Terahertz Generation by Optical Rectification in Uniaxial Birefringent Crystals

The angular dependence of terahertz (THz) emission from birefringent crystals can differ significantly from that of cubic crystals. Here we consider optical rectification in uniaxial birefringent materials, such as chalcopyrite crystals. The analysis is verified in (110)-cut ZnGeP2 and compared to (zincblende) GaP. Although the crystals share the same nonzero second-order tensor elements, the birefringence in chalcopyrite crystals cause the pump pulse polarization to evolve as it propagates through the crystal, resulting in a drastically different angular dependence in chalcopyrite crystals. The analysis is extended to {012}and {114}-cut chalcopyrite crystals and predicts more efficient conversion for the {114} crystal cut over the {012}and {110}-cuts. ©2012 Optical Society of America OCIS codes: (190.5970) Semiconductor nonlinear optics including MQW; (260.1440) Birefringence. References and links 1. J. Federici and L. Moeller, “Review of terahertz and subterahertz wireless communications,” J. Appl. Phys. 107(11), 111101 (2010). 2. P. U. Jepsen, D. Cooke, and M. Koch, “Terahertz spectroscopy and imaging modern techniques and applications,” Laser Photon. Rev. 5(1), 124–166 (2011). 3. K. Yamaguchi, M. Nakajima, and T. Suemoto, “Coherent control of spin precession motion with impulsive magnetic fields of half-cycle terahertz radiation,” Phys. Rev. Lett. 105(23), 237201 (2010). 4. B. Fischer, M. Hoffmann, H. Helm, G. Modjesch, and P. U. Jepsen, “Chemical recognition in terahertz timedomain spectroscopy and imaging,” Semicond. Sci. Technol. 20(7), S246–S253 (2005). 5. J. Hebling, K.-L. Yeh, M. C. Hoffmann, B. Bartal, and K. A. Nelson, “Generation of high-power terahertz pulses by tilted-pulse-front excitation and their application possibilities,” J. Opt. Soc. Am. B 25(7), B6–B19 (2008). 6. D. Creeden, J. C. McCarthy, P. A. Ketteridge, T. Southward, P. G. Schunemann, J. J. Komiak, W. Dove, and E. P. Chicklis, “Compact fiber-pumped terahertz source based on difference frequency mixing in ZGP,” IEEE J. Sel. Top. Quantum Electron. 13(3), 732–737 (2007). 7. J. D. Rowley, J. K. Pierce, A. T. Brant, L. E. Halliburton, N. C. Giles, P. G. Schunemann, and A. D. Bristow, “Broadband terahertz pulse emission from ZnGeP2.,” Opt. Lett. 37(5), 788–790 (2012). 8. C.-W. Chen, T.-T. Tang, S.-H. Lin, J. Y. Huang, C.-S. Chang, P.-K. Chung, S.-T. Yen, and C.-L. Pan, “Optical properties and potential applications of ε-GaSe at terahertz frequencies,” J. Opt. Soc. Am. B 26(9), A58–A65 (2009). 9. A. Schneider, M. Neis, M. Stillhart, B. Ruiz, R. U. A. Khan, and P. Günter, “Generation of terahertz pulses through optical rectification in organic DAST crystals: theory and experiment,” J. Opt. Soc. Am. B 23(9), 1822– 1835 (2006). 10. F. D. Brunner, O.-P. Kwon, S.-J. Kwon, M. Jazbinsek, A. Schneider, and P. Günter, “A hydrogen-bonded organic nonlinear optical crystal for high-efficiency terahertz generation and detection,” Opt. Express 16(21), 16496–16508 (2008). 11. J. L. Shay and J. H. Wernick, Ternary Chalcopyrite Semiconductors: Growth, Electronic Properties, and Applications (Pergamon Press, 1975). 12. Y.-S. Lee, Principles of Terahertz Science and Technology (Springer Science + Business Media, 2009). 13. P. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University Press, 1990). #168061 $15.00 USD Received 7 May 2012; revised 2 Jul 2012; accepted 6 Jul 2012; published 11 Jul 2012 (C) 2012 OSA 16 July 2012 / Vol. 20, No. 15 / OPTICS EXPRESS 16968 14. S. Hargreaves, K. Radhanpura, and R. A. Lewis, “Generation of terahertz radiation by bulk and surface optical rectification from crystal planes of arbitrary orientation,” Phys. Rev. B 80(19), 195323 (2009). 15. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic Press, 1998). 16. K. T. Zawilski, P. G. Schunemann, S. D. Setzler, and T. M. Pollak, “Large aperture single crystal ZnGeP2 for high-energy applications,” J. Cryst. Growth 310(7-9), 1891–1896 (2008). 17. M. H. Rakowsky, W. K. Kuhn, W. J. Lauderdale, L. E. Halliburton, G. J. Edwards, M. P. Scripsick, P. G. Schunemann, T. M. Pollak, M. C. Ohmer, and F. K. Hopkins, “Electron paramagnetic resonance study of a native acceptor in as-grown ZnGeP2,” Appl. Phys. Lett. 64(13), 1615–1617 (1994). 18. K. Vodopyanov, “Optical THz-wave generation with periodically-inverted GaAs,” Laser Photon. Rev. 2(1-2), 11–25 (2008). 19. P. C. M. Planken, H.-K. Nienhuys, H. J. Bakker, and T. Wenckebach, “Measurement and calculation of the orientation dependence of terahertz pulse detection in ZnTe,” J. Opt. Soc. Am. B 18(3), 313–317 (2001). 20. A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (WileyInterscience, 1984).

