Abstract

We present a new approach to defining the “beam quality” of a broadband terahertz beam as it propagates through a first-order optical system. A new definition to beam variance is exhibited, taking into account the spectral weight function of the beam. The near-field far-field propagation product is presented by the general averaged M2, marked as (M¯Broadband)2. This parameter is invariant while the beam propagates through optical systems, and it obeys the rule of (M¯Broadband)21. Moreover, we show that, in the special case of monochromatic radiation, the definitions we made are reduced to the known expressions of single frequency radiation. We demonstrate a practical procedure for measuring the beam quality of a broadband radiation. The importance of this procedure is emphasized by our experimental results.

© 2012 Optical Society of America

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2010

J. Y. Suen, W. Li, Z. D. Taylor, and E. R. Brown, “Characterization and modeling of a terahertz photoconductive switch,” Appl. Phys. Lett. 96, 141103 (2010).
[CrossRef]

2009

2005

2003

R. M. Woodward, V. P. Wallace, R. J. Pye, B. E. Cole, D. D. Arnone, E. H. Linfield, and M. Pepper, “Terahertz pulse imaging of ex vivo basal cell carcinoma,” J. Invest. Dermatol. 120, 72–78 (2003).
[CrossRef]

2002

1997

Q. Cao and X. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

1996

1995

1991

A. E. Siegman, “Defining the effective radius of curvature for a non-ideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

1990

A. E. Siegman, “New development in laser resonators,” Proc. SPIE 1224, 2 (1990).
[CrossRef]

W. Jian, “Propagation of a Gaussian–Schell beam through turbulent media,” J. Mod. Opt. 37, 671–684 (1990).
[CrossRef]

Arnone, D. D.

R. M. Woodward, V. P. Wallace, R. J. Pye, B. E. Cole, D. D. Arnone, E. H. Linfield, and M. Pepper, “Terahertz pulse imaging of ex vivo basal cell carcinoma,” J. Invest. Dermatol. 120, 72–78 (2003).
[CrossRef]

Betz, A. L.

Boreiko, R. T.

Brown, C. T. A.

Brown, E. R.

J. Y. Suen, W. Li, Z. D. Taylor, and E. R. Brown, “Characterization and modeling of a terahertz photoconductive switch,” Appl. Phys. Lett. 96, 141103 (2010).
[CrossRef]

Cao, Q.

Q. Cao and X. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

Chen, Z.

Cole, B. E.

R. M. Woodward, V. P. Wallace, R. J. Pye, B. E. Cole, D. D. Arnone, E. H. Linfield, and M. Pepper, “Terahertz pulse imaging of ex vivo basal cell carcinoma,” J. Invest. Dermatol. 120, 72–78 (2003).
[CrossRef]

Coulombe, M. J.

Danylov, A. A.

Deng, X.

Q. Cao and X. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

Dholakia, K.

Ding, Z.

Fischer, P.

Gatesman, A. J.

Giles, R. H.

Goodhue, W. D.

Goyette, T. M.

Hu, Q.

Jacobsen, R. H.

Jian, W.

W. Jian, “Propagation of a Gaussian–Schell beam through turbulent media,” J. Mod. Opt. 37, 671–684 (1990).
[CrossRef]

Keiding, S. R.

Kumar, S.

Li, P.

P. Li, K. Shi, and Z. Liu, “Manipulation and spectroscopy of a single particle by use of white-light optical tweezers,” Opt. Express 30, 156–158 (2005).
[CrossRef]

Li, W.

J. Y. Suen, W. Li, Z. D. Taylor, and E. R. Brown, “Characterization and modeling of a terahertz photoconductive switch,” Appl. Phys. Lett. 96, 141103 (2010).
[CrossRef]

Linfield, E. H.

R. M. Woodward, V. P. Wallace, R. J. Pye, B. E. Cole, D. D. Arnone, E. H. Linfield, and M. Pepper, “Terahertz pulse imaging of ex vivo basal cell carcinoma,” J. Invest. Dermatol. 120, 72–78 (2003).
[CrossRef]

Liu, Z.

P. Li, K. Shi, and Z. Liu, “Manipulation and spectroscopy of a single particle by use of white-light optical tweezers,” Opt. Express 30, 156–158 (2005).
[CrossRef]

Lopez-Marsical, C.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Martinez-Herrero, R.

Mejias, P. M.

Mittleman, D. M.

