1. Introduction

Terahertz (THz) pulses can be used in many applications in security inspections, spec-troscopies, and other fields[1]. Generation of intense THz pulses is required for real-time measurements[2, 3] and investigation of nonlinear phenomena[4, 5]. Two types of apparatuses have been widely used in laboratory based systems for intense THz pulse generation[6]. One is a large-aperture photoconductive antenna[7, 8, 9, 10], and the other used optical rectification (OR) processes in electro-optic (EO) crystals[11, 12, 13]. Advantages of the use of EO crystals are broad bandwidth of emitted THz waves, high saturation fluence of the pump light, and scalability. THz pulse intensity scaling using a large area mosaic of EO crystals has been reported[11].

Saturation of THz intensity under large pump light fluence have been observed in ZnTe crystals and attributed to two-photon absorption[13, 14]. Loss of the pump light energy due to two-photon absorption and absorption of THz waves by the carriers generated by the two-photon absorption can limit the achievable level of THz intensity. Possible solution for this problem is lowering the two-photon absorption cross-section using other crystals[15]. Other reported methods to obtain intense THz pulses using EO crystals are based on reduction of absorption by lowering the temperature of the crystal[16, 17, 18] or shaping the pump pulses[19]. Recently, a method to achieve phase matching using tilted pulse fronts has been applied to THz pulse generation in lithium niobate and succeded in generating THz pulses of μJ level[20, 21].

Under almost perfect phase matching conditions, more fundamental question arises: i.e., what limits the conversion efficiency. Influence of pump light depletion on the THz generation efficiency is of general and practical interest. Since THz wave generation by OR is a difference frequency generation (DFG) process between optical fields within the pump light spectrum, photon number of the pump light does not change during the THz generation although the spectrum is expected to be shifted to lower frequencies. Manley-Rowe relation[22] for nonlinear optical mixing states that one difference-frequency photon is created via annihilation of one pump photon. This relation leads to the Manley-Rowe limit of conversion efficiency that the emitted photon number of the difference frequency field cannot exceed the photon number of the pump light. Since THz photon energy is about two orders of magnitude smaller than optical photon energy, energy conversion efficiency of less than 1% at maximum is expected. In the OR process, however, the red-shifted pump photon can be reused for the generation of a second THz photon, and the cascaded OR processes can give conversion efficiency much higher than the Manley-Rowe limit[19]. For DFG of two narrow band light fields, possibility of achieving high conversion efficiency using cascaded DFG processes has been analyzed[23, 24].

To investigate the effects of cascading, we conducted numerical simulations using coupled equations of propagation of the pump and THz fields. Simulations were performed based on one-dimensional propagation equations using parameters of a ZnTe crystal pumped by 800 nm femtosecond optical pulses. The purpose of this paper is twofold. One is to study the features of the behaviors of the optical and THz field when cascaded OR occurs to a significant extent. For this perpose, ideal case was studied where the phase matching is perfect and there is no absorption. The other purpose is to search for the optimal condition for generation of the largest THz wave output. For this perpose, more realistic parameters of ZnTe were used. Effects of phase matching and linear absorption was studied and the results were compared with those of ideal conditions free from phase mismatch and absorption. Effects of pump pulse width and multiple pulse pumping were also studied, which can lower the two-photon absorption and narrow the THz wave spectrum. Third-order Kerr-like nonlinearity via cascaded OR processes in EO media has been studied[25, 26, 27, 28]. This type of nonlinearity is automatically included in the present simulations. Time dependent refractive index change generally induces spectrum modification to ultrashort optical pulses. The spectrum changes of the pump light observed in this study, therefore, include third-order and higher-order effects. On the other hand, two-photon absorption, which is another type of direct third-order nonlinearity, was not included in the simulation model explicitly.

