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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 22 — Oct. 25, 2010
  • pp: 23173–23182
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Terahertz white-light pulses from an air plasma photo-induced by incommensurate two-color optical fields

Mark D. Thomson, Volker Blank, and Hartmut G. Roskos  »View Author Affiliations


Optics Express, Vol. 18, Issue 22, pp. 23173-23182 (2010)
http://dx.doi.org/10.1364/OE.18.023173


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Abstract

We demonstrate plasma-THz generation in ambient air with two-color (ω-2ω) optical excitation using input pulses with a bandwidth supporting sub-20-fs duration, yielding a continuous bandwidth exceeding 100 THz with no significant roll-off and a pulse energy of 360 nJ at a 1-kHz repetition rate. The key aspect in achieving this performance is an optimized geometry of the second-harmonic generation (SHG) crystal, producing a 2ω-field detuned from the second-harmonic of the ω-field, which promotes both a high polarization bandwidth and optimal ω-2ω temporal overlap in the plasma, as supported by theoretical results.

© 2010 Optical Society of America

1. Introduction

The generation of intense, broadband terahertz (THz) pulses from femtosecond (fs) photoinduced gas plasmas [1

1. D. J. Cook and R. M. Hochstrasser, “Intense terahertz pulses by four-wave rectification in air,” Opt. Lett. 25, 1210–1212 (2000). [CrossRef]

10

10. A. Houard, Y. Liu, B. Prade, and A. Mysyrowicz, “Polarization analysis of terahertz radiation generated by four-wave mixing in air,” Opt. Lett. 33, 1195–1197 (2008). [CrossRef] [PubMed]

] has shown considerable potential for emerging non-linear [11

11. J. Hebling, K.-L. Yeh, M. C. Hoffmann, and K. A. Nelson, “High-power THz generation, THz nonlinear optics, and THz nonlinear spectroscopy,” IEEE J. Sel. Top. Quant. Electron. 14, 345–353 (2008). [CrossRef]

] and time-domain [12

12. C. Kübler, H. Ehrke, R. Huber, R. Lopez, A. Halabica, J. R. F. Haglund, and A. Leitenstorfer, “Coherent structural dynamics and electronic correlations during an ultrafast insulator-to-metal phase transition in VO2,” Phys. Rev. Lett. 99, 116401 (2007). [CrossRef] [PubMed]

] THz spectroscopies. The requirements are typically peak fields in the MV/cm range and/or continuous spectra with bandwidths reaching well into the mid-infrared (IR). Among the various plasma-THz methods, the approach using two-color (ω-2ω) optical excitation has demonstrated the best performance to date in terms of emission bandwidth and intensity. The THz emission is due to the asymmetric low-frequency transverse polarization component of the ionized electrons, allowing dipole emission in the forward direction. Compared to other broadband, high-field THz generation methods (e.g. second-order non-linear mixing [11

11. J. Hebling, K.-L. Yeh, M. C. Hoffmann, and K. A. Nelson, “High-power THz generation, THz nonlinear optics, and THz nonlinear spectroscopy,” IEEE J. Sel. Top. Quant. Electron. 14, 345–353 (2008). [CrossRef]

, 13

13. C. Kübler, R. Huber, and A. Leitenstorfer, “Ultrabroadband terahertz pulses: generation and field-resolved detection,” Semicond. Sci. Technol. 20, S128–S133 (2005). [CrossRef]

, 14

14. F. Junginger, A. Sell, O. Schubert, B. Mayer, D. Brida, M. Marangoni, G. Cerullo, A. Leitenstorfer, and R. Huber, “Single-cycle multiterahertz transients with peak fields above 10 MV/cm,” Opt. Lett. 35, 2645–2647 (2010). [CrossRef] [PubMed]

]), the plasma-THz emitter has no inherent damage threshold, and does not suffer from the dispersion and absorption issues of non-linear crystals. Moreover, the ionization dynamics are extremely non-linear, producing polarization components of high non-linear order and hence a broader polarization bandwidth for a given pulse duration. While various models have been proposed to account for the THz polarization [15

15. K. Y. Kim, J. H. Glownia, A. J. Taylor, and G. Rodriguez, “Terahertz emission from ultrafast ionizing air in symmetry-broken laser fields,” Opt. Express 15, 4577–4584 (2007). [CrossRef] [PubMed]

18

18. A. A. Silaev and N. V. Vvedenskii, “Quantum-mechanical approach for calculating the residual quasi-dc current in a plasma produced by a few-cycle laser pulse,” Phys. Scr. T135, 014024 (2009). [CrossRef]

], the transient photocurrent model [8

8. K. Y. Kim, A. J. Taylor, J. H. Glownia, and G. Rodriguez, “Coherent control of terahertz supercontinuum generation in ultrafast laser-gas interactions,” Nature Photonics 2, 605–609 (2008). [CrossRef]

, 15

15. K. Y. Kim, J. H. Glownia, A. J. Taylor, and G. Rodriguez, “Terahertz emission from ultrafast ionizing air in symmetry-broken laser fields,” Opt. Express 15, 4577–4584 (2007). [CrossRef] [PubMed]

] appears to provide a reasonable semi-classical description, and predicts that the photo-induced plasma mixes the two-color optical radiation to produce polarization components in the THz and mid-IR range.

One aspect that has only recently emerged [19

19. V. Blank, M. D. Thomson, and H. G. Roskos, “Terahertz radiation with a continuous spectral bandwidth reaching beyond 100 THz from a laser-induced gas plasma,” in Ultrafast Phenomena XVII, M. Chergui, D. Jonas, E. Riedle, B. Schoenlein, and A. Taylor, eds. (Oxford University Press, New York, 2010). To be published.