An example of a positive-uniaxial birefringent THz source is ZGP [6,7], which is a chalcopyrite crystal and a ternary analog of the zincblende structure. Chalcopyrite crystals are of interest as THz sources due to their generally large nonlinearities and in some cases wide band gaps [11], the latter of which limits loss due to multiphoton absorption for near-infrared pumping. A previous comparative study [7] of the THz emission amplitude from ZGP, GaP, and GaAs showed ZGP to have suitable characteristics for excitation by near-infrared sources such as pulsed-fiber or chromium-doped forsterite lasers. However, it has a 2% lattice compression in the [001] direction, producing significant birefringence.
The work presented here explores the orientation dependence of uniaxial birefringent chalcopyrite crystals in order to optimize the phase-matching condition and build understanding of the optical rectification process. Chalcopyrite crystals provide an ideal choice for a study of the effects of birefringence due to their similarity to cubic zincblende structures. Because the second-order nonlinear tensors of these two structures have identical nonzero elements, differences in the respective angular dependence result either from differing values of the nonzero tensor elements or effects due to birefringence, and we argue here that the latter is dominant. First, analysis is presented comparing zincblende and chalcopyrite materials. Second, experimental verification of the analysis is presented. Finally, the analysis is extended to other crystal orientations to demonstrate optimization of the effective nonlinear coefficient.

Analysis
The second-order polarization due to optical rectification [12] is where ε o is the permittivity of free space, ( ) ( ) where E o is the magnitude of the optical field, ω is the optical frequency, and k o and k e are the wavevectors of the ordinary and extraordinary components. For a pump field defined with arbitrary θ orientation Eq. (1) gives the nonlinear polarization in the crystal frame where the relative phase is defined as k d = k e − k o . For d 14 = d 36 and k d = 0, these expressions reduce to those of zincblende crystals [12,14]. For chalcopyrites however, the oscillating cos( ) d k D dependence of (2) X P and (2) Y P leads to generation of periodically interfering THz field components that become increasingly negligible for crystal lengths greater than 2 / d k π .
In contrast, the field components resulting from (2) Z P increase linearly with the crystal length, assuming good velocity-matching of the optical and THz waves. In (110)-cut ZGP, 2 / 30 d k π ≈ µm for 1300 nm excitation [15], which is a pump wavelength of interest. Typically, a real sample will be at least an order of magnitude longer than 2 / d k π , and therefore, it is justified to neglect the (2) ( ) 2 cos sin ,

Experimental verification
The angular dependence of THz emission was measured in as-grown ZGP and undoped GaP samples with thicknesses of 0.33 mm and 0.1 mm respectively. As-grown ZGP single crystals were produced by the horizontal gradient freeze technique [16]. The ZGP sample was cut in the (110) plane and double-side polished for transmission measurements. The orientation of the crystals was verified by electron paramagnetic resonance, ensuring that the 001 direction is in the plane of the sample [17]. The transmission and birefringence homogeneity were verified with polarized transmission imaging. GaP is commercially available.
Measurements were performed at normal pump incidence with the crystal rotated about the axis of the pump wavevector. The generated THz field components were measured in directions parallel ( C THz E ) and perpendicular ( X THz E ) to the pump pulse polarization. Measurements employ ~100 fs pulses from a 1 kHz regenerative amplifier and optical parametric amplifier. For GaP and ZGP the pump pulses were tuned to center wavelengths of 800 nm and 1300 nm respectively. Electro-optic sampling, with linearly polarized 800 nm probe pulses and a (110)-cut ZnTe crystal, was used to detect the THz field. As expected, the measured spectra were broadband (0.5 THz to 3 THz), since the emission from GaP or ZGP is not limited by absorption due to IR-active phonons; see [7]. For ZGP, absorption in this frequency range is known to be smaller than that of ZnTe, GaP and and LiNbO 3 but similar to that of GaAs [18]. The electro-optic crystal detects one polarization component [19]. To acquire C THz E and X THz E , the detection crystal and probe pulse polarization are rotated by 90°. Field amplitudes are obtained by extracting the peak-to-peak voltage in individual THz transients as a function of θ. The resulting angle-dependent field amplitudes for (110)-cut GaP and ZGP are shown in Figs. 1(a) and 1(b) respectively. The data exhibit excellent agreement with the model described above. ZGP is significantly longer than 2 / d k π , hence the only surviving nonzero tensor element in the model is d 36 . Consequently, discussion of the differences between the zincblende and chalcopyrite THz response resulting from the nonlinear tensor is moot, since the signal does not depend on d 14 .

