Investigation of ultra-broadband terahertz generation from sub-wavelength lithium niobate waveguides excited by few-cycle femtosecond laser pulses

B. N. Carnio; A. Y. Elezzabi

doi:10.1364/OE.25.020573

1. Introduction

Terahertz (THz) radiation generation through nonlinear optical rectification in electro-optic media, such as semiconductor crystals (e.g. ZnTe, GaP, GaAs, GaSe, ZnGeP₂) [1–5], inorganic crystals (e.g. LiNbO₃, LiTaO₃) [6, 7], and highly nonlinear organic crystals (e.g. OHQ-T, DAST, DSTMS, HMQ-T, HMQ-TMS) [8–12], provides a promising method for delivering high electric field THz pulses. While all these nonlinear crystals are prime candidates for THz generation, only a few are at the forefront when considering key criteria such as high crystalline quality, excellent chemical and environmental stability, low optical absorption, high second-order nonlinear susceptibility ( $χ^{_{(2)}} (ω)$ ), and high damage threshold. Over the past few years, high quality magnesium doped lithium niobate (LN) crystals have emerged as an exemplary nonlinear material for THz generation, satisfying the above conditions and exhibiting a large nonlinear coefficient of |d₃₃| = 181 pm/V [13]. Its optical damage threshold of ~1 J/cm² [14] renders it the dominant crystal for high power THz generation. However, while LN has been the crystal of choice for many optical rectification research applications, it suffers from one major drawback: low bandwidth of the generated THz signal.

A widely employed configuration is based on a 63° prism-cut MgO:LiNbO₃ crystal pumped with a tilted-front femtosecond laser pulse. While this arrangement has been shown to produce intense THz electric field pulses up to 1.2 MV/cm [15], the emitted THz spectrum is confined to the frequency range below 2.6 THz. The limitation in achieving a broad bandwidth is inherently imposed by the strong LN absorption bands centered at 7.6 and 19 THz and the challenge in maintaining a uniform wave front tilt across all the frequency components encompassed by the ultrashort, intense optical pulse. Another commonly implemented arrangement makes use of intense optical pulses to produce Cherenkov THz radiation in bulk LN crystals [6]. However, due to the THz phase velocity being greater than the optical group velocity ( $v_{T H z}^{p} > v_{o p t}^{g}$ ) in the spectral regions of 8.2-14.6 THz and >19.5 THz, the phase matched Cherenkov generation condition is not satisfied, making it very challenging to produce coherent broadband THz pulses.

Optical rectification of femtosecond pulses propagating within LN waveguides are able to produce THz pulses having powers of a few µWs, as well as improve the THz bandwidth [16–19]. An important advantage of these structures is their ability to confine the optical pump pulse as it propagates, allowing THz generation to occur over larger distances than in the bulk arrangements. However, such structures do not generate THz frequencies beyond the lowest frequency reststrahlen region in LN. We note that core widths having sub-wavelength dimensions with respect to the optical pump pulse can be a compelling method to minimize the strong absorption in the reststrahlen regions, while simultaneously improving the local optical field confinement. Due to the sub-wavelength core dimension, this optical waveguiding structure also ensures that Cherenkov THz radiation is efficiently generated, even in regions of the THz frequency spectrum where the LN phase velocity is greater than the optical pulse group velocity.

Here, we report on ultra-broadband Cherenkov radiation generated from sub-wavelength SiO₂:LN:SiO₂ waveguides. These devices are studied using a 2D finite-difference time-domain (FDTD) analysis, where Maxwell's equations are solved to acquire the field components in the LN nonlinear anisotropic media and the isotropic surroundings. We show that these waveguides generate a THz frequency spectrum spanning between 0.18 and 106 THz when excited by few-cycle optical pulses. Our investigations reveal that waveguides having sub-micron sizes are superior to large-dimension waveguides, due to their ability to generate stronger THz radiation at LN’s reststrahlen bands. When a 500 nm × 2 mm LN core is excited by a 7 fs, 780 nm laser pulse, a kV/cm THz sub-ps electric field pulse is generated at an energy of 520 fJ over a waveguide length of only 300 µm. Since the THz generation efficiency is 2.6 × 10⁻⁴ for this 300 µm long waveguide, it is envisioned that this class of LN waveguides can be realized for integrated on-chip THz devices requiring ultra-broadband and high-field THz radiation sources in a small footprint.