,20

20. T. Balčiūnas, D. Lorenc, M. Ivanov, O. Smirnova, A. Pugžlys, A. M. Zheltikov, D. Dietze, J. Darmo, K. Unterrainer, T. Rathje, G. Paulus, and A. Baltǔska, “Tunability of THz emission originating from sub-cycle electron bursts in a laser induced plasma,” in Ultrafast Phenomena XVIIM. Chergui, D. Jonas, E. Riedle, B. Schoenlein, and A. Taylor, eds. (Oxford University Press, New York, 2010). To be published.

] concerns the effect of using a 2ω-spectrum detuned from twice the center frequency of the ω-field, as depicted in Fig. 1(a). In this case, one expects a dynamic phase between the incommensurate ω- and 2ω-waves (Fig. 1(a), right), such that the mixing products are shifted to higher THz frequencies. In this contribution, we employ an ultra-broadband ω-field which allows one to achieve a significantly detuned 2ω-spectrum via SHG. The optimal conditions for THz emission in our experiments correspond to precisely such an incommensurate situation. We can account for the essential features of the THz-emission experiments using numerical simulations based on plane-wave pulse propagation in both the SHG crystal and plasma, using tunnel ionization and the photocurrent model for the plasma polarization.

Fig. 1 (a) Plasma-THz generation with commensurate (left) and incommensurate (right) optical spectra: optical and resulting THz spectra, two-color fields and plasma polarization components (single FWR term only). (b) Schematic of experimental set-up, including hollow-core fiber compression, plasma generation and FTIR interferometer. (c) Detail of BBO crystal geometry. The optimal geometry for THz emission with the 150-μm-thick 32°-cut BBO crystal corresponded to tilt angles α = 8.1° (horizontal) and β = 2.3° (vertical).

2. Experimental details

The experimental set-up (Fig. 1(b)) is based on a 150-fs Ti:Al2O3 amplifier laser (Clark-MXR CPA-2101, center wavelength 775 nm). These pulses are spectrally broadened in an commercial hollow-core fiber (Femtolasers Kaleidoscope, adapted and distributed for the Clark-MXR CPA-2010 amplifier by Horiba Scientific) filled with Ar gas (2.5 bar) to achieve a bandwidth sufficient to support sub-20 fs pulses, followed by re-compression with a sequence of negative-dispersion mirrors. Correlated energy measurements showed that the ∼0.5%-rms shot-to-shot fluctuations are essentially preserved between the input (10%-maximum spectral width, Δν = 6.2 THz) and output pulses (Δν = 83 THz), with a pulse-energy throughput above 55% (fibre length 1 m). This beam (energy Jopt = 420 μJ, diameter 2w = 9.2 mm, p-polarized) is then focused with a singlet lens (BK-7, f = 200 mm) to form a plasma in ambient air, via a 150-μm-thick β-BBO crystal (Fig. 1(c)) where the 2ω-field is generated. After discarding the optical beams using a 400-μm-thick Si wafer, the plasma-THz radiation is collimated by an off-axis paraboloidal mirror (OAPM, feff = 76.2 mm) and guided to a Michelson interferometer based on a Si-beam splitter (400-μm-thick). The output is focused by another OAPM (feff = 101.6 mm) onto an integrating thermal detector (Golay cell, QMC Instruments OAD-7), yielding the field autocorrelation signal G(τ) = 〈ET(t)ET(tτ)〉. Tests were performed to ensure that no residual optical pump light (e.g. from the long-wavelength tail of the fundamental) leaked through the Si wafers.

3. THz emission

A representative correlation trace G(τ) is shown in Fig. 2(a), with a full-width-half-maximum duration of only ∼ 6.8 fs. The deviation from ideal symmetry and modulation depth are due to minor imperfections in the interferometer (due e.g. to the residual curvature of the Si-beam splitter). For the intensity spectrum estimation this results in a complex Fourier transform {G(τ)}, regardless of temporal center. Hence we take S(ν) = | {G(τ)}| as the best available estimate (which should not overestimate the spectral bandwidth). The resulting intensity spectrum is shown in Fig. 2(b) (blue curve, note the linear scale), with a finite signal extending to beyond 100 THz (10%-maximum).

Fig. 2 (a) Detected field correlation signal G(τ) (pulse energy 420 μJ, averaged over 12 scans, time constant 50 ms). (b) Intensity spectrum S(ν) calculated from (a), as well as that for the case where a 1-mm-thick polystyrene (PS) plate was inserted into the beam. (c) Corresponding power transmission spectrum T (ν) (red), compared to a measurement using a commercial cw-FTIR spectrometer (blue). Shaded regions in (b) and (c) indicate 95%-confidence intervals in S(ν) and T(ν) from a statistical analysis of the spectra from multiple scans, i.e. the signal S(ν) is statistically significant compared to the noise level wherever the lower confidence bound does not reach zero.

