Abstract

We present a polarization shaping technique for few-cycle terahertz (THz) waves. For this, N femtosecond laser pulses are generated from a devised diffractive optical system made of as-many glass wedges, whcih then simultaneously illuminate on various angular positions of a sub-wavelength circular pattern of an indium arsenide thin film, to produce a THz wave of tailor-made polarization state given as a superposition of N linearly-polarized THz pulses. By properly arranging the orientation and thickness of the glass wedges, which determine the polarization and its timing of the constituent THz pulses, we sucessfully generate THz waves of various unconventioal polarization states, such as polarization rotation and alternation between circular polarization states.

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    [CrossRef] [PubMed]
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2011 (3)

J. Kono, “Spintronics: Coherent terahertz control,” Nat. Photonics 5, 5–6 (2011).
[CrossRef]

T. Kampfrath, A. Sell, G. Klatt, A. Pashkin1, S. Mahrlein, T. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and R. Huber, “Coherent terahertz control of antiferromagnetic spin waves,” Nat. Photonics 5, 31–34 (2011).
[CrossRef]

Y. Kim, J. Ahn, B. G. Kim, and D. Yee, “Terahertz birefringence in zinc oxide,” Jpn. J. Appl. Phys. 50, 030203 (2011).
[CrossRef]

2010 (2)

2009 (1)

2008 (1)

G. Matthäus, S. Nolte, R. Hohmuth, M. Voitsch, W. Richter, B. Pradarutti, S. Riehemann, G. Notni, and A. Tünnermann, “Microlens coupled interdigital photoconductive switch,” Appl. Phys. Lett. 93, 091110 (2008).
[CrossRef]

2007 (5)

K. Y. Kim, B. Yellampalle, A. J. Taylor, G. Rodriguez, and J. H. Glownia, “Single-shot terahertz pulse characterization via two-dimensional electro-optic imaging with dual echelons,” Opt. Lett. 32, 1968–1970 (2007).
[CrossRef] [PubMed]

T. Brixner, “Poincaré representation of polarization-shaped femtosecond laser pulses,” Appl. Phys. B 76, 531–540 (2007).
[CrossRef]

R. Piesiewicz, T. Kleine-Ostmann, D. Mittleman, M. Koch, J. Schoebel, N. Krumbholz, and T. Kürner, “Short-range ultra-broadband terahertz communications: concepts and perspectives,” IEEE Antennas Propag. Mag. 49, 24–39 (2007).
[CrossRef]

A. V. Kimel, A. Kirilyuk, F. Hansteen, R. V Pisarev, and Th. Rasing, “Nonthermal optical control of magnetism and ultrafast laser- induced spin dynamics in solids,” J. Phys. Condens. Matter 19, 043201 (2007).
[CrossRef]

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1, 97–105 (2007).
[CrossRef]

2006 (2)

2005 (1)

R. Shimano, H. Nishimura, and T. Sato, “Frequency tunable circular polarization control of terahertz radiation,” Jpn. J. Appl. Phys. 44, 676–678 (2005).
[CrossRef]

2004 (1)

A. V. Kimel, A. Kirilyuk, A. Tsvetkov, R. V. Pisarev, and Th. Rasing, “Laser-induced ultrafast spin reorientation in the antiferromagnet TmFeO3,” Nature 429, 850–853 (2004).
[CrossRef] [PubMed]

2002 (1)

P. H. Siegel, “Terahertz technology,” IEEE Trans. Microwave Theory Tech. 50, 910–928 (2002).
[CrossRef]

2001 (1)

2000 (1)

1998 (1)

D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature 396, 239–242 (1998).
[CrossRef]

1997 (1)

L. A. Nafie, “Infrared and Raman vibrational optical activity: theoretical and experimental aspects,” Annu. Rev. Phys. Chem. 48, 357–386 (1997).
[CrossRef] [PubMed]

1991 (1)

M. Shapiro and P. Brumer, “Controlled photon induced symmetry breaking: chiral molecular products from achiral precursors,” J. Chem. Phys. 95, 8658–8661 (1991).
[CrossRef]

1988 (1)

W. S. Warren, “Effects of pulse shaping in laser spectroscopy and nuclear magnetic resonance,” Science 242, 878–884 (1988).
[CrossRef] [PubMed]

Ahn, J.