2. Waveguide structure

Figure 1 depicts a schematic representation of the LN waveguide, which can have sub-wavelength dimensions with respect to the 780 nm-central wavelength optical pump pulse that is being guided. The center of the structure consists of a thin layer of LN crystal of width, w. In order to confine the 780 nm central-wavelength optical beam, this LN core layer is sandwiched between two SiO₂ cladding layers of 500 nm thickness. SiO₂ is chosen as the cladding because extensive work has been performed to perfect the bonding process between this material and thin film LN [20]. This bonding technique produces high quality, single crystal LN thin films that exhibit refractive index values matching that of bulk LN. As a result of the THz sub-wavelength core width (w << λ_THz), the SiO₂:LN:SiO₂ waveguide can be inserted between two high index dielectric layers to allow the Cherenkov generation condition (i.e. $v_{o p t}^{g, m} > v_{T H z}^{p}$ , where $v_{o p t}^{g, m}$ is the group velocity of the waveguide’s optical mode) to be satisfied at a wide range of THz frequencies. This requirement is met by implementing 47°-cut high-resistivity Si prisms on the waveguide structure, where the angle is chosen to allow the generated THz radiation to be directed at the Si-air interface at normal incidence. Notably, 2D FDTD simulations are performed, such that the waveguide height is much larger than the THz wavelengths (i.e. h>>λ_THz). By performing 2D simulations, both the optical pump pulse energies and the generated THz energies are presented per waveguide height.

Fig. 1 Schematic showing the sub-wavelength SiO₂:LN:SiO₂ waveguide geometry incorporating the Si cladding prisms necessary to achieve phase matching over a wide spectrum. Depicted in the illustration is the waveguide height, h, length, l, and core width, w. The z-polarized optical pulse is coupled into the SiO₂:LN:SiO₂ waveguide and the generated THz radiation exits the Si prisms having an electric field polarization oriented along the z-axis.

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To obtain a realistic picture of the nonlinear optical rectification in the sub-wavelength SiO₂:LN:SiO₂ waveguides, it is essential to accurately model the LN and SiO₂ complex refractive indices across the THz, infrared, and the optical frequency ranges. This information is critical, especially since dispersion and group velocity of the optical pulse propagating in the waveguide core affects the Cherenkov THz generation. The refractive index, n, and extinction coefficient, κ, for both the LN and SiO₂ are presented in Fig. 2(a) and 2(b), respectively. For the LN crystal, the extraordinary (E||c) n and κ curves are extracted from the double Lorentz model,

{(n + i κ)}^{2} = ε_{\infty} + \frac{A^{2}}{B^{2} - ω^{2}} + \sum_{m = 1}^{2} \frac{ε_{L, m} ω_{0, m}^{2}}{ω_{0, m}^{2} - 2 i δ_{0, m} ω - ω^{2}},

where

ε_{\infty}

is the frequency-independent relative permittivity, A and B are the optical dispersion coefficients,

ε_{L, m}

are the Lorentz relative permittivities,

ω_{0, m}

are the angular frequencies of the phonon resonances,

δ_{0, m}

are the damping terms associated with each resonance, and

ω

is the angular frequency. An excellent fit to the combined experimental data from [21–23] is achieved using the parameters