As noise in G(τ) can readily generate a spurious high-frequency offset in S(ν), we analyzed the spectra from each scan to derive confidence intervals for S(ν). Because S(ν) is taken as the modulus of a complex quantity, one must go beyond standard first-order error propagation methods (as these are not valid for S(ν) ∼ 0). Here we first calculated the complex spectrum for each scan, and analyzed the standard error of the real (R(ν)) and imaginary (I(ν)) parts, in order to derive the confidence ellipse in the complex plane. The confidence intervals in S=R2+I2 are then derived from the minimum and maximum modulus on this ellipse. In this way, a given finite positive value S(ν) is statistically significant (relative to the noise level) if the lower confidence bound is above zero. The 95-%-confidence intervals are also shown in Fig. 2(b) (shaded region) and indicate that the signals are statistically significant up to at least 120 THz. As a more direct test, we also performed measurements with various mid-IR-absorbing samples in the beam. The spectrum with a 1-mm-thick polystyrene plate is also included in Fig. 2(b), with the corresponding transmission spectrum in Fig. 2(c) (red curve). The absorption features extending to ∼100 THz are clearly reproduced when compared to a reference measurement with a commercial cw-FTIR spectrometer (Bruker Vector 22, DTGS detector, blue curve). The confidence intervals for T(ν) were calculated similarly to that for S(ν) by analyzing first the ratio of the complex spectra with and without the sample.

Having established the integrity of the ultra-broadband spectrum, it still remains to demonstrate that the THz radiation is indeed coherent emission from the plasma, and not thermal radiation. In Fig. 3(a), we show the measured THz power following an IR-polarizer (Thorlabs KRS-5) vs. rotation angle, which demonstrates that the radiation is polarized, ruling out thermal emission as the radiation source. The residual offset of the data is due to a frequency dependence in the polarization state of the THz emission, as supported by model results given later Sec. 4. As the THz emission is sensitive to the relative ω-2ω-phase in the plasma [3

3. M. Kreß, T. Löffler, S. Eden, M. D. Thomson, and H. G. Roskos, “Terahertz-pulse generation by photoionization of air with laser pulses composed of both fundamental and second-harmonic waves,” Opt. Lett. 29, 1120–1122 (2004). [CrossRef]

, 8

8. K. Y. Kim, A. J. Taylor, J. H. Glownia, and G. Rodriguez, “Coherent control of terahertz supercontinuum generation in ultrafast laser-gas interactions,” Nature Photonics 2, 605–609 (2008). [CrossRef]

, 9

9. J. M. Dai, N. Karpowicz, and X.-C. Zhang, “Coherent Polarization Control of Terahertz Waves Generated from Two-Color Laser-Induced Gas Plasma,” Phys. Rev. Lett. 103, 023001 (2009). [CrossRef] [PubMed]

], we measured the emission vs. the distance between BBO crystal and plasma (which affects this phase due to the dispersion of the intermediate air path), as shown in Fig. 3(b). An oscillation with the predicted peak spacing [7

7. M. D. Thomson, M. Kreß, T. Löffler, and H. G. Roskos, “Broadband THz emission from gas plasmas induced by femtosecond optical pulses: From fundamentals to applications,” Laser Photon. Rev. 1, 349–368 (2007). [CrossRef]

] of ∼28 mm can indeed be resolved. Interestingly, this modulation is much less pronounced than that observed with longer optical pulses [3

3. M. Kreß, T. Löffler, S. Eden, M. D. Thomson, and H. G. Roskos, “Terahertz-pulse generation by photoionization of air with laser pulses composed of both fundamental and second-harmonic waves,” Opt. Lett. 29, 1120–1122 (2004). [CrossRef]

, 15

15. K. Y. Kim, J. H. Glownia, A. J. Taylor, and G. Rodriguez, “Terahertz emission from ultrafast ionizing air in symmetry-broken laser fields,” Opt. Express 15, 4577–4584 (2007). [CrossRef] [PubMed]

]. This can be reconciled by considering the predicted spectral phases after the BBO crystal as a function of frequency across the ω- and 2ω-spectra, as shown in Fig. 4. Here we plot the relative phase Δφ1, Ω2) = φ(2ω0 + Ω2) – 2φ(ω0 + Ω1), as is appropriate for the lowest-order mixing terms Ei(Ω1)Ej(Ω1)Ek*(Ω2), both for the case of (a,b) a model 150-fs Gaussian pulse and (c,d) the case corresponding to the pulses used in our experiments. For the ultra-broadband spectra here, the ω-2ω-phase varies over more than 2π radians across the optical bandwidth, such that a well-defined value no longer exists, washing out the oscillations in Fig. 3(b). This behavior is reproduced in our numerical simulations (see Appendix for details), as shown by the dashed curve in Fig. 3(b).

Fig. 3 (a) THz power following an infrared polarizer (Thorlabs KR-5) vs. rotation angle. Note that no polarization-sensitive optics were placed before the polarizer. (b) THz power vs. position of BBO crystal (including polarization-resolved curves). Model results from a batch of numerical simulations are included (dashed curve). The more pronounced oscillations in the model results are partly due to neglecting the finite beam convergence and hence phase shift variation across the optical beams.
Fig. 4 Relative ω-2ω-phase shift vs. local frequencies Ωn following a 150-μm-thick BBO crystal, calculated using numerical SHG propagation for both sets of polarisation components ωord-2ωext (left graphs) and ωext-2ωext (right graphs). (a,b) For a model 150-fs (FWHM) Gaussian pulse (α = β = 0, transform-limited). (c,d) For the ultra-broadband pulses and crystal orientation (α = 8.1°, β = 2.3°) used in the experiments (using the measured ω-intensity spectrum as input, with a quadratic chirp of +250 fs2). Note the different frequency ranges between (a,b) and (c,d).

The emitted THz pulse energy directly after the plasma was measured with a calibrated pyroelectric detector (Molectron P4-42), as the THz power exceeded the damage threshold of the Golay cell (a cross-calibration was performed at another position in the experiment to ensure both detectors predict the same THz signal). This yielded a pulse energy of J = 360 nJ (i.e. directly after the plasma), which corresponds to an energy conversion efficiency of 8.6 × 10−4.