Y. Kim, J. Ahn, B. G. Kim, and D. Yee, “Terahertz birefringence in zinc oxide,” Jpn. J. Appl. Phys. 50, 030203 (2011).
[CrossRef]

M. Yi, K. Lee, J. Lim, Y. Hong, Y.-D. Jho, and J. Ahn, “Terahertz waves emitted from an optical Fiber,” Opt. Express 18, 13693–13699 (2010).
[CrossRef] [PubMed]

Bartels, A.

Bastian, G.

Beck, M.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, N.Y., 2000), Ch. 13.

Brixner, T.

T. Brixner, “Poincaré representation of polarization-shaped femtosecond laser pulses,” Appl. Phys. B 76, 531–540 (2007).
[CrossRef]

T. Brixner and G. Gerber, “Femtosecond polarization pulse shaping,” Opt. Lett. 26, 557–559 (2001).
[CrossRef]

Brumer, P.

M. Shapiro and P. Brumer, “Controlled photon induced symmetry breaking: chiral molecular products from achiral precursors,” J. Chem. Phys. 95, 8658–8661 (1991).
[CrossRef]

Dadap, J. I.

Dekorsy, T.

T. Kampfrath, A. Sell, G. Klatt, A. Pashkin1, S. Mahrlein, T. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and R. Huber, “Coherent terahertz control of antiferromagnetic spin waves,” Nat. Photonics 5, 31–34 (2011).
[CrossRef]

G. Klatt, F. Hilser, W. Qiao, M. Beck, R. Gebs, A. Bartels, K. Huska, U. Lemmer, G. Bastian, M. B. Johnston, M. Fischer, J. Faist, and T. Dekorsy, “Terahertz emission from lateral photo-Dember currents,” Opt. Express 18, 4939–4947 (2010).
[CrossRef] [PubMed]

Faist, J.

Fiebig, M.

T. Kampfrath, A. Sell, G. Klatt, A. Pashkin1, S. Mahrlein, T. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and R. Huber, “Coherent terahertz control of antiferromagnetic spin waves,” Nat. Photonics 5, 31–34 (2011).
[CrossRef]

Fischer, M.

Gallot, G.

Gebs, R.

Gerber, G.

Glownia, J. H.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company Publishers, Englewood, 2005), Ch. 5.

Hangyo, M.

Hansteen, F.

A. V. Kimel, A. Kirilyuk, F. Hansteen, R. V Pisarev, and Th. Rasing, “Nonthermal optical control of magnetism and ultrafast laser- induced spin dynamics in solids,” J. Phys. Condens. Matter 19, 043201 (2007).
[CrossRef]

Hattori, R.

Hecht, E.

E. Hecht, Optics, 4th ed. (Addison Wesley, 2002), Ch. 8.

Heinz, T. F.

Hilser, F.

Hirota, Y.

Hohmuth, R.

G. Matthäus, S. Nolte, R. Hohmuth, M. Voitsch, W. Richter, B. Pradarutti, S. Riehemann, G. Notni, and A. Tünnermann, “Microlens coupled interdigital photoconductive switch,” Appl. Phys. Lett. 93, 091110 (2008).
[CrossRef]

Hong, Y.

Huber, R.

T. Kampfrath, A. Sell, G. Klatt, A. Pashkin1, S. Mahrlein, T. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and R. Huber, “Coherent terahertz control of antiferromagnetic spin waves,” Nat. Photonics 5, 31–34 (2011).
[CrossRef]

Huska, K.

Jho, Y.-D.

Johnston, M. B.

Kampfrath, T.

T. Kampfrath, A. Sell, G. Klatt, A. Pashkin1, S. Mahrlein, T. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and R. Huber, “Coherent terahertz control of antiferromagnetic spin waves,” Nat. Photonics 5, 31–34 (2011).
[CrossRef]

Kim, B. G.

Y. Kim, J. Ahn, B. G. Kim, and D. Yee, “Terahertz birefringence in zinc oxide,” Jpn. J. Appl. Phys. 50, 030203 (2011).
[CrossRef]

Kim, K. Y.

Kim, Y.

Y. Kim, J. Ahn, B. G. Kim, and D. Yee, “Terahertz birefringence in zinc oxide,” Jpn. J. Appl. Phys. 50, 030203 (2011).
[CrossRef]

Kimel, A. V.

