequencyFluence/ rate energy powerPower

1 GaAs 40 µJ/cm2 10 Hz 770 nm 8×10−7 J 8×10−6 W 0.1 – 3 THz [36]PC antenna peak at 0.5 THz

2 GaAs 90 µJ/cm2 1 kHz 800 nm 4×10−7 J 4×10−4 W 0.1 – 3 THz ∼ 8×10−4 [37]PC antenna peak at 0.3 THz

3 GaAs 400 mW – 780 nm – 2.6×10−6 W 0.85 THz 6.6×10−6 [43]travelling-wave (CW) (CW)photomixer

4 Res. dipole-antenna – 1.55 µm – 1.1×10−5 W 1.04 THz [49]integr. InP/InGaAs (CW) (CW)photodiode

5 n-InAs 10 nJ 76 MHz 800 nm 1.6×10−13 J 1.2×10−5 W 0.1 – 3 THz 1.6×10−5 [61]surface emitter (60 nJ/cm2)in magnetic field

6 p-InAs 926 mW 250 kHz 790 nm ∼ 2×10−11 J 5.7×10−5 W 6.1×10−5 [69]surface emitter

7 TPO in bulk 11.4 mJ 16.7 Hz 1.064 µm 5×10−11 J 8×10−10 W 1.7 THz 4.3×10−9 [94]cooled LiNb03

8 DFG in bulk 17 mJ 500 Hz 1.064 µm 10−11 J 5×10−9 W 1.6 THz 6×10−10 [95]Mg:LiNb03

(CW 250 mW seed)

9 DFG in GaP 1.2 mJ 0.94 µm 5×10−9 J 2.6 THz 2.1×10−6 [105]+ and1.2 mJ 1 µm

10 DFG in GaSe 143 µJ 60 kHz 10.59 µm 4×10−9 J 2.6×10−4 W 0.914 THz 1.4×10−5 [107]+ and133 µJ 10.26 µm

11 DFG in ZnGeP2 4 W 100 kHz 1.064 µm 2×10−8 J 2×10−3 W 2.45 THz 5×10−4 [108]and1.059 µm

12 QPM DFG in GaP 0.3 mJ 10 Hz 1.549 µm 2×10−10 J 2×10−9 W 1 THz 4×10−7 [124]+ and0.2 mJ 1.557 µm

13 QPM DFG in GaAs 0.2 µJ 50 MHz 2.107 µm 2×10−11 J 10−3 W 2.8 THz 4×10−5 [125]+ and0.34 µJ 2.150 µm

14 Collinear OR in 360 µJ 1 kHz 775 nm 5.4×10−10 J 5.4×10−7 W 1.5×10−6 [139]ZnTe

15 Cherenkov OR in 32 µJ 1 kHz 780 nm 3.1×10−9 J 3.1×10−6 W 9.7×10−5 [154]LiNbO3

(line-focusing)

16 Tilted pulse front < 20 mJ 10 Hz 800 nm 10−5 J 10−4 W 6×10−4 [159]OR in Mg:LiNbO3

17 Tilted pulse front 300 µJ 1 kHz 780 nm 1.5×10−7 J 1.5×10−4 W 5×10−4 [158]OR in Mg:LiNbO3 [160]

18 QPM OR in cooled 2 µJ 250 kHz 800 nm 2×10−11 J 5×10−6 W 1.8 THz 10−5 [164]PPLN (500 mW) 2.8×10−4 W/THz (18 GHz band)

19 QPM OR in GaAs 2.3 µJ 1 kHz 4.4 µm 6.6×10−10 J 6.6×10−7 W 2.2 THz 2.9×10−4 [113]

Table 1 Optical-input and terahertz-output characteristics for different generation schemes. The energy efficiency is calculated as theratio between the THz power and the total power of the optical pumping beams

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duration or less), absorption of photons results in thetransition of electrons from the valence to the conductionband. Conduction carriers are accelerated in an externalDC electric bias field and the resultant current surgegives rise to the emission of an electromagnetic pulse.The pulse duration is mostly determined by the lifetimeof conduction carriers in the semiconductor material. Inmaterials with picosecond and sub-picosecond carrierrecombination times, the spectral peak of the pulses hitsthe THz range.

Proposed as early as in the 1980s [29], PC emitters arebeing continuously improved up to now [28,30–45]. It wasfound that typically low THz output power (a few hundrednW) and the optical-to-THz conversion efficiency can beincreased by overcoming the effects of screening out thebias field by the photoexcited carriers [30–33]. The useof large excitation-aperture PC antennas was proposed forthis purpose [30], and the effects of Coulomb and radiationscreening were studied [32,33]. It was shown that for thefurther increase of the THz energy output and optical-to-terahertz conversion efficiency, higher optical powers werenot needed and, in general, led to optical saturation, butthe increase of electrical-bias field was desirable, givinga linear increase of the THz field [30,34–36]. PC anten-nas based on GaAs pumped by regeneratively-amplifiedTi:Sapphire lasers with large area emitters (with electrodesseparation up to 10 mm or above) emit up to 800 nJ [36]and 400 nJ [37] THz pulses at 10 Hz and at 1 kHz repe-tition rates, respectively (Table 1, lines 1,2). These high-power sources generate near unipolar pulses of approxi-mately 500 fs duration. The spectral peak of the pulseslies in the 0.3 – 0.5 THz range, while the whole spectrumranges from near 3 THz and down to the low gigahertzregion. Recently the possibilities for alternating-currenthigh-voltage biasing were considered [38]. To overcomethe slow photo-response time of conventional semiconduc-tors, low-temperature-grown (LTG) technique was sug-gested [39]. LTG epitaxial layers of GaAs [40,41], fabri-cated with the implantation of O, Si, Ga, As, and espe-cially N ions [42–44], exhibit the shortest electron life-time reaching a sub-100-fs duration and high resistivity.The spectrum of the THz radiation emitted by LTG-GaAsreaches up to 5 THz [45].

Apart from the femtosecond pulse pumping, sub-THzand low-frequency THz radiation can be obtained underpumping of the same photoconductive switch, or simply aphotodiode operating at a long-wavelength illumination,by two continous-wave near-IR (780 – 1550 nm) diodelaser beams with different frequencies [46]. The THz ra-diation appears as a result of resonance photo-mixing [46–49]. The absorption edge of GaAs (∼ 0.9 µm) enablesone to use powerful femtosecond radiation of Ti:Sapphirelaser systems for pumping, but this material cannot be acti-vated by solid-state or fiber lasers emitting at longer wave-lengths. Meanwhile, a topical is a THz source capableof coupling by an optical fiber to optical communicationtechniques. For this purpose, other promising materials areconsidered, such as heavy-ion-irradiated In0.53Ga0.47As

[50], LTG Be doped InxGa1−xAs [51], LTG GaBiAs al-loy [52], which demonstrate broadband generation up to2.5 – 3 THz. Semi-insulating GaAs, other semiconductors(such as InP, ErAs-doped GaAs and InGaAs) [48,49] arealso studied with the same purpose. Besides the high car-rier relaxation speed of the photo-mixing material, whichis required for the carriers to follow the beat frequencyin the THz range, the transit- and RC-time of the elec-trode structures are also much important. Special construc-tions of transit-time-limited p-i-n photomixers [43,48],log-periodic antenna-integrated device [49], quasi-ballisticcascaded p-i-n photomixers [53] are developed for increas-ing the conversion efficiency and expanding the genera-tion range to the THz frequencies above 1 THz. High-power THz radiation, tuned within 0.85 – 1.2 THz, was ob-tained recently using the travelling-wave photomixer fab-ricated on high-energy nitrogen-ion-implanted GaAs [43].The output power 2.64 µW was achieved at 0.85 THz forthe input power of 400 mW and 15 V bias voltage. Theoutput power of 10.9 µW was obtained at 1.04 THz witha resonant dipole-antenna integrated uni-travelling-carrierphotodiode [49] (Table 1, lines 3,4).

