Abstract

We describe the use of a two-wavelength beam deflection technique in the measurement of electron density and expansion velocity in a laser-produced plasma. Beam deflection measurements are made with a spatial resolution of 250 μm, temporal resolution of 25 ns, and a dynamic range of 1000. Several techniques for determining the spatial and temporal variation of the electron density from beam deflection measurements are described.

© 1991 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  4. R. A. Lacy, A. C. Nilsson, R. L. Byer, W. T. Silfvast, O. R. Wood, S. Svanberg, “Photoionization-Pumped Gain at 185 nm in a Laser-Ablated Indium Plasma,” J. Opt. Soc. Am. B 6, 1209–1216 (1989).
    [CrossRef]
  5. J. B. Hopkins, P. R. R. Langridge-Smith, M. D. Morse, R. E. Smalley, “Supersonic Metal Cluster Beams of Refractory Metals: Spectral Investigations of Ultracold Mo2,” J. Chem. Phys. 78, 1627–1637 (1983).
    [CrossRef]
  6. M. L. Bortz, R. H. French, “Optical Reflectivity Measurements Using a Laser Plasma Light Source,” Appl. Phys. Lett. 55, 1955–1957 (1989).
    [CrossRef]
  7. M. M. Murnane, H. C. Kapteyn, R. W. Falcone, “High Density Plasmas Produced by Ultrafast Laser Pulses,” Phys. Rev. Lett. 62, 155–158 (1989).
    [CrossRef] [PubMed]
  8. J. A. Trail, R. L. Byer, “Compact Scanning Soft-X-Ray Microscope Using a Laser-Produced Plasma Source and Normal-Incidence Multilayer Mirrors,” Opt. Lett. 14, 539–541 (1989).
    [CrossRef] [PubMed]
  9. P. Gohil, H. Kapoor, D. Ma, M. C. Pekerar, T. J. McIlrath, M. L. Ginter, “Soft X-Ray Lithography Using Radiation from Laser-Produced Plasmas,” Appl. Opt. 24, 2024–2027 (1985).
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  10. S. Maxon et al., “Calculation for Ni-Like Soft X-Ray Lasers: Optimization for W (43.1 Å),” Phys. Rev. Lett. 63, 236–239 (1989).
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  11. A. Richter, H.-J. Scheibe, W. Pompe, K.-W. Brzezinka, I. Muhling, “About the Structure and Bonding of Laser Generated Carbon Films by Raman and Electron Energy Loss Spectroscopy,” J. Non-Cryst. Solids 88, 131–144 (1986); C. B. Collins, F. Davanloo, E. M. Juengerman, W. R. Osborn, D. R. Jander, “Laser Plasma Source of Amorphic Diamond,” Appl. Phys. Lett. 54, 216–218 (1989).
    [CrossRef]
  12. D. Dijkkamp et al., “Preparation of Y–Ba–Cu Oxide Superconductor Thin Films Using Pulsed Laser Evaporation from High Tc Bulk Material,” Appl. Phys. Lett. 51, 619–621 (1987); M. J. Ferrari et al., “Low Magnetic Flux Noise Observed in Laser-Deposited In Situ Films of YB2Cu3O and Implications for High-Tc SQUIDs,” Nature (London) 341, 723–725 (1989).
    [CrossRef]
  13. R. Srinivasan, V. Mayne-Banton, “Self-Developing Photoetching of Poly(ethylene Terephthalate) Films by Far-Ultraviolet Excimer Laser Radiation,” Appl. Phys. Lett. 41, 576–578 (1982); S. Lazare, V. Granier, “Excimer Laser Light Induced Ablation and Reactions at Polymer Surfaces as Measured with a Quartz-Crystal Microbalance,” J. Appl. Phys. 63, 2110–2115 (1988).
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  14. G. W. Faris, R. L. Byer, “Three-Dimensional Beam-Deflection Optical Tomography of a Supersonic Jet,” Appl. Opt. 27, 5202–5212 (1988).
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  15. H. Bergstrom, G. Faris, H. Hallstadius, H. Lundberg, A. Persson, C.-G. Wahlstrom, “Spectroscopy on Laser-Evaporated Boron and Carbon,” in Technical Digest, Eighth International Conference on Laser Spectroscopy, Are, Sweden (22–26 June 1987).