2. Theoretical model

2.1. Material parameters

We use parameters of ZnTe for the EO medium. Since Ti:sapphire laser based amplifier systems are almost the only solution for intense femotosecond pulses although high power fiber laser amplifier systems are being developed recently, we set the pump wavelength at 800 nm. Dispersion of ZnTe in THz region is described by a dielectric function at frequencyν:

with ϵ _el = 7.44, V ₀ = 5.32 THz, Γ₀ = 0.09 THz, and S ₀ = 2.58, as reported by Gallot et al. [16, 29]. For the optical region, we used a Sellmeier equation for refractive index n(λ) [30]:

Here, λ is the wavelength in μm. Refractive index in the THz and optical regions is plotted as a function of frequency in Fig. 1. When group velocity dispersion is neglected within the pump light spectrum, the phase matching condition for the OR process is expressed as[30]

 

Fig. 1. Dispersion relation of ZnTe in the (a) THz and (b) optical regions. Red horizontal line in (a) shows the group refractive index at 375 THz (i.e., 800 nm).

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where ω_T is the THz ferquency and n_g(ω _opt) is the group refractive index at optical frquencies. In Fig. 1(a) is shown the group refractive index at 800 nm by a red horizontal line. The phase matching condition is satisfied at THz wave frequency of 2.1 THz. For the simulations under perfect phase matching conditions, the THz refractive index was set constant at the values of this frequency.

Absorption in the THz region that is not included in the THz dispersion relation of Eq. (1) has been reported[16, 18]. We used α_T = 12 cm^-1 for the absorption coefficient for THz waves. We set it as a constant independent of frequency for simplicity of calculations. Since this value approximately corresponds to the maximum absorption coefficient in the frequency region below 2.5 THz, it may be an overestimate. The strong absorption in the phonon resonance region around 5.3 THz was not taken into account. Thus we used only data below 5.3 THz because we found spurious results in the resonance region in some cases because of strong dispersion. The absorption coefficient for optical waves was set as α_L = 3 cm^-1[23, 31].

When a (110) ZnTe crystal is used for the THz wave generation, the amplitude of the THz wave becomes maximum when the incident light polarization lies in [1̅,1,1] direction. Furthermore, it is the case when the polarization of the optical field generated by the cascaded mixing of the incident light and the THz field becomes parallel to the incident polarization so that cascading can continue further. Consequently, we set the incident light polarization in this direction. The THz wave polarization is also in the same direction. The effective nonlinear optical coefficient d _eff in this configuration is related to the tensor element d ₁₄ as

We derived the value of d ₁₄ for the simulations using the reported electro-optic constant r ₄₁ as

where ϵ ₀ is the vacuum dielectric constant, n ₀ is the refractive index of light at the center wavelength, and r ₄₁ = 3.9 pm/V[29], which yields d ₁₄/ϵ ₀ = 129 pm/V. We set the nonlinear optical coefficient as constant and neglected the dispersion since our simulations do not deal with the phonon resonance region, where large dispersion exists in the nonlinear optical coefficient.

2.2. Propagation equations

One-dimensional coupled wave equations of the optical and THz waves were derived using the wave equation[22]:

Here,

is the electric field composed of an optical frequency component, E_L(z,t), and a THz frequency component, E_T(z,t), ϵ is the dielectric constant, μ is the magnetic permeability, which is assumed to be that of vacuum and independent of frequency, σ is the conductivity, which is related to the absorption coefficient, and

is the nonlinear polarization composed of optical and THz components. Both optical and THz waves are assumed to propagate in z direction.

The electric fields of the pump light are expressed using a slowly varying envelope e_L(z,t) as

where ω ₀ is the center frequency of the light spectrum. Envelope e_L(z,t) is expressed in terms of Fourier components as

The THz waves are expressed as

Here, k(ω+ω ₀) and k(ω_T) are the real wavevectors at respective frequencies, which are determined by the dispersion relation. The integration for the optical waves is carried out over positive frequencies of the spectrum of the light. The THz Fourier component, on the other hand, is set to be centered at zero frequency, and the integration is performed over positive and negative frequencies.

