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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 6 — Mar. 20, 2006
  • pp: 2263–2276
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Optical generation of narrow-band terahertz packets in periodically-inverted electro-optic crystals: conversion efficiency and optimal laser pulse format

Konstantin L. Vodopyanov  »View Author Affiliations


Optics Express, Vol. 14, Issue 6, pp. 2263-2276 (2006)
http://dx.doi.org/10.1364/OE.14.002263


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Abstract

We explore optical-to-terahertz conversion efficiencies which can be achieved with femto- and picosecond optical pulses in electro-optic crystals with periodically inverted sign of second-order susceptibility. Optimal crystal lengths, pulse durations, pulse formats and focusing are regarded. We show that for sufficiently short (femtosecond) optical pulses, with a pulsewidth much shorter than the inverse terahertz frequency, conversion efficiency does not depend on pulse duration. We also show that by mixing two picosecond pulses (bandwidth-limited or chirped), one can achieve conversion efficiency, which is the same as in the case of femtosecond pulse with the same pulse energy. Additionally, when the group velocity dispersion of optical pulses is small, one can substantially exceed Manley‒Rowe conversion limit due to cascaded processes.

© 2006 Optical Society of America

1. Introduction

Optical rectification (OR) of ultrashort laser pulses is an established way of generating broadband terahetz (THz) radiation. THz output in this case is produced in an electro-optic (EO) medium via difference-frequency mixing between Fourier components of the same optical pulse. First demonstrated with picosecond pulses in ZnTe, ZnSe, CdS and quartz [1

1. T. Yajima and N. Takeuchi, “Far-infrared difference-frequency generation by picosecond laser pulses,“ Jpn. J. Appl. Phys. 9, 1361–1371 (1970). [CrossRef]

] crystals, as well as in LiNbO3 [1

1. T. Yajima and N. Takeuchi, “Far-infrared difference-frequency generation by picosecond laser pulses,“ Jpn. J. Appl. Phys. 9, 1361–1371 (1970). [CrossRef]

,2

2. K.H. Yang, P.L. Richards, and Y.R. Shen, “Generation of far-infrared radiation by picosecond light pulses in LiNbO3,” Appl. Phys. Lett. 19, 320–323 (1971). [CrossRef]

], this technique was later extended to femtosecond laser pulses [3

3. L. Xu, X.-C. Zhang, and D. H. Auston, “Terahertz beam generation by femtosecond optical pulses in electro-optic materials,” Appl. Phys. Lett. 61, 1784–6 (1992) [CrossRef]

] which allowed generation of much broader bandwidths in terahertz region, [4

4. B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nature Materials 1, 26–33 (2002). [CrossRef]

] and even reaching mid-IR freqiencies of 40 to 50 THz [5

5. A. Bonvalet, M. Joffre, J.-L. Martin, and A. Migus, “Generation of ultrabroadband femtosecond pulses in the mid-infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate,” Appl. Phys. Lett. 67, 2907–2909 (1995). [CrossRef]

,6

6. R. A. Kaindl, F. Eickemeyer, M. Woerner, and T. Elsaesser, “Broadband phasematched difference frequency mixing of femtoseconds pulses in GaSe: Experiment and theory,” Appl. Phys. Lett. 75, 1060–1062 (1999). [CrossRef]

].

Typically, to generate broadband THz transients via OR, thin (~1mm or less) electro-optic crystals are used, because of phase matching constraints as well as high absorption in conventional crystals (LiNbO3, ZnTe) at THz frequencies. Optical-to-THz conversion efficiencies achieved so far by optical rectification methods are low [7

7. Peter H. Siegel, “Terahertz Technology,“ IEEE Transactions on Microwave Theory and Techniques 50, 910–28 (2002). [CrossRef]

], typically 10-6 - 10-9, even with femtosecond pump pulse energies as high as 10 mJ [8

8. T. J. Carrig, G. Rodriguez, T. S. Clement, and A. J. Taylor, “Scaling of terahertz radiation via optical rectification in electro-optic crystals,” Appl. Phys. Lett. 66, 121–3 (1995). [CrossRef]

]. However, recently conversion efficiency of 5×10-4 was reported. The authors used OR in LiNbO3 with tilted pulse front excitation and energy per optical pulse 500 μJ [9

9. A. G. Stepanov, J. Kuhl, I. Z. Kozma, E. Riedle, G. Almási, and J. Hebling, “Scaling up the energy of THz pulses created by optical rectification,” Optics Express 13, 5762–68 (2005). [CrossRef] [PubMed]

].

In order to enhance the optical-to-THz conversion efficiency, larger interaction lengths with collinear interaction of THz and optical waves is desirable. The idea of using quasi-phase-matched (QPM) materials with periodically-inverted sign of second-order susceptibility χ (2) for THz optical rectification was proposed by Lee and coauthors [10

10. Y.-S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–7 (2000). [CrossRef]

]. Multi-cycle narrow-band terahertz radiation was produced in periodically poled lithium niobate (PPLN) crystal. The authors used femtosecond pulses at 800 nm and cryogenically cooled (18K) PPLN crystal to reduce THz absorption, and achieved 10-5 conversion efficiency. OR in this case gives rise to a THz waveform which corresponds to the domain structure of the PPLN. Recently, efficient narrow-band terahertz radiation was demonstrated in another QPM material, orientation-patterned gallium arsenide (OP-GaAs) [11

11. K. L. Vodopyanov, M. M. Fejer, D. M. Simanovskii, V.G. Kozlov, and Y.-S. Lee, “Terahertz-wave generation in periodically-inverted GaAs,” Conference on Lasers and Electro Optics, May 2005, Baltimore MD, Technical Digest (Optical Society of America, Washington DC, 2005), paper CWM1.

], which is extremely promising for THz generation because of intrinsically small THz absorption in GaAs.

THz wave generation in QPM crystals is illustrated in the time-domain animation of Fig. 1. Each inverted domain of a nonlinear crystal contributes a half-cycle of the THz pulse [10

10. Y.-S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–7 (2000). [CrossRef]

] and thus the THz wave packet (provided that THz-wave attenuation is low) has as many oscillation cycles as the number of quasi-phase-matched periods over the length of the crystal.