  16. C. L. Enloe, R. M. Gilgenbach, J. S. Meachum, “Fast, Sensitive Laser Deflection System Suitable for Transient Plasma Analysis,” Rev. Sci. Instrum. 58, 1597–1600 (1987).
    [CrossRef]
  17. G. W. Faris, R. L. Byer, “Quantitative Three-Dimensional Optical Toiographic Imaging of Supersonic Flows,” Science 238, 1700–1702 (1987).
    [CrossRef] [PubMed]
  18. G. W. Faris, R. L. Byer, “Beam-Deflection Optical Tomography of a Flame,” Opt. Lett. 12, 155–157 (1987).
    [CrossRef] [PubMed]
  19. I. C. Potter, I. S. Falconer, W. I. B. Smith, “Measurement of the Electron Density Distribution in Plasmas from the Bending of a Gas Laser Beam,” J. Phys. E 5, 910–914 (1972).
    [CrossRef]
  20. P. W. Schreiber, A. M. Hunter, D. R. Smith, “The Determination of Plasma Electron Density from Refraction Measurements,” Plasma Phys. 15, 635–646 (1973).
    [CrossRef]
  21. H. Schmidt, B. Ruckle, “Beam Deviation Method as a Diagnostic Tool for the Plasma Focus,” Appl. Opt. 17, 1275–1279 (1978).
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  22. M. A. Greenspan, K. V. Reddy, “A Laser Deflection Technique for Sensitive Measurements of a Reduced-Density Channel in Neutral Gas,” Appl. Phys. Lett. 40, 576–578 (1982).
    [CrossRef]
  23. J. M. Green, W. T. Silfvast, O. R. Wood, “Evolution of a CO2-Laser-Produced Cadmium Plasma,” J. Appl. Phys. 48, 2753–2761 (1977).
    [CrossRef]
  24. G. Koren, “Observation of Shock Waves and Cooling Waves in the Laser Ablation of Kapton Films in Air,” Appl. Phys. Lett. 51, 569–571 (1987).
    [CrossRef]
  25. S. Petzoldt, A. P. Elg, M. Reighling, J. Reif, E. Matthias, “Surface Laser Damage Thresholds Determined by Photo-acoustic Deflection,” Appl. Phys. Lett. 53, 2005–2007 (1988).
    [CrossRef]
  26. J. A. Sell, D. M. Heffelfinger, P. Vantzek, R. M. Gilgenbach, “Laser Beam Deflection as a Probe of Laser Ablation of Materials,” Appl. Phys. Lett. 55, 2435–2437 (1989).
    [CrossRef]
  27. F. C. Jahoda, G. A. Sawyer, “Optical Refractivity of Plasmas,” in Plasma PhysicsR. H. Lovberg, H. R. Griem, Eds. (Academic, New York, 1971), Part B, Vol. 9, pp. 1–48.
    [CrossRef]
  28. W. L. Wiese, G. A. Martin, “Atomic Transition Probabilities,” in Handbook of Chemistry and Physics, R. C. Weast, M. J. Astle, Eds. (CRC Press, West Palm Beach, FL, 1978), pp. 112–140.
  29. F. J. Allen, “Production of High-Energy Ions in Laser-Produced Plasmas,” J. Appl. Phys. 43, 2169–2175 (1972).
    [CrossRef]
  30. A. V. Gurevich, L. V. Pariiskaya, L. P. Pitaevskii, “Self-Similar Motion of a Rarefied Plasma,” Sov. Phys. JETP 22, 449–454 (1966); P. Mora, “Self-Similar Expansion of a Plasma into a Vacuum,” Phys. Fluids 22, 2300–2304 (1979).
    [CrossRef]
  31. P. T. Rumsby, J. W. M. Paul, “Temperature and Density of an Expanding Laser-Produced Plasma,” Plasma Phys. 16, 247–260 (1974).
    [CrossRef]
  32. G. J. Tallents, “On the Fitting of Displaced Maxwellians to Laser-Produced Plasma Ion Velocity Distributions,” Opt. Commun. 37, 108–112 (1981).
    [CrossRef]
  33. F. Keilmann, “An Infrared Schlieren Interferometer for Measuring Electron Density Profiles,” Plasma Phys. 14, 111–122 (1972).
    [CrossRef]
  34. Y. A. Bykovskii, V. G. Degtyarev, N. N. Degtyarenko, V. F. Elesin, I. D. Laptev, V. N. Nevolin, “Ion Energies in a Laser-Produced Plasma,” Sov. Phys. Tech. Phys. 17, 517–520 (1972).
  35. J. Pawliszyn, “LEDs and Laser Diodes in Schlieren Optics Methods,” Rev. Sci. Instrum. 58, 245–248 (1987).
    [CrossRef]
  36. G. W. Faris, H. M. Hertz, “Tunable Differential Interferometer for Optical Tomography,” Appl. Opt. 28, 4662–4667 (1989).
    [CrossRef] [PubMed]