ε_{\infty}

= 0.95, A = 2.027 × 10¹⁶ rad/s, B = 1.068 × 10¹⁶ rad/s,

ε_{L, 1}

= 16.6,

ε_{L, 2}

= 2.6,

ω_{0, 1}

= 4.7 × 10¹³ rad/s,

ω_{0, 2}

= 1.19 × 10¹⁴ rad/s,

δ_{0, 1}

= 2.3 × 10¹² rad/s, and

δ_{0, 2}

= 2.8 × 10¹² rad/s. Clearly, the strong phonon absorption bands centered at 7.6 and 19 THz, as well as the optical dispersion (see inset of Fig. 1(a)), are well-represented by this model. The use of these bulk refractive index values represents the actual index of the LN thin films when grown using highly optimized techniques such as those discussed in [20]. Using a multi-coefficient fitting routine [24], n and κ for the SiO₂ material are found to provide an excellent fit to the experimental data provided in [25]. Importantly, the strong phonon resonance features at 14.1 and 32.7 THz, as well as the weaker mode at 23.7 THz, are fit with a high degree of precision. As discussed in [26], bulk refractive index values well-describe those for SiO₂ layers having thicknesses larger than 20 nm. It is also critical to model the Si complex refractive index at THz and infrared frequencies, which extend over regions dominated by two phonon absorption processes. As shown in Fig. 2(c), the absorption region between 10 and 50 THz and the material dispersion are both very accurately modelled, where experimental data is obtained from [25, 27]. Notably, multiphoton absorption, self phase modulation, and Raman effects in LN occur at pump intensities >100 GW/cm² [28]. The optical damage threshold of LN is ~10 TW/cm² [14], such that pump pulse intensities in our simulations are restricted to values <44 GW/cm² to avoid these effects.

Fig. 2 (a) LN, (b) SiO₂, and (c) Si refractive index and extinction coefficient. The experimental data is obtained from [21–23] for LN, [25] for SiO₂, and [25,27] for Si. The inset in (a) displays the LN refractive index at frequencies of 100-500 THz, which shows the optical dispersion.

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To maximize THz radiation generation, the optical pump pulse electric field polarization is aligned along the c-axis of the LN crystal (z-axis in Fig. 1). This permits optical photons having angular frequencies of $ω_{1}$ and $ω_{2}$ to generate a THz polarization in the LN at an angular frequency of $ω_{3}$ . The resulting z-directed polarization is described by the equation,

P_{z} (ω_{3} : ω_{1}, ω_{2}) = χ^{(2)} (ω_{3}) E_{z} (ω_{1}) E_{z} (ω_{2})

where

χ^{(2)} (ω_{3})

is the nonlinear susceptibility and

E_{z} (ω_{1, 2})

represent the z-directed electric field components present in the optical pump pulse. The nonlinear susceptibility of LN can be written as [29],

χ^{(2)} (ω_{3}) = δ_{(e, e, e)} χ_{e}^{3} + χ_{e}^{2} \sum_{m = 1}^{2} δ_{(e, e, i m)} χ_{i m} (ω_{3})

where

χ_{e}

is the electronic contribution to the linear susceptibility,

χ_{i m}

are the ionic contributions to the linear susceptibility associated with the ionic resonance m,

δ_{(e, e, e)}

is the Miller coefficient due purely to electronic contribution, and

δ_{(e, e, i m)}

are the Miller coefficients associated with the ionic resonance m. The

χ_{e}

and

χ_{i}

terms can be inferred from Eq. (1), while the

δ

’s must be determined from experimental data. Here, both ionic resonances are described by the same scaling coefficient, such that

δ_{(e, e, i 1)} = δ_{(e, e, i 2)} = δ_{(e, e, i)}

. Figure 3 shows the experimental data for the LN nonlinear susceptibility [13, 30–36], along with the results from Eq. (3) plotted using

δ_{(e, e, e)} = - 0.9436

and

δ_{(e, e, i)} = - 1.0693

. Due to the lack of explicit frequency information, the low frequency measurements are plotted at 150 GHz, where

χ^{(2)}

dispersion is negligible. Clearly, these

δ

parameters allow very good modelling of the available measurement data.

Fig. 3 Magnitude and phase of the second order susceptibility of LN. The experimental data is obtained from [13, 30–36].

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3. Nonlinear THz generation

THz generation is demonstrated for an l = 100 µm LN waveguide that has a w = 500 nm. The waveguide is excited by an optical pulse having a duration, τ, of 10 fs, that is centered at a wavelength of 780 nm, and having a ξ_opt (optical energy per waveguide height) of 10 nJ/cm. Figure 4(a) displays the generated THz time-domain electric field pulse, which has a bipolar shape and short duration of <1 ps. At this ξ_opt, the THz pulse has a peak-to-peak electric field strength of 1.56 kV/cm, which is a considerably large electric field value obtained from a sub-wavelength optical waveguide.

