Abstract

The propagation characteristics of a beam generated by driving an aperture with an ultrashort, hence ultrawide-bandwidth, space–time Gaussian pulse are considered. It is shown analytically with an approximate form of the solution that the beam intensity and the beam energy have different diffraction lengths and rates of beam spread in the far field. These beam properties are also discussed for a derivative receiver system. The analytical results are supported with numerical simulations of the exact pulsed-beam solution.

© 1992 Optical Society of America

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References

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  1. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989).
  2. A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  3. R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3984 (1991).
    [CrossRef] [PubMed]
  4. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).
  5. R. W. Ziolkowski, “Properties of electromagnetic beams generated by ultra-wide bandwidth pulse-driven arrays,”IEEE Trans. Antennas Propag. 40, 888–905 (1992).
    [CrossRef]
  6. M. van Exeter, D. R. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microwave Theory Tech. 38, 1684–1691 (1990).
    [CrossRef]
  7. M. van Exeter, Ch. Fattinger, D. R. Grischkowsky, “High-brightness terahertz beams characterized with an ultrafast detector,” Appl. Phys. Lett. 55, 337–339 (1989).
    [CrossRef]
  8. Y. Pastol, G. Arjavalingam, J.-M. Halbout, G. V. Kopcsay, “Coherent broadband microwave spectroscopy using picosecond optoelectronic antennas,” Appl. Phys. Lett. 54, 307–309 (1989).
    [CrossRef]
  9. C. R. Lutz, A. P. deFonzo, “Far-field characteristics of optically pulsed millimeter wave antennas,” Appl. Phys. Lett. 54, 2186–2188 (1989).
    [CrossRef]
  10. R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
    [CrossRef] [PubMed]
  11. R. W. Ziolkowski, D. K. Lewis, “Verification of the localized-wave transmission effect,” J. Appl. Phys. 68, 6083–6086 (1990).
    [CrossRef]
  12. X.-C. Zhang, B. B. Hu, J. T. Darrow, D. H. Auston, “Generation of femtosecond electromagnetic pulses from semiconductor surfaces,” Appl. Phys. Lett. 56, 1011–1013 (1990).
    [CrossRef]
  13. X.-C. Zhang, J. T. Darrow, B. B. Hu, D. H. Auston, M. T. Schmidt, P. Tham, E. S. Yang, “Optically induced electromagnetic radiation from semiconductor surfaces,” Appl. Phys. Lett. 56, 2228–2230 (1990).
    [CrossRef]
  14. N. Froberg, M. Mack, B. B. Hu, X.-C. Zhang, D. H. Auston, “500 GHz electrically steerable photoconducting antenna array,” Appl. Phys. Lett. 58, 446–448 (1991).
    [CrossRef]

1992 (1)

R. W. Ziolkowski, “Properties of electromagnetic beams generated by ultra-wide bandwidth pulse-driven arrays,”IEEE Trans. Antennas Propag. 40, 888–905 (1992).
[CrossRef]

1991 (2)

R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3984 (1991).
[CrossRef] [PubMed]

N. Froberg, M. Mack, B. B. Hu, X.-C. Zhang, D. H. Auston, “500 GHz electrically steerable photoconducting antenna array,” Appl. Phys. Lett. 58, 446–448 (1991).
[CrossRef]

1990 (4)

R. W. Ziolkowski, D. K. Lewis, “Verification of the localized-wave transmission effect,” J. Appl. Phys. 68, 6083–6086 (1990).
[CrossRef]

X.-C. Zhang, B. B. Hu, J. T. Darrow, D. H. Auston, “Generation of femtosecond electromagnetic pulses from semiconductor surfaces,” Appl. Phys. Lett. 56, 1011–1013 (1990).
[CrossRef]

X.-C. Zhang, J. T. Darrow, B. B. Hu, D. H. Auston, M. T. Schmidt, P. Tham, E. S. Yang, “Optically induced electromagnetic radiation from semiconductor surfaces,” Appl. Phys. Lett. 56, 2228–2230 (1990).
[CrossRef]

M. van Exeter, D. R. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microwave Theory Tech. 38, 1684–1691 (1990).
[CrossRef]

1989 (4)