1989 (7)

M. L. Bortz, R. H. French, “Optical Reflectivity Measurements Using a Laser Plasma Light Source,” Appl. Phys. Lett. 55, 1955–1957 (1989).
[CrossRef]

M. M. Murnane, H. C. Kapteyn, R. W. Falcone, “High Density Plasmas Produced by Ultrafast Laser Pulses,” Phys. Rev. Lett. 62, 155–158 (1989).
[CrossRef] [PubMed]

S. Maxon et al., “Calculation for Ni-Like Soft X-Ray Lasers: Optimization for W (43.1 Å),” Phys. Rev. Lett. 63, 236–239 (1989).
[CrossRef] [PubMed]

J. A. Sell, D. M. Heffelfinger, P. Vantzek, R. M. Gilgenbach, “Laser Beam Deflection as a Probe of Laser Ablation of Materials,” Appl. Phys. Lett. 55, 2435–2437 (1989).
[CrossRef]

R. A. Lacy, A. C. Nilsson, R. L. Byer, W. T. Silfvast, O. R. Wood, S. Svanberg, “Photoionization-Pumped Gain at 185 nm in a Laser-Ablated Indium Plasma,” J. Opt. Soc. Am. B 6, 1209–1216 (1989).
[CrossRef]

J. A. Trail, R. L. Byer, “Compact Scanning Soft-X-Ray Microscope Using a Laser-Produced Plasma Source and Normal-Incidence Multilayer Mirrors,” Opt. Lett. 14, 539–541 (1989).
[CrossRef] [PubMed]

G. W. Faris, H. M. Hertz, “Tunable Differential Interferometer for Optical Tomography,” Appl. Opt. 28, 4662–4667 (1989).
[CrossRef] [PubMed]

1988 (3)

G. W. Faris, R. L. Byer, “Three-Dimensional Beam-Deflection Optical Tomography of a Supersonic Jet,” Appl. Opt. 27, 5202–5212 (1988).
[CrossRef] [PubMed]

H. Bergstrom, G. W. Faris, H. Hallstadius, H. Lundberg, A. Persson, C.-G. Wahlstrom, “Radiative Lifetime and Hyperfine-Structure Studies on Laser-Evaporated Boron,” Z. Phys. D 8, 17–23 (1988).
[CrossRef]

S. Petzoldt, A. P. Elg, M. Reighling, J. Reif, E. Matthias, “Surface Laser Damage Thresholds Determined by Photo-acoustic Deflection,” Appl. Phys. Lett. 53, 2005–2007 (1988).
[CrossRef]

1987 (6)

G. Koren, “Observation of Shock Waves and Cooling Waves in the Laser Ablation of Kapton Films in Air,” Appl. Phys. Lett. 51, 569–571 (1987).
[CrossRef]

D. Dijkkamp et al., “Preparation of Y–Ba–Cu Oxide Superconductor Thin Films Using Pulsed Laser Evaporation from High Tc Bulk Material,” Appl. Phys. Lett. 51, 619–621 (1987); M. J. Ferrari et al., “Low Magnetic Flux Noise Observed in Laser-Deposited In Situ Films of YB2Cu3O and Implications for High-Tc SQUIDs,” Nature (London) 341, 723–725 (1989).
[CrossRef]

C. L. Enloe, R. M. Gilgenbach, J. S. Meachum, “Fast, Sensitive Laser Deflection System Suitable for Transient Plasma Analysis,” Rev. Sci. Instrum. 58, 1597–1600 (1987).
[CrossRef]

G. W. Faris, R. L. Byer, “Quantitative Three-Dimensional Optical Toiographic Imaging of Supersonic Flows,” Science 238, 1700–1702 (1987).
[CrossRef] [PubMed]

G. W. Faris, R. L. Byer, “Beam-Deflection Optical Tomography of a Flame,” Opt. Lett. 12, 155–157 (1987).
[CrossRef] [PubMed]

J. Pawliszyn, “LEDs and Laser Diodes in Schlieren Optics Methods,” Rev. Sci. Instrum. 58, 245–248 (1987).
[CrossRef]

1986 (1)