Fig. 4 (a) Time-domain THz electric field generated from a SiO₂:LN:SiO₂ waveguide having w = 500 nm, l = 100 µm, and pumped by a 10 fs, 10 nJ/cm, 780 nm optical pulse. The time-domain signal is recorded after exiting the Si prisms. (b) NESD of the THz radiation measured after exiting the prisms. (c-f) Frequency-domain THz electric field waves at frequencies of 10, 20, 30, and 40 THz propagating outwards from the waveguide. Electric field values in (e) and (f) are scaled by a factor of 5.

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Of great interest is the spectral content of the THz pulse. Figure 4(b) shows the THz energy spectral density normalized to the device height (NESD). Remarkably, the spectrum is ultra-broad, extending from 0.18 to 81 THz when using a 40 dB dynamic range (which is chosen as a conservative estimated based on the 50 dB range achieved in [17]). Over this bandwidth, the generated THz signal has an energy per waveguide height, ξ_THz, of 1.35 pJ/cm, such that the optical-to-THz conversion efficiency for the l = 100 µm structure is 1.35 × 10⁻⁴. There is reduced generation at frequencies of 13 and 28.3 THz, which correspond to regions in the spectrum where the electronic and ionic contributions of the second order susceptibility destructively interfere [37]. The latter reduced generation range falls below the 40 dB dynamic range, leading to a null within the generation spectrum when using this bandwidth definition. Also noted in the frequency spectrum is the strongest phonon resonance of the SiO₂ layer occurring at 32.7 THz.

Particularly important is the spatial distribution of the emitted THz radiation. When considering a bulk LN crystal, the THz and optical refractive indices permit Cherenkov emission (where $v_{o p t}^{g} > v_{T H z}^{p}$ ) at frequencies <8.2 THz and in the range of 14.6-19.5 THz, but not between 8.2 and 14.6 THz and >19.5 THz (where in these regions $v_{o p t}^{g} < v_{T H z}^{p}$ ). In the Cherenkov frequency emission ranges, the generated THz wave has a planar wave front and travels at an angle, $θ = \cos^{- 1} (n_{o p t}^{g} / n_{T H z}^{p})$ , with respect to the optical pulse driving the nonlinear dipoles, where $n_{o p t}^{g}$ is the optical group refractive index and $n_{T H z}^{p}$ is the THz radiation phase refractive index. Notably, in the frequency ranges of 8.2-14.6 THz and >19.5 THz, where $n_{o p t}^{g} > n_{T H z}^{p}$ , the criterion for $θ$ is not satisfied and the THz radiation is emitted as highly diverging cylindrical waves. However, when driving the w<<λ_THz SiO₂:LN:SiO₂ waveguide structure with a femtosecond laser pulse, the excited traveling dipole wave emits a THz electric field that experiences an effective THz phase refractive index, $n_{T H z}^{p, e f f}$ , dominated by the Si layers. As a result, the Cherenkov emission angle becomes, $θ = {cos}^{- 1} (n_{o p t}^{g, m} / n_{T H z}^{p, e f f})$ , where $n_{o p t}^{g, m}$ is the group refractive index of the waveguide’s optical mode, allowing $θ$ to be satisfied across the entire 0.18-81 THz bandwidth of the generated THz electric field pulse. This ultra-broadband Cherenkov effect is clearly observed in the frequency-domain electric field waves depicted in Fig. 4(c-f). Notably, since the refractive index of the Si prisms varies only by 0.009 within the frequency range between 0.18 and 81 THz, the theoretically calculated angle of THz radiation emission varies between θ = 47.2° to 47.4° over this frequency interval (i.e. $n_{o p t}^{g, m}$ = 2.32 in the LN core and $n_{T H z}^{p, e f f}$ = 3.417-3.426 in the Si prisms). At representative frequencies of 10, 20, 30, and 40 THz, Fig. 4(c-f) show this nearly constant THz radiation emission angle.