4. Role of spectral detuning of the SH field

We now address the spectral detuning of the 2ω-field and its consequences for the THz generation. In order to target significant spectral detunings, we investigated the THz emission over a wide range of tilt angles (α, β) of the BBO crystal (along with the rotation angle ψ, see Fig. 1(c)). This affects the internal angle θ between the ω-beam and the optical axis, and hence the phase-matching wavelength λm. A well-defined geometry was found where the THz emission was maximal (α = 8.1°, β = 2.3°, ψ = 38°), corresponding to θ = 28.0° and hence λm = 834 nm (strongly red-shifted from the center wavelength λ0 = 776 nm). We measured the dependence of both the optical spectra and THz-pulse energy on ψ at this optimum tilt geometry, which, due to the tilt of the crystal, leads to a variation in θ and hence the 2ω-detuning. The results are summarized in Fig. 5. The dependence of the SH pulse energy (Fig. 5(a)) with two asymmetric peaks can be reproduced using our numerical model with the measured input ω-spectrum and accounting for the detailed ray/polarization geometry in the BBO crystal (solid curve). The corresponding THz signal is shown in Fig. 5(b), and features two maxima at ψ = 38° and 128°. The most striking feature is that the maximum THz energy is obtained for an orientation significantly shifted from that for maximum SHG. To investigate this further, we plot the measured ω- and 2ω-spectra for three representative values of ψ vs. their respective local frequencies. As expected, the maximum SH power (ψ = +10°) corresponds to well-centered ω- and 2ω-spectra, i.e. λm = λ0. However, for the case of the peak THz signal (ψ = +38°), the 2ω-spectrum is red-shifted (by ∼ −39 THz), due to the shift of λm towards lower frequencies (the same situation applies to the second THz peak at ψ = 120°).

Fig. 5 Experimental and model (a) 2ω- and (b) THz-pulse energy vs. BBO rotation angle ψ for optimized crystal tilt geometry. Upper panels show the ω- and 2ω-spectra vs. their respective frequency detuning (Ωn = ω0, n = 1, 2, where ω0 is the center frequency of the ω-spectrum) for three values ψ = −10°, +10° (maximum SHG) and +38° (maximum THz). The asymmetric dependence on ψ is due to the oblique incidence of the optical beam on the crystal, such that the angle between the ω-ray and the optical axis (o) varies with ψ. The model curve in (b) corresponds to the sum of the THz energies for two values of the relative phase between ω- and 2ω-fields, Δφ = 0,π/2.

We performed a set of simulations of the THz emission vs. ψ, yielding the model curve for the THz pulse energy (Fig. 5(b), solid curve) which exhibits good agreement with the experimental peaks. To understand this dependence, in Fig. 6 we show further results from the simulations for the three values of ψ: (a) the model temporal intensity components entering the plasma, (b) the time-dependent plasma density, (c) the resulting predicted far-field THz temporal emission, and (d) the corresponding THz intensity spectrum. Two key factors emerge to explain the condition for optimal broadband THz generation at ψ = 38°. Firstly, the red-shifted 2ω-pulse enters the plasma with the smallest group delay with respect to the ω-pulse, such that it co-propagates close in time to the point of maximum ionization rate ρ̇(t) (which is mainly determined by the ω-pulse). As the frequency mixing between ρ(t) and the optical field E(t) is only efficient when ρ̇(t) ≠ 0 (see Appendix), this situation results in the largest polarization derivative (t). For ψ = 10° (centered SHG) and ψ = −10° (blue-shifted SHG), the 2ω-group delay progressively increases, and the 2ω-pulse lags behind this optimal time interval, reducing the mixing efficiency and THz emission. Although the coherent drift component of the polarization P(t) persists on a much longer time-scale [7

7. M. D. Thomson, M. Kreß, T. Löffler, and H. G. Roskos, “Broadband THz emission from gas plasmas induced by femtosecond optical pulses: From fundamentals to applications,” Laser Photon. Rev. 1, 349–368 (2007). [CrossRef]

, 15

15. K. Y. Kim, J. H. Glownia, A. J. Taylor, and G. Rodriguez, “Terahertz emission from ultrafast ionizing air in symmetry-broken laser fields,” Opt. Express 15, 4577–4584 (2007). [CrossRef] [PubMed]

], only the ultrafast transient (t) is relevant for the emitted THz field, which explains why THz pulses considerably shorter than the optical pulse can be achieved in the plasma.

Fig. 6 Simulation results for the three values of BBO rotation angle ψ in Fig. 5. (a) Temporal intensities In(z = 0, t) of the ω- (n = 1) and 2ω-pulses (n = 2) on each polarization axis of the BBO crystal (ordinary, extraordinary). For the relative plasma densities (∼1%) used in the model calculations, only a small spectral blue-shift occurs for each field during propagation through the plasma region [21, 22], and these temporal intensity profiles do not change significantly. (b) Time-dependent relative plasma density ρ(z = 0, t)/ρ0. (c) Predicted low-frequency far-field emission ET(L,t)0Ldz2P(z,t)/t2 on each polarization axis. Note that in all cases the total field area is indeed zero, as expected for a propagating wave. The apparent missing contribution is contained in a very weak decaying tail (on the time scale of the scattering time τs), although the integrated intensity in this tail ( (ET2) is negligible in comparison to the main peaks. (d) Corresponding THz intensity spectra calculated from (c) (left graph for ψ = −10° with 10× magnification).