A. Richter, H.-J. Scheibe, W. Pompe, K.-W. Brzezinka, I. Muhling, “About the Structure and Bonding of Laser Generated Carbon Films by Raman and Electron Energy Loss Spectroscopy,” J. Non-Cryst. Solids 88, 131–144 (1986); C. B. Collins, F. Davanloo, E. M. Juengerman, W. R. Osborn, D. R. Jander, “Laser Plasma Source of Amorphic Diamond,” Appl. Phys. Lett. 54, 216–218 (1989).
[CrossRef]

1985 (1)

1983 (1)

J. B. Hopkins, P. R. R. Langridge-Smith, M. D. Morse, R. E. Smalley, “Supersonic Metal Cluster Beams of Refractory Metals: Spectral Investigations of Ultracold Mo2,” J. Chem. Phys. 78, 1627–1637 (1983).
[CrossRef]

1982 (2)

R. Srinivasan, V. Mayne-Banton, “Self-Developing Photoetching of Poly(ethylene Terephthalate) Films by Far-Ultraviolet Excimer Laser Radiation,” Appl. Phys. Lett. 41, 576–578 (1982); S. Lazare, V. Granier, “Excimer Laser Light Induced Ablation and Reactions at Polymer Surfaces as Measured with a Quartz-Crystal Microbalance,” J. Appl. Phys. 63, 2110–2115 (1988).
[CrossRef]

M. A. Greenspan, K. V. Reddy, “A Laser Deflection Technique for Sensitive Measurements of a Reduced-Density Channel in Neutral Gas,” Appl. Phys. Lett. 40, 576–578 (1982).
[CrossRef]

1981 (2)

G. J. Tallents, “On the Fitting of Displaced Maxwellians to Laser-Produced Plasma Ion Velocity Distributions,” Opt. Commun. 37, 108–112 (1981).
[CrossRef]

P. K. Carroll, E. T. Kennedy, “Laser-Produced Plasmas,” Contemp. Phys. 22, 61–96 (1981).
[CrossRef]

1978 (1)

1977 (1)

J. M. Green, W. T. Silfvast, O. R. Wood, “Evolution of a CO2-Laser-Produced Cadmium Plasma,” J. Appl. Phys. 48, 2753–2761 (1977).
[CrossRef]

1974 (1)

P. T. Rumsby, J. W. M. Paul, “Temperature and Density of an Expanding Laser-Produced Plasma,” Plasma Phys. 16, 247–260 (1974).
[CrossRef]

1973 (1)

P. W. Schreiber, A. M. Hunter, D. R. Smith, “The Determination of Plasma Electron Density from Refraction Measurements,” Plasma Phys. 15, 635–646 (1973).
[CrossRef]

1972 (4)

I. C. Potter, I. S. Falconer, W. I. B. Smith, “Measurement of the Electron Density Distribution in Plasmas from the Bending of a Gas Laser Beam,” J. Phys. E 5, 910–914 (1972).
[CrossRef]

F. Keilmann, “An Infrared Schlieren Interferometer for Measuring Electron Density Profiles,” Plasma Phys. 14, 111–122 (1972).
[CrossRef]

Y. A. Bykovskii, V. G. Degtyarev, N. N. Degtyarenko, V. F. Elesin, I. D. Laptev, V. N. Nevolin, “Ion Energies in a Laser-Produced Plasma,” Sov. Phys. Tech. Phys. 17, 517–520 (1972).

F. J. Allen, “Production of High-Energy Ions in Laser-Produced Plasmas,” J. Appl. Phys. 43, 2169–2175 (1972).
[CrossRef]

1966 (1)

A. V. Gurevich, L. V. Pariiskaya, L. P. Pitaevskii, “Self-Similar Motion of a Rarefied Plasma,” Sov. Phys. JETP 22, 449–454 (1966); P. Mora, “Self-Similar Expansion of a Plasma into a Vacuum,” Phys. Fluids 22, 2300–2304 (1979).
[CrossRef]

Allen, F. J.

F. J. Allen, “Production of High-Energy Ions in Laser-Produced Plasmas,” J. Appl. Phys. 43, 2169–2175 (1972).
[CrossRef]

Bergstrom, H.

H. Bergstrom, G. W. Faris, H. Hallstadius, H. Lundberg, A. Persson, C.-G. Wahlstrom, “Radiative Lifetime and Hyperfine-Structure Studies on Laser-Evaporated Boron,” Z. Phys. D 8, 17–23 (1988).
[CrossRef]

H. Bergstrom, G. Faris, H. Hallstadius, H. Lundberg, A. Persson, C.-G. Wahlstrom, “Spectroscopy on Laser-Evaporated Boron and Carbon,” in Technical Digest, Eighth International Conference on Laser Spectroscopy, Are, Sweden (22–26 June 1987).