M. van Exeter, Ch. Fattinger, D. R. Grischkowsky, “High-brightness terahertz beams characterized with an ultrafast detector,” Appl. Phys. Lett. 55, 337–339 (1989).
[CrossRef]

Y. Pastol, G. Arjavalingam, J.-M. Halbout, G. V. Kopcsay, “Coherent broadband microwave spectroscopy using picosecond optoelectronic antennas,” Appl. Phys. Lett. 54, 307–309 (1989).
[CrossRef]

C. R. Lutz, A. P. deFonzo, “Far-field characteristics of optically pulsed millimeter wave antennas,” Appl. Phys. Lett. 54, 2186–2188 (1989).
[CrossRef]

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

Arjavalingam, G.

Y. Pastol, G. Arjavalingam, J.-M. Halbout, G. V. Kopcsay, “Coherent broadband microwave spectroscopy using picosecond optoelectronic antennas,” Appl. Phys. Lett. 54, 307–309 (1989).
[CrossRef]

Auston, D. H.

N. Froberg, M. Mack, B. B. Hu, X.-C. Zhang, D. H. Auston, “500 GHz electrically steerable photoconducting antenna array,” Appl. Phys. Lett. 58, 446–448 (1991).
[CrossRef]

X.-C. Zhang, B. B. Hu, J. T. Darrow, D. H. Auston, “Generation of femtosecond electromagnetic pulses from semiconductor surfaces,” Appl. Phys. Lett. 56, 1011–1013 (1990).
[CrossRef]

X.-C. Zhang, J. T. Darrow, B. B. Hu, D. H. Auston, M. T. Schmidt, P. Tham, E. S. Yang, “Optically induced electromagnetic radiation from semiconductor surfaces,” Appl. Phys. Lett. 56, 2228–2230 (1990).
[CrossRef]

Cook, B. D.

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

Darrow, J. T.

X.-C. Zhang, J. T. Darrow, B. B. Hu, D. H. Auston, M. T. Schmidt, P. Tham, E. S. Yang, “Optically induced electromagnetic radiation from semiconductor surfaces,” Appl. Phys. Lett. 56, 2228–2230 (1990).
[CrossRef]

X.-C. Zhang, B. B. Hu, J. T. Darrow, D. H. Auston, “Generation of femtosecond electromagnetic pulses from semiconductor surfaces,” Appl. Phys. Lett. 56, 1011–1013 (1990).
[CrossRef]

deFonzo, A. P.

C. R. Lutz, A. P. deFonzo, “Far-field characteristics of optically pulsed millimeter wave antennas,” Appl. Phys. Lett. 54, 2186–2188 (1989).
[CrossRef]

Fattinger, Ch.

M. van Exeter, Ch. Fattinger, D. R. Grischkowsky, “High-brightness terahertz beams characterized with an ultrafast detector,” Appl. Phys. Lett. 55, 337–339 (1989).
[CrossRef]

Froberg, N.

N. Froberg, M. Mack, B. B. Hu, X.-C. Zhang, D. H. Auston, “500 GHz electrically steerable photoconducting antenna array,” Appl. Phys. Lett. 58, 446–448 (1991).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Grischkowsky, D. R.

M. van Exeter, D. R. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microwave Theory Tech. 38, 1684–1691 (1990).
[CrossRef]

M. van Exeter, Ch. Fattinger, D. R. Grischkowsky, “High-brightness terahertz beams characterized with an ultrafast detector,” Appl. Phys. Lett. 55, 337–339 (1989).
[CrossRef]

Halbout, J.-M.

Y. Pastol, G. Arjavalingam, J.-M. Halbout, G. V. Kopcsay, “Coherent broadband microwave spectroscopy using picosecond optoelectronic antennas,” Appl. Phys. Lett. 54, 307–309 (1989).
[CrossRef]

Hu, B. B.

