Abstract

To determine the spatio-temporal field distribution of freely propagating terahertz bandwidth pulses, we measure the time-resolved electric field in two spatial dimensions with high resolution. The measured, phase-coherent electric-field distributions are compared with an analytic model in which the radiation from a dipole antenna near a dielectric interface is coupled to free space through a spherical lens. The field external to the lens is limited by reflection at the lens–air dielectric interface, which is minimized at Brewster’s angle, leading to an annular field pattern. Field measurements compare favorably with theory. Propagation of terahertz beams is determined both by assuming a TEM0,0 Gaussian profile as well as expanding the beam into a superposition of Laguerre–Gauss modes. The Laguerre–Gauss model more accurately describes the beam profile for free-space propagation and after propagating through a simple optical system. The accuracy of both models for predicting far-field beam patterns depend upon accurately measuring complex field amplitudes of terahertz beams.

© 2003 Optical Society of America

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References

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    [CrossRef]
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2002 (1)

2001 (3)

2000 (3)

R. W. McGowan, R. A. Cheville, and D. Grischkowsky, “Direct observation of the Gouy phase shift in THz impulse ranging,” Appl. Phys. Lett. 76, 670–672 (2000).
[CrossRef]

D. R. Grischkowsky, “Optoelectronic characterization of transmission lines and waveguides by terahertz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 6, 1122–1135 (2000).
[CrossRef]

A. Gurtler, C. Winnewisser, H. Helm, and P. U. Jepsen, “Terahertz pulse propagation in the near field and the far field,” J. Opt. Soc. Am. A 17, 74–83 (2000).
[CrossRef]

1999 (2)

1998 (4)

1997 (4)

H. Harde, R. A. Cheville, and D. Grischkowsky, “Terahertz studies of collision-broadened rotational lines,” J. Phys. Chem. A 101, 3646–3660 (1997).
[CrossRef]

T. I. Jeon and D. Grischkowsky, “Nature of conduction in doped silicon,” Phys. Rev. Lett. 78, 1106–1109 (1997).
[CrossRef]

D. You and P. H. Bucksbaum, “Propagation of half-cycle FIR pulses,” J. Opt. Soc. Am. B 14, 1651–1655 (1997).
[CrossRef]

J. Bromage, S. Radic, G. P. Agrawal, C. R. Stroud, P. M. Fauchet, and R. Sobolewski, “Spatiotemporal shaping of terahertz pulses,” Opt. Lett. 22, 627–629 (1997).
[CrossRef] [PubMed]

1996 (2)

1995 (1)

1993 (1)

1991 (1)

D. C. Edelstein, R. B. Romney, and M. Scheuermann, “Rapid programmable 300 ps optical delay scanner and signal-averaging system for ultrafast measurements,” Rev. Sci. Instrum. 62, 579–583 (1991).
[CrossRef]

1990 (1)

1989 (1)

M. van Exter, C. Fattinger, and G. Grischkowsky, “High-brightness terahertz beams characterized with an ultrafast detector,” Appl. Phys. Lett. 55, 337–339 (1989).
[CrossRef]

1984 (1)

D. H. Auston, K. P. Cheung, and P. R. Smith, “Picosecond photoconducting Hertzian dipoles,” Appl. Phys. Lett. 45, 284–286 (1984).
[CrossRef]

1982 (1)

D. B. Rutledge and M. S. Muha, “Imaging antenna arrays,” IEEE Trans. Antennas Propag. 30, 535–540 (1982).
[CrossRef]

1979 (1)

1978 (1)

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust. 2, 105–112 (1978).
[CrossRef]

1966 (1)

Agrawal, G. P.

Auston, D. H.

D. H. Auston, K. P. Cheung, and P. R. Smith, “Picosecond photoconducting Hertzian dipoles,” Appl. Phys. Lett. 45, 284–286 (1984).
[CrossRef]

Besieris, I. M.

Bokor, J.

Bromage, J.

Bucksbaum, P. H.

Budiarto, E.

Cheung, K. P.

D. H. Auston, K. P. Cheung, and P. R. Smith, “Picosecond photoconducting Hertzian dipoles,” Appl. Phys. Lett. 45, 284–286 (1984).
[CrossRef]

Cheville, R. A.