Fig. 1. Illustration of optical rectification in a media with periodically inverted χ (2) sign, using femtosecond pump pulses. The static picture shows optical and THz electric fields, while the animation shows the nonlinear driving polarization and the THz electric field (779kB).

The purpose of this paper is to evaluate maximum optical-to-terahertz conversion efficiency which can be achieved by quasi-phase-matched down conversion process and to find optimal conditions in terms of pump wavelength, pump pulse format, crystal length, and focusing.

2. Plane wave analysis, femtosecond pulses (optical rectification).

Consider as an optical pump, bandwidth-limited ultrashort (e.g., femtosecond) laser pulses propagating along z in the form of infinite plane waves, with the gaussian time envelope of the electric field

E(t)=Re{E0exp(t2τ2)exp[i(ω0t)]}=12{E0exp(t2τ2)exp[i(ω0t)]+c.c.},
(1)

where ω0 is the central frequency and τ is the pulsewidth.

Intensity envelope is thus I(t) ~ exp(-2t 2/τ 2) and the pulse duration at full width of half-maximum is τ FWHM=(2ln2)1/2 τ= 1.18τ. Using Fourier transform pair in the form

f(t)=f(ω)exp(iωt)
(2)
f(ω)=12πf(t)exp(iωt)dt,
(3)

and taking into account that E(t) is real, we can get (z=0) a transform of the electric field (1) in the form 12{E(ω)+E*(ω)}, where E(ω) is a one-sided (ω > 0) Fourier component given by

E(ω)=E0τ2πexp(τ2(ωω0)24)
(4)

Similar form of one-sided Fourier representation will be used for other fields below in the text. For arbitrary z,

Eωz=E(ω)exp[ik(ω)z],
(5)

where k(ω) is a module of the wave vector.

The one-dimensional equation (scalar form) for the Fourier component of the THz field at the angular frequency Ω,E(Ω,z), follows directly from Maxwell’s equations and is given, in the slowly varying envelope approximation and in the limit of no absorption, by [12

12. A. Yariv, Quantum Electronics, (Wiley, New York, 3rd edition, 1988), Chapter 16.

]:

dEΩzdz=iμ0Ωc2n1PNL(Ω)exp(iΔkz),
(6)

where the Fourier component of nonlinear polarization, PNL (Ω), can be expressed through the material nonlinear susceptibility χ (2) as

PNL(Ω)=ε0χ(2)E(ω+Ω)E*(ω)
(7)

Here ε0 and μ0 are respectively the permittivity and permeability of free space, c is the speed of light in vacuum, n 1 is the THz refractive index. From Eqs. (4) and (7) it follows that

PNL(Ω)=ε0χ(2)E02τ22πexp(τ2Ω28)
(8)

and (6) becomes

dEΩzdz=iΩχ(2)E02τ42πcn1exp(τ2Ω28)exp(iΔkz).
(9)

The k- vector mismatch Δk is given by the following relation

Δk=k(Ω)+k(ω)k(ω+Ω)2πʌ.
(10)

Since Ω ≪ ω, we can replace k(ω + Ω) - k(ω) by (∂k/∂ω)opt Ω. [1

1. T. Yajima and N. Takeuchi, “Far-infrared difference-frequency generation by picosecond laser pulses,“ Jpn. J. Appl. Phys. 9, 1361–1371 (1970). [CrossRef]

,13

13. A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–23 (1996). [CrossRef]

] and obtain Δk in the form

Δk=ΩnTHzc(dk)optΩ2πʌ=Ωc(nTHznoptgr)2πʌ,
(11)

where n THz is the phase refractive index for the THz wave, noptgr is the optical group velocity refractive index, and ʌ is the QPM orientation-reversal period. We assumed here that the optical group velocity dispersion 2kω2 is negligible (which will be justified later). With the undepleted pump approximation, Eq. (9) can be integrated to get the power spectrum of the THz field

EΩL2=Ω2deff2E04τ28πc2n12L2exp(τ2Ω24)sinc2(ΔkL2).
(12)

Here we assumed χ (2) = 2d eff, where d eff = (2/π)d OR is an effective QPM nonlinear coefficient; d OR corresponds to the optical rectification process d OR(0=ω-ω) and is derived from electro-optic coefficient rijk using the relation [12

12. A. Yariv, Quantum Electronics, (Wiley, New York, 3rd edition, 1988), Chapter 16.

] d jkl= - n 4/4r jlk, where n is the optical refractive index. In GaAs, for example, r 14=1.5pm/V [14

14. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, (Springer, Berlin, 1997).

], corresponding to the nonlinear coefficient d OR = 47 pm/V.

If the nonlinear crystal is long enough, terahertz radiation will be emitted in the form of narrow-band spectrum centered at Ω0, corresponding to Δk=0 condition

Ω0=2πcΛΔn,λTHz=ΛΔn,
(13)

where Δn = nTHz - noptgr is the index mismatch. By differentiating (11) we get d(Δk)dΩ=Δnc and the phase-matching acceptance bandwidth based on the condition ΔkL/2 = π

ΔΩaccept=cΔnΔkaccept=2πcLΔn
(14)

We assumed that nTHz is nearly constant, which is true for frequencies well below the lowest phonon resonance. For GaAs, this resonance is at 8.1 THz [15

15. W.J. Moore and R.T. Holm, “Infrared dielectric constant of gallium arsenide,” J. Appl. Phys. 80, 6939–42 (1996). [CrossRef]

], nTHz ≈3.6 [16

16. D. Grischkovsky, 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]

], noptgr =3.41 (for the 2.1-μm optical pump) [17

17. T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, and M. M. Fejer, “Determination of GaAs refractive index and its temperature dependence, with application to quasi-phasematched nonlinear optics,” J. of Appl. Phys. 94, 6447–55 (2003). [CrossRef]

], Δn=0.19, and acceptance bandwidth for L=1cm crystal is ΔνTHzaccept =53 cm-1. In the absence of quasi phase-matching, interaction between the optical and THz waves is limited to the coherence length

lc=πcΩΔn,
(15)

corresponding to the Δklc = π condition.