N. Froberg, M. Mack, B. B. Hu, X.-C. Zhang, D. H. Auston, “500 GHz electrically steerable photoconducting antenna array,” Appl. Phys. Lett. 58, 446–448 (1991).
[CrossRef]

X.-C. Zhang, J. T. Darrow, B. B. Hu, D. H. Auston, M. T. Schmidt, P. Tham, E. S. Yang, “Optically induced electromagnetic radiation from semiconductor surfaces,” Appl. Phys. Lett. 56, 2228–2230 (1990).
[CrossRef]

X.-C. Zhang, B. B. Hu, J. T. Darrow, D. H. Auston, “Generation of femtosecond electromagnetic pulses from semiconductor surfaces,” Appl. Phys. Lett. 56, 1011–1013 (1990).
[CrossRef]

Kopcsay, G. V.

Y. Pastol, G. Arjavalingam, J.-M. Halbout, G. V. Kopcsay, “Coherent broadband microwave spectroscopy using picosecond optoelectronic antennas,” Appl. Phys. Lett. 54, 307–309 (1989).
[CrossRef]

Lewis, D. K.

R. W. Ziolkowski, D. K. Lewis, “Verification of the localized-wave transmission effect,” J. Appl. Phys. 68, 6083–6086 (1990).
[CrossRef]

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

Lutz, C. R.

C. R. Lutz, A. P. deFonzo, “Far-field characteristics of optically pulsed millimeter wave antennas,” Appl. Phys. Lett. 54, 2186–2188 (1989).
[CrossRef]

Mack, M.

N. Froberg, M. Mack, B. B. Hu, X.-C. Zhang, D. H. Auston, “500 GHz electrically steerable photoconducting antenna array,” Appl. Phys. Lett. 58, 446–448 (1991).
[CrossRef]

Pastol, Y.

Y. Pastol, G. Arjavalingam, J.-M. Halbout, G. V. Kopcsay, “Coherent broadband microwave spectroscopy using picosecond optoelectronic antennas,” Appl. Phys. Lett. 54, 307–309 (1989).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Schmidt, M. T.

X.-C. Zhang, J. T. Darrow, B. B. Hu, D. H. Auston, M. T. Schmidt, P. Tham, E. S. Yang, “Optically induced electromagnetic radiation from semiconductor surfaces,” Appl. Phys. Lett. 56, 2228–2230 (1990).
[CrossRef]

Siegman, A.

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

Tham, P.

X.-C. Zhang, J. T. Darrow, B. B. Hu, D. H. Auston, M. T. Schmidt, P. Tham, E. S. Yang, “Optically induced electromagnetic radiation from semiconductor surfaces,” Appl. Phys. Lett. 56, 2228–2230 (1990).
[CrossRef]

van Exeter, M.

M. van Exeter, D. R. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microwave Theory Tech. 38, 1684–1691 (1990).
[CrossRef]

M. van Exeter, Ch. Fattinger, D. R. Grischkowsky, “High-brightness terahertz beams characterized with an ultrafast detector,” Appl. Phys. Lett. 55, 337–339 (1989).
[CrossRef]

Yang, E. S.

X.-C. Zhang, J. T. Darrow, B. B. Hu, D. H. Auston, M. T. Schmidt, P. Tham, E. S. Yang, “Optically induced electromagnetic radiation from semiconductor surfaces,” Appl. Phys. Lett. 56, 2228–2230 (1990).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989).

Zhang, X.-C.

N. Froberg, M. Mack, B. B. Hu, X.-C. Zhang, D. H. Auston, “500 GHz electrically steerable photoconducting antenna array,” Appl. Phys. Lett. 58, 446–448 (1991).
[CrossRef]

X.-C. Zhang, J. T. Darrow, B. B. Hu, D. H. Auston, M. T. Schmidt, P. Tham, E. S. Yang, “Optically induced electromagnetic radiation from semiconductor surfaces,” Appl. Phys. Lett. 56, 2228–2230 (1990).
[CrossRef]

X.-C. Zhang, B. B. Hu, J. T. Darrow, D. H. Auston, “Generation of femtosecond electromagnetic pulses from semiconductor surfaces,” Appl. Phys. Lett. 56, 1011–1013 (1990).
[CrossRef]

Ziolkowski, R. W.