2.2. Semiconductor surface emitters

Formation of a macroscopic current is not necessary forelectromagnetic field generation. When illuminated byfemtosecond visible laser pulses, most semiconductor sur-faces emit stronger or weaker electromagnetic transientwith characteristic frequency in the THz range [54]. Af-ter photoexcitation of the carriers to the conduction band,surfaces of bulk semiconductors generate THz radiationdue to ultrafast charge transport and resonant optical recti-fication [55–66]. The charge transport can be driven by theintrinsic surface electric field of the semiconductor [57,58]or by a difference in the mobilities of the electrons andholes (photo-Dember effect [59,60]). The drift current dueto the surface field is dominant for GaAs and InP, photo-Dember effect is dominant for narrow-gap semiconductorssuch as InAs and InSb. Terahertz pulse generation fromInAs surface can be enhanced several times by a magneticfield [61,62] (Table 1, line 5). Additional continuous-wavelaser illumination was found to be effective for THz radi-ation from the surfaces of semi-insulating GaAs and InP[63]. Modification of the surfaces of GaAs and GaAsNcrystals by heavy ion irradiation [64,65] also leads to anincrease of the THz output.

Apart from the charge transport, emission of IR-activecoherent phonons and phonon polaritons under intrabandrelaxation processes, which proceed on the picosecond andfemtosecond scales after photoexcitation of the carriers,also lead to the generation of THz waves. In the frame-work of the phenomenological approach, these processesare considered as resonance optical rectification of fem-tosecond pulses [55,66,67]. An average THz output power56.5 µW has been obtained in this regime for an average



pump power of 926 mW under 180 fs pulse irradiation ofInN thin films [68,69] (Table 1, line 6).

Comparison between different THz-emitter concepts,based on the best semiconductor material, GaAs, wasmade recently [28]. Capabilities of surface emitters, con-tactless p-i-n emitters [70] and PC emitters with variousparameters were studied by means of Monte-Carlo simu-lations. It was found that the THz energy radiated by thephotoconductive emitter essentially exceeds the THz en-ergy radiated by surface and p-i-n emitters.

2.3. Intramolecular photoinduced-chargetransfer

Recently, a new method was proposed to produce a macro-scopic current that radiates a THz field, based on the pho-toinduced charge transfer along aligned molecules [27].Alignment can take place due to incorporation of themolecules into a crystal lattice [71], oriented deposition ona surface [72], or due to an external voltage as in the caseof PC emitters [73]. Produced by femtosecond laser ex-citation, microscopic currents through individual alignedmolecules add up to form a macroscopic current that gen-erates a THz-range electromagnetic wave. This effect issuccessfully used for the monitoring of the charge trans-port within the constituent molecules [71–74] in solutions,organic molecular crystals, semiconducting polymers. Asa method of THz generation, it is perspective, but the cur-rently achieved efficiencies are not high in comparisonwith the other techniques.

2.4. Ionized gases

Intrinsic vibrations of laser plasma, produced under thegas ionization by high-power laser pulses, lead to po-tentially high-power and scalable THz radiation. Differ-ent mechanisms of this type of generation were studiedover the past years, such as ponderomotive electron ac-celeration [75], crossing of the plasma-vacuum boundaryby a laser-plasma accelerated bunch [76,77], dynamics ofplasma density [78], four-wave mixing of the fundamentalpump field and its second harmonic [79–83], transition-Cherenkov mechanism [84]. In addition to the fundamen-tal pump, fields of other frequencies can take part in theprocess, in particular, the static electric bias field, or har-monics of the fundamental laser field. The mixing processis treated as a four-wave optical rectification via the reso-nant third-order susceptibility of the laser-induced plasmain a gas medium [79].

Recently [83], peak THz field amplitudes of more than400 kV/cm in the frequency range 0.3 – 7 THz were ob-tained for the air irradiated by the fundamental and thesecond harmonic of 25 fs pulses from a Ti:Sapphire am-plifier. The observed linear increase of the THz field withthe incident pulse energy promises further scaling of the

THz output. The remarkable feature of the gas emission isthe possibility to emit THz radiation within a large spec-tral range, due to the absence of phonon resonances. Inaddition, there are no reflections of optical and THz wavesinside the gas medium. A THz spectrometer covering theentire gap 0.3 – 10 THz was designed [82], based on ion-ized gases used as the emitter and sensor media.

3. Non-resonance polarization effects

This group of methods provides THz radiation as a resultof parametric down-conversion-type non-linear interactionof coherent optical waves due to non-resonant second-order optical-to-terahertz (electrooptical) susceptibility ofthe source medium. This can be photomixing of two sep-arate narrow-band (continuous-wave) optical beams withTHz-shifted frequencies (difference-frequency parametricgeneration), or THz emission in a terahertz parametric os-cillator (TPO), or interaction of different spectral com-ponents of a broadband optical laser radiation within thesame beam (optical rectification of femtosecond pulses).Although the medium operates in a non-inversion regime,sharply collimated narrow-band coherent THz radiationemerges at its output. Under special conditions, the sig-nal is constructively collected from the whole bulk ofthe source material and super-µJ-scale energy pulses areachievable.

3.1. Difference frequency generation and THzparametric oscillation

Parametric interaction between two plane waves of fre-quencies ω1 and ω2 in a non-linear medium with non-resonance real second-order susceptibility χ(2) results inthe generation of the third wave of difference frequency,Ω = ω1 − ω2, whose amplitude is maximal if the phasematching condition is fulfilled: ∆k = 0. In principle, suchwaves can be generated in two opposite directions, back-ward and forward [85]. The phase mismatch for the for-ward generation ∆kf ≡ k1 − k2 − kTHz can be madeto be zero, but the phase mismatch for the backward gen-eration ∆kb ≡ k2 − k1 − kTHz is usually far from zero(∆kb is never zero at least in all topical schemes basedon spatially uniform transparent non-linear media wherethe both pumping waves have the same polarization typeand the birefringence effects are insufficient). A detaileddescription is usually performed considering the generalwave equation (1) in one-dimensional geometry. It takesthe following form for the amplitude of the THz waveE(Ω, x) exp(−iΩt):

∂2E(Ω, x)∂x2

+ ε(Ω)ω2

c2E(Ω, x) = (2)

= −4πΩ2

c2P (nl)(Ω, x) ,




where ε(Ω) is the dielectric function of the medium,P (nl)(Ω, x) is the non-linear polarization,

P (nl)(Ω, x) = χ(2)E1(ω1, x)[E2(ω2, x)

]∗.