It is worth mentioning that backward emission of THz wave is also possible. In this case, the phase-matching condition becomes

Ωc(nTHz+noptgr)2πΛ=0
(16)

Optical-to-THz energy fluence efficiency for plane waves (PW) is

ηTHzPW=Fluence(THz)Fluence(pump).
(17)

The pump fluence is given by

Fpu=cε0n22Et02dt=π2cε0n22E02τ,
(18)

where n 2 is the optical refractive index. The THz fluence is

FTHz=cε0n12ETHztL2dt=cε0n122π0EΩL2dΩ,
(19)

where we used Parseval’s theorem: f(t)2dt=2πf(Ω)2dΩ.

From Eqs. (12) and (18–19) we get

ηTHzPW=Ω2deff2E02τ22πc2n1n2L20exp(τ2Ω24)sinc2[Δn(ΩΩ0)L2c]dΩ
(20)

where we replaced ΔkL/2 with Δn(Ω - Ω0)L/2c. For Ll c, which is equivalent to the condition ΔΩaccept ≪ Ω0, sinc2 function under the integral dominates and we obtain

ηTHzPW=g12Ω03deff2πε0c3n1n22LlcFpu=g12Ω03deff2Lε0c3n1n22ΔnFpu
(21)

where Ω0 is given by Eq. (13). The reduction factor

g1=exp((τΩ02)2)=exp((πνTHzτ)2)
(22)

reflects the fact that the optical pulse should be short enough, so that its spectrum span is larger than the THz frequency Ω0. For very short optical pulses, τΩ0 < 1, g 1≈1 and optical-to-THz conversion efficiency depends on pulse fluence only, not intensity. Figure 2 shows the reduction factor g 1 as a function of the product νThzτ. For example at νTHzτ=0.1, g 1= 0.91, close to unity, and experiences little change as the pulse duration is further decreased.

Fig. 2. Reduction factor g 1 as a function of vTHzT for the case of femtosecond pump pulses (νTHz=Ω/2π).

3. Plane wave analysis, picosecond pulses

Let us now regard as a pump, bandwidth-limited pulses with longer (pico- or nanosecond) duration τ, such that νTHzτ>1. In this case, the spectrum of a single pulse is narrow and to generate THz output, two different pump pulses need to be mixed to achieve difference frequency generation (DFG). Assume that two gaussian bandwidth-limited optical pulses (plane waves) at frequencies ω2 and ω3 with equal pulse widths propagate collinearly and generate THz wave centered at Ω03 - ω2. Assume that the electric fields (i=2,3) are in the form

Ei(t)=Re{Eiexp(t2τ2)exp[i(ωit)]},
(23)

then, similar to Eq. (4), Fourier transform is given (ω>0) by

Ei(ω)=Eiτ2πexp(τ2(ωωi)24).
(24)

The Fourier component of nonlinear polarization at THz frequency Ω is

PNL(Ω)=ε0χ(2)0E3(ω+Ω)E2*(ω)=ε0χ(2)E2E3τ22πexp(τ2(ΩΩ0)28).
(25)

Integrating Eq. (6) in the limit of no absorption and no pump depletion, we get the power spectrum of the THz field

EΩL2=Ω2deff2E22E32τ28πc2n12L2exp(τ2(ΩΩ0)24)sinc2(ΔkL2),
(26)

where Δk is given by (11).

Suppose that the QPM orientation-reversal period Ό is such that the sinc2 peak is centered exactly at Ω03 - ω2. In this case ΔkL/2 can be replaced by (Ω-Ω0nL/2c. Using Parseval’s theorem, we find the optical-to-THz energy fluence conversion efficiency with respect to one of the two pump pulses (at ω2)

ηTHzPW=Fluence(THz)Fluence(ω2)=Ω2deff2E32τ22πc2n1n2L20exp(τ2Ω'24)sinc2[ΔnΩ'L2c]dΩ'
(27)

where Ω’=Ω-Ω0

The temporal walk-off length between the optical and THz pulses can be introduced as

lw=πΔn,
(28)

which is proportional to the length in a crystal at which the optical pulse walks away in time, with respect to THz wave, by its width τ, due to difference in propagation velocities. It is analogous to the Boyd-Kleinmann’s [18

18. G.D. Boyd and D.A. Kleinman, “Parametric interaction of focussed gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968). [CrossRef]

] birefringent aperture length in the theory of second harmonic generation la = √πw 0/ρ (w 0 is beam radius, and ρ is the birefringent walk-off angle). Now (27) can be expressed as

ηTHzPW=Ω2deff2E32τ22πc2n1n2L20exp(τ2Ω'24)sinc2[π2(Llw)τΩ']dΩ'
(29)

In the limit of long pulses, l wL, exponential function under the integral dominates and (29) becomes

ηTHz=Ω2deff2E32L22c2n1n2=2Ω2deff2L2ε0c2n1n2n3I32,
(30)

where n 2 and n 3 are refractive indices at ω2 and ω3 and I 3 is the peak pump intensity. This formula is similar to the well-know expression [19

19. R.L. Byer and R.L. Herbst, “Parametric oscillation and mixing,” in Topics in Applied Physics: Nonlinear Infrared Generation, ed. by Y.R. Shen (Springer, Berlin, 1977), vol. 16, p. 81–137. [CrossRef]

] for the CW difference frequency generation, with I 3/√2 playing the role of time-averaged pump intensity.

In the limit of short pulses, l wL, (29) becomes

ηTHz=Ω2deff2E32Llw2c2n1n2=2Ω2deff2Llwε0c2n1n2n3I32,
(31)

Thus, for l wL, the L 2 term is replaced by Ll w, in full analogy with the case of second harmonic generation with the spatial walk-off [18

18. G.D. Boyd and D.A. Kleinman, “Parametric interaction of focussed gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968). [CrossRef]

], where L 2 is replaced by I a for l aL. Also, conversion efficiency can be rewritten in terms of coherence length l cc/ΩΔn and energy fluence F 3

ηTHz=2Ω3deff2Llcπε0c3n1n2n3F3,
(32)

This shows that at l wL, terahertz conversion efficiency is a function of fluence only. Besides, it is equal to the conversion efficiency for the case of a femtosecond pulses with νTHzτ≪1. The only difference is that in a two-color picosecond case, in order to get the same energy per THz pulse, one needs to have twice total energy (U 0 in each of the beams), as compared to U 0 in the femtosecond case.