M. T. Reiten, K. McClatchey, D. Grischkowsky, and R. A. Cheville, “Incidence-angle selection and spatial reshaping of terahertz pulses in optical tunneling,” Opt. Lett. 26, 1900–1902 (2001).
[CrossRef]

R. W. McGowan, R. A. Cheville, and D. Grischkowsky, “Direct observation of the Gouy phase shift in THz impulse ranging,” Appl. Phys. Lett. 76, 670–672 (2000).
[CrossRef]

R. A. Cheville, R. W. McGowan, and D. Grischkowsky, “Time resolved measurements which isolate the mechanisms responsible for terahertz glory scattering from dielectric spheres,” Phys. Rev. Lett. 80, 269–272 (1998).
[CrossRef]

H. Harde, R. A. Cheville, and D. Grischkowsky, “Terahertz studies of collision-broadened rotational lines,” J. Phys. Chem. A 101, 3646–3660 (1997).
[CrossRef]

Edelstein, D. C.

D. C. Edelstein, R. B. Romney, and M. Scheuermann, “Rapid programmable 300 ps optical delay scanner and signal-averaging system for ultrafast measurements,” Rev. Sci. Instrum. 62, 579–583 (1991).
[CrossRef]

Eth, Z. S.

Fattinger, C.

D. Grischkowsky, S. Keiding, M. van Exter, and C. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
[CrossRef]

M. van Exter, C. Fattinger, and G. Grischkowsky, “High-brightness terahertz beams characterized with an ultrafast detector,” Appl. Phys. Lett. 55, 337–339 (1989).
[CrossRef]

Fauchet, P. M.

Feng, S.

Grischkowsky, D.

M. T. Reiten, K. McClatchey, D. Grischkowsky, and R. A. Cheville, “Incidence-angle selection and spatial reshaping of terahertz pulses in optical tunneling,” Opt. Lett. 26, 1900–1902 (2001).
[CrossRef]

R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. 26, 846–848 (2001).
[CrossRef]

R. W. McGowan, R. A. Cheville, and D. Grischkowsky, “Direct observation of the Gouy phase shift in THz impulse ranging,” Appl. Phys. Lett. 76, 670–672 (2000).
[CrossRef]

R. A. Cheville, R. W. McGowan, and D. Grischkowsky, “Time resolved measurements which isolate the mechanisms responsible for terahertz glory scattering from dielectric spheres,” Phys. Rev. Lett. 80, 269–272 (1998).
[CrossRef]

T. I. Jeon and D. Grischkowsky, “Nature of conduction in doped silicon,” Phys. Rev. Lett. 78, 1106–1109 (1997).
[CrossRef]

H. Harde, R. A. Cheville, and D. Grischkowsky, “Terahertz studies of collision-broadened rotational lines,” J. Phys. Chem. A 101, 3646–3660 (1997).
[CrossRef]

D. Grischkowsky, S. Keiding, M. van Exter, and C. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
[CrossRef]

Grischkowsky, D. R.

D. R. Grischkowsky, “Optoelectronic characterization of transmission lines and waveguides by terahertz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 6, 1122–1135 (2000).
[CrossRef]

Grischkowsky, G.

M. van Exter, C. Fattinger, and G. Grischkowsky, “High-brightness terahertz beams characterized with an ultrafast detector,” Appl. Phys. Lett. 55, 337–339 (1989).
[CrossRef]

Gurtler, A.

Harde, H.

H. Harde, R. A. Cheville, and D. Grischkowsky, “Terahertz studies of collision-broadened rotational lines,” J. Phys. Chem. A 101, 3646–3660 (1997).
[CrossRef]

Hellwarth, R. W.

Helm, H.

Hunsche, S.

Ippen, E. P.

Jacobsen, R. H.

Jeon, T. I.

T. I. Jeon and D. Grischkowsky, “Nature of conduction in doped silicon,” Phys. Rev. Lett. 78, 1106–1109 (1997).
[CrossRef]

Jeong, S.

Jepsen, P. U.

Johnson, J. L.

Kaplan, A. E.

Keiding, S.

Keiding, S. R.

Khazan, M. A.

Kogelnik, H.

Kroupa, J.

Kuzel, P.

Leitenstorfer, A.

Li, T.

Lukosz, W.

McClatchey, K.

McGowan, R. W.

R. W. McGowan, R. A. Cheville, and D. Grischkowsky, “Direct observation of the Gouy phase shift in THz impulse ranging,” Appl. Phys. Lett. 76, 670–672 (2000).
[CrossRef]

R. A. Cheville, R. W. McGowan, and D. Grischkowsky, “Time resolved measurements which isolate the mechanisms responsible for terahertz glory scattering from dielectric spheres,” Phys. Rev. Lett. 80, 269–272 (1998).
[CrossRef]

Mendis, R.