Bortz, M. L.

M. L. Bortz, R. H. French, “Optical Reflectivity Measurements Using a Laser Plasma Light Source,” Appl. Phys. Lett. 55, 1955–1957 (1989).
[CrossRef]

Brzezinka, K.-W.

A. Richter, H.-J. Scheibe, W. Pompe, K.-W. Brzezinka, I. Muhling, “About the Structure and Bonding of Laser Generated Carbon Films by Raman and Electron Energy Loss Spectroscopy,” J. Non-Cryst. Solids 88, 131–144 (1986); C. B. Collins, F. Davanloo, E. M. Juengerman, W. R. Osborn, D. R. Jander, “Laser Plasma Source of Amorphic Diamond,” Appl. Phys. Lett. 54, 216–218 (1989).
[CrossRef]

Byer, R. L.

Bykovskii, Y. A.

Y. A. Bykovskii, V. G. Degtyarev, N. N. Degtyarenko, V. F. Elesin, I. D. Laptev, V. N. Nevolin, “Ion Energies in a Laser-Produced Plasma,” Sov. Phys. Tech. Phys. 17, 517–520 (1972).

Carroll, P. K.

P. K. Carroll, E. T. Kennedy, “Laser-Produced Plasmas,” Contemp. Phys. 22, 61–96 (1981).
[CrossRef]

Degtyarenko, N. N.

Y. A. Bykovskii, V. G. Degtyarev, N. N. Degtyarenko, V. F. Elesin, I. D. Laptev, V. N. Nevolin, “Ion Energies in a Laser-Produced Plasma,” Sov. Phys. Tech. Phys. 17, 517–520 (1972).

Degtyarev, V. G.

Y. A. Bykovskii, V. G. Degtyarev, N. N. Degtyarenko, V. F. Elesin, I. D. Laptev, V. N. Nevolin, “Ion Energies in a Laser-Produced Plasma,” Sov. Phys. Tech. Phys. 17, 517–520 (1972).

Dijkkamp, D.

D. Dijkkamp et al., “Preparation of Y–Ba–Cu Oxide Superconductor Thin Films Using Pulsed Laser Evaporation from High Tc Bulk Material,” Appl. Phys. Lett. 51, 619–621 (1987); M. J. Ferrari et al., “Low Magnetic Flux Noise Observed in Laser-Deposited In Situ Films of YB2Cu3O and Implications for High-Tc SQUIDs,” Nature (London) 341, 723–725 (1989).
[CrossRef]

Elesin, V. F.

Y. A. Bykovskii, V. G. Degtyarev, N. N. Degtyarenko, V. F. Elesin, I. D. Laptev, V. N. Nevolin, “Ion Energies in a Laser-Produced Plasma,” Sov. Phys. Tech. Phys. 17, 517–520 (1972).

Elg, A. P.

S. Petzoldt, A. P. Elg, M. Reighling, J. Reif, E. Matthias, “Surface Laser Damage Thresholds Determined by Photo-acoustic Deflection,” Appl. Phys. Lett. 53, 2005–2007 (1988).
[CrossRef]

Enloe, C. L.

C. L. Enloe, R. M. Gilgenbach, J. S. Meachum, “Fast, Sensitive Laser Deflection System Suitable for Transient Plasma Analysis,” Rev. Sci. Instrum. 58, 1597–1600 (1987).
[CrossRef]

Falcone, R. W.

M. M. Murnane, H. C. Kapteyn, R. W. Falcone, “High Density Plasmas Produced by Ultrafast Laser Pulses,” Phys. Rev. Lett. 62, 155–158 (1989).
[CrossRef] [PubMed]

Falconer, I. S.

I. C. Potter, I. S. Falconer, W. I. B. Smith, “Measurement of the Electron Density Distribution in Plasmas from the Bending of a Gas Laser Beam,” J. Phys. E 5, 910–914 (1972).
[CrossRef]

Faris, G.

H. Bergstrom, G. Faris, H. Hallstadius, H. Lundberg, A. Persson, C.-G. Wahlstrom, “Spectroscopy on Laser-Evaporated Boron and Carbon,” in Technical Digest, Eighth International Conference on Laser Spectroscopy, Are, Sweden (22–26 June 1987).