A. V. Kimel, A. Kirilyuk, F. Hansteen, R. V Pisarev, and Th. Rasing, “Nonthermal optical control of magnetism and ultrafast laser- induced spin dynamics in solids,” J. Phys. Condens. Matter 19, 043201 (2007).
[CrossRef]

A. V. Kimel, A. Kirilyuk, A. Tsvetkov, R. V. Pisarev, and Th. Rasing, “Laser-induced ultrafast spin reorientation in the antiferromagnet TmFeO3,” Nature 429, 850–853 (2004).
[CrossRef] [PubMed]

Kirilyuk, A.

A. V. Kimel, A. Kirilyuk, F. Hansteen, R. V Pisarev, and Th. Rasing, “Nonthermal optical control of magnetism and ultrafast laser- induced spin dynamics in solids,” J. Phys. Condens. Matter 19, 043201 (2007).
[CrossRef]

A. V. Kimel, A. Kirilyuk, A. Tsvetkov, R. V. Pisarev, and Th. Rasing, “Laser-induced ultrafast spin reorientation in the antiferromagnet TmFeO3,” Nature 429, 850–853 (2004).
[CrossRef] [PubMed]

Klatt, G.

T. Kampfrath, A. Sell, G. Klatt, A. Pashkin1, S. Mahrlein, T. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and R. Huber, “Coherent terahertz control of antiferromagnetic spin waves,” Nat. Photonics 5, 31–34 (2011).
[CrossRef]

G. Klatt, F. Hilser, W. Qiao, M. Beck, R. Gebs, A. Bartels, K. Huska, U. Lemmer, G. Bastian, M. B. Johnston, M. Fischer, J. Faist, and T. Dekorsy, “Terahertz emission from lateral photo-Dember currents,” Opt. Express 18, 4939–4947 (2010).
[CrossRef] [PubMed]

Kleine-Ostmann, T.

R. Piesiewicz, T. Kleine-Ostmann, D. Mittleman, M. Koch, J. Schoebel, N. Krumbholz, and T. Kürner, “Short-range ultra-broadband terahertz communications: concepts and perspectives,” IEEE Antennas Propag. Mag. 49, 24–39 (2007).
[CrossRef]

Koch, M.

R. Piesiewicz, T. Kleine-Ostmann, D. Mittleman, M. Koch, J. Schoebel, N. Krumbholz, and T. Kürner, “Short-range ultra-broadband terahertz communications: concepts and perspectives,” IEEE Antennas Propag. Mag. 49, 24–39 (2007).
[CrossRef]

Kono, J.

J. Kono, “Spintronics: Coherent terahertz control,” Nat. Photonics 5, 5–6 (2011).
[CrossRef]

Krumbholz, N.

R. Piesiewicz, T. Kleine-Ostmann, D. Mittleman, M. Koch, J. Schoebel, N. Krumbholz, and T. Kürner, “Short-range ultra-broadband terahertz communications: concepts and perspectives,” IEEE Antennas Propag. Mag. 49, 24–39 (2007).
[CrossRef]

Kürner, T.

R. Piesiewicz, T. Kleine-Ostmann, D. Mittleman, M. Koch, J. Schoebel, N. Krumbholz, and T. Kürner, “Short-range ultra-broadband terahertz communications: concepts and perspectives,” IEEE Antennas Propag. Mag. 49, 24–39 (2007).
[CrossRef]

Lee, K.

Leitenstorfer, A.

T. Kampfrath, A. Sell, G. Klatt, A. Pashkin1, S. Mahrlein, T. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and R. Huber, “Coherent terahertz control of antiferromagnetic spin waves,” Nat. Photonics 5, 31–34 (2011).
[CrossRef]

Lemmer, U.

Lim, J.

Mahrlein, S.

T. Kampfrath, A. Sell, G. Klatt, A. Pashkin1, S. Mahrlein, T. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and R. Huber, “Coherent terahertz control of antiferromagnetic spin waves,” Nat. Photonics 5, 31–34 (2011).
[CrossRef]

Masson, J.

Matthäus, G.

