We detail the design and characterization of a terahertz range achromatic quarter-wave plate based on a stack of aligned variable thickness birefringent sapphire discs. The disc thicknesses and relative rotations of the discs are determined through a basin-hopping Monte Carlo thermal annealing routine. The basin-hopping scheme allows an improved refinement of the required thicknesses and rotations to give a predicted phase error from the ideal π/2 of only 0.5%, which is a factor of approximately 6 better than previous efforts. Moreover, the large contrast between extraordinary and ordinary axes of sapphire allows us to greatly decrease the overall optical path length of our wave plate design by approximately a factor of 10 over similar designs based on quartz discs. However, this very same contrast requires very precise tolerances in the milled thicknesses of the discs and their assembly. We detail a method to compensate for differences in the thickness from their calculated ideal values. We have constructed one of our designs and found it similar in performance to other configurations, but with our much more compact geometry.
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Martin C. Nuss and Joseph Orenstein. Terahertz time-domain spectroscopy. In: Millimeter and submillimeter wave spectroscopy of solids, pages 7–50. Springer, 1998.
Robert A. Kaindl, Marc A. Carnahan, D. Hägele, R. Lövenich, and Daniel S. Chemla. Ultrafast terahertz probes of transient conducting and insulating phases in an electron–hole gas. Nature, 423(6941):734–738, 2003.
James N. Heyman, Roland Kersting, and Karl Unterrainer. Time-domain measurement of intersubband oscillations in a quantum well. Applied physics letters, 72(6):644–646, 1998.
Christiaan Richter and Charles A. Schmuttenmaer. Exciton-like trap states limit electron mobility in TiO2 nanotubes. Nature Nanotechnology, 5(11):769–772, 2010.
Philip F. Taday. Applications of terahertz spectroscopy to pharmaceutical sciences. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 362(1815):351–364, 2004.
Min Ki Choi, Alan Bettermann, and D.W. Van Der Weide. Potential for detection of explosive and biological hazards with electronic terahertz systems. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 362(1815):337–349, 2004.
M.R. Leahy-Hoppa, M.J. Fitch, X. Zheng, L.M. Hayden, and R. Osiander. Wideband terahertz spectroscopy of explosives. Chemical Physics Letters, 434(4):227–230, 2007.
B.B. Hu and M.C. Nuss. Imaging with terahertz waves. Optics letters, 20(16):1716–1718, 1995.
Jon L. Johnson, Timothy D. Dorney, and Daniel M. Mittleman. Enhanced depth resolution in terahertz imaging using phase-shift interferometry. Applied Physics Letters, 78(6):835–837, 2001.
Daniel M. Mittleman, Stefan Hunsche, Luc Boivin, and Martin C. Nuss. T-ray tomography. Optics letters, 22(12):904–906, 1997.
S. Wang, B. Ferguson, D. Abbott, and X.-C. Zhang. T-ray imaging and tomography. Journal of Biological Physics, 29(2-3):247–256, 2003.
Wai Lam Chan, Jason Deibel, and Daniel M. Mittleman. Imaging with terahertz radiation. Reports on progress in physics, 70(8):1325, 2007.
John Corson, R. Mallozzi, J. Orenstein, J.N. Eckstein, and I. Bozovic. Vanishing of phase coherence in underdoped Bi2Sr2CaCu2O8+δ. Nature, 398(6724):221–223, 1999.
L. S. Bilbro, R. Valdés Aguilar, G. Logvenov, O. Pelleg, I. Bozović, and N. P. Armitage. Temporal correlations of superconductivity above the transition temperature in La2 − x SrxCuO4 probed by terahertz spectroscopy. Nature Physics, 7(4):298–302, 2011.
R. Valdés Aguilar, A. V. Stier, W. Liu, L.S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A.G. Markelz, S. Oh, et al. Terahertz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3. Physical review letters, 108(8):087403, 2012.
R. Valdés Aguilar, L. Wu, A.V. Stier, L.S. Bilbro, M. Brahlek, N. Bansal, S. Oh, and N.P. Armitage. Aging and reduced bulk conductance in thin films of the topological insulator Bi2Se3. Journal of Applied Physics, 113(15):153702, 2013.