In the case of intermediate pulse durations, we can use (32) with a reduction factor g 2(l w/L),

ηTHz=g22Ω3deff2Llcπε0c3n1n2n3F3=g22Ω2deff2Lε0c2n1n2n3ΔnF3,
(33)

where

g2(x)=1π0exp(x2μ2π)sinc2(μ).
(34)

Figure 3 is the plot of the reduction factor g 2 as a function of l w/L. In many cases, it is desirable to have longer pulses to suppress high-order nonlinear optical effects, even at the expense of some loss in efficiency. Thus setting l w/L=1 (g 2=0.69) might be a good compromise between efficiency and pump intensity. For an L = 1cm GaAs and pump at 2.1 μm, the l w/L =1 condition corresponds to the pulse duration of 3.6 ps. At longer pulses, the THz efficiency will decline; however it will not be improved dramatically if the pulses are made shorter.

Fig. 3. Reduction factor g 2 as a function of l w/L for the case of picosecond pump pulses.

4. Chirped stretched femtosecond pulses

Narrow-band THz radiation can also be generated by mixing two linearly chirped optical pulses (Fig. 4). Overlap of two relatively delayed optical pulses produces a constant beat frequency, which is proportional to the time delay and the amount of frequency chirp. Weling et al. [20

20. A. S. Weling, B. B. Hu, N.M. Froberg, and D.H. Auston, “Generation of tunable narrow-band THz radiation from large aperture photoconducting antennas,” Appl. Phys. Lett. 64, 137 (1994). [CrossRef]

] demonstrated this with photoconducting antennas and 23-ps-long pulses and showed that the generated THz frequency exhibited a linear dependence on the time delay. The same technique can be applied to THz generation using optical rectification.

Assume that the two pulses are of equal intensities. The electric field of the first optical pulse is in the form

E2(t)=Re{E0exp(t2τ2)exp[i(ω0t+bt2)]}
(35)

and that of the second (delayed) pulse is

E3(t)=E2(t+Δt)=Re{E0exp((t+Δt)2τ2)exp[i(ω0(t+Δt)+b(t+Δt)2)]}
(36)

Then the nonlinear polarization at the THz beat frequency is

PNL(t)=Re{ε0χ(2)(E2(t)+E3(t))2ω=0}
=Re{ε0χ(2)E02exp[2τ2(t2+Δtt+Δt22)+i(2bΔtt+bΔt2)]}
(37)

Its Fourier transform (Ω>0)

PNL(Ω)=ε0χ(2)E022πexp[2τ2(t2+Δtt+Δt22)+i(2bΔtt+bΔt2Ωt)]dt
=ε0χ(2)E02τ22πexp(τ2Ω'2812Δt2τ2)exp(iφ1)
(38)

where Ω’=2bΔt - Ω and φ1 = -ΔtΩ’2+bΔt 2.

Similar to Eq. (29), we get THz fluence conversion efficiency with respect to one of the two pump pulses

ηTHzPW=Ω2deff2E02τ22πc2n1n2L20exp(τ2Ω'24Δt2τ2)sinc2[Δn(Ω'+Ω")L2c]dΩ'
(39)

Exponential function here peaks at Ω’=0, that is when Ω=2bΔt. If we assume that the sinc2 function peaks at the same frequency (Ω”=0), then Eq. (39) becomes

ηTHz=g2exp(Δt2τ2)2Ω2deff2Llcπε0c3n1n2n3F3,
(40)
Fig. 4. Scheme to generate tunable THZ radiation from the overlap of two linearly chirped pulses.

Here g 2 is the reduction factor given by Eq. (34); the exponential term can be regarded as another reduction factor, associated with the temporal overlap of the two pulses, one of which is delayed by Δt. Suppose that the two chirped pulses with the pulsewidth τ=τ2 are created by stretching in time a much shorter (femtosecond) pulse with the pulsewidth τ1. It is easy to show [21

21. A. E. Siegman, Lasers, (University Science Books, Mill Valley, 1986), Ch.9.

] that in this case b=1/τ1τ2 and exp(-(Δt2)2)= exp(-(Ωτ1/2)2). Interestingly, this reduction factor is the same as g 11) in (22); thus (40) becomes

ηTHz=g1g22Ω2deff2Llcπε0c3n1n2n3F3.
(41)

We see that in the case of linearly chirped optical pulses, THz conversion efficiency is, again, a function of fluence only, provided that pulse durations τ1 and τ2 are short enough.

5. Optimal length of the EO crystal

From the simple plane-wave analysis above, we see that the optical-to-THz conversion efficiency, in the optimized case, is proportional to L. If we take into account THz absorption in the crystal (which is usually much larger than the optical absorption) but still neglect pump depletion, we get

ηTHz(L)1α[1exp(αL)]=Leff,
(42)

where α is the THz intensity absorption coefficient and L eff is an effective length. When L → ∞, L eff → 1/α. Choosing L=1/α will give us L eff=0.63/α. In general, one can introduce another reduction factor, associated with the absorption and write

ηTHz(L)g3L;g3=1αL[1exp(αL)]
(43)

For L=1/α, g 3=0.63.