G. Matthäus, S. Nolte, R. Hohmuth, M. Voitsch, W. Richter, B. Pradarutti, S. Riehemann, G. Notni, and A. Tünnermann, “Microlens coupled interdigital photoconductive switch,” Appl. Phys. Lett. 93, 091110 (2008).
[CrossRef]

Meshulach, D.

D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature 396, 239–242 (1998).
[CrossRef]

Mittleman, D.

R. Piesiewicz, T. Kleine-Ostmann, D. Mittleman, M. Koch, J. Schoebel, N. Krumbholz, and T. Kürner, “Short-range ultra-broadband terahertz communications: concepts and perspectives,” IEEE Antennas Propag. Mag. 49, 24–39 (2007).
[CrossRef]

Nafie, L. A.

L. A. Nafie, “Infrared and Raman vibrational optical activity: theoretical and experimental aspects,” Annu. Rev. Phys. Chem. 48, 357–386 (1997).
[CrossRef] [PubMed]

Nelson, K. A.

Nishimura, H.

R. Shimano, H. Nishimura, and T. Sato, “Frequency tunable circular polarization control of terahertz radiation,” Jpn. J. Appl. Phys. 44, 676–678 (2005).
[CrossRef]

Nolte, S.

G. Matthäus, S. Nolte, R. Hohmuth, M. Voitsch, W. Richter, B. Pradarutti, S. Riehemann, G. Notni, and A. Tünnermann, “Microlens coupled interdigital photoconductive switch,” Appl. Phys. Lett. 93, 091110 (2008).
[CrossRef]

Notni, G.

G. Matthäus, S. Nolte, R. Hohmuth, M. Voitsch, W. Richter, B. Pradarutti, S. Riehemann, G. Notni, and A. Tünnermann, “Microlens coupled interdigital photoconductive switch,” Appl. Phys. Lett. 93, 091110 (2008).
[CrossRef]

Pashkin1, A.

T. Kampfrath, A. Sell, G. Klatt, A. Pashkin1, S. Mahrlein, T. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and R. Huber, “Coherent terahertz control of antiferromagnetic spin waves,” Nat. Photonics 5, 31–34 (2011).
[CrossRef]

Piesiewicz, R.

R. Piesiewicz, T. Kleine-Ostmann, D. Mittleman, M. Koch, J. Schoebel, N. Krumbholz, and T. Kürner, “Short-range ultra-broadband terahertz communications: concepts and perspectives,” IEEE Antennas Propag. Mag. 49, 24–39 (2007).
[CrossRef]

Pisarev, R. V

A. V. Kimel, A. Kirilyuk, F. Hansteen, R. V Pisarev, and Th. Rasing, “Nonthermal optical control of magnetism and ultrafast laser- induced spin dynamics in solids,” J. Phys. Condens. Matter 19, 043201 (2007).
[CrossRef]

Pisarev, R. V.

A. V. Kimel, A. Kirilyuk, A. Tsvetkov, R. V. Pisarev, and Th. Rasing, “Laser-induced ultrafast spin reorientation in the antiferromagnet TmFeO3,” Nature 429, 850–853 (2004).
[CrossRef] [PubMed]

Pradarutti, B.

G. Matthäus, S. Nolte, R. Hohmuth, M. Voitsch, W. Richter, B. Pradarutti, S. Riehemann, G. Notni, and A. Tünnermann, “Microlens coupled interdigital photoconductive switch,” Appl. Phys. Lett. 93, 091110 (2008).
[CrossRef]

Qiao, W.

Rasing, Th.

A. V. Kimel, A. Kirilyuk, F. Hansteen, R. V Pisarev, and Th. Rasing, “Nonthermal optical control of magnetism and ultrafast laser- induced spin dynamics in solids,” J. Phys. Condens. Matter 19, 043201 (2007).
[CrossRef]

A. V. Kimel, A. Kirilyuk, A. Tsvetkov, R. V. Pisarev, and Th. Rasing, “Laser-induced ultrafast spin reorientation in the antiferromagnet TmFeO3,” Nature 429, 850–853 (2004).
[CrossRef] [PubMed]

Richter, W.

G. Matthäus, S. Nolte, R. Hohmuth, M. Voitsch, W. Richter, B. Pradarutti, S. Riehemann, G. Notni, and A. Tünnermann, “Microlens coupled interdigital photoconductive switch,” Appl. Phys. Lett. 93, 091110 (2008).
[CrossRef]

Riehemann, S.

