1. INTRODUCTION

Femtosecond pulse shaping techniques have been used for ultrafast optical control over electronic and vibrational responses of atoms, molecules, and crystals [1, 2, 3, 4, 5, 6, 7]. Complex temporal patterns of an electric field have been generated to control phenomena such as two-photon absorption [3], high harmonic generation [4], control of molecular processes [2], and terahertz (THz) wave generation [1, 5, 6]. The use of tailored THz waveforms will extend the flexibility of THz technology in applications including telecommunications, signal processing, and spectroscopy. Feurer et al. showed programmable automated THz polariton control in lithium niobate crystals by controlling spatial and temporal profiles of optical pulses [7]. In addition, tunable THz waveforms have been generated by using various pulse shaping techniques in lithium niobate [8], photoconductive antenna devices [9], and nonlinear ZnTe crystals [1, 5]. However, the controllability of THz waveforms varies in different types of emitters as a result of different THz generation mechanisms, and different kinds of THz emitters deserve separate studies.

THz wave generation in air has attracted the recent attention of researchers due to simultaneous broad bandwidths and strong electric fields [10, 11, 12, 13, 14, 15, 16, 17], which are highly desirable for chemical/biological sensing. A THz field greater than $400\mathrm{kV}∕\mathrm{cm}$ generated using short $25\mathrm{fs}$ pump pulses has been reported by Bartel et al. [18] Karpowicz et al. reported the generation and detection of broadband THz pulses in air covering the spectral range of $0.3\text{to}10\mathrm{THz}$ with 10% or larger of the maximum electric field [17]. THz wave generation mechanisms in air were originally attributed to the four-wave rectification process through the third-order optical nonlinearity of air [11]. However, it has been suggested that plasma formation plays an important role in the THz wave generation process [12]. The four-wave rectification mechanism was thoroughly tested by Xie et al. by controlling the relative phase, amplitude, and polarization of the fundamental and second-harmonic pulses, and very good agreement was shown between the four-wave mixing (FWM) model and the experiments [14]. Chen et al. showed a strong correlation between the ionization potential energy and the THz wave generation efficiency in a series of noble gases [19]. Kim et al. proposed a transient photocurrent model based on rapid ionization by femtosecond laser pulses and following driving motions of electrons by an asymmetric electric field [20].

In this paper, we use a femtosecond pulse shaping technique to generate a series of shaped optical excitation pulses with different temporal profiles to study the effects on the generation of THz waveforms. Femtosecond pulse shaping is achieved by using a phase-only pulse shaper. Phase shaping is particularly useful over amplitude shaping for THz wave generation where the power of the source remains an issue for many of the applications using THz radiation. In the case of THz generation in gas, both ionization and the following driving processes of electron motion should be affected by the change of the temporal profile of excitation pulses. In our experiment, we used a genetic optimization procedure [4, 21, 22] to increase THz power by using optical pulse shaping techniques. Direct optimization of THz power by using the THz wave power as a fitness function and indirect optimization by using the intensity of the second harmonics of the excitation pulse were compared. The control of the THz waveform shape was examined by using a series of periodically alternated phases in Fourier space.

2. EXPERIMENT

The experimental configuration is shown in Fig. 1 . The pulse shaping apparatus consisted of a pair of $1800\text{\hspace{0.17em} line}∕\mathrm{mm}$ gratings placed at the focal planes of a pair of cylindrical lenses with a $70\mathrm{mm}$ focal length. The optical pulse design was accomplished by a programmable one-dimensional liquid crystal spatial light modulator (SLM) array (SLM-128, Cambridge Research and Instrumentation, Inc.) placed at the Fourier plane of the shaper and used as a programmable filter for spectral manipulation of the incoming pulses. In this Fourier plane, the frequency components of the optical pulse were spatially displaced from each other, and the frequency-dependent phase delay could be given by controlling the refractive index of the liquid crystal at the position where a particular frequency component of the optical pulse was passing through. The width of each pixel was $100\mu \mathrm{m}$, and the interpixel gap was $2\mu \mathrm{m}$. A Ti:sapphire laser provided the incident $100\mathrm{fs}$, $800\mathrm{nm}$ pulses at a $76\mathrm{MHz}$ repetition rate, which were directed into the SLM setup. The shaped pulses were used as seed pulses for a regenerative femtosecond laser amplifier. The pulse energy from the amplifier was $1\mathrm{mJ}$/pulse and the repetition rate was $1\mathrm{kHz}$. Single-shot autocorrelation was used to characterize the output pulses.