Solving of the wave equation (2) yields two differentwaves,

E(Ω, x) = Af (Ω, x) exp(ikTHz,xx − iΩt)+

+Ab(Ω, x) exp(−ikTHz,xx − iΩt) ,

with the opposite projections kTHz,x of the wave vectorskTHz(Ω). If pump depletion is neglected, the amplitudesof the forward THz wave at the output of the non-linearmedium, Af , and the backward THz wave at its input, Ab,are calculated as [86,87]

Af,b =i2πΩ2L

kTHzc2exp

(−αL

4

)Tf,b(Ω)C(ω1, ω2) (3)

at the initial low-gain stage of the process. Here, L is thelength of the non-linear sample, α ≡ 2k′′

THz,x is equalto the absorption coefficient at the THz frequency Ω forcollinear interaction, C(ω1, ω2) = A1(ω1)[A2(ω2)]∗ isdetermined by the pump amplitudes, Tf (Ω) and Tb(Ω)are nonlinear transfer functions (forward and backward“T-functions”, correspondingly) of the sample [87]. Ex-pressed in terms of the dimensionless mismatches,

∆f ≡ (k1x − k2x − kTHz,x)L , (4)

∆b ≡ (−k1x + k2x − kTHz,x)L ,

the backward and forward T-functions of uniform mediahave the same form, T (∆b,f ) = χ(2)f(∆b,f ), where

f(∆) ≡ 1L

L2∫

−L2

exp(

ix∆

L

)dx (5)

for the general case of a complex ∆. Each T-function ismaximal under phase-matching conditions, when the realpart of the corresponding ∆ is zero. In a uniform losslessmedium, wave vectors kTHz(Ω) are real, and T-functionsare determined simply by the sinc-functions of the corre-sponding real phase mismatches ∆f,b,

Tf,b(Ω) = χ(2) sinc[∆f,b(Ω)

2

]. (6)

Dispersion of the group velocity vgr for the optical pump-ing waves is usually negligibly small. So, in the case whereboth pumping waves are of the same polarization,

∆f (Ω) ≈[

Ω

vgr− kTHz,x(Ω)

]L , (7)

∆b(Ω) ≈[− Ω

vgr− kTHz,x(Ω)

]L .

(a)

(b)

Opticalpump Crystal

Terahertzoutput

x

θ

θ

k1

k2 kTHz

Figure 1 (online color at www.lphys.org) Difference-frequencygeneration in a bulk non-linear crystal. Schematic diagram (a)and orientations of the wave-vectors inside the crystal (b)

One can see from Eq. (7) that the phase matching condition∆ = 0 is satisfied in spectral regions with normal opticaldispersion only for the forward wave. Indeed, if vgr > 0,the backward mismatch ∆b(Ω) is always negative. Ac-cording to Eq. (7), the condition ∆f (Ω) = 0 for collinearforward-generating processes turns into the matching con-dition between the group velocity of the optical wavesvgr and the phase velocity of the THz wave vph(Ω) ≡Ω/kTHz ,

vgr = vph(Ω) . (8)

The above consideration is appropriate for low-gain DFGprocesses, which have been now realized experimentallyin a lot of schemes with low quantum efficiency, where theratio between the numbers of generated THz photons andinput pumping photons is small, ηq ≡ nTHz/n1 1. Inhigh-gain schemes the energy exchange between opticaland THz waves becomes substantial and this approxima-tion is not valid. Nevertheless, the main requirements tothe medium parameters are clearly seen from Eq. (3) andremain true in any case. Thus, there are three conditionsimportant for effective difference-frequency THz genera-tion: high second-order susceptibility χ(2), low THz ab-sorption, and optimal phase matching.

Among perspective materials satisfying the first tworequirements, there are bulk dielectric crystals of LiNb03,and LiTaO3, semiconductor crystals of CdTe, ZnTe, GaAs,GaSe, GaP, ZnGeP2, polymer materials such as DAST(4-dimethylamino-N-methyl-4-stilbazolium-tosylate [88])and others (see also Table 1 in [89]). At the same time, allexcellent crystal parameters become meaningful for bulkuniform samples only if the real part ∆k can be madeclose to zero. In the first works on THz difference fre-quency generation (DFG), started in the end of the 60-s[90,91], this problem was solved using non-collinear DFG(Fig. 1) in LiNbO3 crystals. At that time there was only asingle pumping wave at the input of the sample, the sec-ond pumping (“seed”) wave was generated inside via stim-ulated phonon-polariton scattering. By tuning the pump



angle of incidence on the crystal, the frequency of thegenerated THz signal was tuned from 1.26 THz up to6 THz. It was shown that using high-power Q-switchedlaser radiation, it is possible to get THz signals from long(L = 3.3 cm) crystal samples in the regime of more than50% depletion of the pump beam. The exponential gainfor the THz wave was given by [94]

gTHz = (9)

=α

2

⎡⎣

√1 + 16 cos θ

πΩω2χ(2)I1

2c3nTHzn1n2

(χ(2)

α

)2

− 1

⎤⎦ .

In Eq. (9), θ is the angle between the wavevectors k1 andkTHz in the crystal, ni (i = 1,2, or “THz”) are refractiveindices of the crystal at corresponding frequencies.

In the 1990-s the efficiency of this method of THz gen-eration under pumping by optical Q-switched lasers wasimproved substantially [92–94]. In terahertz parametricoscillators (TPO), pumped by Q-switched Nd:YAG lasers,a LiNbO3 crystal was placed into a resonator for idlerwaves, and the THz radiation walked out the crystal via aspecial output-coupling system, a monolithic grating cou-pler [92] or a prism coupler [93], mounted at the lateralcrystal surface. The best output energies of 50 pJ/pulsewere obtained under pumping at 11.4 mJ/pulse and cool-ing of the crystal down to 78K (Table 1, line 7). This datacorresponds to a rather low energy efficiency η∼ 4×10−9.Nevertheless, compactness and low-composition of suchdevices are still attractive. Recently [95,96] a wavelength-agile THz DFG, with frequency tuning from 0.6 to 2.4 THzand a narrow linewidth of 50 MHz, using two cascade bulkMg-doped LiNbO3 crystals at room temperature (Table 1,line 8), was developed and used for spectroscopy applica-tions. The maximal energy efficiency in this device waseven lower by an order of magnitude, but this did not pre-vent one from carrying out the measurements of absorp-tion spectra of water vapor and polar gases with high res-olution and high signal-to-noise ratio.

The well-known disadvantage of noncollinear wave in-teraction is weak field coupling between the pump andTHz waves. In a congruent LiNb03 crystal the angle be-tween pump and THz waves, for example, at 1 THz is63.6. A possible way to prevent this was proposed in [97],where waveguide confinement of the interacting waveswas used in 0.5 – 1 mm thick and 4.5 mm long slabs of z-cut congruent LiNb03. A direct way to improve the THzefficiency is to realize collinear wave interaction. How-ever, the group-phase matching condition (8), necessaryfor effective collinear eee-type interaction, cannot be ful-filled in most cases. For example, in LiNb03 and LiTa03

crystals, for extraordinarily polarized waves, vgr is almosttwice as much as vph(Ω) in the whole THz transparencyrange of these materials. Finite mismatches ∆kb,f deter-mine the coherence lengths of the crystal,

lforwcoh ≡ lcoh ≡ π

|∆kf | =πc

Ω|ngropt − nTHz| , (10)

lbackcoh ≡ π

|∆kb| =πc

Ω|ngropt + nTHz| ,

where ngropt ≡ c/vgr, nTHz is the refractive index for the

THz wave. In this case, no matter how long and transpar-ent is the non-linear sample, most part of the THz signaldetected in the forward direction originates only in the thinlayer of thickness lforw

coh at the output of the crystal. Cor-respondingly, the THz signal detected in the backward di-rection originates in the thin layer of thickness lback

coh at itsinput. lcoh is too small for a lot of non-linear materials dueto a high mismatch of Eq. (8), lback

coh is even smaller, so theTHz signals usually are weak.