6. Optimal focusing

To maximize the THz efficiency, one needs to focus pump beams. In the near field approximation, when the focusing is loose and diffraction can be neglected, the optical-to-THz conversion efficiency (OR, femtosecond pulses) with respect to the pump pulse energy U pu can be obtained by integrating Eq. (21) over the transverse coordinate r(Epu (r) ~ exp(-r 2/w 2))

ηTHz=UTHzUpu=g1g32Ω2deff2Lε0c3n1n22ΔnUpuπw2,
(44)

where w is the gaussian pump beam size. Following Ch.5 of [18

18. G.D. Boyd and D.A. Kleinman, “Parametric interaction of focussed gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968). [CrossRef]

], which considers the DFG case, we can characterize the focusing strength by a focusing parameter ξ = L/k 1 w 2 = λ 1 L/2πn 1 w 2, where λ 1 is the THz wavelength and n 1 is the THz refractive index (Boyd-Kleinman’s theory itself [18

18. G.D. Boyd and D.A. Kleinman, “Parametric interaction of focussed gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968). [CrossRef]

] is not applicable here since THz field is not a resonant field and its distribution is not defined a priori by an optical cavity). Morris and Shen [22

22. J.R. Morris and Y.R. Shen, “Theory of far infrared generation by optical mixing,” Phys. Rev. A 15, 1143–56 (1977). [CrossRef]

] developed a theory of far-infrared generation by optical mixing of focused laser beams, based on Fourier analysis with respect to transverse k-vector components, and have found that focusing of the pump beams appreciably enhances the far-infrared output despite the strong far-infrared diffraction. For example in a 1-cm-long GaAs crystal and an output wavelength 100μm, the optimal focal-spot size (for the optimized phase-matching condition) was found to be around w=20 μm (ξ=110), that is less than the THz wavelength.

Fig. 5. Enhancement factor h as a function of the focusing parameter ξ. Solid curve is based on ref. 22. Dashed curve ‒ plane-wave approximation. Dots represent our calculations based on the Green’s function method. Inset: far field THz intensity profiles at different ξ for a 1-cm-long GaAs.

THz conversion efficiency for OR with femtosecond pulses and focused beams can be expressed in the form

ηTHz=UTHzUpu=g1g32Ω3deff2πε0c3n1n22ΔnUpuh(ξ),
(45)

and, similarly, for DFG with picosecond pulses, by

ηTHz=UTHzU2=g2g32Ω3deff2πε0c3n2n3ΔnU3h(ξ),
(46)

Here h(ξ), by analogy with [18

18. G.D. Boyd and D.A. Kleinman, “Parametric interaction of focussed gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968). [CrossRef]

], is an enhancement factor due to pump focusing, such that h(ξ)=ξ, at ξ≪1 (plane waves). Enhancement factor h(ξ) obtained from the calculations of Morris and Shen [22

22. J.R. Morris and Y.R. Shen, “Theory of far infrared generation by optical mixing,” Phys. Rev. A 15, 1143–56 (1977). [CrossRef]

] is shown on Fig. 5, along with the enhancement factor for plane-waves. Fig. 5 (dots) also shows results of our calculations of the enhancement factor, based on the Green’s function method [23

23. Y.R. Shen, “Far-infrared generation by optical mixing,” Prog. Quant. Electr. 4, 207–232 (1976) [CrossRef]

,24

24. S. Guha, “Focusing dependence of the efficiency of a singly resonant optical parametric oscillator,” Appl. Phys. B 66, 663–675 (1998). [CrossRef]

], where the far field solution was obtained by integration over the nonlinear polarization distribution in real space; these results are in excellent agreement with the Morris-Shen curve.

There are however limitations on the very tight focusing. (i) When pump beam waist w is too small, THz output extends over a large span of angles θ to the normal of the crystal; when θ exceeds the total internal reflection angle θmax of the material, THz transmission falls to zero. This leads to the condition √2λ/πn 1 w < 2θmax. For GaAs, for example, θmax = 16°, and for 1.5 THz frequency, this corresponds to w > 45 μm. (ii) In the case of tight focusing, different Fourier components of the transverse k- vector have different phase-matching conditions and it appears [22

22. J.R. Morris and Y.R. Shen, “Theory of far infrared generation by optical mixing,” Phys. Rev. A 15, 1143–56 (1977). [CrossRef]

] that the THz output is maximized when the far field distribution is in the form of a hollow cone. The phase-matching condition for each angle θ is

Δkz=k1z(k3k2)=k1cosθ(k3k2)Δk0k12θ2=0
(47)

where Δk 0 is a collinear wave-vector mismatch. For the high beam quality (solid vs hollow cone) we can require that variation of Δk z due to variation of θ is small, so that δ(Δk z)L<π (corresponding to half of the QPM acceptance bandwidth). This leads to the condition k 1θd2 L/2 ≤ π where θd is given by the diffraction: θd = √2λ1/2πnw. Thus we get

ξ=λ1L2πn1w2<π.
(48)

The far field intensity profiles of THz radiation at different ξ(GaAs, L=1cm) based on our calculations using the Green’s function method is shown in the inset to Fig. 5 and is consistent with Eq. (48); ξ≈1 can be regarded as a good compromise between conversion efficiency and the beam quality.

From Eqs. (45–46) one can see that the THz conversion efficiency does not depend on the crystal length L. The length affects however the spectral width of the THz pulse; also, it is more advantageous to have longer crystals (on the order of 1/α), since in this case the beam waist will be larger and the peak intensity smaller. This is important from the viewpoint of reducing unwanted higher order nonlinear effects (see below).

As a numerical example, regard a 100-fs-long optical pump pulse at 2.1 μm, 1.5 THz output frequency, GaAs crystal length L=1/α =9.5 mm (α=1.05 cm-1), and pump beam waist w=290 μm (ξ=1). In this case reduction factors are g 1=0.8, g 3=0.63, h(ξ)=0.82 and from (45) we obtain ηTHz= 3.95×10-4/ μJ. For the 2.5 THz output (α=4.3 cm-1), for L =2.3 mm and w=110 μm (ξ=1), g 1=0.54, g 3=0.63, h(ξ)=0.82, and ηTHz=1.23×10-3 / μJ.

7. Cascading and red shift

THz wave generation via OR is a parametric process of self-mixing in which a photon from the blue (high-frequency) wing of the femtosecond optical pulse decays into a THz photon plus a red-shifted photon, corresponding to the low-frequency wing of the same optical pulse. From the photon energy conservation argument, it follows (if we neglect losses) that the center of weight of the optical pulse spectrum will be red-shifted by Δω/ω0 ~ ηTHz , where ω0 is the central optical frequency. When optical-to-THz photon conversion efficiency approaches 100%, the red shift will be on the order of THz frequency Ω. Once the optical pulse becomes red-shifted, it can still contribute to THz generation ‒ the same process of cascaded optical down-conversion continues to transfer optical energy to lower frequencies, as long as the phase mismatch Δk in Eq. (11) is small. Accordingly, cascaded down-conversion will be the most efficient when the pump wavelength is close to the point of zero group velocity dispersion. Quantitatively, the number of cascading cycles can be expressed as N= ½ (acceptance bandwidth) / (teraherz frequency). Here acceptance bandwidth is with respect to the pump frequency and can be found by differentiating (11): dk)/pump = (Ω/c)dngr /, from which we get

Δωpumpaccept=2πcLωpumpΩ(λdngr)1
(49)

Fig. 6. Number of THz cascading cycles as a function of THz frequency and pump wavelength for GaAs, L=1cm.