Mittleman, D. M.

Muha, M. S.

D. B. Rutledge and M. S. Muha, “Imaging antenna arrays,” IEEE Trans. Antennas Propag. 30, 535–540 (1982).
[CrossRef]

Nuss, M. C.

Pu, N.-W.

Radic, S.

Reiten, M. T.

Romney, R. B.

D. C. Edelstein, R. B. Romney, and M. Scheuermann, “Rapid programmable 300 ps optical delay scanner and signal-averaging system for ultrafast measurements,” Rev. Sci. Instrum. 62, 579–583 (1991).
[CrossRef]

Rutledge, D. B.

D. B. Rutledge and M. S. Muha, “Imaging antenna arrays,” IEEE Trans. Antennas Propag. 30, 535–540 (1982).
[CrossRef]

Scheuermann, M.

D. C. Edelstein, R. B. Romney, and M. Scheuermann, “Rapid programmable 300 ps optical delay scanner and signal-averaging system for ultrafast measurements,” Rev. Sci. Instrum. 62, 579–583 (1991).
[CrossRef]

Shaarawi, A. M.

Sheppard, C. J. R.

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust. 2, 105–112 (1978).
[CrossRef]

Smith, P. R.

D. H. Auston, K. P. Cheung, and P. R. Smith, “Picosecond photoconducting Hertzian dipoles,” Appl. Phys. Lett. 45, 284–286 (1984).
[CrossRef]

Sobolewski, R.

Stroud, C. R.

van Exter, M.

D. Grischkowsky, S. Keiding, M. van Exter, and C. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
[CrossRef]

M. van Exter, C. Fattinger, and G. Grischkowsky, “High-brightness terahertz beams characterized with an ultrafast detector,” Appl. Phys. Lett. 55, 337–339 (1989).
[CrossRef]

Van Rudd, J.

Wilson, T.

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust. 2, 105–112 (1978).
[CrossRef]

Winful, H. G.

Winnewisser, C.

You, D.

Ziolkowski, R. W.

Appl. Opt. (1)

Appl. Phys. Lett. (3)

M. van Exter, C. Fattinger, and G. Grischkowsky, “High-brightness terahertz beams characterized with an ultrafast detector,” Appl. Phys. Lett. 55, 337–339 (1989).
[CrossRef]

D. H. Auston, K. P. Cheung, and P. R. Smith, “Picosecond photoconducting Hertzian dipoles,” Appl. Phys. Lett. 45, 284–286 (1984).
[CrossRef]

R. W. McGowan, R. A. Cheville, and D. Grischkowsky, “Direct observation of the Gouy phase shift in THz impulse ranging,” Appl. Phys. Lett. 76, 670–672 (2000).
[CrossRef]

IEE J. Microwaves, Opt. Acoust. (1)

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust. 2, 105–112 (1978).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

D. R. Grischkowsky, “Optoelectronic characterization of transmission lines and waveguides by terahertz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 6, 1122–1135 (2000).
[CrossRef]

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE J. Sel. Top. Quantum Electron. 2, 679–692 (1996).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

D. B. Rutledge and M. S. Muha, “Imaging antenna arrays,” IEEE Trans. Antennas Propag. 30, 535–540 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (7)

J. Phys. Chem. A (1)

H. Harde, R. A. Cheville, and D. Grischkowsky, “Terahertz studies of collision-broadened rotational lines,” J. Phys. Chem. A 101, 3646–3660 (1997).
[CrossRef]

Opt. Lett. (6)

Phys. Rev. Lett. (2)

T. I. Jeon and D. Grischkowsky, “Nature of conduction in doped silicon,” Phys. Rev. Lett. 78, 1106–1109 (1997).
[CrossRef]

R. A. Cheville, R. W. McGowan, and D. Grischkowsky, “Time resolved measurements which isolate the mechanisms responsible for terahertz glory scattering from dielectric spheres,” Phys. Rev. Lett. 80, 269–272 (1998).
[CrossRef]

Rev. Sci. Instrum. (1)

D. C. Edelstein, R. B. Romney, and M. Scheuermann, “Rapid programmable 300 ps optical delay scanner and signal-averaging system for ultrafast measurements,” Rev. Sci. Instrum. 62, 579–583 (1991).
[CrossRef]

Other (7)

R. W. P. King and G. S. Smith, Antennas in Matter (MIT Press, Cambridge, 1981).

J. T. Verdeyen, Laser Electronics (Prentice-Hall, Englewood Cliffs, N.J., 1995).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communications Electronics (Wiley, New York, 1984).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, New York, 1999).