The mismatch between optical group velocity and theTHz phase velocity can be minimized using a non-linearmaterial with the proper phonon-polariton dispersion [98]in the THz range. In all materials considered so far, THzwaves propagate as phonon polaritons. Their dispersion isgiven by the lower polariton branch and depends most ofall on the frequencies and oscillator strengths of the low-est polar phonons [99–101]. It was found that the phase-group mismatch is minimal in a number of zinc-blende-type semiconductors [102,103], due to their typical THzphonon-polariton dispersion, described by a single phononmode. The fairly extraordinary condition (8) is valid inZnTe crystals for Ω = 2.3 THz under pumping at a wave-length of 0.8 µm; in ZnTe and ZnSe the group-phase mis-matches remain small within the widest known spectralrange, so that THz waves can be effectively generated inthe collinear regime at frequencies from 0.1 to 3 – 4 THz orhigher [103]. Coherence lengths for 2 THz generation ex-ceed 3 mm in CdTe, GaP, InP, GaAs crystals if the pump-ing wavelength is properly chosen in the near-IR range[104]. Reports have recently been published on efficientDFG of THz waves from zinc-blende-type semiconduc-tors with long coherence lengths, such as ZnTe, GaAs,GaSe, ZnGeP2, and GaP [105–108]. Tunable THz-wavegeneration from 0.5 to 4.5 THz was achieved by DFG in a2 cm long, bulk GaP crystal with pump dual wavelengthsof around 1 µm [105,106]. THz output energy of 5 nJ/pulsewas detected when the pump energy was 1.2 mJ/pulse ineach pumping beam, so that the energy efficiency withrespect to the sum of the pumping energies was about2×10−6 at 2.6 THz (Table 1, line 9). In a number of recentreports, collinear phase-matched DFG was obtained un-der quasi-CW IR pumping. Frequency-mixing lines of twoshort-pulse repetition-frequency-excited waveguide CO2-lasers near 10 µm in GaSe crystals (Table 1, line 10)resulted in the average THz output power of 260 µW[107] with the energy efficiency of 1.4×10−5. Zinc ger-manium phosphide (ZnGeP2) crystal pumped by 0.7 nspulses of two semiconductor diodes pre-amplified by afiber-amlification system produced an average THz powerof 2 mW in the range of 2 – 3 THz (Table 1, line 11) [108]with even higher efficiency 5×10−4.

Among other advantages of these crystals is their lowabsorption in the THz frequency range. For example, ab-sorption coefficient αTHz in GaAs changes only from




0.5 cm−1 to 5.7 cm−1 as the frequency increases from1 THz to 3.3 THz [109]. At the same time, in LiNbO3

αTHz ∼ 12 cm−1 at 1 THz and it increases up to 170 cm−1

or higher (depending on the composition and doping[110,111]) as the frequency approaches 2.5 THz. As forthe dispersion of the wavevector real part, the spectralvicinity of the lowest polar phonon, its oscillator strength,and additionally, its damping constant play the most im-portant role for the absorption behavior. For LiNbO3 it isthe lowest A1-symmetry phonon mode at 256 cm−1 (withthe damping constant γ varying at room temperatures from7.5 cm−1 up to 14 cm−1, depending on the crystal compo-sition and doping) [112], which corresponds to 7.68 THz.For GaAs it is the phonon mode at 269 cm−1 (with thedamping constant γ = 2.4 cm−1), whose frequency is about8.2 THz [104]. In addition to phonon damping, variousother mechanisms also have influence on the absorptiondispersion, but in the case of LiNbO3 they do not changeit drastically [99–101,110]. Unfortunately, for the case ofsemiconductor crystals it is not the case. Non-linear ef-fects of two-photon absorption hinder the application ofthese crystals in schemes with high-power pumping in thenear-infrared or visible ranges. Not only optical absorp-tion increases, but also large THz losses appear due tothe generation of free carriers. The necessity to use long-wave pump sources makes additional difficulties. For ex-ample, to avoid two-photon absorption, pump wavelengthsin GaAs should be larger than 1.74 µm [113].

The problem of absorption will be discussed againlater concerning schemes based on optical rectification.Coming back to phase matching, let us note that what-ever long the coherence length may be, it limits the overallactive length of a bulk crystal. This terminates further in-crease of the THz output and further narrowing of the THzspectral line, which could be possible by extending thenon-linear sample under exact phase-matching conditions.The problem is avoided by using crystal structures witha periodic inversion of the crystal axes. Spatial periodicvariation of the sign of second-order optical susceptibilitytakes place in such structures and gives rise to new condi-tions for the efficient frequency conversion, the so-calledquasi-phase matching (QPM) conditions [114–116]. Themost well-known structures of this type are periodicallypoled ferroelectric crystals with regular one-dimensionaldomain gratings. Different poling techniques are devel-oped now for crystals of LiNbO3, LiTaO3, KTiOPO4

(KTP), RbTiOAsO4 (RTA), KNbO3, including poling di-rectly under the growth procedure [117–119] or after it[120–123]. The idea of quasi-phase matching has been firstrealized in THz generation via optical rectification (seenext section), namely in periodically poled LiNbO3 crystal(PPLN). Nevertheless, later an efficient narrow-band DFGwas obtained not in ferroelectric domain gratings, but inperiodic stacks of inverted layers of GaP [124] and GaAs[125] semiconductor crystals. The QPM structures werefabricated here by alternating 〈111〉 direction of crystalplates cut normally to 〈110〉.

(a)

(c)

(b)

(d)

Opticalpump

Opticalpump

Terahertzoutput

Terahertzoutput

x

x

kTHz

kTHz

qm

q-m

k1, k2

k1, k2

qmqm

q-mq-m

Periodically poledcrystal

kTHz - qm=k1 - k2≈-ΩTHzvgr

Periodically poledcrystal

-kTHz + q-m=k1 - k2≈-ΩTHzvgr

Figure 2 (online color at www.lphys.org) QPM frequencydown-conversion in a periodically poled crystal. Schematic di-agram (a,c) and orientation of the wave-vectors inside the crystal(b,d) for forward (a,b) and backward (c,d) THz generation

For the subsequent treatment of the three-wave non-linear interaction in a quasi-phase matching (QPM) struc-ture, one needs to solve the wave equation (2) with an ac-count for the spatial dependence χ(2)(x). As it was shownin [87], the final solution for the THz wave can be pre-sented in the same form as in Eq. (3), but the dependenceof the nonlinear transfer function on the phase mismatchis changed:

Tf,b(Ω) =∞∑

m=−∞χmf(∆f,b + 2πm) . (11)

Summation is made over all Fourier harmonics χm ofthe second-order susceptibility spatial variation χ(2)(x)within the medium length L along the x-direction:

χm =1L

L2∫

−L2

χ(2)(x) exp(−im

2π

Lx

)dx .