While this aspect of the emission mechanism already promotes a large THz bandwidth, the spectral range is further enhanced by the incommensurate character of the ω- and 2ω-fields. This leads to a shift of the peak emission frequency away from zero, as can be seen in Fig. 6(d) for both ψ = −10° and ψ = 38°, in contrast to the commensurate case (ψ = 10°). This peak shift widens the spectral range covered by the THz pulse. The model THz spectrum in Fig. 6(d) (right) is in good agreement with the experimental measurement shown in Fig. 2(b). Note that the predicted temporal far-field emission in Fig. 6(c) corresponds to an almost transform-limited pulse.

5. Conclusion

In conclusion, we have demonstrated the emission of ultra-broadband THz pulses from a plasma with incommensurate two-color excitation, with a continuous bandwidth of over 100 THz, and a pulse energy of 360 nJ at a 1-kHz repetition rate. In contrast to tunable phase-matched generation methods in this range, the continuous spectrum can in principle support single-cycle pulses. The use of a plane-wave pulse propagation model can account for the key aspects of the THz emission and the experimental geometry used which produces the optimal bandwidth and pulse energy. We anticipate that this THz source will be well suited to ultra-broadband time-domain and non-linear THz spectroscopy.

Appendix: Theoretical propagation model

The theoretical treatment of the two-color plasma-THz emission used here is based on the photocurrent model [8

8. K. Y. Kim, A. J. Taylor, J. H. Glownia, and G. Rodriguez, “Coherent control of terahertz supercontinuum generation in ultrafast laser-gas interactions,” Nature Photonics 2, 605–609 (2008). [CrossRef]

, 15

15. K. Y. Kim, J. H. Glownia, A. J. Taylor, and G. Rodriguez, “Terahertz emission from ultrafast ionizing air in symmetry-broken laser fields,” Opt. Express 15, 4577–4584 (2007). [CrossRef] [PubMed]

], where one considers the (number density) ionization rate in the plasma, ρ̇(t) = W(t)(ρ0ρ(t)) (where W(t) is the ionization probability rate due to the total optical field, E(t) = A1(t)eiω0t + A2(t)e2iω0t + c.c.) and the subsequent electronic motion in this field. Here we employ static tunneling ionization [7

7. M. D. Thomson, M. Kreß, T. Löffler, and H. G. Roskos, “Broadband THz emission from gas plasmas induced by femtosecond optical pulses: From fundamentals to applications,” Laser Photon. Rev. 1, 349–368 (2007). [CrossRef]

], W(t) = a/Ê (t)exp{−b/Ê (t)}, where Ê = |E|/ Ea is the optical field strength in atomic units, treating the air medium as pure N2 gas at 1 bar. The velocity of an electron, ionized at time t0, at a subsequent time t (prior to scattering) is x˙(t;t0)=e/mt0tdt'E(t'). Assuming first-order decay of the coherent motion due to scattering (time constant τs), the effective displacement is x¯(t;t0)=t0tdtx˙(t;t0)R(tt0), where R(t) = exp(−t/τs)Θ(t) is the survival function. The total polarization at time t is then P(t)=etdt0ρ˙(t0)x¯(t;t0). Substituting (t;t0), re-ordering the integration and integrating analytically the survival function term yields (after re-ordering integration again and taking the derivative),
Pt=e2met/τstdt'Iρ(t')E(t'),
where Iρ(t)=tdt'ρ˙(t')e+t'/τs. Hence the far-field emission source (t) is,
2Pt2=e2met/τs[Iρ(t)E(t)1τstdt'Iρ(t')E(t')].
(1)
This expression is used in the numerical simulations to efficiently calculate the low-frequency (i.e. sub-optical) THz emission. For the present situation where τs (several 100 fs in ambient air [23

23. Z. Mics, P. Kužel, P. Jungwirth, and S. E. Bradforth, “Photoionization of atmospheric gases studied by time-resolved terahertz spectroscopy,” Chem. Phys. Lett. 465, 20–24 (2008). [CrossRef]

]) is much longer than the optical pulse duration, Iρ (t) → ρ (t) and the far-field source term (t) → (e2/m)ρ(t)E(t). Given that ρ̇(t) = f(E2(t)), i(ω) (i = x,y) contains mixing terms such as E1jE1kE2l* (and higher-order) which correspond to optical rectification and occur in the low-frequency (far- and mid-IR) range. Note that in a time-interval where ρ(t) would be constant, only a very weak emission contribution can result (as the cycle averaged field is essentially zero for a multi-cycle optical pulse).

The optical propagation in the plasma is based on a plane-wave treatment of the two-color pulse field envelopes An(z, T) in a co-moving time frame T = tz/c by integrating the equations [24

24. E. Priori, G. Cerullo, M. Nisoli, S. Stagira, S. D. Silvestri, P. Villoresi, L. Poletto, P. Ceccherini, C. Altucci, R. Bruzzese, and C. de Lisio, “Nonadiabatic three-dimensional model of high-order harmonic generation in the few-optical-cycle regime,” Phys. Rev. A 61, 063801 (2000). [CrossRef]

]:
An(z,Ω)z=i2c(nω0+Ω){ωp2(z,T)An(z,T)},
where ωp=ρe2/mɛ0 is the instantaneous plasma frequency, and denotes a Fourier transformation. In order to calculate the plasma polarization (Eq. 1) for each z-step, one requires the real optical field on a sub-cycle time scale. In the simulations, this is achieved by interpolating the slowly-varying complex field envelopes onto a fine time-base (here with 32 points per optical cycle), calculating ρ̇(T) and Iρ (T) and then low-passing the results. The model plasma was taken with a nominal length of L = 1 mm and step size δz = 5 μm. The output THz temporal field is then taken as ET(z=L,T)0LdzP¨(z,T) (i.e. neglecting any interaction between the plasma and THz near-field). The peak optical intensity was chosen such that the final relative plasma density ρ(+∞)/ρ0 ∼ 0.01 (where ρ0 = 2.7 · 1019 cm−3 is the molecular density). While the use of a plane-wave model neglects the detailed spatio-temporal nature of the propagation, its use is partially justified by the fact that the axial peak intensities should be effectively clamped by plasma defocussing [7