G. Matthäus, S. Nolte, R. Hohmuth, M. Voitsch, W. Richter, B. Pradarutti, S. Riehemann, G. Notni, and A. Tünnermann, “Microlens coupled interdigital photoconductive switch,” Appl. Phys. Lett. 93, 091110 (2008).
[CrossRef]

Rodriguez, G.

Sato, T.

R. Shimano, H. Nishimura, and T. Sato, “Frequency tunable circular polarization control of terahertz radiation,” Jpn. J. Appl. Phys. 44, 676–678 (2005).
[CrossRef]

Schoebel, J.

R. Piesiewicz, T. Kleine-Ostmann, D. Mittleman, M. Koch, J. Schoebel, N. Krumbholz, and T. Kürner, “Short-range ultra-broadband terahertz communications: concepts and perspectives,” IEEE Antennas Propag. Mag. 49, 24–39 (2007).
[CrossRef]

Sell, A.

T. Kampfrath, A. Sell, G. Klatt, A. Pashkin1, S. Mahrlein, T. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and R. Huber, “Coherent terahertz control of antiferromagnetic spin waves,” Nat. Photonics 5, 31–34 (2011).
[CrossRef]

Shan, J.

Shapiro, M.

M. Shapiro and P. Brumer, “Controlled photon induced symmetry breaking: chiral molecular products from achiral precursors,” J. Chem. Phys. 95, 8658–8661 (1991).
[CrossRef]

Shimano, R.

R. Shimano, H. Nishimura, and T. Sato, “Frequency tunable circular polarization control of terahertz radiation,” Jpn. J. Appl. Phys. 44, 676–678 (2005).
[CrossRef]

Siegel, P. H.

P. H. Siegel, “Terahertz technology,” IEEE Trans. Microwave Theory Tech. 50, 910–928 (2002).
[CrossRef]

Silberberg, Y.

D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature 396, 239–242 (1998).
[CrossRef]

Tani, M.

Taylor, A. J.

Tonouchi, M.

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1, 97–105 (2007).
[CrossRef]

Tsvetkov, A.

A. V. Kimel, A. Kirilyuk, A. Tsvetkov, R. V. Pisarev, and Th. Rasing, “Laser-induced ultrafast spin reorientation in the antiferromagnet TmFeO3,” Nature 429, 850–853 (2004).
[CrossRef] [PubMed]

Tünnermann, A.

G. Matthäus, S. Nolte, R. Hohmuth, M. Voitsch, W. Richter, B. Pradarutti, S. Riehemann, G. Notni, and A. Tünnermann, “Microlens coupled interdigital photoconductive switch,” Appl. Phys. Lett. 93, 091110 (2008).
[CrossRef]

Voitsch, M.

G. Matthäus, S. Nolte, R. Hohmuth, M. Voitsch, W. Richter, B. Pradarutti, S. Riehemann, G. Notni, and A. Tünnermann, “Microlens coupled interdigital photoconductive switch,” Appl. Phys. Lett. 93, 091110 (2008).
[CrossRef]

Wakeham, G. P.

Warren, W. S.

W. S. Warren, “Effects of pulse shaping in laser spectroscopy and nuclear magnetic resonance,” Science 242, 878–884 (1988).
[CrossRef] [PubMed]

Wolf, M.

T. Kampfrath, A. Sell, G. Klatt, A. Pashkin1, S. Mahrlein, T. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and R. Huber, “Coherent terahertz control of antiferromagnetic spin waves,” Nat. Photonics 5, 31–34 (2011).
[CrossRef]

Yee, D.

Y. Kim, J. Ahn, B. G. Kim, and D. Yee, “Terahertz birefringence in zinc oxide,” Jpn. J. Appl. Phys. 50, 030203 (2011).
[CrossRef]

Yellampalle, B.

Yi, M.

Annu. Rev. Phys. Chem. (1)

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Figures (5)

Fig. 1
Fig. 1

(Color online) (a) Two half-cycle pulses of mutually orthogonal polarizations overlapped with a 1-ps time delay. (a) The temporal profile of the combined pulse (blue) represented in a three-dimensioal coordinate space of Ex, and Ey; and its projections to each plane, respectively. (b) The temporal profiles of the right (blue) and the left (red) circular polarization amplitudes. (c) Calculated time-varying polarization ellipticity. (For the definition, see the text.) The arrows in (b) and (c) represent the peak positions of the individual half-cycle pulses.