Eq. (11) is valid for all kinds of spatial variations χ(2)(x),including non-periodic ones, and the Fourier series forχ(2)(x) is taken in the general form:

χ(2)(x) =∞∑

m=−∞χm exp

(im

2π

Lx

), (12)



and χm = χ∗−m. In a lossless medium,

Tf,b(Ω) =∞∑

m=−∞χm sinc

(∆f,b

2+ πm

). (13)

Dependencies of both T-functions on the correspondingphase mismatches consist of a number of shifted sinc-functions, so that the amplitude of each sinc peak dependson the value of the corresponding Fourier harmonic χm.The frequencies of the peaks are determined by QPM con-ditions

∆kf,b + qm = 0 , (14)

where qm ≡ 2πm/L (Fig. 2). In a spatially uniform bulkcrystal there can be only one appreciable peak correspond-ing to m = 0 for the forward-generated wave. If there is aperiodic spatial modulation of χ(2) with the period d, ad-ditional peaks appear in T-functions corresponding to non-zero harmonics with numbers m = (L/d)m0 ≡ nm0,where m0 is an integer, n is the number of periods in thecrystal. The peak intensities are distributed according tothe relations between different-order χm0n. For a stepwisemodulation of χ(2), caused by the change of its sign at theinverted layers, these relations are described by [123]

χnm0 =iχ(2)(−1)m0n

πm0

[1 − exp(i2πm0D)

]. (15)

Here, χ(2)is the module of the effective second-order sus-ceptibility for each layer (the convolution of the second-order susceptibility tensor with signal, idler, and pump unitpolarization vectors), D is the duty-cycle of the structure,characterizing the thickness of one-type layer with respectto the whole period: D ≡ l(+)/d. First-order (m0 =±1)QPM in periodically poled structures with equal lengths ofinverted domains has the largest amplitude. In this case thelength of each domain is equal to the coherence length ofthe medium. For a large number of domains with invertedpolarization, constructive interference occurs between sig-nals generated in subsequent domains. Unlike in a bulkuniform crystal, increasing of the structure length effec-tively leads to the amplification of the THz output. In ac-cordance with Eq. (14), the frequency of the first-order for-ward generation satisfies the QPM condition kTHz(Ω) =Ω/vgr +2π/d, the frequency of the backward one is deter-mined through kTHz(Ω) = −Ω/vgr + 2π/d. Generationof the backward wave becomes also realistic in QPM struc-tures (Fig. 3) and has been observed in PPLN [86,126–128].

The same approach, extended to the general case of3-dimentional arbitrary spatial variation of χ(2), gives thegeneralized QPM condition in the vector form:

k1 − k2 + qm = kTHz,f , (16)

k2 − k1 + qm = kTHz,b ,

where

qm =

2πmx

Lx,2πmy

Ly,2πmz

Lz

kTHz(Ω)-kTHz(Ω)

qm

q-m

k

Ω

Ωforw

Ωback

k = k1 - k2 ≈-Ω

vgr

Figure 3 (online color at www.lphys.org) The graphic schemefor the determination of THz frequencies for forward and back-ward generation under collinear QPM process. The lower polari-ton branches for opposite directions are shown by brown curves;black arrows represent the non-linear grating vectors compensat-ing for the group-phase mismatches

can be treated as eigenvectors of the inverse non-lineargrating, mx,my, mz are integer numbers of non-zero spa-tial Fourier harmonics of χ(2).

Zinc-blende semiconductors GaAs and GaP show agreat potential for QPM terahertz generation. A THz-wave source was realized employing DFG from a periodi-cally inverted GaP stack pumped with two 1.55 µm pumpsources with slightly shifted wavelengths [124]. The opti-cal loss in GaP caused by two-photon absorption is negli-gible at 1.55 µm. Due to alternating of the 〈111〉 orienta-tion in the stack of the GaP plates, the sign of the largestcomponent of second-order nonlinear susceptibility wasmodulated. The energy efficiency of 4×10−7 was achieved(Table 1, line 12). Afterward a room-temperature THzsource was demonstrated [125] using QPM GaAs with1 mW average output power at 2.8 THz, based on intra-cavity difference-frequency mixing between the two res-onating optical waves of a synchronously pumped doublyresonant optical parametric oscillator. The DFG processwas studied in two different QPM structures: diffusion-bonded stacked GaAs [129] and epitaxially grown orien-tation patterned GaAs [130]. High efficiency was obtainedfor the first-type structure, being of order of η∼ 4×10−5

in terms of energy transfer from the two optical beamsto the THz wave (Table 1, line 13). At the same time,pulse energies of the optical pumping waves were not high,0.2 – 0.4 µJ/pulse. It seems that further progress could beachieved within this scheme after increasing the ampli-tudes of the optical waves, would the problem of pump-induced absorption be solved.




Recently a new type of dual pumping was proposedfor efficient intracavity DFG. The semiconductor materialwith high non-resonant bulk nonlinearity, or the nonlin-earity due to the resonant intersubband transitions in quan-tum wells, is placed inside a mode-locked dual-wavelengthheterolaser [131,132]. These monolithically integrated de-vices combine the advantages of electrically pumped semi-conductor lasers and nonlinear optical sources. A perspec-tive compact system producing terahertz output at 5 THz,with a dual-wavelength quantum cascade laser, generatingat λ1 = 7.6 µm and λ2 = 8.7 µm, integrated with an activenonlinear region on the basis of coupled quantum wells,was demonstrated in [131].

3.2. Optical rectification

Under nonlinear-optical generation of THz radiation, thefrequencies of each two optical waves differ from eachother to a very small extent – to a value of a THz frequency.Thus, instead of two monochromatic pumping beams, suchas the “pump” and the “seed” in DFG, in many cases itis quite enough to have one quasi-monochromatic high-power beam with a properly broad spectral bandwidth.The spectral bandwidth of a sub-picosecond laser is appro-priate for the generation of THz waves under difference-frequency interaction between its spectral components.This idea underlies all tabletop schemes based on the op-tical pumping of optically transparent non-linear materi-als by femtosecond lasers. Difference-frequency interac-tion takes place under optical rectification (OR) of fem-tosecond pulses here. As early as in 1962 optical recti-fication in electrooptic media was reported for the firsttime [133]. Developing of OR schemes for THz gener-ation was started just with the first explorations of sub-picosecond and femtosecond lasers in the 1970-s [134–136] and is steadily continued up to now. Due to a con-siderable progress achieved, devices based on this generalprinciple are considered to be the best for a lot of importantpractical applications, especially for the THz time-domainspectroscopy [1,2] and imaging [3–5,108].

Conventionally named as OR schemes, this vast groupof schemes imply non-centrosymmetric crystals with highreal part of non-resonance second-order optical suscepti-bility χ(2). For the classification of THz – generation de-vices, it is essential that these OR processes take place inspectral regions far from intrinsic resonances of the non-linear medium, so that the non-linear sample is transparentfor the pump and can be quite large. Femtosecond pump-ing is also used in large-aperture photoconductive anten-nas, semiconductor surface emitters, and other schemesconsidered in Sec. 2, and sometimes the generation pro-cess there is treated as the resonant OR of femtosecondpulses [27]. Note that there is a fundamental differencebetween devices of this type and the conventional ORschemes. Resonant OR cannot be attributed to paramet-ric processes. Due to the strong pump and THz absorp-tion, these processes occur only in thin near-surface area

of the medium, while non-resonant OR takes place in thebulk of the sample. Usually simply named as OR, non-resonant OR is characterized by practically instantaneousTHz response, preserves phase relations between the in-teracting optical and THz waves, and can provide coherenttightly collimated THz radiation. Among the different op-tical techniques of THz generation, OR has turned out tobe extraordinarily well suited for the generation of highTHz frequencies and powers.