The entire setup was purged with dry nitrogen gas, and optically shaped laser pulses were used for THz wave generation in nitrogen. The fundamental $\left(\omega \right)$ beam and its second harmonic $\left(2\omega \right)$ generated via passing through a $100\mu \mathrm{m}$ thick type I beta barium borate (BBO) crystal were focused in nitrogen gas to produce plasma using an achromatic lens with a $100\mathrm{mm}$ focal length. The generated THz wave was measured using a combination of a THz Michelson interferometer and pyroelectric detector (Spectrum Detector, Inc.). The THz Michelson interferometer consisted of an Si beam splitter and two gold mirrors, one of which was mounted on a high-precision delay stage to control the relative beam path lengths for the split THz waves. The pump beam was mechanically chopped at the frequency of $73\mathrm{Hz}$ and used for the reference signal in the lock-in detection for the pyroelectric detector.

3. OPTIMIZATION OF OPTICAL PULSE FOR TERAHERTZ WAVE GENERATION

Figure 2a shows the broadband THz pulse excited in nitrogen at ambient pressure with optimized optical excitation pulses, obtained by phase-only pulse shaping with an SLM. In this experiment, collinearly propagating fundamental and second-harmonic pulses are focused in gas to generate ionized gas plasma near the focal point, and a THz wave is generated from the plasma. The optical fundamental and second-harmonic electric field dependence of the THz wave amplitude generated from the gas plasma has been described by the FWM model [11, 14], and the generated THz field in the presence of additional phase is given as

The optimal pulse shape for the generation of the THz wave from nitrogen was evaluated by using an evolutionary search procedure [21]. The temporal profile of optical excitation pulses affects both the ionization process and the following driving process of electron motion by the electric field. In this experiment, we optimize the emission of a THz wave from gas plasma by shaping the optical excitation pulses with a phase-only pulse shaper. For the optimization algorithm, we use two kinds of feedback parameters (fitness functions) for the genetic optimization procedure: THz power and optical second-harmonic intensity. In the first procedure, THz power measured by the pyroelectric detector is used as the fitness function, and the voltages applied to each of the 128 channels in the SLM, which determines the phase delay of each frequency component of optical excitation pulse, are the optimization parameters. This is a direct optimization procedure of THz power. In the second procedure, an optical excitation pulse was optimized so that the generation of second-harmonic intensity was maximized. In this procedure, THz power was measured after the optimization process. We started with a set of randomly selected initial frequency-dependent phases of the excitation pulses. The algorithm searches for a phase, which produces a pulse shape to maximize the signal. Signal intensities corresponding to 100 phase combinations (population) for the optical excitation pulses were measured at the beginning of the process. A pair of phase combinations is selected with a probability weighted by the signal intensity produced by the phase combination, then 3% of the phase combinations are randomly chosen and altered (mutation), partially exchanged between the pair, and used as a pair of new phase combinations in the next generation. The same procedure is repeated for another 49 pairs. The newly generated 100 phase combinations are used for the second generation, and the procedure repeats itself until signal intensity is optimized. Details of the genetic optimization procedure are found elsewhere in literature [21].

Figure 3a shows direct optimization of THz power as a function of the iteration of the genetic optimization algorithm. As a random phase is used for the initial conditions, the THz power drops from the power generated by pulses with a constant phase at the beginning of the optimization procedure. The THz power increases rapidly at the beginning and stabilizes after 5–10 iterations when THz power is used as a fitness function. Solid and open marks in Fig. 3a correspond to the different runs of the optimization procedure. Similar optimization progresses were observed. The observed enhancement of the generated THz power was about 20% compared to the THz power with constant phases. Figure 3b shows the second-harmonic intensity of the optical excitation pulses. In this experiment, optical excitation pulses were optimized to produce higher intensity of the second harmonics by using the genetic optimization procedure. The progress of the optimization showed a similar trend as the THz power optimization process, but it approached the saturation more slowly and the saturation value was reached at around 20–30 iterations. The THz power after this second-harmonic optimization process was also increased and reached about 25% larger than the value obtained with the constant phase. For ideal SHG optimization, the optimal pulse conversion should occur for the transform-limited pulses. When a constant phase was added to the excitation pulse by the SLM, the pulse durations just after the SLM and at the position of the BBO crystal were about $90\mathrm{fs}$ and $105\mathrm{fs}$, respectively. This broadening is due to the group velocity dispersion in the laser amplifier and optical components. For the SH optimization process, pulse duration converges to $90\mathrm{fs}\pm 2\mathrm{fs}$ at the position of the BBO, which matches to the transform limit pulse width.