For a constant pump pulse energy, the intensity of the optical mode, and therefore the THz signal strength, depends on w. Figure 5(a) shows the NESD for a waveguide having w = 300 nm to 5 µm, l = 100 µm, and excited by a 10 fs, 10 nJ/cm optical pulse. The reststrahlen bands at 7.6 and 19 THz clearly exhibit stronger absorption with increasing core widths. This is a result of the generated THz radiation needing to propagate larger distances in the waveguide core before exiting the LN. Interestingly, at frequencies $≳$ 84 THz, waveguides having core widths of 300 and 500 nm have worse generation characteristics than the w = 1 µm waveguide. This is a result of the optical pump mode dispersion increasing with reducing core width, since more of the optical mode begins to leak into the SiO₂ layers. Group velocity dispersion (GVD) of the optical mode causes a reduction in the length over which the high and low frequency components in the optical pulse exhibit the spatial overlap necessary to generate frequencies $≳$ 84 THz. The phase ( $n_{o p t}^{p, m}$ ) and group ( $n_{o p t}^{g, m}$ ) refractive indices of the optical mode are shown in Fig. 5(b) at core widths of 300 nm, 500 nm, and 5 µm. Dispersion of the optical mode in the w = 5 µm waveguide configuration is determined mainly by the LN refractive index, since the optical mode is confined to the LN core. The optical mode begins to leak into the SiO₂ cladding regions at core widths of 300 and 500 nm, such that the optical mode dispersion is influenced by the waveguide dimensions in these sub-wavelength structures. Importantly, at the optical pulse central frequency of 385 THz (i.e. 780 nm), sub-wavelength effects cause $n_{o p t}^{g, m}$ to increase from 2.28 (w = 5 µm waveguide) to 2.32 (w = 500 nm waveguide).

Fig. 5 (a) NESD of the THz radiation emitted from a waveguide having w = 300 nm to 5 µm, l = 100 µm, and being pumped by a 10 fs, 10 nJ/cm optical pulse. (b) $n_{o p t}^{p, m}$ and $n_{o p t}^{g, m}$ at core widths of 300 nm, 500 nm, and 5 µm. (c) ξ_THz contained in the frequency components. The inset shows the intensity profiles of the modes.

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All investigated core widths allow for the emission of 120 THz frequency components with a NESD of ~10⁻⁴ fJ/(cm THz). However, reducing w to submicron dimensions permits drastic improvement in the total generation energy, as can be seen in Fig. 5(c). Notably, reducing w from 5 µm to 300 nm results in a 22 times improvement in ξ_THz. This is due to higher optical field confinement in the LN core and lower absorption of the generated THz frequencies within the reststrahlen bands. The intensity profiles displayed in the inset of Fig. 5(c) show that the optical mode is mainly confined to the LN core, and negligibly intensity values exist in the Si layers. Here, a 500 nm waveguide width is chosen as a compromise between a large ξ_THz and a dimension that would be easily fabricated.

The THz bandwidth and NESD dependence on τ is shown in Fig. 6(a) for a SiO₂:LN:SiO₂ waveguide having an l = 100 µm, w = 500 nm, and pumped at τ = 7-100 fs and ξ_opt= 10 nJ/cm. Here, the τ = 7 fs curve exhibits the largest THz bandwidth, producing frequency components between 0.18 and 106 THz when considering a dynamic range of 40 dB. Alternatively, the radiation generated by the 10, 20, 50, and 100 fs optical pump pulses have bandwidths that extend up to 81, 42, 21, and 11 THz, respectively. Dispersion experienced by the 7 fs optical pump pulse (having full-width half-maximum (FWHM) bandwidth of 63 THz) is more pronounced than dispersion of the 100 fs pump pulse (FWHM bandwidth of 4.4 THz), which is evident from the waveguide dispersion data presented in Fig. 5(b). To achieve an ultra-wide spectrum that generates THz radiation above the 19 THz LN phonon mode, it is essential to excite the LN waveguide with $≲$ 20 fs optical pulses. As such, short waveguides (having lengths on the order of hundreds of microns or less) are ideal to minimize dispersion and achieve ultra-broad THz radiation generation. Interestingly, as shown in Fig. 6(b), reducing the pump pulse duration from 100 fs to 7 fs allows for a 2300% improvement in the THz generation energy. This is due to the increase in optical intensity that arises from shortening the optical pulse (while retaining the optical energy) and the additional high frequency THz generation that results from the shorter pulse.