Under the slowly-varying envelope approximation, the one-dimensional coupled propagation equations are derived as

Here, Δk_L and Δk_T are phase mismatches defined by

The first term in Eq. (12) describes the effects of linear absorption and the second term that of sum-fequency (for ω_T > 0) and difference-frequency (for ω_T < 0) generation between the optical and THz photons on the propagation of the optical field. The first term in Eq. (13) describes the effects of linear absorption and the second term that of difference-frequency generation between two optical photons on the propagation of the THz field. Effects of nonlinear mixing between THz photons were omitted since they are weaker by a factor of ω_T /ω ₀ even if the THz field grows to an amplitude comparable with that of the optical field. In simulations for perfect phase matching conditions, these phase mismatches were set zero. Simulations for the lossless case and for a realistic absorption case were conducted. For the lossless case we set α_L =α_T = 0.

2.3. Analytical expressions

When the pump light is weak and effects of cascading can be neglected, we can drop the second term in the right hand side of Eq. (12), and obtain the THz field as

where E_T(z = 0,ω_T) = 0 is assumed, and E_L(ω) = E_L(z = 0,ω) is the incident light spectrum. When further assuming that the phase matching is perfect and there are no losses, this equation becomes simply

where n_T(ω_T) is the refractive index at a THz frequency ω_T. When we neglect the dispersion in the THz frequencies, Eq. (17) leads to a simpler expression in time domain,

which is helpful to get basic physical insight of the processes under consideration. Here, c is the speed of light in vacuum.

In the simulation, the incident light pulses were assumed to have a Gaussian temporal shape as

where δt is the pulse width, which is defined by the full width at half maximum of the intensity pulse shape. The Fourier transform of the incident optical wave becomes

The energy conversion efficiency, r_E, and photon conversion ratio, r_p, from optical to THz waves are expressed as

Now we derive the propagation distance, r_c, where the photon conversion ratio reaches unity under the ideal case of perfect phase matching and no losses, using Eq. (17). This length, which we refer to a “cascading length,” is a measure of the magnitude of the nonlinear interaction. By assuming a Gaussian pulse shape as given in Eq. (19), we obtain

and the numerator of Eq. (22) becomes

when the dispersion in THz region is neglected. The denominator of Eq. (22), on the other hand, is

Here, the approximation

is used since the spectrum width is much narrower than the center frequency. Equations (24) and (25) lead to the propagation distance z_c where the condition of r_p = 1 is satisfied as

For analytical evaluation of the effects of absorption, we define an absorption length, z _a, as the propagation distance where the THz field becomes maximum when under perfect phase matching and no losses. Using Eq. (16), we obtain

3. Simulation results

Numerical integrations of Eqs. (12) and (13) were conducted using the Runge-Kutta method with a step of 0.01 mm.

3.1. Δk = 0, α= 0 case

First we conducted numerical calculations under conditions of perfect phase matching (Δk _L = Δk_T = 0) and no absorption (α_T = α_L = 0) in order to understand the basic physical nature of the cascaded OR processes. When ZnTe is used as the EO medium, perfect phase matching is achieved only around 2 THz. Since the linear optical properties of ZnTe are isotropic, phase matching condition cannot be controlled by changing the orientation of the crystal. Broadband phase matching, however, can be achieved by using other EO media such as LiNbO ₃ and applying, if necessary, the tilted pulse front method. The aim of the simulations described in this subsection, therefore, is to understand the ideal case.

 

Fig. 2. Power spectra of (a) THz and (b) optical fields under perfect phase matching and no absorption conditions at propagation distances in the ZnTe crystal from 0 to 5 mm by a step of 0.25 mm. Each spectrum is shifted upward by 0.1.

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Fig. 3. Peak frequency of the optical spectrum as a function of propagation distance obtained under perfect phase matching and no absorption conditions.

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Fig. 4. Black line shows peak field of THz pulses in time domain as a function of propagation distance under perfect phase matching and no absorption conditions. Red line is a straight line that is obtained assuming no pump light change.

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Fig. 5. Photon conversion ratio from optical to THz photons under perfect phase matching and no absorption conditions. Red line shows the level of unity, which corresponds to the Manley-Rowe limit.

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It was assumed that the incident pump light pulse has a Gaussian temporal shape of 180 fs duration and peak intensity of 300 MW/cm², which corresponds to a pump fluence of 57 μJ/cm². The cascading length, z_c, as defined by Eq. (27) is 1.76 mm at this incident fluence.