The red shift of the optical pump pulse can also be understood from a different standpoint. Gustafson et al. [26

26. T.K. Gustafson, J.-P.E. Taran, P.L. Kelley, and R.Y. Chiao, “Self-modulation of picosecond pulses in electro-optic crystals,” Opt. Commun. 2, 17–21 (1970). [CrossRef]

] have pointed out that rectified field which is generated during the propagation of optical pulses through electro-optic crystals can produce considerable phase modulation of the optical pulse itself, especially when the rectified pulse travels at the same speed as the optical pulse within the crystal. This cascaded Kerr-like nonlinearity occurs because of back action of the optical rectified field upon the pump wave through the electro-optical effect [27

27. J.-P. Caumes, L. Videau, C. Touyez, and E. Freysz, “Kerr-like nonlinearity induced via terahertz generation and the electro-optic effect in zinc blende crystals,” Phys. Rev. Letters 89, 047401 (2002). [CrossRef]

,28

28. Y.J. Ding, “Efficient generation of high-power quasi-single-cycle terahertz pulses from a single infrared beam in a second-order nonlinear medium,” Opt. Lett. 29, 2650–52 (2004). [CrossRef] [PubMed]

].

Fig. 7. (a) The optical pulse intensity profile, (b) rectified field profile, (c) phase profile across the optical pulse after traveling the crystal, and (d) frequency shift of the optical pulse.

Figure 7 illustrates the process of red shift for the case of single-cycle terahertz pulse generation. Through the EO effect, rectified field (Fig. 7(b)) produces a change in the refractive index of the medium δn ~ r ij E THz(t,z) which causes the time-dependent phase shift δϕ(t,z) (Fig. 7(c)). The amplitude of this phase shift is proportional to the integral of the rectified field over the crystal length; since the rectified field grows linearly with the length, the phase shift is proportional to the crystal length squared [26

26. T.K. Gustafson, J.-P.E. Taran, P.L. Kelley, and R.Y. Chiao, “Self-modulation of picosecond pulses in electro-optic crystals,” Opt. Commun. 2, 17–21 (1970). [CrossRef]

]. The frequency shift δω=-∂δϕ(t,z)/∂t (Fig. 7(d)) is also proportional to the crystal length squared. The central portion of the optical pulse experiences red shift, independent on the sign of EO coefficient. This ‘microscopic’ consideration of the red shift gives the same dependence as before: it scales as THz conversion efficiency (proportional to the EO coefficient squared and crystal length squared) for the index-matched single-cycle terahertz pulse generation. In the case of multicycle THz generation with periodically inverted crystals, the optical pulse will see in average the same induced phase shift as shown in Fig. 7(b), since the relative phase between the optical and THz pulse is corrected at regular intervals via a structural periodicity of the nonlinear medium.

8. The role of higher order nonlinear effects

Two-photon absorption (2PA), three-photon absorption (3PA), and nonlinear refraction index (NRI), n2, can severely limit the maximum THz conversion efficiency in the OR or DFG process: in the case of 2PA and 3PA, the pump beam will create free carriers which strongly absorb THz radiation; NRI, on the other hand, affects the pump pulse itself via self-phase modulation and self-focusing. As a necessary condition for achieving cascading effects in THz generation, one needs to be close to 100% in a single-stage optical-to-THz photon efficiency, which requires for GaAs pump intensities on the order of 100-1000 GW/cm2 (fs pulses) and 1–100 GW/cm2 (ps pulses). It is very advantageous to use longer pump wavelengths to avoid 2PA effects which are usually strong in semiconductors (e.g. for GaAs, β=26 cm/GW at 1.06 μm [29

29. A.A. Said, M. Sheik-Bahae, D.J. Hagan, T.H. Wei, J. Wang, J. Young, and E.W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405–414 (1992). [CrossRef]

]). This dictates that pump wavelength should be above 1.74 μm (GaAs) in which case the dominant nonlinearities are 3PA and NRI (theoretically predicted values for GaAs are correspondingly γ≈0.2 cm3/GW2 [30

30. B.S. Wherrett, “Scaling rules for multiphoton interband absorption in semiconductors,” J. Opt. Soc. Am. B 1, 67–72 (1984). [CrossRef]

] and n2≈2·10-4 cm2/GW [31

31. M. Sheik-Bahae, D.C. Hutchings, D.J. Hagan, and E.W. Van Stryland, “Dispersion of bound electronic nonlinear refraction in solids,” IEEE J. of Quant. Electron. 27, 1296–1309 (1991). [CrossRef]

]). Also, it is more likely that the highest THz conversion efficiencies (> 100% by photons, with cascading) will be achieved in the picosecond, rather than femtosecond regime: since THz conversion efficiency is fluence-sensitive, peak intensity in the former case will be much smaller, and 3PA and NRI effects less pronounced.

9. Conclusion

We show that optically pumped periodically-inverted EO crystals have potential for generating narrow-bandwidth THz wave packets with high conversion efficiency. For sufficiently short (femtosecond) single-beam optical pulses with τΩ<1 (optical rectification), THz conversion efficiency scales with the pulse energy and does not depend on pulse duration. In the case of mixing two picosecond pulses (bandwidth-limited or chirped), with pulse durations such that the temporal walk-off length is smaller than the length of the crystal. again, conversion efficiency is pulse-energy dependent, and has the same scaling factor as in the case of femtosecond pulses. Also, cascaded THz frequency down conversion is possible, allowing achieving > 100% optical-to-THz photon conversion efficiency.

Acknowledgments

References and links

1.