Fig. 6 (a) NESD of the THz electric field emitted from a waveguide structure having a w = 500 nm, l = 100 µm, and pumped by a 7-100 fs, 10 nJ/cm optical pulse. (b) ξ_THz values that are contained in the frequency components.

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In this waveguide configuration, it is essential to understand the relationship by which the THz pulse NESD changes with l. Figure 7(a) shows the THz NESD for waveguides having l = 100-300 µm, w = 500 nm, and driven by a τ = 7 fs optical pump pulse at ξ_opt = 10 nJ/cm. NESD values at frequencies $≲$ 13 THz continually improve by increasing l up to 300 µm, while higher frequency components experience little or no benefit from increasing l. GVD of the optical mode causes the frequency components in the optical pulse to spatially separate during propagation, therefore negating high frequency THz generation. This reduced generation causes ξ_THz to saturate with increasing l, as shown in Fig. 7(b), with the energy improvement mainly arising from additional generation at frequencies $≲$ 13 THz.

Fig. 7 (a) NESD and (b) ξ_THz emitted from waveguides having l = 100-300 µm, w = 500 nm, and pumped by a 7 fs, 10 nJ/cm pulse.

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As in all nonlinear generation devices, it is necessary to understand the impact of pump pulse energies. The THz NESD and ξ_THz are shown in Fig. 8(a) and 8(b), respectively, for waveguides having l = 100 µm, w = 500 nm, and excited by a τ = 7 fs optical pump pulse at ξ_opt = 2-10 nJ/cm. Both the spectrum and ξ_THz show the expected second order improvement with increasing ξ_opt. This is witnessed by the 25 times enhancement in ξ_THz that occurs when ξ_opt is increased from 2 to 10 nJ/cm. This second order increase in the generation energy occurs because the optical pump pulse energy is restricted to values below the threshold for multiphoton absorption, self phase modulation, and Raman effects.

Fig. 8 (a) NESD and (b) ξ_THz generated using waveguides having l = 100 µm, w = 500 nm, and pumped by 7 fs optical pulses having ξ_opt = 2-10 nJ/cm.

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4. Conclusion

A nonlinear FDTD analysis is performed to observe ultra-broadband THz radiation generation from sub-wavelength SiO₂:LN:SiO₂ waveguides when pumped by a few-cycle optical pulse. The generated spectral components extend over the ultra-broad range of 0.18-106 THz for a dynamic range of 40 dB. By reducing the LN core width to sub-micron dimension, we minimize the LN’s reststrahlen band absorption and maximize the optical pump pulse confinement in the waveguide core. Cherenkov emission is achieved at an angle of ~47° over the entire ultra-broad frequency spectrum (including within the LN reststrahlen bands), allowing the sub-wavelength LN waveguides to outperform the bulk LN generation structures in terms of bandwidth and THz generation efficiency. For a waveguide device having a length of 300 µm, a core width of 500 nm, and pumped by a 7 fs, 780 nm pulse having an energy (per waveguide height) of 10 nJ/cm, the generated ultra-broadband THz pulses have an energy (per waveguide height) of 2.6 pJ/cm. This class of SiO₂:LN:SiO₂ waveguides could be fabricated and integrated as on-chip sources having the ability to generate high electric field pulses containing an ultra-broadband range of THz frequencies. To take this structure further for on-chip integration and be able to extract the THz radiation out of the chip, we envision the incorporation of micromachined Si prisms or metasurfaces.

Funding

Natural Sciences and Engineering Research Council of Canada (NSERC).

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Investigation of ultra-broadband terahertz generation from sub-wavelength lithium niobate waveguides excited by few-cycle femtosecond laser pulses

Abstract

1. Introduction

2. Waveguide structure

3. Nonlinear THz generation

4. Conclusion

Funding

References and links

Cited By

Figures (8)

Equations (3)

Optics Express