Morris, J. E.

Nelson, J. S.

Nixon, W. E.

Pepper, M.

R. M. Woodward, V. P. Wallace, R. J. Pye, B. E. Cole, D. D. Arnone, E. H. Linfield, and M. Pepper, “Terahertz pulse imaging of ex vivo basal cell carcinoma,” J. Invest. Dermatol. 120, 72–78 (2003).
[CrossRef]

Pye, R. J.

R. M. Woodward, V. P. Wallace, R. J. Pye, B. E. Cole, D. D. Arnone, E. H. Linfield, and M. Pepper, “Terahertz pulse imaging of ex vivo basal cell carcinoma,” J. Invest. Dermatol. 120, 72–78 (2003).
[CrossRef]

Qian, X.

Ren, H.

Reno, J. L.

Shi, K.

P. Li, K. Shi, and Z. Liu, “Manipulation and spectroscopy of a single particle by use of white-light optical tweezers,” Opt. Express 30, 156–158 (2005).
[CrossRef]

Sibbett, W.

Siegman, A. E.

A. E. Siegman, “Defining the effective radius of curvature for a non-ideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

A. E. Siegman, “New development in laser resonators,” Proc. SPIE 1224, 2 (1990).
[CrossRef]

Suen, J. Y.

J. Y. Suen, W. Li, Z. D. Taylor, and E. R. Brown, “Characterization and modeling of a terahertz photoconductive switch,” Appl. Phys. Lett. 96, 141103 (2010).
[CrossRef]

Taylor, Z. D.

J. Y. Suen, W. Li, Z. D. Taylor, and E. R. Brown, “Characterization and modeling of a terahertz photoconductive switch,” Appl. Phys. Lett. 96, 141103 (2010).
[CrossRef]

Uhd Jepsen, P.

Van Rudd, J.

Waldman, J.

Wallace, V. P.

R. M. Woodward, V. P. Wallace, R. J. Pye, B. E. Cole, D. D. Arnone, E. H. Linfield, and M. Pepper, “Terahertz pulse imaging of ex vivo basal cell carcinoma,” J. Invest. Dermatol. 120, 72–78 (2003).
[CrossRef]

Williams, B. S.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Woodward, R. M.

R. M. Woodward, V. P. Wallace, R. J. Pye, B. E. Cole, D. D. Arnone, E. H. Linfield, and M. Pepper, “Terahertz pulse imaging of ex vivo basal cell carcinoma,” J. Invest. Dermatol. 120, 72–78 (2003).
[CrossRef]

Wright, E. M.

Zhao, Y.

Appl. Phys. Lett.

J. Y. Suen, W. Li, Z. D. Taylor, and E. R. Brown, “Characterization and modeling of a terahertz photoconductive switch,” Appl. Phys. Lett. 96, 141103 (2010).
[CrossRef]

IEEE J. Quantum Electron.

A. E. Siegman, “Defining the effective radius of curvature for a non-ideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

J. Invest. Dermatol.

R. M. Woodward, V. P. Wallace, R. J. Pye, B. E. Cole, D. D. Arnone, E. H. Linfield, and M. Pepper, “Terahertz pulse imaging of ex vivo basal cell carcinoma,” J. Invest. Dermatol. 120, 72–78 (2003).
[CrossRef]

J. Mod. Opt.

W. Jian, “Propagation of a Gaussian–Schell beam through turbulent media,” J. Mod. Opt. 37, 671–684 (1990).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

Q. Cao and X. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

A. E. Siegman, “New development in laser resonators,” Proc. SPIE 1224, 2 (1990).
[CrossRef]

Other

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

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Figures (6)

Fig. 1.
Fig. 1.

Schematic description of a transmission THz pulsed system based on a PC switch.

Fig. 2.
Fig. 2.

Schematic description of the beam-radius measurement at the near field. “A” is the Fourier plane of the THz source. “B” is the focal plane where we measure the near-field beam radius.

Fig. 3.
Fig. 3.

Schematic description of the beam-radius measurement at the far field. “A” is the Fourier plane of the THz source. “B” is a plane located at a distance four times the focal lens from “A,” where we measure the far-field beam radius.

Fig. 4.
Fig. 4.

Experimental results involving measuring the properties of the THz pulse: (a) THz pulse as measured in our system, (b) THz spectrum of the pulse in (a).

Fig. 5.
Fig. 5.