R. W. Ziolkowski, “Properties of electromagnetic beams generated by ultra-wide bandwidth pulse-driven arrays,”IEEE Trans. Antennas Propag. 40, 888–905 (1992).
[CrossRef]

R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3984 (1991).
[CrossRef] [PubMed]

R. W. Ziolkowski, D. K. Lewis, “Verification of the localized-wave transmission effect,” J. Appl. Phys. 68, 6083–6086 (1990).
[CrossRef]

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

Appl. Phys. Lett. (6)

M. van Exeter, Ch. Fattinger, D. R. Grischkowsky, “High-brightness terahertz beams characterized with an ultrafast detector,” Appl. Phys. Lett. 55, 337–339 (1989).
[CrossRef]

Y. Pastol, G. Arjavalingam, J.-M. Halbout, G. V. Kopcsay, “Coherent broadband microwave spectroscopy using picosecond optoelectronic antennas,” Appl. Phys. Lett. 54, 307–309 (1989).
[CrossRef]

C. R. Lutz, A. P. deFonzo, “Far-field characteristics of optically pulsed millimeter wave antennas,” Appl. Phys. Lett. 54, 2186–2188 (1989).
[CrossRef]

X.-C. Zhang, B. B. Hu, J. T. Darrow, D. H. Auston, “Generation of femtosecond electromagnetic pulses from semiconductor surfaces,” Appl. Phys. Lett. 56, 1011–1013 (1990).
[CrossRef]

X.-C. Zhang, J. T. Darrow, B. B. Hu, D. H. Auston, M. T. Schmidt, P. Tham, E. S. Yang, “Optically induced electromagnetic radiation from semiconductor surfaces,” Appl. Phys. Lett. 56, 2228–2230 (1990).
[CrossRef]

N. Froberg, M. Mack, B. B. Hu, X.-C. Zhang, D. H. Auston, “500 GHz electrically steerable photoconducting antenna array,” Appl. Phys. Lett. 58, 446–448 (1991).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

R. W. Ziolkowski, “Properties of electromagnetic beams generated by ultra-wide bandwidth pulse-driven arrays,”IEEE Trans. Antennas Propag. 40, 888–905 (1992).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

M. van Exeter, D. R. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microwave Theory Tech. 38, 1684–1691 (1990).
[CrossRef]

J. Appl. Phys. (1)

R. W. Ziolkowski, D. K. Lewis, “Verification of the localized-wave transmission effect,” J. Appl. Phys. 68, 6083–6086 (1990).
[CrossRef]

Phys. Rev. A (1)

R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3984 (1991).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

Other (3)

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989).

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

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Figures (12)

Fig. 1
Fig. 1

Energy-distribution envelopes. Diffraction affects the higher-order coherence properties of a beam generated by an ultrawide-bandwidth pulse-driven aperture more slowly than it affects the lower-order coherence properties. The evelopes of the energy distribution in a pulsed beam are defined by their diffraction lengths and the rate of spread of the beam in the region beyond their diffraction lengths.

Fig. 2
Fig. 2

Time history of the initial Gaussian pulse.

Fig. 3
Fig. 3

Fourier spectrum of the initial Gaussian pulse.

Fig. 4
Fig. 4

Contour map of the pulsed Gaussian beam field as observed in the planes (a) z = 15.0 μm, (b) z = 30.0 μm, (c) z = 45.0 μm, and (d) z = 60.0 μm. The transverse distance from the propagation axis is given along the vertical axis.

Fig. 5
Fig. 5

Pulsed Gaussian beam’s time signals along the propagation axis at distances (a) z = 0.0 μm, (b) z = 2.0 μm, (c) z = 4.0 μm, and (d) z = 16.0 μm. As the field propagates from the initial aperture, it evolves into a time derivative of the initial field.

Fig. 6
Fig. 6

Intensity and energy profiles of the cw Gaussian beam (solid curve) versus distance along the direction of propagation. The rate of spread of the intensity and energy are identical. The cw frequency is f = frad = 2.19 × 1014 Hz, and the initial waist of the Gaussian amplitude taper is w0 = 2.5 μm. The oblique dashed line represents the asymptote given by relation (7).

Fig. 7
Fig. 7

Rates of spread of the pulsed Gaussian (PG) beam’s maximum-intensity profile for the one-time-derivative (1TD) and two-time-derivative (2TD) systems versus distance along the direction of propagation. These curves are obtained directly from numerical simulations of the beam fields defined by Eq. (9) and its time derivative.