T. Yajima and N. Takeuchi, “Far-infrared difference-frequency generation by picosecond laser pulses,“ Jpn. J. Appl. Phys. 9, 1361–1371 (1970). [CrossRef]

2.

K.H. Yang, P.L. Richards, and Y.R. Shen, “Generation of far-infrared radiation by picosecond light pulses in LiNbO3,” Appl. Phys. Lett. 19, 320–323 (1971). [CrossRef]

3.

L. Xu, X.-C. Zhang, and D. H. Auston, “Terahertz beam generation by femtosecond optical pulses in electro-optic materials,” Appl. Phys. Lett. 61, 1784–6 (1992) [CrossRef]

4.

B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nature Materials 1, 26–33 (2002). [CrossRef]

5.

A. Bonvalet, M. Joffre, J.-L. Martin, and A. Migus, “Generation of ultrabroadband femtosecond pulses in the mid-infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate,” Appl. Phys. Lett. 67, 2907–2909 (1995). [CrossRef]

6.

R. A. Kaindl, F. Eickemeyer, M. Woerner, and T. Elsaesser, “Broadband phasematched difference frequency mixing of femtoseconds pulses in GaSe: Experiment and theory,” Appl. Phys. Lett. 75, 1060–1062 (1999). [CrossRef]

7.

Peter H. Siegel, “Terahertz Technology,“ IEEE Transactions on Microwave Theory and Techniques 50, 910–28 (2002). [CrossRef]

8.

T. J. Carrig, G. Rodriguez, T. S. Clement, and A. J. Taylor, “Scaling of terahertz radiation via optical rectification in electro-optic crystals,” Appl. Phys. Lett. 66, 121–3 (1995). [CrossRef]

9.

A. G. Stepanov, J. Kuhl, I. Z. Kozma, E. Riedle, G. Almási, and J. Hebling, “Scaling up the energy of THz pulses created by optical rectification,” Optics Express 13, 5762–68 (2005). [CrossRef] [PubMed]

10.

Y.-S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–7 (2000). [CrossRef]

11.

K. L. Vodopyanov, M. M. Fejer, D. M. Simanovskii, V.G. Kozlov, and Y.-S. Lee, “Terahertz-wave generation in periodically-inverted GaAs,” Conference on Lasers and Electro Optics, May 2005, Baltimore MD, Technical Digest (Optical Society of America, Washington DC, 2005), paper CWM1.

12.

A. Yariv, Quantum Electronics, (Wiley, New York, 3rd edition, 1988), Chapter 16.

13.

A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–23 (1996). [CrossRef]

14.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, (Springer, Berlin, 1997).

15.

W.J. Moore and R.T. Holm, “Infrared dielectric constant of gallium arsenide,” J. Appl. Phys. 80, 6939–42 (1996). [CrossRef]

16.

D. Grischkovsky, 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]

17.

T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, and M. M. Fejer, “Determination of GaAs refractive index and its temperature dependence, with application to quasi-phasematched nonlinear optics,” J. of Appl. Phys. 94, 6447–55 (2003). [CrossRef]

18.

G.D. Boyd and D.A. Kleinman, “Parametric interaction of focussed gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968). [CrossRef]

19.

R.L. Byer and R.L. Herbst, “Parametric oscillation and mixing,” in Topics in Applied Physics: Nonlinear Infrared Generation, ed. by Y.R. Shen (Springer, Berlin, 1977), vol. 16, p. 81–137. [CrossRef]

20.

A. S. Weling, B. B. Hu, N.M. Froberg, and D.H. Auston, “Generation of tunable narrow-band THz radiation from large aperture photoconducting antennas,” Appl. Phys. Lett. 64, 137 (1994). [CrossRef]

21.

A. E. Siegman, Lasers, (University Science Books, Mill Valley, 1986), Ch.9.

22.

J.R. Morris and Y.R. Shen, “Theory of far infrared generation by optical mixing,” Phys. Rev. A 15, 1143–56 (1977). [CrossRef]

23.

Y.R. Shen, “Far-infrared generation by optical mixing,” Prog. Quant. Electr. 4, 207–232 (1976) [CrossRef]

24.

S. Guha, “Focusing dependence of the efficiency of a singly resonant optical parametric oscillator,” Appl. Phys. B 66, 663–675 (1998). [CrossRef]

25.

M. Cronin-Golomb, “Cascaded nonlinear difference-frequency generation of enhanced terahertz wave production,” Opt. Lett. 29, 2046–48 (2004). [CrossRef] [PubMed]

26.

T.K. Gustafson, J.-P.E. Taran, P.L. Kelley, and R.Y. Chiao, “Self-modulation of picosecond pulses in electro-optic crystals,” Opt. Commun. 2, 17–21 (1970). [CrossRef]

27.

J.-P. Caumes, L. Videau, C. Touyez, and E. Freysz, “Kerr-like nonlinearity induced via terahertz generation and the electro-optic effect in zinc blende crystals,” Phys. Rev. Letters 89, 047401 (2002). [CrossRef]

28.

Y.J. Ding, “Efficient generation of high-power quasi-single-cycle terahertz pulses from a single infrared beam in a second-order nonlinear medium,” Opt. Lett. 29, 2650–52 (2004). [CrossRef] [PubMed]

29.

A.A. Said, M. Sheik-Bahae, D.J. Hagan, T.H. Wei, J. Wang, J. Young, and E.W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405–414 (1992). [CrossRef]

30.

B.S. Wherrett, “Scaling rules for multiphoton interband absorption in semiconductors,” J. Opt. Soc. Am. B 1, 67–72 (1984). [CrossRef]

31.

