Quasicrystals are structural forms that are both ordered and nonperiodic (with perfect long-range order, but with no three-dimensional translational periodicity). In the past decades, due to their fasnating characteristics, the artificial quasicrystals (or quasiperiodic structure materials) have attracted a lot of interests. In particular, much effort has been devoted to the photonic applications of quasiperiodic structures. For example, a strong suppression of the optical transmission in quasiperiodic dielectric multilayer stacks of SiO2 and TiO2 thin films has been observed [1]. Study of wave packet dynamics in quasi-one-dimensional metal–halogen complex has been reported [2] and quasiperiodic envelope solitons has been introduced [3]. Photonic band gap formation and optimization, as well as multiple scattering in photonic quasicrystals have been studied intensively [4–6]. For quasi-phase-matching grating, a nonlinear quasiperiodic optical superlattice, multicolor second harmonic generation has been observed and measured [7]. The quasiperiodic optical superlattice can also be used for efficient direct third harmonic generation [8]. In addition, it has been shown theoretically that two-dimensional photonic quasicrystals are highly advantageous to nonlinear optical frequency conversion [9].

Recently, spurred by the staggering interest in terahertz (THz) science and technology, studies on interactions between electromagnetic wave and photonic quasicrystals are being extended from optical range to THz range, becoming one of the hot topics in fundamental and applied sciences. The sustained growth of THz science and technology requires considerable progress in the development of effective THz sources as many molecular excitations in condensed matter systems fall into the THz frequency range, giving rise to a great demand for narrow-band THz sources.

Depending on the specific THz range in the electromagnetic spectrum, it can be generated from both the optical and the microwave sides. From the optical side, there are two categories of photonic techniques developed for the generation of THz radiations. One approach consists of free electron lasers [10], photoconductive switching [11] and dipolar antennas [12], in which the THz emission is derived from a change in polarization that follows the transport of excited carriers in an applied, external or surface electric field. Another approach, which is most interesting currently, uses nonlinear dielectric crystals. It was reported in 1970s that pulses in the far infrared could be produced by the rectification of picosecond pulses in LiNbO₃ [13]. Recently, it was demonstrated that optical rectification or optical difference frequency generation (DFG) in periodically poled LiNbO₃ (PPLN) could be a very efficient method for THz generation [14]. However, it is known that THz generation in periodically poled materials consists of only single frequency mode radiation, which really limits the applications of THz sources. In most of applications frequency tunability is important and essential [15–16]. In this letter, THz generation and manipulation in engineered nonlinear photonic quasicrystals has been studied. We report the experimental measurement of narrow-band terahertz radiation with multi-frequency modes in an optical quasistructural (Fibonacci) superlattice. A canonical pump–probe experimental technique was used to realize the in-situ detection. Different from the classical THz detection method, THz wave in forward and backward propagation direction would be generated and can be detected simultaneously in this scheme. This is an important advantage of the proposed pump–probe detection method.

THz radiations are generated via optical rectification in the pre-engineered domain structure of poled ferroelectric material. More accurately this effect can be described by a difference frequency generation process: when short laser pulses with a broad frequency spectrum are incident on a nonlinear material, at each point z of the medium the difference-frequency mixing between the closely spaced different spectral components of the pump wave E_P(ω, z) induces a second-order nonlinear polarization:

where ω and Ω lie, respectively, in the optical and THz frequency ranges. This radiation at frequencies Ω contains frequencies from 0 to several THz and χ ⁽²⁾(z)=χ ⁽²⁾g(x) is the spatially modulated second-order optical susceptibility, where g(x) is +1 and -1 in positive or negative domains. The amplitude of each spectral THz component E_T(Ω, z) is computed by solving the nonlinear Maxwell equation in the spatial and frequency Fourier domains:

with H(Ω)=-(Ω²/c²)χ ⁽²⁾(Ω)C(Ω), where C(Ω) is the Fourier transformation of input optical pulse, v_g is the group velocity of the optical pulse. G_m is the reciprocal vectors induced by the spatially modulated second-order optical susceptibility. The solution of second-order Maxwell equation yields the forward and backward propagating THz field at each point in the crystal:

As previously shown [17], if the dispersion of χ ⁽²⁾(Ω) is neglected, the strong modification of the THz spectrum at the exit of the crystal is basically the product of three contributions: the Ω² dependence of the radiative efficiency, the initial power spectrum of the rectified input laser pulse C(Ω), and the phase-matching condition represented by the sinc function.