Fig. 8
Fig. 8

Rates of spread of the pulsed Gaussian (PG) beam’s energy profile for the one-time-derivative (1TD) and two-time-derivative (2TD) systems versus distance along the direction of propagation. These curves are obtained directly from numerical simulations of the beam fields defined by Eq. (9) and its time derivative.

Fig. 9
Fig. 9

Maximum-intensity profile of various beams versus distance along the direction of propagation. The rate of spread in the far field of the maximum intensity is generated with Eq. (22’) for the one-time-derivative (1TD) pulsed Gaussian (PG) beam field and with relation (42) for the two-time-derivative (2TD) pulsed Gaussian beam field. The solid curves represent the beam-spread values for the corresponding cw Gaussian field with ωcw = ωrad. The dashed line represents the asymptote to the cw curve.

Fig. 10
Fig. 10

Energy profile of various beams versus distance along the direction of propagation. The rate of spread in the far field of the energy is generated with Eq. (27′) for the one-time-derivative (1TD) pulsed Gaussian (PG) beam field and with relation (43) for the two-time-derivative (2TD) pulsed Gaussian beam field. The solid curves represent the beam-spread values for the corresponding cw Gaussian field with ωcw = ωrad. The dashed line represents the asymptote to the cw curve.

Fig. 11
Fig. 11

Intensity and energy profiles of the pulsed Gaussian beam in the plane z = 200.0 μm. Both profiles have been normalized to unity.

Fig. 12
Fig. 12

Contour map of the pulsed Gaussian beam field measured in the plane z = 45.0 μm with a one-time-derivative receiving (measurement) system. The transverse distance from the propagation axis is given along the vertical axis.

Equations (57)