Liang Wu, M. Brahlek, R. Valdés Aguilar, A.V. Stier, C.M. Morris, Y. Lubashevsky, L.S. Bilbro, N. Bansal, S. Oh, and N.P. Armitage. A sudden collapse in the transport lifetime across the topological phase transition in (Bi1−xInx)2Se3. Nature Physics, 9(7):410–414, 2013.
Jason N. Hancock, Jacobus Lodevicus Martinu van Mechelen, Alexey B. Kuzmenko, Dirk Van Der Marel, Christoph Brüne, Elena G. Novik, Georgy V. Astakhov, Hartmut Buhmann, and Laurens W. Molenkamp. Surface state charge dynamics of a high-mobility three-dimensional topological insulator. Physical review letters, 107(13):136803, 2011.
LiDong Pan, Se Kwon Kim, A Ghosh, Christopher M. Morris, Kate A. Ross, Edwin Kermarrec, Bruce D. Gaulin, S.M. Koohpayeh, Oleg Tchernyshyov, and N.P. Armitage. Low-energy electrodynamics of novel spin excitations in the quantum spin ice Yb2Ti2O7. Nature communications, 5, 2014.
C.M. Morris, R. Valdés Aguilar, A. Ghosh, S.M. Koohpayeh, J. Krizan, R.J. Cava, O. Tchernyshyov, T.M. McQueen, and N.P. Armitage. Hierarchy of bound states in the one-dimensional ferromagnetic ising chain CoNb2O6 investigated by high-resolution time-domain terahertz spectroscopy. Physical review letters, 112(13):137403, 2014.
N.J. Laurita, J. Deisenhofer, LiDong Pan, C.M. Morris, M. Schmidt, M. Johnsson, V. Tsurkan, A. Loidl, and N.P. Armitage. Singlet-Triplet Excitations and Long-Range Entanglement in the Spin-Orbital Liquid Candidate FeSc2S4. Physical Review Letters, 114(20):207201, 2015.
G. Bossé, L.S. Bilbro, R. Valdés Aguilar, LiDong Pan, Wei Liu, A.V. Stier, Y. Li, L.H. Greene, J. Eckstein, and N.P. Armitage. Low energy electrodynamics of the kondo-lattice antiferromagnet cecu2ge2. Physical Review B, 85(15):155105, 2012.
A. Little, Liang Wu, P. Lampen-Kelley, A. Banerjee, S. Patankar, D. Rees, C.A. Bridges, J.-Q. Yan, D. Mandrus, S.E. Nagler, et al. Antiferromagnetic Resonance and Terahertz Continuum in α-RuCl3. Physical Review Letters, 119(22):227201, 2017.
Jing Xia, Yoshiteru Maeno, Peter T. Beyersdorf, M.M. Fejer, and Aharon Kapitulnik. High resolution polar kerr effect measurements of sr 2 ruo 4: evidence for broken time-reversal symmetry in the superconducting state. Physical review letters, 97(16):167002, 2006.
Xiao-Liang Qi, Taylor L. Hughes, and Shou-Cheng Zhang. Topological field theory of time-reversal invariant insulators. Physical Review B, 78(19):195424, 2008.
Rahul Nandkishore, L.S. Levitov, and A.V. Chubukov. Chiral superconductivity from repulsive interactions in doped graphene. Nature Physics, 8(2):158–163, 2012.
Wang-Kong Tse and A.H. MacDonald. Magneto-optical faraday and kerr effects in topological insulator films and in other layered quantized hall systems. Physical Review B, 84(20):205327, 2011.
Joseph Maciejko, Xiao-Liang Qi, H. Dennis Drew, and Shou-Cheng Zhang. Topological quantization in units of the fine structure constant. Physical review letters, 105(16):166803, 2010.
N.P. Armitage. Constraints on Jones transmission matrices from time-reversal invariance and discrete spatial symmetries. Physical Review B, 90(3):035135, 2014.
Enrique Castro-Camus. Polarization-resolved terahertz time-domain spectroscopy. Journal of Infrared, Millimeter, and Terahertz Waves, 33(4):418–430, 2012.
Hiroyuki Makabe, Yuichi Hirota, Masahiko Tani, and Masanori Hangyo. Polarization state measurement of terahertz electromagnetic radiation by three-contact photoconductive antenna. Optics express, 15(18):11650–11657, 2007.