M. Sheik-Bahae, D.C. Hutchings, D.J. Hagan, and E.W. Van Stryland, “Dispersion of bound electronic nonlinear refraction in solids,” IEEE J. of Quant. Electron. 27, 1296–1309 (1991). [CrossRef]

OCIS Codes
(190.2620) Nonlinear optics : Harmonic generation and mixing
(190.7110) Nonlinear optics : Ultrafast nonlinear optics
(260.3090) Physical optics : Infrared, far

ToC Category:
Nonlinear Optics

History
Original Manuscript: December 6, 2005
Manuscript Accepted: March 8, 2006
Published: March 20, 2006

Citation
Konstantin 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)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-6-2263


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References

  1. 1. T. Yajima, and N. Takeuchi, "Far-infrared difference-frequency generation by picosecond laser pulses," Jpn. J. Appl. Phys. 9, 1361-1371 (1970). [CrossRef]
  2. K. H. Yang, P. L. Richards, and Y. R. Shen, "Generation of far-infrared radiation by picosecond light pulses in LiNbO3," Appl. Phys. Lett. 19, 320-323 (1971). [CrossRef]
  3. L. Xu, X.-C. Zhang, and D. H. Auston, "Terahertz beam generation by femtosecond optical pulses in electro-optic materials," Appl. Phys. Lett. 61, 1784-6 (1992). [CrossRef]
  4. B. Ferguson, and X.-C. Zhang, "Materials for terahertz science and technology," Nat. Mater. 1, 26-33 (2002). [CrossRef]
  5. A. Bonvalet, M. Joffre, J.-L. Martin, and A. Migus, "Generation of ultrabroadband femtosecond pulses in the mid-infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate," Appl. Phys. Lett. 67, 2907-2909 (1995). [CrossRef]
  6. R. A. Kaindl, F. Eickemeyer, M. Woerner, and T. Elsaesser, "Broadband phasematched difference frequency mixing of femtoseconds pulses in GaSe: Experiment and theory," Appl. Phys. Lett. 75, 1060-1062 (1999). [CrossRef]
  7. PeterH. Siegel, "Terahertz technology," IEEE Transactions on Microwave Theory and Techniques 50, 910-28 (2002). [CrossRef]
  8. T. J. Carrig, G. Rodriguez, T. S. Clement, and A. J. Taylor, "Scaling of terahertz radiation via optical rectification in electro-optic crystals," Appl. Phys. Lett. 66, 121-3 (1995). [CrossRef]
  9. A. G. Stepanov, J. Kuhl, I. Z. Kozma, E. Riedle, G. Almási and J. Hebling, "Scaling up the energy of THz pulses created by optical rectification," Opt. Express 13, 5762 - 68 (2005). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-15-5762 [CrossRef] [PubMed]
  10. Y.-S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, "Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate," Appl. Phys. Lett. 76, 2505-2507 (2000). [CrossRef]
  11. K. L. Vodopyanov, M. M. Fejer, D. M. Simanovskii, V. G. Kozlov, and Y.-S. Lee, "Terahertz-wave generation in periodically-inverted GaAs," Conference on Lasers and Electro Optics, May 2005, Baltimore MD, Technical Digest (Optical Society of America, Washington DC, 2005), paper CWM1.
  12. A. Yariv, Quantum Electronics, (Wiley, New York, 3rd edition, 1988), Chap. 16.
  13. A. Nahata, A. S. Weling, and T. F. Heinz, "A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling," Appl. Phys. Lett. 69, 2321-23 (1996). [CrossRef]
  14. V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, (Springer, Berlin, 1997).
  15. W. J. Moore, and R. T. Holm, "Infrared dielectric constant of gallium arsenide," J. Appl. Phys. 80, 6939-6942 (1996). [CrossRef]
  16. D. Grischkovsky, 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]
  17. T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, and M. M. Fejer, "Determination of GaAs refractive index and its temperature dependence, with application to quasi-phasematched nonlinear optics," J. of Appl. Phys. 94, 6447-6455 (2003). [CrossRef]
  18. G. D. Boyd, D. A. Kleinman, "Parametric interaction of focussed gaussian light beams," J. Appl. Phys. 39, 3597-3639 (1968). [CrossRef]
  19. R. L. Byer, and R. L. Herbst, "Parametric oscillation and mixing," in Topics in Applied Physics: Nonlinear Infrared Generation, ed. by Y.R. Shen (Springer, Berlin, 1977), Vol. 16, p. 81-137. [CrossRef]
  20. A. S. Weling, B. B. Hu, N. M. Froberg, and D. H. Auston, "Generation of tunable narrow-band THz radiation from large aperture photoconducting antennas," Appl. Phys. Lett. 64, 137 (1994). [CrossRef]
  21. A. E. Siegman, Lasers, (University Science Books, Mill Valley, 1986), Chap. 9.
  22. J. R. Morris, and Y. R. Shen, "Theory of far infrared generation by optical mixing," Phys. Rev. A 15, 1143-56 (1977). [CrossRef]
  23. Y. R. Shen, "Far-infrared generation by optical mixing," Prog. Quantum Electron. 4, 207-232 (1976). [CrossRef]
  24. S. Guha, "Focusing dependence of the efficiency of a singly resonant optical parametric oscillator," Appl. Phys. B 66, 663-675 (1998). [CrossRef]
  25. M. Cronin-Golomb, "Cascaded nonlinear difference-frequency generation of enhanced terahertz wave production," Opt. Lett. 29, 2046-2048 (2004). [CrossRef] [PubMed]
  26. T. K. Gustafson, J.-P.E. Taran, P. L. Kelley, and R. Y. Chiao, "Self-modulation of picosecond pulses in electro-optic crystals," Opt. Commun. 2, 17-21 (1970). [CrossRef]
  27. J.-P. Caumes, L. Videau, C. Touyez, E. Freysz, "Kerr-line nonlinearity induced via terahertz generation and the electro-optic effect in zinc blende crystals," Phys. Rev. Lett. 89, 047401 (2002). [CrossRef]
  28. Y. J. Ding, "Efficient generation of high-power quasi-single-cycle terahertz pulses from a single infrared beam in a second-order nonlinear medium," Opt. Lett. 29, 2650-52 (2004). [CrossRef] [PubMed]
  29. A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, "Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe," J. Opt. Soc. Am. B 9, 405-414 (1992). [CrossRef]
  30. B. S. Wherrett, "Scaling rules for multiphoton interband absorption in semiconductors," J. Opt. Soc. Am. B 1, 67-72 (1984). [CrossRef]
  31. M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, "Dispersion of bound electronic nonlinear refraction in solids," IEEE J. Quantum Electron. 27, 1296-1309 (1991). [CrossRef]

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