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F ( t ) = exp ( - p t 2 ) .
E ˜ ( r , ϕ , z , ω ) = E 0 ω 0 ω ( z ) exp { - i [ k z - η ( z ) ] - r 2 [ 1 w 2 ( z ) + i k 2 R ( z ) ] } ,
w 2 ( z ) = w 0 2 [ 1 + ( λ z π w 0 2 ) 2 ] = w 0 2 ( 1 + z 2 z 0 2 ) ,
R ( z ) = z [ 1 + ( π w 0 2 λ z ) 2 ] = z ( 1 + z 0 2 z 2 ) ,
η ( z ) = tan - 1 ( λ z π w 0 2 ) = tan - 1 ( z z 0 ) ,
z 0 π w 0 2 λ = ( w 0 2 2 c ) ω .
θ cw = tan - 1 [ w ( z ) 2 z ] tan - 1 ( λ 2 π w 0 ) 2 ( λ 2 π w 0 ) .
F ˜ ( ω ) = ( π / p ) 1 / 2 exp ( - ω 2 / 4 p ) .
E ( r , ϕ , z , t ) = 1 2 π - E ˜ ( r , ϕ , z , ω ) F ˜ ( ω ) exp ( + i ω t ) d ω .
E ˜ ( r , ϕ , z , ω ) E 0 exp ( - i k z ) exp ( - r 2 / w 0 2 )             ( z z 0 ) .
w ( z ) w 0 ( z / z 0 ) ,
E ˜ ( r , ϕ , z , ω ) E 0 z 0 z exp { - i ( k z - π 2 ) - r 2 [ ( z 0 w 0 z ) 2 + i k 2 z ] } = i E 0 w 0 2 2 c z ω exp [ - i ω ( z c + r 2 2 c z ) - ω 2 ( w 0 r 2 c z ) 2 ]             ( z z 0 ) .
E ( r , ϕ , z , t ) E 0 exp ( - r 2 / w 0 2 ) exp [ - p ( z / c - t ) 2 ]
E ( r , ϕ , z , t ) E 0 2 π ( π p ) 1 / 2 w 0 2 2 c z - i ω exp { - ω 2 [ 1 4 p + ( w 0 r 2 c z ) 2 ] } × exp { - i ω [ ( z c - t ) + r 2 2 c z ] } d ω = E 0 2 π ( π p ) 1 / 2 w 0 2 2 c z t ( - exp { - ω 2 [ 1 4 p + ( w 0 r 2 c z ) 2 ] } × exp { - i ω [ ( z c - t ) + r 2 2 c z ] } d ω ) = E 0 p w 0 2 c z [ 1 1 + p ( w 0 r / c z ) 2 ] 3 / 2 ( z c - t + r 2 2 c z ) × exp [ - p ( z / c - t + r 2 / 2 c z ) 2 1 + p ( w 0 r / c z ) 2 ]
I ( r , ϕ , z , t ) E ( r , ϕ , z , t ) 2 = E 0 2 exp ( - 2 r 2 / w 0 2 ) exp [ - 2 p ( z / c - t ) 2 ]
I ( r , ϕ , z , t ) E 0 2 ( p w 0 2 c z ) 2 [ 1 1 + p ( w 0 r / c z ) 2 ] 3 ( z c - t + r 2 2 c z ) 2 × exp [ - 2 p ( z / c - t + r 2 / 2 c z ) 2 1 + p ( w 0 r / c z ) 2 ] .
t max = z c + r 2 2 c z ± ( 1 2 p + w 0 2 r 2 2 c 2 z 2 ) 1 / 2
I max ( r , ϕ , z ) = E 0 2 exp ( - 2 r 2 / w 0 2 )
I max ( r , ϕ , z ) 1 2 e E 0 2 ( w 0 2 c z ) 2 p [ 1 + p ( w 0 r c z ) 2 ] - 2
Γ int , ( r = 0 , ϕ , z ) = def I max ( r = 0 , ϕ , z ) I max ( r = 0 , ϕ , z = 0 ) = { 1.0 ( near field ) p 2 e ( w 0 2 z c ) 2 ( far field ) .
Γ int , ( r , ϕ , z ) = I max ( r , ϕ , z ) I max ( r = 0 , ϕ , z ) = { exp ( - 2 r 2 / w 0 2 ) ( near field ) ( 1 + p w 0 2 r 2 c 2 z 2 ) - 2 ( far field ) .
θ int = def r z = [ ( R int - 1 / 2 - 1 ) c 2 p w 0 2 ] 1 / 2 .
E ( r , ϕ , z ) - E 0 2 exp ( - 2 r 2 / w 0 2 ) exp [ - 2 p ( z / c - t ) 2 ] d t = E 0 2 exp ( - 2 r 2 / w 0 2 ) ( π 2 p ) 1 / 2
E ( r , ϕ , z ) E 0 2 [ p w 0 2 / ( c z ) ] 2 { 1 + p [ w 0 r / ( c z ) ] 2 } 3 - ( z c - t + r 2 2 c z ) 2 × exp [ - 2 p ( z / c - t + r 2 / 2 c z ) 2 1 + p ( w 0 r / c z ) 2 ] d t = E 0 2 ( w 0 2 2 c z ) 2 ( π 2 p ) 1 / 2 [ 1 + p ( w 0 r c z ) 2 ] - 3 / 2
Γ enrg , ( r = 0 , ϕ , z ) = def E ( r = 0 , ϕ , z ) E ( r = 0 , ϕ , z = 0 ) = { 1.0 ( near field ) p ( w 0 2 / 2 c z ) 2 ( far field ) .