Fig. 2
Fig. 2

(Color online) Experimental setup for the generation of polarization-shaped THz waves. (Inset) Laser beam is refracted from a single glass wedge to the edge of an InAs disk pattern, where the azimuthal angle θ and the thickness l of the glass wedge determine the polarization and timing of the generated THz pulse, respectively. When a set of glass wedges of different azimuthal angles and thicknesses is used at the same time, as-many THz pulses are generated simultaneously from the various locations in the InAs disk and the combined THz wave is made with time-varying polarization.

Fig. 3
Fig. 3

(Color online) Electric filed E⃗(t) of a linearly-polarized THz pulse measured by a polarization-sensitive detector as a function of the azimuthal angle θ of a glass wedge: (a) The measured x-polarized component, x̂ · E⃗(t); (b) The measured y-polarization component, ŷ · E⃗(t); (c) The the THz peak amplitudes of (b) and (c) plotted as a function of θ, where the dotted lines are cosθ (red) and sinθ (blue); (d) The polarization angle tan−1[x̂ · Ê (t)/ŷ· E⃗(t)] calculated from (c); (e) Time-domain waveform of a single linearly-polarized THz pulse.

Fig. 4
Fig. 4

(Color online) (a) THz pulse shaping with four glass wedges. The inset shows the orientation of the glass wedges. (For the detail, see the text.) Four THz pulses which are linearly polarized are weaved to make circularly polarized THz wave. (b) Calculated right-circular polarization amplitude |R(t)| (blue) and left-circular polarization amplitude |L(t)| (Red). (c) Calculated polarization elipticity ε(t). The dotts represent the experimental data and the solid line the numerical simulation. The dashed lines along ε(t) = ±π/2 indicate perfect circular polarization.

Fig. 5
Fig. 5

(Color online) Examples of THz pulse shaping with six glass wedges. (a–c) THz pulse shaping for polarization rotation: (a) The electric field E⃗(t) is plotted in a three-dimensional space defined by Ex, Ey, and time. The dots represent the experimental data while the solid line the simulation. The inset shows the glass wedge arrangement, where the arrows indicate the glass wedge orientation, and the ordering number indicates the glass thickeess. (b) Amplitude absolutes of the calculated right- and left-circular polarization components, |R(t)| (blue) and |L(t)| (red), respectively. (c) Polarization ellipticity ε(t), which shows that polarization changes from linear, right-circular, and back to linear in one-cycle time duration. The dashed line at ε(t) = π/2 (−π/2) indicates perfect left (right) circular polarization state. (d–f) THz pulse shaping for polarization alternation: (d) Electric field vector, (e) circular polarization ampltude absolutes, (f) polarization ellipticity. Polarization state changes from right circular to left circular.

Equations (13)

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E ˜ x , y ( t ) = E x , y ( t ) + i H [ E x , y ( t ) ] ,
H ( E ( t ) ) = 1 π P E ( t ) t t d t ,
E ˜ R , L ( t ) = 1 2 ( E ˜ x ( t ) ± i E ˜ y ( t ) ) , .
ε ( t ) = tan 1 | E ˜ L ( t ) | | E ˜ R ( t ) | | E ˜ L ( t ) | + | E ˜ R ( t ) | ,
ε ( t ) = tan 1 | E ˜ L ( t ) | | E ˜ R ( t ) | π 4 ,
E ( t ) = y ^ e ( t t 1 ) 2 / τ 2 + x ^ e ( t t 2 ) 2 / τ 2 ,
I ( X , Y ) = | I ( x , y ) w ( x , y ) e i k f ( x X + y Y ) d x d y | 2 ,
w ( x , y ) = e i k sin ϕ ( x cos θ + y sin θ ) ,
I ( X , Y ) = | I ( x , y ) e i k f [ x ( X X w ) + y ( Y Y w ) ] d x d y | 2 ,
E ( t ) = θ ^ E one ( t ) ,
E total ( t ) = n = 1 N θ ^ n E one ( t t n ) ,
d = 2 f λ L ,
η = 1 2 J 1 ( π f # D / 2 λ T H z ) π f # D / 2 λ T H z

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