Power spectra of THz and optical waves obtained by the calculation are shown in Fig. 2 as a function of propagation distance in ZnTe from 0 to 5 mm. The THz spectra grows almost linearly as a function of propagation distance, and little change in the shape is observed. On the other hand, the optical spectra show red shifting and the speed of the shift becomes faster as the pump pulse propagates.

We plot the frequency of the peak of the optical spectra as a function propagatin distance in Fig. 3. The red shifting of the optical spectra shows quadratic dependence on the propagation distance, except for a kink around 1.75 mm. Since the THz amplitude grows linearly in the ideal case, as expressed in Eq. (17), this quadratic dependence is understood based on linear dependence of the speed of the red shifting or quadratic dependence of energy conversion efficiency on the THz field amplitude. In Fig. 2, it is seen that the central peak at 375 THz decreases and a red-shifted peak around 370 THz gradually grows in the propagation distance region between 0 and 2 mm. The height of the red-shifted peak exceeds around 1.75 mm, where the spectrum is not single-peaked. A blue-shifted peak is also seen to be generated at this distance, which is attributed to a sum frequency generation process. Propagation distance of 1.75 mm almost agrees with the cascading length of 1.76 mm. More blue-shifted peaks are generated at longer distances and they are also shifted to lower frquencies through propagation in the EO medium. The observation above on the simulation results shows that the cascading length, z _c, corresponds to a propagation distance where significant red shifting, change in the main peak shape, and generation of a blue-shifted peak occur in the spectrum of the pump pulses. These are regarded as the features showing that cascaded OR processes start to occur.

 

Fig. 6. Black lines: Temporal waveforms of THz field obtained under perfect phase matching and no absorption conditions. Red line: Temporal shape of the incident pump light pulse intensity.

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Interestingly, growth of the THz spectrum is influenced little by these changes in the optical spectrum. The initial THz spectrum shape is determined only by the pump pulse width in the case of Δk = 0 and α = 0. Change in the spectrum shape during propagation is small, and the amplitude grows almost linearly. We plot the peak field of the THz pulse in time domain as a function of propagation distance in Fig. 4. The red straight line in the figure shows the result obtained using Eq. (17), which assumes that the pump light field does not change during propagation. The dependence shown in the figure is almost linear even at propagation distances larger than z_c. Below z_c, the simulation result almost overlaps with that of the linear dependence calculated from Eq. (17), and at and above z_c, small deviation from the linear dependence is seen while qualitative changes as seen in the optical waves shown in Figs. 2(b) and 3 are not observed.

In Fig. 5, we plot the photon conversion ratio, r _p, as calculated using Eq. (22). We observe a smooth quadratic growth, and r_p reaches seven times the Manley-Rowe limit at a propagation distance of 5 mm. Energy conversion efficiency, r_E, was 3.5% at this distance. Temporal waveforms of the THz pulses are plotted in Fig. 6. They show waveforms almost propartional to the time derivative of the pump pulse shape, as expressed in Eq. (18), and linear dependence of the amplitude on the propagation distance. Although the assumption of no change in the pump field is theoretically justified only in the limit of small conversion efficiency, the present results show that some predictions obtained using this assumption are valid even at conversion efficiencies higher than the Manley-Rowe limit.

 

Fig. 7. Power spectra of (a) THz and (b) optical fields under realistic phase mismatches and no absorption conditions at propagation distances in the ZnTe crystal from 0 to 5 mm by a step of 0.25 mm. Each spectrum is shifted upward by 0.1.

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3.2. Δk ≠ 0, α= 0 case

Now we examine the effects of phase mismatches on the cascaded OR processes by taking into account of finite phase mismatches, Δk_L and Δk_T. We keep the absorption at zero at this stage. Parameters of the pump pulses are the same as in the previous case. THz and optical spectra obtained by simulations are shown in Fig. 7 as a function of propagation distance. The peak of the THz spectra grows almost linearly, but the spectrum width becomes narrower as the propagation distance increases due to the phase matching condtion. The optical spectrum shows red shifting but the speed of the shift is much smaller than the perfectly phase matched case. Small shift of the peak frequency of the THz spectrum is observed. This is attributed to the shift in the THz frequency satisfying the phase matching condition because of the red shifting of the pump spectrum.