7. M. D. Thomson, M. Kreß, T. Löffler, and H. G. Roskos, “Broadband THz emission from gas plasmas induced by femtosecond optical pulses: From fundamentals to applications,” Laser Photon. Rev. 1, 349–368 (2007). [CrossRef]

]. This simplified model reproduces the essential features of the experimental THz emission vs. various parameters.

The optical input fields An(0, T) were determined by using the experimental intensity spectrum to predict the ω-field (with a quadratic spectral chirp of 250 fs2, determined from inspection of the experimental 2ω-spectra), and applying numerical propagation in the BBO crystal in the birefringent eigenaxes (ord, ext) reference frame. The ωord- and 2ωext-fields were propagated using split-step integration of the non-linear coupled differential equations for Type-I SHG [25

25. J. Y. Zhang, J. Y. Huang, H. Wang, K. S. Wong, and G. K. Wong, “Second-harmonic generation from regeneratively amplified femtosecond laser pulses in BBO and LBO crystals,” J. Opt. Soc. Am. B 15, 200–209 (1998). [CrossRef]

] with the full broadband dispersion k(nω0 + Ωn) for each field (neglecting loss and third-order effects), with linear propagation for the ωext-field. For a given set of crystal orientation angles, the refraction, Fresnel transmission, field projections and birefringence and effective non-linear coefficients were calculated.

Acknowledgments

We would like to thank R. Müller-Werkmeister for assisting with the cw-FTIR measurements, and the Deutsche Forschungsgemeinschaft for financial support.

References and links

1.

D. J. Cook and R. M. Hochstrasser, “Intense terahertz pulses by four-wave rectification in air,” Opt. Lett. 25, 1210–1212 (2000). [CrossRef]

2.

T. Löffler and H. G. Roskos, “Gas-pressure dependence of terahertz-pulse generation in a laser-generated nitrogen plasma,” J. Appl. Phys. 91, 2611–2614 (2002). [CrossRef]

3.

M. Kreß, T. Löffler, S. Eden, M. D. Thomson, and H. G. Roskos, “Terahertz-pulse generation by photoionization of air with laser pulses composed of both fundamental and second-harmonic waves,” Opt. Lett. 29, 1120–1122 (2004). [CrossRef]

4.

T. Bartel, P. Gaal, K. Reimann, M. Woerner, and T. Elsaesser, “Generation of single-cycle THz transients with high electric-field amplitudes,” Opt. Lett. 30, 2805–2807 (2005). [CrossRef] [PubMed]

5.

M. Kreß, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Nat. Phys. 2, 327–331 (2006). [CrossRef]

6.

Y. Q. Chen, M. Yamaguchi, M. F. Wang, and X.-C. Zhang, “Terahertz pulse generation from noble gases,” Appl. Phys. Lett. 91, 251116 (2007). [CrossRef]

7.

M. D. Thomson, M. Kreß, T. Löffler, and H. G. Roskos, “Broadband THz emission from gas plasmas induced by femtosecond optical pulses: From fundamentals to applications,” Laser Photon. Rev. 1, 349–368 (2007). [CrossRef]

8.

K. Y. Kim, A. J. Taylor, J. H. Glownia, and G. Rodriguez, “Coherent control of terahertz supercontinuum generation in ultrafast laser-gas interactions,” Nature Photonics 2, 605–609 (2008). [CrossRef]

9.

J. M. Dai, N. Karpowicz, and X.-C. Zhang, “Coherent Polarization Control of Terahertz Waves Generated from Two-Color Laser-Induced Gas Plasma,” Phys. Rev. Lett. 103, 023001 (2009). [CrossRef] [PubMed]

10.

A. Houard, Y. Liu, B. Prade, and A. Mysyrowicz, “Polarization analysis of terahertz radiation generated by four-wave mixing in air,” Opt. Lett. 33, 1195–1197 (2008). [CrossRef] [PubMed]

11.

J. Hebling, K.-L. Yeh, M. C. Hoffmann, and K. A. Nelson, “High-power THz generation, THz nonlinear optics, and THz nonlinear spectroscopy,” IEEE J. Sel. Top. Quant. Electron. 14, 345–353 (2008). [CrossRef]

12.

C. Kübler, H. Ehrke, R. Huber, R. Lopez, A. Halabica, J. R. F. Haglund, and A. Leitenstorfer, “Coherent structural dynamics and electronic correlations during an ultrafast insulator-to-metal phase transition in VO2,” Phys. Rev. Lett. 99, 116401 (2007). [CrossRef] [PubMed]

13.

C. Kübler, R. Huber, and A. Leitenstorfer, “Ultrabroadband terahertz pulses: generation and field-resolved detection,” Semicond. Sci. Technol. 20, S128–S133 (2005). [CrossRef]

14.

F. Junginger, A. Sell, O. Schubert, B. Mayer, D. Brida, M. Marangoni, G. Cerullo, A. Leitenstorfer, and R. Huber, “Single-cycle multiterahertz transients with peak fields above 10 MV/cm,” Opt. Lett. 35, 2645–2647 (2010). [CrossRef] [PubMed]

15.