Predictions
The possibility of efficient generation from other crystal orientations of ZGP were explored using the above analytical approach which gives ∝ + − +   (9) for the C and X polarized field amplitudes in the (012) and (114) planes. Here, the azimuthal angle θ is generalized as the angle between the projection of [001] onto the plane of the crystal cut and the incident linear polarization direction. Figures 1(c) and 1(d) show the predictions of the angle-dependent THz generation for uniaxial birefringent chalcopyrite crystals cut in the (012) and (114) planes respectively [Equivalent planes in zincblende crystals would be (011) and (112).] Each of these planes has a different near-infrared index of refraction for the e wave, but neither value varies as a function of θ. Each crystal cut then exhibits unique phase matched wavelengths (not shown).
To understand the results it helps to inspect the crystal structures and orientations; see Fig. 2. For zincblende structures in the (110) plane the bonds are pointing ± 54.7° away from the [001] projection; this is the angle at which C THz E is maximized; see Fig. 1(a). For ZGP the angle is similar, if the 2% compression is ignored, but the birefringence strongly suppresses the THz generation for pump fields oriented at that angle. Reorienting the chalcopyrite to the (012) plane effectively rotates the bonds and the THz response by 90° relative to the [001] projection, as seen by comparing Figs. 1(b) and 1(c). At the rotation angle for maximum efficiency in each of these planes the phase-matching is best achieved for cross-polarization, i.e. ooe for (110) cut and eeo for (012) cut crystals, where the first two waves correspond to the pump and the third corresponds to the emission. This can be considered as a relaxation of the singular phase matching condition observed in zincblende crystals. For chalcopyrite crystals, our analysis shows other efficient configurations. In fact, stronger emission should be observed for (114)-cut crystals. All of the angle dependences in Fig. 1 are plotted on the same relative scale, considering only the effects of birefringence with identical physical parameters; for a (114)-cut crystal, recovery of the maximum zincblende effective nonlinear coefficient 2/3 1/2 d 14 (assuming d 36 = d 14 ) is predicted; see Fig. 1(d). For this orientation the crystal is rotated such that the bonds discussed above are now parallel to the [001] projection onto the crystal surface. The most efficient phase-matching condition is then eee, where all fields are collinearly polarized along this projection. For this orientation the crystal birefringence plays no role.
To extend the predictions to arbitrary crystal planes comprehensively, the analysis of Hargreaves et al. [14] for THz generation in zincblende crystals may be applied, taking into account the evolving pump polarization by numerically considering the crystal to be comprised of many thin slices. For a given θ the polarization state and orientation of the pump pulse are uniquely defined in the crystal frame by the relative phase between the o and e waves and the major elliptical axis. Optical rectification is then modeled at each distance D by considering field amplitudes along the major (a) and minor (b) axes of the elliptical polarization state also defined by δ at θ. This is illustrated for an arbitrary instantaneous elliptical polarization state in a (110)-cut crystal in Fig. 3(a). As D increases the polarization state evolves, oscillating between two linear polarization extremes about the optical axis of the crystal, as shown in Fig. 3(b) for various values of δ. At each position the major and minor components are used as individual instantaneous pump sources for THz generation within the analysis of [14]. Using the Hargreaves notation in this calculation, β is transformed to the lab frame angle Φ(D) with ' φ = 0 and the incident pump polarization fixed along ˆ'' z . Then Eq. (18) of [14] gives instantaneous polarizations resulting from the a or b field amplitudes, with

Conclusion
Previously, chalcopyrite crystals in the form of ZGP have been shown to be useful materials for THz generation by near infrared laser pulses [7]. The experimental and modeled results performed here show that birefringence modifies the angular dependence of the THz emission compared to the zincblende binary analogs. Birefringence is shown to reduce the maximum effective nonlinear coefficient for the commonly use (110) crystal orientation, which is often chosen based solely on the second-order tensor. In contrast it is predicted that (114) recovers the maximum effective nonlinear coefficient and the THz generation, because this orientation is not affected by birefringence.
This work highlights how birefringence plays a role in relaxing the phase-matching condition for THz generation in efficient emitters, without significant detrition in the efficiency. The model can easily be extended to other birefringent materials, given the appropriate second-order nonlinear optical susceptibility tensor.