Temporal waveforms of the THz pulses are plotted in Fig. 8. The horizontal axis of the figure shows the local time traveling at the group velocity of the pump light. Spectral narrowing during propagation is clearly seen. Peak field in time domain grows up to a propagation distance of 2 mm. After that point, the peak field does not grow but the spectrum height grows because the pulse duration becomes longer and the spectrum is narrowed. Photon conversion ratio is plotted in Fig. 9. Photon conversion ratio larger than unity is obtained and there are no spurious behaviors at the crossover of r_p = 1.

The simulation results of this case shows that because of the dispersion of the EO medium, the phase matching condition can have a significant effect on the cascaded OR processes for THz pulse generation.

3.3. Δk ≠ 0, α≠ 0 case

Finite absorption coefficients of α_T = 12 cm^-1 and α_L = 3 cm^-1 are also included in the simulations now. In Fig. 10(a) and (b), we show the THz and optical spectra at the pump intensity same as that of the previous case of 300 MW/cm². Results obtained at a larger pump intensity of 800 MW/cm2 are shown in Fig. 10(c) and (d). In both cases, the THz spectra become maximum at a propagation distance of 2.25 mm, which agrees with z_a = 2.3 mm obtained using Eq. (28). Narrowing of the THz spectra is observed as seen in Fig. 7. Optical spectra show small red shift at 300 MW/cm² pump intensity. At 800 MW/cm², a significant red shift is observed. These results show that there is a competition between the nonlinear interaction and linear absorption of THz and optical fields, and cascaded OR processes occur at a high enough pump intensity.

 

Fig. 8. Temporal waveforms of THz pulses obtained under realistic phase mismatches and no absorption conditions. Temporal shape of the incident pump pulse intensity is also shown.

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Fig. 9. Photon conversion ratio obtained under realistic phase mismatches and no absorption conditions. Red line shows the level of unity, which corresponds to the Manley-Rowe limit.

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Fig. 10. Power spectra of THz and optical fields under realistic phase mismatches and absorption at propagation distances in the ZnTe crystal from 0 to 5 mm by a step of 0.25 mm. Each spectrum is shifted upward by 0.1. (a) THz spectra at 300 MW/cm² pump; (b) optical spectra at 300 MW/cm² pump; (c) THz spectra at 800 MW/cm² pump; and (d) optical spectra at 800 MW/cm² pump.

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Fig. 11. Propagation distance dependence of the peak field of the THz temporal waveform at pump intensities of 300 to 800 MW/cm₂.

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In Fig. 11 is plotted the peak field of the THz temporal waveform as a function of propagation distance at incident pump intensities of 300 to 800 MW/cm². The field depends on the pump intensity almost linearly. The peak field is maximized at a propagation distance around 1.5 mm at all pump intensities. This is shorter than the propagation distance of 2.25 mm, where the THz spectrum reaches the maximum. This fact can be explained as follows. At the THz frequency where the phase matching condition is satisfied, the propagation behavior depends only on the absorption, thus the optimum distance agrees with z_a. In time domain, in contrast, the THz waveform is affected by both the phase matching and absorption. As seen in Fig. 8, growth of peak height in time domain is suppressed by the spectral narrowing due to phase matching condition. Thus the optimum propagation distance for the peak field in time domain is less than that for the spectrum peak.

3.4. Effects of pump pulse shape

As seen in the previous subsection, higher pump intensity increase the efficiency of THz pulse generation via cascaded OR processes. In actual experiments, however, two-photon absorption, which is not included in the present simulation model, increases at higher pump intensities. Consequently, methods to obtain larger THz amplitudes at less pump intensities are required. We study here effects of the pump pulse width and multiple pulse pumping.