For THz power optimization, phase effects on the ionization rate, plasma dynamics, and phase matching inside the plasma, in addition to the intensity of the SH pulse, may contribute to the optimization process. Figure 3c shows the optical autocorrelation signal of the optical excitation pulses before and after the optimization of the generated THz wave power when the THz power was used as a fitness function. The excitation pulse before the optimization has constant phase at the SLM position, and the amplifier and optical components in the optical path will add group-velocity dispersion and broaden the pulse width. Relative group-delay dispersion (GDD) at $795\mathrm{nm}$ in a BBO crystal ($100\mu \mathrm{m}$, $7.54{\mathrm{fs}}^2$) and KDP ($100\mu \mathrm{m}$, $6.71{\mathrm{fs}}^2$) used in the autocorrelator would cause the difference of pulse width less than $1\mathrm{fs}$, and the effect is negligible. The ratio of the peak intensity of the optical autocorrelation before and after optimization is increased about 18%. As the THz power is proportional to the second-harmonic intensity, this value is within the error of the prediction of FWM. The convergence of the search procedure tends to converge faster when THz power is used for the optimization. The final THz power is about 5% lower when THz power is used as a fitness function and small shoulder peaks in autocorrelation signal are often seen, while we did not observed such peaks when SHG intensity was used as a fitness function. These shoulder peaks are below the ionization threshold and would not contribute to the THz generation. They are “dark” peaks to the optimization procedure and may contribute to the difference of the optimization by the second harmonics and THz power. We have observed that when the mutation probability is lower than 1%, the optimized value is slightly smaller and also a longer iteration (40 iterations or more) was necessary for the optimization. In an ideal case, the global optimum of THz power when the THz power is directly optimized should be equal to or higher than the one obtained by the SH optimization process, since THz optimization process can optimize the SH intensity, ionization, and other parameters. One possible reason for the lower optimized THz power for the direct THz optimization process is the higher level of noise of the pyroelectric detector (NEP of ${10}^{-9}\mathrm{W}∕\sqrt{\mathrm{Hz}}$) compared to that of the Si-PIN photodiode (NEP of ${10}^{-14}\mathrm{W}∕\sqrt{\mathrm{Hz}}$) used for the SH optimization. It has been reported that the achievable optimum can be lowered, depending on the noise level in the field noise [24, 25]. Separate controls of the SHG efficiency and plasma emission dynamics processes may produce higher THz power since the SHG process requires a transform-limited pulse, which may not be the optimal pulse for the ionization and THz emission process from gas plasma.

4. CONTROL OF THE NUMBER OF PULSES IN TIME DOMAIN BY USING PERIODIC PHASE

The relationship between the phase and temporal profiles of the optical excitation pulse and the generated THz waveform has been investigated further by using relatively simple frequency-dependent phase patterns of a periodically alternated phase in Fourier space (hereafter the rectangular phase). The oscillation periods of the rectangular phase were sequentially changed to control the temporal profile of the THz waveforms. Figure 4a shows the rectangular phase pattern used in this experiment, and these phase modulations were added to the phases of optical excitation pulses by using the pulse shaper. Autocorrelation signals of the optical excitation pulse with these phase modulations are given in Fig. 4b. The autocorrelation signals show multiple peak features except for the signal from the compressed pulse [(A) in Fig. 4b]. The effect of the rectangular phase can be understood in the following way. The Fourier components of the optical excitation pulses just above and below the rising and trailing edges of the rectangular phase should interfere destructively. The modulation depth of the rectangular phase determines the degree of destructive interference in the spectrum. On the other hand, the period of the rectangular phase determines the period of destructive interference in the Fourier spectrum and hence determines the separation of pulses in the time domain. By changing the period of the rectangular phase, the side peak positions in autocorrelation signals are sequentially shifted to larger values. The peak positions for $\mathrm{t}>0$ in Fig. 4b are 0.59, 0.73, 0.90, and $1.22\mathrm{ps}$ for B, C, D, and E, respectively. The THz waves generated by using these excitation pulses are shown in Fig. 4c. The waveform in Fig. 4c has side peaks in addition to the main peak around $\mathrm{t}=0$ as indicated by arrows, and the peak positions determined as the zero crossing point of the signals are 0.55, 0.71, 0.88, and $1.18\mathrm{ps}$, respectively. These positions match the side peak positions of the excitation pulses, and each peak in the autocorrelation signal generates THz waveform under these conditions. Both Fig. 4b and Fig. 4c show three peaks. However, the optical and THz waveforms are measured as autocorrelation signals, and both optical and THz electric fields should have two-peak features rather than three-peak features. The separation of these pulse trains is well controlled in this range of the rectangular phase period (Fig. 4c). The largest separation of THz pulse train $\left(1.18\mathrm{ps}\right)$ is generated with the shortest period of oscillation (E) with alternating the voltage of every second pixel in the SLM. To generate THz wave trains with larger separation, a higher resolution of phase modulation is necessary. On the other hand, when the modulation period becomes too large, the THz subpeaks generated by the side peaks of the excitation pulses begin to overlap, and the separation control can no longer be achieved. When the period of the rectangular phase is in the proper range, we have demonstrated that the separation of the THz pulses generated in gas can be controlled by changing the period of rectangular phases. Figure 4d shows the Fourier spectra of respective waveforms in Fig. 4c. These spectra show periodic modulations, and the modulation periods correspond to the inverses of the subpulse separations in the time domain signal. The peak frequency is about $4.5\mathrm{THz}$ for the transform-limited pulse, and the other spectra exhibit a peak around $3\mathrm{THz}$. The low intensity of the spectrum below $2\mathrm{THz}$ is due to the lower sensitivity of the pyrodetector in this frequency range.