Experimental results involving measuring the THz beam in the near field: (a) measurement of the beam at the near field, (b) horizontal cross section of the beam at the near field (a) across the peak.

Fig. 6.
Fig. 6.

Experimental results involving measuring the THz beam in the far-field: (a) measurement of the beam at the far field, (b) horizontal cross section of the beam at the far field (a) across the peak.

Equations (31)

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E(x,z,λ)=iΔzλeikΔz+E(x0,z0,λ)eik2Δz(xx0)2dx0.
E(x,z,ω)=iωΔz2πc0eiωc0Δz+E(x0,z0,ω)eiω2c0Δz(xx0)2dx0.
E*(x,z,ω)=iiωΔz2πc0eiωc0Δz+E*(x0,z0,ω)eiω2c0Δz(xx0)2dx0.
I(x,z,ω)E(x,z,ω)E*(x,z,ω)=ωΔz2πc0+dx1+dx2E*(x1,z0,ω)E(x2,z0,ω)eiω2c0Δz[(xx2)2(xx1)2].
P(z,ω)+I(x,z,ω)dx.
P(z,ω)=P(z0,ω)=P(ω).
Pω+P(ω)dω.
σ¯x2(z)+x2I(x,z,ω)dxdω+I(x,z,ω)dxdω.
σ¯x2(z)=1Pωω2πc0Δzdω+x2E*(x1,z0,ω)E(x2,z0,ω)eiωc0Δzx(x2x1)eiω2c0Δz(x22x12)dxdx1dx2.
σ¯x2(z)=1Pω+(2π)2c02Δz2ω2dx2dω{ω2c02Δz2x22|E(x2,z0,ω)|2+|E(x2,z0,ω)|2iωc0Δzx2[E*(x2,z0,ω)E(x2,z0,ω)C.C]}.
σ¯x2(z0)=(2π)2Pω+x22|E(x2,z0,ω)|2dx2dω.
[(2π)2Pωc02+|E(x2,z0,ω)|2ω2dx2dω]Δz2.
σsx2(ω,z0)+Sx2|E˜(sx,z0,ω)|2dsx+|E˜(sx,z0,ω)|2dsx=+|E(x2,z0,ω)|2dx2+|E(x2,z0,ω)|2dx2=+|E(x2,z0,ω)|2dx2P(ω).
θ¯Sx2(z0)(2π)2c02Pω+σSx2(ω,z0)P(ω)ω2dω.
A(z0)(2π)2Pω2c0ωx2Im(E*(x2,z0,ω)E(x2,z0,ω))dx2dω
σ¯x2(z)=σ¯x2(z0)+A(z0)Δz+θ¯Sx2(z0)Δz2.
σ¯x2(z)=σ¯x2(z0)+θ¯Sx2(z0)Δz2.
Pω=+P(ω)δ(ωω)dω=P(ω).
θSx2(z0)(2π)2c02(ω)2σSx2=(λ)2σSx2.
σx2(z)=σx2(z0)+(λ)2σSx2(z0)Δz2.
σ¯x2(z)=σ¯x2(z0)+[(2π)2c02Pω+σSx2(ω)P(ω)ω2dω]Δz2.
Mx2(ω)=4πσx(z0)σSx(ω,z0).
W¯x2(z)=W¯x2(z0)+[4PωC02ω2Mx4(ω)W¯x2(z0)P(ω)dω]Δz2.
2W0(λ)θ0(λ)=4λπ.
Mx2(λ)=2σm,x(λ)2θm,x(λ)4λπyieldsM2(ω)=σm,x(ω)θm,x(ω)ω2C0.
W¯x2(z)=W¯x2(z0)+[1PωW¯x2(z0)σm,x2(ω,z0)θm,x2(ω,z0)P(ω)dω]Δz2,
W¯x2(z)=W¯x2(z0)+[1Pω(flensimaging)2σm,x2(ω,z)P(ω)dω]Δz2.
W¯x2(z)=W¯x2(z0)+[4PωC02ω21W¯x2(z0)P(ω)dω]Δz2.
(Φ¯IdealBroadband)2=[4PωC02ω21W¯x2(z0)P(ω)dω].
(Φ¯RealBroadband)2=[1Pω(flensimaging)2σm,x2(ω)P(ω)dω].
(M¯Broadband)2=(Φ¯RealBroadband)2(Φ¯IdealBroadband)2.

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