Pump pulse width dependence of the power spectra of generated THz waves are calculated using Eq. (17), which is obtained under perfect phase matching and no loss conditions. Results are plotted in Fig. 12 for pulse widths from 100 to 300 fs. Peak incident intensity was kept constant in the calculations. It is seen from the figure that spectrum peak heights do not depend much on pump pulse width, and that peak frequencies and widths are in inverse proportion to pump pulse width. This can be explained by the fact that the influence of the pump pulse width comes from the integral in Eq. (17), which is the autocorrelation of the pump spectrum. When be centered at 2 THz, where the phase matching condition is satisfied. The optimum pump pulse width that gives maximum THz output is about 200 fs, as seen from the figure. From calculations under the realistic phase mismatches and absorption conditions, it was found that the optimum pulse width depends on what quantity should be maximized. Time-domain peak field of the THz pulse was maximized at pump pulse width of 225 fs, and photon conversion ratio at 170 fs.

 

Fig. 12. Pulse width dependence of the THz power spectra under perfect phase matching and no loss conditions.

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Fig. 13. Temporal waveform of THz pulse obtained at 1.66 mm when pumped by a 180 fs single pulse. Time zero is set at the peak time.

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By pumping the EO medium using multiple pulses, we can obtain larger THz wave amplitudes while avoiding undisirable nonlinear effects such as two-photon absorption. Effects of multiple pulse pumping were investigated by examining the temporal peak field of THz pulse at a propagation distance of 1.66 mm, where the peak field was maximum. Incident pump intensity and pulse width were set at 300 MW/cm² and 180 fs. Effects of phase mismatches and absorption were included in the calculations. In Fig. 13 is shown the waveform of THz pulses obtained when pumped by a single pump pulse. After the main peak at time zero, peaks with less amplitudes are observed at 0.50, 0.93, 1.34, and 1.72 ps. THz waveforms obtained by multiple pulse pumping are expected to be time-shifted summations of this waveform. The optimum delay times, therefore, of additional pump pulses should be -0.50, -0.93, -1.34, and -1.72 ps in this order. In the calculations, the electric field of the pump light was expressed as a summation of time-shifted replicas of the original pulse shape. Peak intensity and pulse width for each pulse were kept at 300 MW/cm² and 180 fs.

 

Fig. 14. Peak THz field obtained by multiple pulse pumping. Black circles and green triangles show results obtained by pumping with in-phase and out-of-phase pulse trains, respectively. Red squares are summations of peaks of the THz waveform obtained by single pulse pumping.

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Obtained peak THz fields are plotted in Fig. 14. Since there are overlaps between adjacent pump pulses, obtained THz peak fields depended on the relative phase of pump pulses. When phases of adjacent pump pulses are inverted, pump light intensity in the time region between the pulses become small because of destructive interference, and maximum THz output was obtained. The peak THz fields obtained by time-shifted summation of the waveform of single-pulse pumping case, as shown in Fig. 13, were between those obtained by in-phase and out-of-phase multiple-pulse pumping. Since the height of peaks in the trailing part of the THz waveform becomes less and less, effects of multiple pulse pumping on attainable peak field in time domain are limited, and maximum peak field of less than twice of that of single pulse pumping case was obtained at five pump pulses.

Effects of multiple pulse pumping on THz spectra are more pronounced. In Fig. 15, we plot temporal waveforms and power spectra of THz pulses obtained by one to five pump pulses. Fields of all the pump pulses were set to have the same phase in the calculations. It is seen from the figure that by multiple pulse pumping, the spectrum is narrowed to a peak frequency of 2.1 THz, and the enhancement in spectrum peak is more than 18 times from one to five pump pulses. The results show that multiple pulse pumping is very effective if large intensity of THz waves at a specific frequency is required.

4. Conclusion

We have conducted mumerical simulations based on one-dimensional coupled propagation equations of optical and THz waves for intense THz pulse generation using optical rectification in EO crystals. Effects of cascaded optical rectification processes are examined under perfect phase matching and no loss case and also under realistic phase matching or absorption conditions using parameters of ZnTe. Under ideal conditions, conversion efficiencies much higher than the Manley-Rowe limit were observed. Competition between cascaded optical rectifications and absorption was shown to be expressed in terms of characteristic lengths describing these effects. Red shifting of the pump light spectrum has been observed which exhibits the occurrence of cascaded processes. Effects of pump pulse width and multiple pulse pumping also have been studied.

 

Fig. 15. Waveforms and power spectra of THz pulses obtained by multiple pulse pumping.

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