The same wave equation (2) as for the DFG processdescribes the THz output under OR, but the expression forthe non-linear polarization should be taken in the integralform,

P (nl)(Ω, x) =χ(2)(x)

2π

∞∫−∞

|Ep(x, t)|2 exp(iΩt) dt = (17)

= χ(2)(x)

∞∫−∞

Ep(x, ω1)E∗p(x, ω1 − Ω) dω1 .

Here, Ep(x, t) is the time-dependent electric field of thepumping pulse and Ep(x, ω1) is its temporal Fourier trans-form,

Ep(x, t) =

∞∫−∞

Ep(x, ω1) exp(−iω1t) dω1 .

The amplitudes of THz waves generated under optical rec-tification of a short-pulse pump at the input (x =−L/2)and output (x = +L/2) of the crystal sample are

ATHz,f

(Ω,

L

2

)= (18)

=i2πΩ2L

kTHzc2exp

(−αL

4

)T (∆f )C(Ω) ,

ATHz,b

(Ω,−L

2

)=

i2πΩ2L

kTHzc2exp

(−αL

4

)T (∆b)C(Ω) ,

at each THz frequency Ω/2π at the initial low-gain stageof the non-linear interaction [87]. The only difference withthe DFG-case Eq. (3) is in the C-factor, which takes intoaccount the pump amplitude distribution. Now, this factorrepresents the Fourier transform of the pump pulse inten-sity:

C(Ω) =

∞∫−∞

Ap(x,Ω′)A∗p(x,Ω′ − Ω) dΩ′ = (19)

=12π

∞∫−∞

|Bp(t)|2 exp(iΩt) dt .

Here, Bp(t) is the time-domain envelope of the pump field

Ep(x, t) ≡ Bp(t) exp[ − iω0t + ik(ω0)x

].



x

Opticalfemtosecond

pump

Crystal

Terahertzoutput

k1 - k2 = kTHz,||

k1, k2

npump

∆k⊥

θcher

kTHz

-= kTHzcos θcherΩTHzvgr

θcher

Figure 4 (online color at www.lphys.org) Schematic diagramand orientation of the wave-vectors for optical rectification in abulk non-linear crystal under Cherenkov-type geometry

Potential resources of this method for THz generationwere studied recently [137] by numerical simulations ofthe cascaded processes which occur at longer distances.It was shown that under ideal conditions of perfect phasematching and no absorption the efficiency of generationexceeds the Manley-Rowe limit, so that the quantum ef-ficiency ηq formally becomes several times higher than100%. It is connected with the red line shifting and dis-tortion of the femtosecond pump spectrum caused by in-tensive energy exchange between the pump spectral com-ponents and the THz wave. If the matching condition (8)between the group velocity of the optical waves vgr andthe phase velocity of the THz waves is not satisfied, the op-tical and THz pulses walk off each other as they propagatethrough the crystal. Due to the walk-off effect the growthof the THz signal is slowing down. Numerical calculationsmade neglecting absorption for ZnTe crystal under 0.8 µmpumping at 300 MW/cm2 show that the time-domain THzpeak grows up only to a propagation distance of 2 mm.After that the growth of the total photon conversion ratiostill proceeds but becomes sufficiently weaker and turns tosaturation at a level of 150% at a distance of 5 mm (insteadof 750% at 5 mm and quadratic growth under ideal phasematching condition). The numerical simulations show alsothat the undepleted pump approximation is quite valid upto distances of 2 – 2.5 mm. Further, for the realistic con-ditions of absorption losses, a competition between cas-caded OR and absorption was studied for pump intensitiesas high as 300 – 800 MW/cm2, so that the optimal distancewas found to be about 1.5 – 2 mm.

Different materials, such as ferroelectric LiNbO3,LiTaO3 [138,141], semiconductor ZnSe [103], ZnTe[139], CdS, DAST [140–142], N -benzyl derivative of 2-methyl-4-nitroaniline (MNA) [143], GaAs [104], GaSe

[144], ZnGeP2 [108], CdTe [145], and others [89,146] areconsidered to be advantageous for THz generation via ORprocesses in different spectral regions. However, for a longtime optical-to-terahertz conversion efficiencies achievedby optical rectification methods were too low [139,141],typically 10−6 – 10−9 (as for example, see Table 1, line14). Even though materials with the highest known realχ(2) are used for OR, their non-resonant non-linear sus-ceptibilities are sufficiently lower than the absolute valuesof the resonance χ(2), which determine the efficiency ofthe resonance schemes considered in the previous section.Under OR processes the benefit should be derived fromthe bulk character of the processes involved. To realize anopportunity of constructive collection of THz signals fromlong bulk crystals, the two points are of key importance (aswell as in the case of DFG). In addition to high χ(2), it isfirstly the fulfilling of phase-matching conditions, or anyother conditions for constructive interference of the THzwaves generated from different parts of the medium, andsecondly, the low attenuation of the THz waves in the non-linear sample. A number of approaches solving these tasksare listed below.

3.2.1. Cherenkov geometry for bulk crystals

Typically for electro-optic crystals (LiNbO3, LiTaO3, KTPand others), the phase velocity vph in the THz range ismuch smaller than the group velocity vgr in the opticalregion, so that collinear phase matching is not possibleand coherence lengths (the same as in case of DFG, seeEq. (10)) are too small. For example, in LiNbO3 ngr

opt ≈ 2.3within the visible and NIR ranges, nTHz ≈ 5.17 at 0.67 –1.7 THz [147], so that the forward coherence length is∼ 50 µm, the backward one is several times smaller. How-ever, a sort of phase-matching can still be accomplishedwith these materials via a Cherenkov type geometry -the analogue of the Cherenkov radiation from superlu-minal charged particles for intense light pulses predictedby Askaryan in 1962 [148]. It was this approach thatwas first proposed to solve the problem of phase match-ing [136,149]. In this geometry, the THz wave is gener-ated at an angle θcher to the direction of the laser pump(Fig. 4). The spatial phase matching condition k1 − k2 =kTHz is fulfilled along the pump direction only, so thatk1 − k2 = kTHz,‖ = kTHz cos θcher and the Cherenkovangle is θcher = arccos(vTHz/vgr). This means that thelongitudinal crystal length L can be as large as neces-sary for the efficient energy exchange, many times ex-ceeding the forward coherence length lcoh. Nevertheless,there remains a non-zero mismatch in the perpendicu-lar direction, ∆k⊥ = kTHz,⊥ = kTHz sin θcher. So,collection of the signals generated from shifted in thetransverse direction different parts of the sample will beconstructive only if the transverse length is smaller thanthe corresponding transverse coherence length, lcoh,⊥ ≡π/(∆k⊥) = π/(kTHz sin θcher). Usually the transversecoherence length is on order of the THz wavelength, and




one has to arrange a sharp focusing of the pump beaminto a spot of this diameter. It was shown subsequently in[150] for non-collinear parametric generation taking intoaccount diffraction of THz waves that the waist diame-ter of the focused optical beam should be smaller than theTHz wavelength for the effective generation.