5. CONCLUSION

We have studied THz wave generation in nitrogen gas using optically shaped pulses. Different phase configurations were introduced using an SLM, and their effects on generating THz wave were examined. Generated THz waveforms showed a very strong dependence on the phase of the optical excitation pulses, even though the bandwidth of the optical excitation pulses was not particularly wide $(\mathrm{FWHM}=12\mathrm{nm})$. Optimization of THz wave generation was explored by shaping the optical excitation pulses with a genetic optimization algorithm [21]. Two kinds of fitness functions were used for the algorithm: direct optimization of THz power and indirect optimization of second-harmonic intensity. Both optimization processes produced power enhancement of the generated THz waves: 20% for THz power and 25% for second-harmonic intensity. Control of a THz waveform using a series of periodic rectangular phases for optical excitation pulses has been shown, and a series of THz pulses with different separations has been generated. A THz waveform generated in gas is sensitive to the optical phase of the excitation pulses, and phase-only pulse shaping of the pump pulses may serve as a useful tool for further investigation of the THz wave generation mechanism in gaseous media.

ACKNOWLEDGMENTS

This research was supported in part by the Office of Naval Research (ONR).

 

Fig. 1 Experimental setup for THz wave generation from nitrogen gas with pulse shaper. The pulse shaper is placed between the Ti-sapphire oscillator and the regenerative laser amplifier. Abbreviations are: GM, gold mirror; CL, cylindrical lens; G, grating; SLM, spatial light modulator; M, mirror; BS, THz beam splitter; F, Si filter; BBO, beta barium borate crystal; PC, personal computer; PM, parabolic mirror. The SLM is controlled by a PC.

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Fig. 2 (a) Optimized THz waveform measured by Michelson interferometer and (b) its Fourier power spectrum.

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Fig. 3 Optimization of THz power using the genetic optimization algorithm. (a) Direct optimization by using THz power measured by a pyroelectric detector as a fitness function for the optimization procedure. Solid and open dots are different runs of the optimization process. (b) Indirect optimization using second harmonics intensity as a fitness function. Solid and open dots are different runs of the optimization process. (c) Autocorrelation signal of the optical excitation pulse before and after optimization using THz power as a fitness function.

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Fig. 4 THz waveform control with a periodically alternated phase in Fourier space. (a) Phase configuration given to the SLM. The modulation depths of the rectangular phases are $1.34\pi$, and the periods the rectangular phases are infinite (constant phase), $1.2\mathrm{nm}$, $1.0\mathrm{nm}$, and $0.7\mathrm{nm}$ for A, B, C, and D, respectively. (b) Experimentally measured autocorrelation signal of optical excitation pulses with the phases given in (a). (c) Experimentally measured THz waveform generated by the excitation pulses given in (b). (d) Fourier spectrum of THz waveforms in (c). Traces marked A, B, C, and D in (b), (c) and (d) have optical phases given in (a) with the same character.

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