Γ enrg , ( r , ϕ , z ) = E ( r , ϕ , z ) E ( r = 0 , ϕ , z ) = { exp ( - 2 r 2 / w 0 2 ) ( near field ) [ 1 + p ( w 0 2 r 2 / c 2 z 2 ) ] - 3 / 2 ( far field ) .
θ enrg = def r z = [ ( R enrg - 2 / 3 - 1 ) c 2 p w 0 2 ] 1 / 2 .
ω rad 2 = def A d S - d t t F ( r , t ) 2 A d S - d t F ( r , t ) 2 = - d t t F ( t ) 2 - d t F ( t ) 2 = p .
λ rad = 2 π c ω rad = 2 π c p ,
L rad = π w 0 2 λ rad .
Γ int , ( r = 0 , ϕ , z ) = 2 e ( π w 0 2 λ rad z ) 2 = 2 e ( L rad z ) 2 ( L int z ) 2 ,
θ int = ( R int - 1 / 2 - 1 ) 1 / 2 λ rad 2 π w 0 ,
Γ enrg , ( r = 0 , ϕ , z ) = ( π w 0 2 λ rad z ) 2 = ( L rad z ) 2 ( L enrg z ) 2 ,
θ enrg = ( R enrg - 2 / 3 - 1 ) 1 / 2 λ rad 2 π w 0 .
L int L enrg = ( 2 e ) 1 / 2 = 0.858 ,
θ int θ enrg = ( R int - 1 / 2 - 1 R enrg - 2 / 3 - 1 ) 1 / 2 = 0.827.
ϒ rad = def max t A d S t F ( r , t ) 2 A d S - d t t F ( r , t ) 2 = 4 e ( p 2 π ) 1 / 2 = 2 2 e π ω rad ,
Γ ˜ int , ( r = 0 , ϕ , z ) = def I max ( r = 0 , ϕ , z ) ω rad E ( r = 0 , ϕ , z = 0 ) = 2 2 e π ( π w 0 2 λ rad z ) 2 ϒ rad ω rad ( L rad z ) 2 ,
Λ = 1 4 p + ( w 0 r 2 c z ) 2 .
E meas ( r , ϕ , z , t ) = def t E ( r , ϕ , z , t ) E 0 p w 0 2 8 c z [ ( z / c - t + r 2 / 2 c z ) 2 - 2 Λ 2 Λ 5 / 2 ] × exp [ - ( z / c - t + r 2 / 2 c z ) 2 4 Λ ] .
I max meas ( r , ϕ , z ) = E 0 2 p ( w 0 2 8 c z ) 2 4 e 3 Λ - 3 ,
E meas ( r , ϕ , z ) = E 0 2 p ( w 0 2 8 c z ) 2 3 2 π 4 Λ - 5 / 2 .
ω meas 2 = def A d S - d t t 2 F ( r , t ) 2 A d S - d t t F ( r , t ) 2 = 3 p = 3 ω rad 2 ,
ϒ meas = d e f max t A d S t 2 F ( r , t ) 2 A d S - d t t 2 F ( r , t ) 2 = 32 3 e 3 ( p 2 π ) 1 / 2 = 32 3 e 3 2 π ω rad .
Γ int , meas ( r , ϕ , z ) = def I max meas ( r , ϕ , z ) I max meas ( r = 0 , ϕ , z ) = ( 1 + p w 0 2 r 2 c 2 z 2 ) - 3 ,
Γ enrg , meas ( r , ϕ , z ) = def E meas ( r , ϕ , z ) E meas ( r = 0 , ϕ , z ) = ( 1 + p w 0 2 r 2 c 2 z 2 ) - 5 / 2 ,
θ int meas ( R int - 1 / 3 - 1 ) 1 / 2 λ rad 2 π w 0 ,
θ enrg meas ( R enrg - 2 / 5 - 1 ) 1 / 2 λ rad 2 π w 0 ,
Γ ˜ enrg , meas ( r = 0 , ϕ , z ) = def E meas ( r = 0 , ϕ , z ) ω rad 2 E ( r = 0 , ϕ , z = 0 ) = ( ω meas ω rad ) 2 ( L rad z ) 2 ( L enrg meas z ) 2 ,
Γ ˜ int , meas ( r = 0 , ϕ , z ) = def I max meas ( r = 0 , ϕ , z ) ω rad 3 E ( r = 0 , ϕ , z = 0 ) = ϒ meas ω rad ( ω meas ω rad ) 2 ( L rad z ) 2 ( L int meas z ) 2 ,
L enrg meas = def ( ω meas ω rad ) L rad ,
L int meas = def ( ϒ meas ω rad ) 1 / 2 ( ω meas ω rad ) L rad .
L enrg meas = 1.73 L rad = 1.73 L cw ,
L int meas = 0.797 L rad = 0.797 L cw ,
θ int meas = 0.780 θ int rad = 0.444 θ int cw ,
θ enrg meas = 0.718 θ enrg rad = 0.496 θ enrg cw .
A 1 / e × π ( θ enrg meas ) 2 = π 2 w 0 2 2 ( R int - 1 / 3 - 1 ) ( λ rad 2 π w 0 ) 2 = 9 ( e 2 / 3 - 1 ) 8 λ meas 2 = 1.07 λ meas 2 λ meas 2 .

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