The Cherenkov-type geometry also enables one to de-crease absorption losses if the lateral side of the sample isused for lifting-off the THz beam (Fig. 4). In this case theinclined THz waves will have the shortest possible pathinside the crystal. Special samples with a wedged surface(in the case of LiNbO3, the wedge angle is about 26) ora silicon prism [151] were prepared to decrease the lossesrelated to the THz lift-off [152].

The main disadvantage of the Cherenkov-type geom-etry is the limitation of the permitted pump power. Dueto sharp focusing, the pump intensity increases, but at thesame time it should not exceed the damage threshold ofthe non-linear material. This condition limits the power ofthe input optical beam and, correspondingly, the achievedTHz power is lower. One way for scaling up the THz pulseenergy in this geometry was proposed [153] and recentlydemonstrated experimentally [154]. Line focusing of thepump beam by a cylindrical lens enables one to preserve anecessarily narrow beam width in the plane of the opticaland terahertz wave vectors, but the total area of the focalspot remains rather large [155]. Maximal efficiency (some-what less than 10−4) and maximal THz pulses (with theenergy of 3.1 nJ) were obtained under OR of 32 µJ/150-fs-pulse Ti:Sapphire amplifier (Table 1, line 15). Furtherincrease of the pump energy resulted in a drop of the THzoutput.

3.2.2. Tilted pulse front

Considering all the OR schemes realized experimen-tally from the viewpoint of generating the most power-ful THz pulses, the set-up using the bulk uniform near-stoichiometric 0.6% Mg-doped LiNbO3 crystal and thetilted pulse front [89,154–161] should be recognized as themost advantageous. Recently, THz pulses of 10 µJ energywere generated in this scheme with the pump pulses of20 mJ energy and 10 Hz repetition rate from a Ti:Sapphireamplifier system [158–160] at room temperatures (Table 1,line 16).

In the geometry of the tilted pulse front [162], thepump beam propagates at an adjustable angle θ to the prop-agation direction of its wave front (Fig. 5). It is made us-ing an optical dispersing element, a prism or a grating.The THz beam is emitted in the direction normal to thepump front, in accordance with the Huygens-Fresnel prin-ciple [156]. The optimal angle of pulse front tilting θ cor-responds to the case where the projection of the opticalpulse velocity on the direction of the THz beam, vgr cos θ,coincides with the phase velocity vTHz of the THz wave,so that again, θ = arccos(vTHz/vgr) = θcher. It is alsoimportant, that the phase matching condition between the

x

Opticalfemtosecond

pump

Terahertzoutput

kTHzk1

k2

npump

θ

-≈ |kTHz|ΩTHzvgr cos θθ

Tilted pulsefront

θ

θ

Figure 5 (online color at www.lphys.org) Schematic diagramand orientation of the wave-vectors for optical rectification in abulk non-linear crystal under tilted pulse front geometry

THz wave and each of the two pump spectral componentsshifted to the necessary THz frequency, is completely sat-isfied in this case, as it is shown in Fig. 5. Due to the dis-persion of the non-linear material, the pulse front tiltingangle is slightly dependent on the frequency of each com-ponent of the pump, and the angle appearing between thewave vectors of the shifted components is just necessaryfor the non-collinear phase matching [156,157]. The samesituation could be achieved using two femtosecond pumpbeams incident on the sample at different angles. But thelongitudinal spatial overlap between the two pump fieldswould be limited in this case [163]. The tilted pulse frontscheme is also advantageous over the Cherenkov geom-etry, since the sharp focusing is not necessary, and suffi-ciently higher pulse energies may be applied simply in-creasing the transverse size of the pumping area. Due tothe two-dimensional phase-matching, the THz fields gen-erated in different parts of the pumped volume, no matterwhether it is shifted along or across the pumping direc-tion, interfere constructively. This results in higher con-version efficiencies for the frequencies where absorptionis not large.

The energy efficiencies obtained in this scheme arealso among the top values (Table 1, line 17). Under 1 kHzrepetition rate and 300 µJ pump pulse energy, an effi-ciency of 5×10−4 was achieved [158] at room tempera-tures. Further improvement of the attainable THz pulse en-ergy was forecasted in [161] on the base of model calcula-tions for LiNbO3 of stoichiometric and congruent compo-sition, GaP, GaSe, and ZnTe in a tilted pulse front scheme,taking into account the pump pulse duration, the THz fre-quency, the crystal length and possible cooling of the crys-tal. It was shown that the most promising material for gen-erating up to 2 THz (and for pump duration of 100 fs orlonger) is still the stoichiometric LiNbO3 crystal.



3.2.3. Quasi-phase matching

Narrow-band multicycle THz waves can be generated un-der quasi-phase matching conditions in periodically orspecial aperiodically poled crystals (Fig. 2). In this casethe walk-off effect between the optical and THz waves atdistances larger than the coherence length is compensateddue to a periodic reverse of the nonlinear polarization ofthe crystal. The THz spectrum is determined by the squareof amplitudes given by Eq. (18), so that all the peaks of thecrystal transfer functions described by Eq. (11), which arein the range of crystal transparency and within the pumpenvelope C(Ω), appear in the spectrum. While absorptionis negligibly small, the width of each peak is inversely pro-portional to the length of the crystal sample, and can bemade on order of 0.1 – 0.05 THz or even less.

As well as for QPM DFG, the mostly high-powerisolated narrow lines are achievable under OR in peri-odic one-dimensional QPM structures, at THz frequen-cies that satisfy conditions kTHz(Ω) = ±Ω/vgr + 2π/d(Fig. 3). This was demonstrated at room temperatures firstby Lee et al. [126,164–166] for periodically poled congru-ent LiNbO3 crystals, then for the Mg-doped periodicallypoled LiNbO3 with the lower absorption [86,127]. For fur-ther reduction of the THz absorption, cryogenically cooledPPLN crystal was also used [164], and 10−5 conversionefficiency was achieved with 200 fs pulses at 800 nm (Ta-ble 1, line 18). A very narrow, 18 GHz, line at 1.8 THzwas obtained in a 7.2 mm long sample with the periodd = 60 µm. Although the total power was not the high-est one within the results achieved, obviously, the spectralpower density was extremely high, 2.8×10−4 W/THz. Forcomparison, in bulk velocity-matched ZnTe, the spectralpower density was 1.7×10−6 W/THz at the same THz fre-quency [164].

Terahertz-wave generation with higher energy effi-ciency but lower absolute output was demonstrated re-cently in QPM GaAs using the optical rectification of fem-tosecond pulses. In [113], a widely tunable source of ter-ahertz radiation based on OR in a QPM GaAs structuregenerated 0.66 nJ of output at 2.2 THz with 2.3 µJ ofpump pulse energy at the wavelength more than 3 µm.This corresponds to a high optical-to-terahertz conversionefficiency of 2.9×10−4 (Table 1, line 19). But non-linearabsorption effects dramatically increase the optical andTHz absorption of periodically inverted GaAs structures[113,167] as in case of DFG in this material. Higher pumpenergies are not permitted yet, as well as shorter NIR pumpwavelengths. Up to now, these promising structures are notappropriate for the efficient generation of high-energy THzpulses.