K. Y. Kim, J. H. Glownia, A. J. Taylor, and G. Rodriguez, “Terahertz emission from ultrafast ionizing air in symmetry-broken laser fields,” Opt. Express 15, 4577–4584 (2007). [CrossRef] [PubMed]

16.

H.-C. Wu, J. M. ter Vehn, and Z.-M. Sheng, “Phase-sensitive terahertz emission from gas targets irradiated by few-cycle laser pulses,” New J. Phys. 10, 043001 (2008). [CrossRef]

17.

N. Karpowicz and X.-C. Zhang, “Coherent terahertz echo of tunnel ionization in gases,” Phys. Rev. Lett. 102, 093001 (2009). [CrossRef] [PubMed]

18.

A. A. Silaev and N. V. Vvedenskii, “Quantum-mechanical approach for calculating the residual quasi-dc current in a plasma produced by a few-cycle laser pulse,” Phys. Scr. T135, 014024 (2009). [CrossRef]

19.

V. Blank, M. D. Thomson, and H. G. Roskos, “Terahertz radiation with a continuous spectral bandwidth reaching beyond 100 THz from a laser-induced gas plasma,” in Ultrafast Phenomena XVII, M. Chergui, D. Jonas, E. Riedle, B. Schoenlein, and A. Taylor, eds. (Oxford University Press, New York, 2010). To be published.

20.

T. Balčiūnas, D. Lorenc, M. Ivanov, O. Smirnova, A. Pugžlys, A. M. Zheltikov, D. Dietze, J. Darmo, K. Unterrainer, T. Rathje, G. Paulus, and A. Baltǔska, “Tunability of THz emission originating from sub-cycle electron bursts in a laser induced plasma,” in Ultrafast Phenomena XVIIM. Chergui, D. Jonas, E. Riedle, B. Schoenlein, and A. Taylor, eds. (Oxford University Press, New York, 2010). To be published.

21.

W. M. Wood, C. W. Siders, and M. C. Downer, “Measurement of femtosecond ionization dynamics of atmospheric density gases by spectral blueshifting,” Phys. Rev. Lett. 67, 3523–3526 (1991). [CrossRef] [PubMed]

22.

S. C. Rae and K. Burnett, “Detailed simulations of plasma-induced spectral blueshifting,” Phys. Rev. A 46, 1084–1090 (1992). [CrossRef] [PubMed]

23.

Z. Mics, P. Kužel, P. Jungwirth, and S. E. Bradforth, “Photoionization of atmospheric gases studied by time-resolved terahertz spectroscopy,” Chem. Phys. Lett. 465, 20–24 (2008). [CrossRef]

24.

E. Priori, G. Cerullo, M. Nisoli, S. Stagira, S. D. Silvestri, P. Villoresi, L. Poletto, P. Ceccherini, C. Altucci, R. Bruzzese, and C. de Lisio, “Nonadiabatic three-dimensional model of high-order harmonic generation in the few-optical-cycle regime,” Phys. Rev. A 61, 063801 (2000). [CrossRef]

25.

J. Y. Zhang, J. Y. Huang, H. Wang, K. S. Wong, and G. K. Wong, “Second-harmonic generation from regeneratively amplified femtosecond laser pulses in BBO and LBO crystals,” J. Opt. Soc. Am. B 15, 200–209 (1998). [CrossRef]

OCIS Codes
(190.7110) Nonlinear optics : Ultrafast nonlinear optics
(350.5400) Other areas of optics : Plasmas
(300.6495) Spectroscopy : Spectroscopy, teraherz

ToC Category:
Nonlinear Optics

History
Original Manuscript: August 9, 2010
Revised Manuscript: September 16, 2010
Manuscript Accepted: September 22, 2010
Published: October 19, 2010

Citation
Mark D. Thomson, Volker Blank, and Hartmut G. Roskos, "Terahertz white-light pulses from an air plasma photo-induced by incommensurate two-color optical fields," Opt. Express 18, 23173-23182 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-23173