In the general scheme, THz generation and detection are realized in different nonlinear crystals. The pump-induced THz waves are detected by reverse nonlinear parametric interaction, in which the energy transfer from the THz to optical range occurs. Using this detection scheme, THz waves in forward and backward propagation directions should be detected separately. However, through a modification of the conventional pump–probe technique, THz generation process and the reverse nonlinear parametric process can be realized in the same nonlinear crystal. Theoretically the reverse nonlinear parametric process can be considered as the parametric interaction between the pump-induced THz wave and the probe wave in the same nonlinear crystal. The sum and difference frequency mixing contributions lead to the modulation of probe beam, and the transmittance of the probe beam will be modulated in the time domain. For any delay time between pump and probe pulses, the transmittance change of probe beam can be detected using a lock-in amplifier, which reflect the pump-induced THz wave generated in the crystal.

As the femtosecond optical pulses propagate through the domain-reversal crystal, a THz nonlinear polarization is generated by optical rectification. Each domain in the crystal contributes a half cycle to the radiated THz field. The frequency of THz wave is determined essentially by

where c is the light velocity, for a periodic structure, Λ is the period of domain-reversal crystal, n _T is the refractive index of THz wave and vg is the group velocity of the optical pump wave, respectively. m is integer index and m=1 represents principle value of THz frequencies which corresponds to the most intense mode in spectral domain. The negative and positive signs in the equation correspond to the THz generation propagated in forward and backward directions, respectively.

The above formula can be extended from periodic structure to quasiperiodic structure and aperiodic structure. In analogy to periodic structure, it is expected that their THz radiation will reflect the quasiperiodicity and aperiodicity. The quasiperiodic superlattice used for our experiment was fabricated by high voltage poling technique, which is the same as the one introduced and studied previously [18–19]. It has two building blocks A and B of length l _A and l _B, respectively, which are ordered in a Fibonacci sequence. Each block has a domain of length l _A1 (l _B1) with positive ferroelectric domain (black) and a domain of length l _A2 (l _B2) with negative ferroelectric domain. In our design, l _A1 is set to be equal to l _B1. We have chosen l _A=70.4 µm and l _B=43.2 µm and the ratio of length scales l _A and l _B as (1+√5)/2, so-called golden ratio τ, respectively. The width of positive domain in both blocks A and B is 25 µm. The average parameter D=τl _A+l _B of quasiperiodical Fibonacci grating. For comparison, Λ_1,1=D/(1+τ) is designed as 60 µm, which equals to the period Λ of the periodic structure. The refractive index of THz radiation and the group velocity of the optical pulse are obtained as n _T=6.5 and v_g=1.34×10⁸m/s, respectively [20]. The samples were cut normal to the x-axis and polished. The structure consists of 12 building blocks A and 8 building blocks B.

For domain-reversal crystal with the Fibonacci sequence, the multi-frequency modes THz radiations are given by quasiperiodicities:

where m and n are integer indices of the quasiperiodicities and D is the average lattice parameter.

 

Fig. 1. Diagrammatic layout of canonical pump–probe experiment and the configuration of quasi-phase-matching: k_O1 and k_O2 are the wave vectors of two spectral components of pump wave, respectively. k_T is the wave vector of THz radiation.

Download Full Size | PDF

THz measurement was carried out on a canonical pump-probe setup [21]. The schematic diagram of experimental method and wave vector compensation are shown in Fig. 1. Here k_O1 and k_O2 are the wave vectors of two spaced spectral components of pump wave, respectively. A femtosecond laser pulse was delivered from a Ti: Sapphire laser (Mira, Coherent) with pulse duration of about 70 fs, repetition of 76 MHz, and a center wavelength at 800 nm. The laser beam was divided into pump (~90%) and probe (~10%) beams by a beamsplitter. The pump beam 50 mW, with its polarization parallel to z-axis of the crystal, was chopped at 1.7 kHz and passed through an optical delayed line monitored by a computer-controlled step-motor (resolution 20.8fs). A quarter-wave plate and a polarizer were inserted into the probe beam path for adjusting the polarization of probe beam with respect to that of pump beam freely. Two beams were focused on the same spot of a sample by a lens of f=30 mm.