Among the materials possessing high nonlinear sus-ceptibility and suitable for QPM, periodically poledlithium niobate crystals still occupy leading positions[150,152] due to their transparency for visible and near-infrared pumping [168–171] and well-developed polingtechniques [116–122]. Frequency tuning of the radiationgenerated by QPM structures can be obtained by chang-

x

Opticalpump

Terahertzoutput

k1, k2

npump

- kTHz,⊥ = ∆k⊥ = = qm,⊥

kTHzqm

kTHz,|| + qm,||=k1 - k2≈-ΩTHzvgr

qmqm

Figure 6 (online color at www.lphys.org) Schematic diagramand orientation of the wave-vectors for optical rectification in atilted periodically poled crystal

ing the poling period and the crystal dispersion via tem-perature. Besides, using different discrete poling periods,continuous tuning of the effective poling period was pro-posed by rotation of the periodically poled crystal un-der non-collinear OR in a one-dimensional QPM structure[172]. However, THz absorption of LiNbO3 crystals in-creases very rapidly with the THz frequency and becomestoo large above 2 THz. Up to now the low-gain approx-imation is still valid for the description of experimentalOR schemes. Far from the cascading regime, absorptionplays a crucial role and only thin layers near the outputand input surfaces of the PPLN sample emit THz radia-tion out of the crystal under collinear QPM. That is whythe number of working periods of a PPLN sample is de-creased when the frequency is increased. At frequencieshigher than 2 – 3 THz, collinear QPM interaction becomesinefficient at room temperatures. Doping by Mg, prepar-ing crystals of near-stoihiometric Li/Nb composition, orcryogenic cooling which is more effective but more incon-venient, decrease the absorption [111] and can slightly ex-pand the frequency range. Nevertheless, the spectral rangeremains lower than in such semiconductors as GaAs andGaP.

The problem of high THz absorption in lithium nio-bate can be solved by using non-collinear geometries,where the grating vector, optical wave vectors (mutuallycollinear under OR of femtosecond pulses) and the THzwave vector are directed at proper angles to each other.In this case it becomes possible to let the THz radiationout through a lateral crystal surface, so that the path ofthe THz wave inside the crystal can be made very short,while the longitudinal length of the interaction still con-tains a large number of domains. A number of originalnon-collinear schemes for tunable narrow-band generationin PPLN are described in [150,152]. In a surface-emission




scheme experimentally demonstrated by Weiss et al. [173],the THz frequency is tuned by changing the angle of ob-servation from the lateral surface. In [152] the signal wascollected from a 8-mm long PPLN with the poling period127 mm by using this sideways geometry. As in the case ofCherenkov-type OR, a non-compensated transverse phasemismatch exists in this scheme. QPM in the longitudi-nal direction should be accompanied by a sharp focus-ing in the transverse direction to make the pump spot sizesmaller than the expected wavelength of the THz radia-tion. This drawback is prevented by using a tilted PPLN[150,152,174] with the domains oriented at an angle tothe crystal edges (Fig. 6). The transverse projection of thevector of a 1-dimensional domain grating compensates forthe transverse wave mismatch and sharp pump focusingbecomes not necessary. Tilted PPLN crystals have beensuccessfully used for surface-emitted THz generation un-der difference-frequency mixing of nanosecond- and sub-nanosecond duration optical pulses [174,175] and for theOR of 100-fs pulses at 0.78 µm [152]. The same idea is in-corporated in the use of two-dimensional chessboard typePPLN [150,152] for non-collinear THz emission, wherethe transverse wave mismatch is compensated for by atransverse eigenvector of a 2-dimemsional domain grating.

Aperiodically poled crystals are of special interest dueto the possibility to control the frequency or angular bandsof the emitted THz radiation, to convert spatial imagesfrom THz to optical ranges, to emit temporarily shapedTHz pulses. According to Eq. (18), engineering of theshapes of THz signals is possible via two different ap-proaches: by spatial or temporal structuring of the pumpprofile with the necessary C-factor or by engineering of anoptimal non-linear grating with the necessary T-function[87]. The first method was realized by arranging spa-tial masks between the pump beam and a PPLN crystal[176], or using a pair of temporarily separated femtosec-ond pulses [177]. In these cases the QPM structure wasprepared in a fanned-out form available for continuouslytunable narrowband THz generation. In the framwork ofthe second approach, special quasi-periodic domain grat-ings, such as Fibonacci-like fractal structures [178,179],pre-designed structures consisting of uniform gratings andmultiple phase-reversal sequences [180] were proposed formulti-frequency THz generation. The temporal terahertzwaveform maps out [165] the crystal domain structure inall types of aperiodically poled crystals [87].

4. Conclusions

The field of laser-induced terahertz photonics develops sorapidly that, inevitably, any conclusions concerning its per-spectives have a short-term validity. As one can see fromthe brief review given above, the numerous methods pro-posed for terahertz generation are based on quite differentphysical effects, and new materials and new design ideaskeep appearing putting in the forefront newer or older ap-proaches.

In Table 1 there are presented several brief resultsachieved recently within the main directions. Comparingthe energy efficiency of the schemes (not the quantum effi-ciency, which seems to be not a well-representative charac-teristic under cascaded down-conversion [137,181]), onecan conclude that the best efficiencies obtained up to noware on the order of 0.05%. From this viewpoint, among theleading schemes there are still GaAs – based PC anten-nas, and a number of non-resonant schemes: difference-frequency generation in high-efficient materials such asZnGeP2, optical rectification in near-stoichiometric 0.6%Mg-doped LiNbO3 bulk crystal in the tilted pulse frontscheme, and optical rectification in quasi-phase matchedGaAs structures. Non-resonance schemes are advanta-geous due to the high angle directivity of the emitted radia-tion. The materials involved in these non-resonance down-conversion schemes (QPM GaAs, ZnGeP2) also demon-strate best results on the mean powers achieved, on theorder of 1 – 2 mW. However, the absolute values of thegenerated pulse energy in QPM GaAs structures are stilllow due to the limitations caused by the induced absorp-tion effects. The record THz peak intensity and pulse en-ergy 10−5 J were obtained in OR tilted pulse front scheme.At the same time, this energy is distributed within a widespectral range, whereas DFG and QPM OR schemes en-able one to obtain a high power within narrow spectrally-tuned lines which is more important for applications inspectroscopy.

Among other valuable characteristics there are attain-able spectral range of generation, compactness and pos-sibility of CW operation. The spectral range is limited inall schemes, mainly due to phonon absorption in the caseof non-resonance methods, and due to the limited carri-ers response time in the resonance methods. ResonanceOR in laser-ionized gases promises to be among the mostperspective methods for generation above 5 THz. Com-pactness and possibility of CW operation is demonstratedby perspective modern devices based on resonance photo-mixing; non-resonance photo-mixing processes in intra-cavity DFG devices are also promising from this view-point.

In summary, a brief review of laser-based methodsfor generation in terahertz range is presented. Results ob-tained by the methods of resonance optical excitation ofcharge transfer in photoconductive materials, molecules,ionized gases and surface emitters, are compared withthe progress achieved in the field of non-resonance opti-cal rectification and difference-frequency generation. Ad-vantages of tilted pulse front, quasi-phase matching, line-focusing Cherenkov-type schemes, as well as competitionbetween dielectric, polymer and semiconductor non-linearmaterials are discussed for the non-resonant types of thelaser excitation.

Acknowledgements The author is grateful to M.V. Chekhova forvaluable remarks, and acknowledges support from the RussianFoundation for Basic Research (grant No. 08-02-00555a) and the



Program of supporting the leading scientific groups of Russia(grant No. 796.2008.02).

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