E. Hecht, Optics (Addison-Wesley, San Francisco, Calif., 2002).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

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Figures (13)

Fig. 1
Fig. 1

(a) Detail of silicon-lens system used to model field pattern at waist. (b) The experimental setup for measuring the spatially and temporally resolved electric fields. (c) Imaging system used to measure field pattern.

Fig. 2
Fig. 2

Calculated electric field amplitude (a) just inside the silicon-lens surface and (b) just exterior to the silicon lens. The S plane and P planes relative to the dipole orientation are shown in Fig. 1(c).

Fig. 3
Fig. 3

Measured time-resolved electric field at d=16 mm from source silicon lens. Measured field is shown for (a) the xˆzˆ and (b) the yˆzˆ planes.

Fig. 4
Fig. 4

Spatially resolved electric field amplitudes at frequencies of (a) 0.18 THz, (b) 0.38 THz, and (c) 0.58 THz. Data are measured near the silicon-lens surface at distance d=16 mm. The insets to the figure are the calculated beam profiles.

Fig. 5
Fig. 5

Spatially resolved electric field phase corresponding to the amplitude profiles of Fig. 7: (a) 0.18 THz, (b) 0.38 THz, and (c) 0.58 THz. The insets to the figure are the calculated phase distributions.

Fig. 6
Fig. 6

Measured THz beam amplitude profiles at frequency-independent waist size image plane, Fig. 1(c). The insets show the theoretically calculated lens patterns after propagating a distance of 1 mm.

Fig. 7
Fig. 7

Measured time-resolved fields using a reduced-aperture silicon lens in the (a) xz and (b) yz planes shown in Fig. 1 at a distance d=116 mm from the silicon lens.

Fig. 8
Fig. 8

Spatially resolved electric field amplitudes at frequencies of (a) 0.38 THz, (b) 0.58 THz, and (c) 0.78 THz. Data are measured at distances d=116 mm (left) and d=216 mm (right). The insets to the figure are the calculated beam profiles.

Fig. 9
Fig. 9

Measured waist radius (points) as a function of frequency at (a) 116 mm and (b) 216 mm. The solid curve is the best fit to TEM00 Gaussian beam obtained with fit parameters (a) dfit=63 mm and (b) dfit=147 mm. The dashed curve is the best fit obtained for the experimentally measured distances, d.

Fig. 10
Fig. 10

Comparison of measured electric field amplitude distributions (left column) with amplitude distribution determined from a decomposition into Laguerre–Gauss modes (right column.) Data are the same as Fig. 4, d=16 mm for frequencies of (a) 0.18 THz, (b) 0.38 THz, and (c) 0.58 THz.

Fig. 11
Fig. 11

Amplitude of Laguerre–Gauss modes from Fig. 10. The dark bar represents the zeroth angular mode, while the light-colored bar is total amplitudes of angular modes from 1 to 4. The insets are the modal decomposition of the corresponding theoretically calculated field distributions (insets of Figs. 4 and 5).

Fig. 12
Fig. 12

(a) THz beam amplitude profiles at 0.58 THz generated from experimentally determined Laguerre–Gauss modes, Anl, at d=116 mm of Fig. 10. (b) These modes are used to predict the amplitude profile at d=216 mm. (c) The measured amplitude profile at d=216 mm.

Fig. 13
Fig. 13

Beam radius from propagated multimode and single mode compared with measured data at d=216 mm. The extracted TEM00 mode was determined by Laguerre–Gauss decomposition. The inset shows the fitting figure of merit as determined by the least-squares method.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

Enl(r, ϕ, ω)
=2n!π(n+|l|)exp{j(2n+l+1)[ψ(z)-ψo]}w(z)×2rw(z)lLnl2r2w(z)2exp-jk r22q(z)+ilϕ.
Anl=Enl*(r, ϕ, wo, ω)Emeas(x, y, ω)rϕ.
Ecalc(x, y, ω)=n=0Nl=0LAnl(ω)Enl(r, ϕ, wo, ω)×exp(-jkz).

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