R. Shimano, Y. Ikebe, K.S. Takahashi, M. Kawasaki, N. Nagaosa, and Y. Tokura. Terahertz faraday rotation induced by an anomalous hall effect in the itinerant ferromagnet srruo3. EPL (Europhysics Letters), 95(1):17002, 2011.
M. Neshat and N.P. Armitage. Improved measurement of polarization state in terahertz polarization spectroscopy. Optics letters, 37(11):1811–1813, 2012.
Christopher M. Morris, R. Valdés Aguilar, Andreas V. Stier, and N Peter Armitage. Polarization modulation time-domain terahertz polarimetry. Optics express, 20(11):12303–12317, 2012.
Deepu K. George, Andreas V. Stier, Chase T. Ellis, Bruce D. McCombe, John Černe, and Andrea G. Markelz. Terahertz magneto-optical polarization modulation spectroscopy. JOSA B, 29(6):1406–1412, 2012.
Nikesh Koirala, Matthew Brahlek, Maryam Salehi, Liang Wu, Jixia Dai, Justin Waugh, Thomas Nummy, Myung-Geun Han, Jisoo Moon, Yimei Zhu, et al. Record surface state mobility and quantum hall effect in topological insulator thin films via interface engineering. Nano letters, 15(12):8245–8249, 2015.
Liang Wu, Wang-Kong Tse, M. Brahlek, C.M. Morris, R. Valdés Aguilar, N. Koirala, S. Oh, and N.P. Armitage. High-resolution Faraday rotation and electron-phonon coupling in surface states of the bulk-insulating topological insulator Cu0.02Bi2Se3. Physical review letters, 115(21):217602, 2015.
Liang Wu, R.M. Ireland, M. Salehi, B. Cheng, N. Koirala, S. Oh, H.E. Katz, and N.P. Armitage. Tuning and stabilizing topological insulator Bi2Se3 in the intrinsic regime by charge extraction with organic overlayers. Applied Physics Letters, 108(22):221603, 2016.
Liang Wu, M. Salehi, N. Koirala, J. Moon, S. Oh, and N.P. Armitage. Quantized Faraday and Kerr rotation and axion electrodynamics of a 3D topological insulator. Science, 354(6316):1124–1127, 2016.
N. J. Laurita, Yi Luo, Rongwei Hu, Meixia Wu, S. W. Cheong, O. Tchernyshyov, and N. P. Armitage. Asymmetric Splitting of an Antiferromagnetic Resonance via Quartic Exchange Interactions in Multiferroic Hexagonal HoMnO3. Phys. Rev. Lett., 119:227601, Dec 2017.
C.D. West and A.S. Makas. The spectral dispersion of birefringence, especially of birefringent plastic sheets. JOSA, 39(9):791–794, 1949.
G. Destriau and J. Prouteau. Realisation d’un quart d’onde quasi achromatique par juxtaposition de deux lames cristallines de même naturé. J. Phys. Radium, 10(2):53–55, 1949.
Jean-Baptiste Masson and Guilhem Gallot. Terahertz achromatic quarter-wave plate. Optics letters, 31(2):265–267, 2006.
Zhenqin Li and Harold A Scheraga. Monte carlo-minimization approach to the multiple-minima problem in protein folding. Proceedings of the National Academy of Sciences, 84(19):6611–6615, 1987.
Elijah Polak and Gerard Ribiere. Note on the convergence of m ’e conjugate direction methods é es. ESAIM: Mathematical Modeling and Numerical Analysis-Model ’e lisation Math é matique et Analyze Num ’e rique, 3(R1).
We thank M. Neshat for sharing data on the index of refraction of sapphire.
The THz instrumentation development was funded by the Gordon and Betty Moore Foundation through Grant GBMF2628 to NPA and the NSF through DMR-1508645. T.M. was supported by NSF DMR-1352373. LW was also supported by a seed grant from the NSF supported Materials Research Science and Engineering Center (MRSEC) at Penn (DMR-1720530).
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Wu, L., Farid, A., Laurita, N.J. et al. A Compact Broadband Terahertz Range Quarter-Wave Plate. J Infrared Milli Terahz Waves 41, 642–654 (2020). https://doi.org/10.1007/s10762-020-00686-2
- Time domain terahertz spectroscopy
- Millimeter wave spectroscopy
- Terahertz waveplates