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References

  1. . D. J. Cook and R. M. Hochstrasser, “Intense terahertz pulses by four-wave rectification in air,” Opt. Lett. 25, 1210–1212 (2000). [CrossRef]
  2. . T. Löffler and H. G. Roskos, “Gas-pressure dependence of terahertz-pulse generation in a laser-generated nitrogen plasma,” J. Appl. Phys. 91, 2611–2614 (2002). [CrossRef]
  3. M. Kreß, T. Löffler, S. Eden, M. D. Thomson, and H. G. Roskos, “Terahertz-pulse generation by photoionization of air with laser pulses composed of both fundamental and second-harmonic waves, ” Opt. Lett. 29, 1120–1122 (2004). [CrossRef]
  4. T. Bartel, P. Gaal, K. Reimann, M. Woerner, and T. Elsaesser, “Generation of single-cycle THz transients with high electric-field amplitudes,” Opt. Lett. 30, 2805–2807 (2005). [CrossRef] [PubMed]
  5. M. Kreß, T. Löffler, M. D. Thomson, R. Dörner, H. Gimpel, K. Zrost, T. Ergler, R. Moshammer, U. Morgner, J. Ullrich, and H. G. Roskos, “Determination of the carrier-envelope phase of few-cycle laser pulses with terahertz-emission spectroscopy,” Nat. Phys. 2, 327–331 (2006). [CrossRef]
  6. Y. Q. Chen, M. Yamaguchi, M. F. Wang, and X.-C. Zhang, “Terahertz pulse generation from noble gases,” Appl. Phys. Lett. 91, 251116 (2007). [CrossRef]
  7. Q2. M. D. Thomson, M. Kreß, T. Löffler, and H. G. Roskos, “Broadband THz emission from gas plasmas induced by femtosecond optical pulses: From fundamentals to applications,” Laser Photon. Rev. 1, 349–368 (2007). [CrossRef]
  8. K. Y. Kim, A. J. Taylor, J. H. Glownia, and G. Rodriguez, “Coherent control of terahertz supercontinuum generation in ultrafast laser-gas interactions,” Nature Photonics 2, 605–609 (2008). [CrossRef]
  9. . J. M. Dai, N. Karpowicz, and X.-C. Zhang, “Coherent Polarization Control of Terahertz Waves Generated from Two-Color Laser-Induced Gas Plasma, ” Phys. Rev. Lett. 103, 023001 (2009). [CrossRef] [PubMed]
  10. A. Houard, Y. Liu, B. Prade, and A. Mysyrowicz, “Polarization analysis of terahertz radiation generated by four-wave mixing in air,” Opt. Lett. 33, 1195–1197 (2008). [CrossRef] [PubMed]
  11. Q4. J. Hebling, K.-L. Yeh, M. C. Hoffmann, and K. A. Nelson, “High-power THz generation, THz nonlinear optics, and THz nonlinear spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 14, 345–353 (2008). [CrossRef]
  12. C. Kübler, H. Ehrke, R. Huber, R. Lopez, A. Halabica, J. R. F. Haglund, and A. Leitenstorfer, “Coherent structural dynamics and electronic correlations during an ultrafast insulator-to-metal phase transition in VO2,” Phys. Rev. Lett. 99, 116401 (2007). [CrossRef] [PubMed]
  13. C. Kübler, R. Huber, and A. Leitenstorfer, “Ultrabroadband terahertz pulses: generation and field-resolved detection,” Semicond. Sci. Technol. 20, S128–S133 (2005). [CrossRef]
  14. F. Junginger, A. Sell, O. Schubert, B. Mayer, D. Brida, M. Marangoni, G. Cerullo, A. Leitenstorfer, and R. Huber, “Single-cycle multiterahertz transients with peak fields above 10 MV/cm,” Opt. Lett. 35, 2645–2647 (2010). [CrossRef] [PubMed]
  15. K. Y. Kim, J. H. Glownia, A. J. Taylor, and G. Rodriguez, “Terahertz emission from ultrafast ionizing air in symmetry-broken laser fields,” Opt. Express 15, 4577–4584 (2007). [CrossRef] [PubMed]
  16. H.-C. Wu, J. M. ter Vehn, and Z.-M. Sheng, “Phase-sensitive terahertz emission from gas targets irradiated by few-cycle laser pulses,” New J. Phys. 10, 043001 (2008). [CrossRef]
  17. N. Karpowicz and X.-C. Zhang, “Coherent terahertz echo of tunnel ionization in gases,” Phys. Rev. Lett. 102, 093001 (2009). [CrossRef] [PubMed]
  18. A. A. Silaev and N. V. Vvedenskii, “Quantum-mechanical approach for calculating the residual quasi-dc current in a plasma produced by a few-cycle laser pulse,” Phys. Scr. T 135, 014024 (2009). [CrossRef]
  19. V. Blank, M. D. Thomson, and H. G. Roskos, “Terahertz radiation with a continuous spectral bandwidth reaching beyond 100 THz from a laser-induced gas plasma,” in “Ultrafast Phenomena XVII,”, M. Chergui, D. Jonas, E. Riedle, B. Schoenlein, and A. Taylor, eds., (Oxford University Press, New York, 2010). To be published.
  20. T. Balčinas, D. Lorenc, M. Ivanov, O. Smirnova, A. Puğzlys, A. M. Zheltikov, D. Dietze, J. Darmo, K. Unterrainer, T. Rathje, G. Paulus, and A. Baltˇuska, “Tunability of THz emission originating from sub-cycle electron bursts in a laser induced plasma,” in “Ultrafast Phenomena XVII,”, M. Chergui, D. Jonas, E. Riedle, B. Schoenlein, and A. Taylor, eds., (Oxford University Press, New York, 2010). To be published.
  21. W. M. Wood, C. W. Siders, and M. C. Downer, “Measurement of femtosecond ionization dynamics of atmospheric density gases by spectral blueshifting,” Phys. Rev. Lett. 67, 3523–3526 (1991). [CrossRef] [PubMed]
  22. S. C. Rae and K. Burnett, “Detailed simulations of plasma-induced spectral blueshifting,” Phys. Rev. A 46, 1084–1090 (1992). [CrossRef] [PubMed]
  23. Z. Mics, P. Kuzel, P. Jungwirth, and S. E. Bradforth, “Photoionization of atmospheric gases studied by timeresolved terahertz spectroscopy,” Chem. Phys. Lett. 465, 20–24 (2008). [CrossRef]
  24. E. Priori, G. Cerullo, M. Nisoli, S. Stagira, S. D. Silvestri, P. Villoresi, L. Poletto, P. Ceccherini, C. Altucci, R. Bruzzese, and C. de Lisio, “Nonadiabatic three-dimensional model of high-order harmonic generation in the few-optical-cycle regime,” Phys. Rev. A 61, 063801 (2000). [CrossRef]
  25. J. Y. Zhang, J. Y. Huang, H. Wang, K. S. Wong, and G. K. Wong, “Second-harmonic generation from regeneratively amplified femtosecond laser pulses in BBO and LBO crystals,” J. Opt. Soc. Am. B 15, 200–209 (1998). [CrossRef]

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