 

Fig. 2. The Fourier transform of experimental THz wave forms in the quasiperiodic Fibonacci domain structure. Inset: experimental time-domain THz wave forms. The structure parameters are designed as l_A=70.4 µm and l _B=43.2 µm (Λ_1,1=60 µm), respectively. The width of positive domain in block A and B is 25 µm, The structure consists of 12 building blocks A and 8 building blocks B.

Download Full Size | PDF

 

Fig. 3. The Fourier transform of experimental THz wave forms in the periodic domain structure. Inset: experimental time-domain THz wave forms. The period of structure is set as Λ=60 µm, the sample length L=1.2mm. The structure consists of 20 periodic building blocks.

Download Full Size | PDF

The transmittance change of the probe beam modulated by generated THz wave can be obtained when polarizations of both pump and the probe beams were set to be parallel to z-axis of the sample. In our experiment, the transmitted probe signal was detected with a photodiode connected to a lock-in amplifier. According to our experimental configuration, the signal (see inset of Fig.2) obtained from lock-in amplifier was contributed only by the THz-wave-modulated probe beam. The corresponding measurement in the sample with periodic structure is also shown in the inset of Fig.3 for comparison. It is obvious that the fine structures of THz wave forms appear in quasiperiodical Fibonacci grating.

The corresponding Fourier transform of experimental results are shown in Fig.2 (Fibonacci domain-inverted structure) and Fig.3 (periodic domain-inverted structure). It is seen that the frequencies of the most pronounced modulation are at 1.18, 0.78, 0.59 and, 0.37 THz, respectively. In spectrum (Fig.2) 1.18 and 0.59 THz generations are marked by (m, n)=(1, 1); which correspond to forward and backward propagation, respectively. Meanwhile 0.78, 0.37 THz generations propagate in forward and backward, respectively, labed by (0, 1). The other THz generations labed by (2, 1) and so on cannot be observed due to the stronger crystal absorption in the upper-frequency range. If the crystal would be transparent at all THz frequencies, more THz waves in forward and backward propagation were observable.

 

Fig. 4. The window Fourier transform of experimental THz wave forms (A) in the quasiperiodic Fibonacci domain structure with parameters designed as l_A=70.4 µm and l _B=43.2 µm (Λ1,1=60 µm), respectively. The window Fourier transform of experimental THz wave forms (B) in the periodic domain structure with period Λ=60 µm, the sample length L=1.2mm.

Download Full Size | PDF

In order to investigate the spatial THz characteristics inside superlattice, we used windows Fourier and wavelet transforms to analyze the measured time-delay-dependent THz electric field. Figs. 4(A) and 4(B) show the windows Fourier spectrum of the THz pulses, corresponding to quasi-periodic and periodic superlattice, respectively. Apparently the THz intensities inside the quasi-periodic superlattice are stronger than those inside the periodic superlattice, indicating the better performance of quasi-periodic superlattice. It is also shown in Fig.4 that the THz intensity distribution strongly depends on the time delay between the optical pump beam and probe beam. The THz intensity decreases when the time delay gets to be longer, which result from the two factors below. Actually, for less time delay, optical pump beam and probe beam propagates almost synchronously, the probe beam modulated by the THz generation in the whole of superlattice; for the larger time delay, the mixing of the probe and THz waves occurs in the part of the superlattice only. In addition, the crystal absorption of THz wave should be taken into account in this situation as THz wave will travel the certain distance before the mixing with probe beam. Both factors contribute to the decrease of the THz intensity. Therefore, strictly speaking, the windows Fourier spectrum actually reflect the distributions of THz wave inside the superlattice and the dispersion and damping of the THz wave, in general, can be achieved based on the windows Fourier spectrum.

In conclusion, by employing a canonical pump-probe experimental technique, we have demonstrated that the multi-frequency modes THz wave generation and detection can be carried out in a LiTaO₃ superlattice with Fibonacci domain-inverted structure. The well pronounced THz waves in both forward and backward directions at frequencies of 1.18, 0.78, 0.59, 0.37 THz were experimentally detected with precise profile and good sensitivity. The experimental results agree with theoretical prediction given by equations 4 and 5. Pump-probe experimental technique provides the possibilities to detect the two THz waves in forward and backward propagations simultaneously. The THz sources with multi-frequency modes are of potential interest in medical diagnoses as well as biomedical imaging applications.