Faris, G. W.

G. W. Faris, H. M. Hertz, “Tunable Differential Interferometer for Optical Tomography,” Appl. Opt. 28, 4662–4667 (1989).
[CrossRef] [PubMed]

G. W. Faris, R. L. Byer, “Three-Dimensional Beam-Deflection Optical Tomography of a Supersonic Jet,” Appl. Opt. 27, 5202–5212 (1988).
[CrossRef] [PubMed]

H. Bergstrom, G. W. Faris, H. Hallstadius, H. Lundberg, A. Persson, C.-G. Wahlstrom, “Radiative Lifetime and Hyperfine-Structure Studies on Laser-Evaporated Boron,” Z. Phys. D 8, 17–23 (1988).
[CrossRef]

G. W. Faris, R. L. Byer, “Beam-Deflection Optical Tomography of a Flame,” Opt. Lett. 12, 155–157 (1987).
[CrossRef] [PubMed]

G. W. Faris, R. L. Byer, “Quantitative Three-Dimensional Optical Toiographic Imaging of Supersonic Flows,” Science 238, 1700–1702 (1987).
[CrossRef] [PubMed]

French, R. H.

M. L. Bortz, R. H. French, “Optical Reflectivity Measurements Using a Laser Plasma Light Source,” Appl. Phys. Lett. 55, 1955–1957 (1989).
[CrossRef]

Gilgenbach, R. M.

J. A. Sell, D. M. Heffelfinger, P. Vantzek, R. M. Gilgenbach, “Laser Beam Deflection as a Probe of Laser Ablation of Materials,” Appl. Phys. Lett. 55, 2435–2437 (1989).
[CrossRef]

C. L. Enloe, R. M. Gilgenbach, J. S. Meachum, “Fast, Sensitive Laser Deflection System Suitable for Transient Plasma Analysis,” Rev. Sci. Instrum. 58, 1597–1600 (1987).
[CrossRef]

Ginter, M. L.

Gohil, P.

Green, J. M.

J. M. Green, W. T. Silfvast, O. R. Wood, “Evolution of a CO2-Laser-Produced Cadmium Plasma,” J. Appl. Phys. 48, 2753–2761 (1977).
[CrossRef]

Greenspan, M. A.

M. A. Greenspan, K. V. Reddy, “A Laser Deflection Technique for Sensitive Measurements of a Reduced-Density Channel in Neutral Gas,” Appl. Phys. Lett. 40, 576–578 (1982).
[CrossRef]

Gurevich, A. V.

A. V. Gurevich, L. V. Pariiskaya, L. P. Pitaevskii, “Self-Similar Motion of a Rarefied Plasma,” Sov. Phys. JETP 22, 449–454 (1966); P. Mora, “Self-Similar Expansion of a Plasma into a Vacuum,” Phys. Fluids 22, 2300–2304 (1979).
[CrossRef]

Hallstadius, H.

H. Bergstrom, G. W. Faris, H. Hallstadius, H. Lundberg, A. Persson, C.-G. Wahlstrom, “Radiative Lifetime and Hyperfine-Structure Studies on Laser-Evaporated Boron,” Z. Phys. D 8, 17–23 (1988).
[CrossRef]

H. Bergstrom, G. Faris, H. Hallstadius, H. Lundberg, A. Persson, C.-G. Wahlstrom, “Spectroscopy on Laser-Evaporated Boron and Carbon,” in Technical Digest, Eighth International Conference on Laser Spectroscopy, Are, Sweden (22–26 June 1987).

Heffelfinger, D. M.

J. A. Sell, D. M. Heffelfinger, P. Vantzek, R. M. Gilgenbach, “Laser Beam Deflection as a Probe of Laser Ablation of Materials,” Appl. Phys. Lett. 55, 2435–2437 (1989).
[CrossRef]

Hertz, H. M.

Hopkins, J. B.

J. B. Hopkins, P. R. R. Langridge-Smith, M. D. Morse, R. E. Smalley, “Supersonic Metal Cluster Beams of Refractory Metals: Spectral Investigations of Ultracold Mo2,” J. Chem. Phys. 78, 1627–1637 (1983).
[CrossRef]

Hughes, T. P.

T. P. Hughes, Plasmas and Laser Light (Hilger, Bristol, 1975).

Hunter, A. M.

P. W. Schreiber, A. M. Hunter, D. R. Smith, “The Determination of Plasma Electron Density from Refraction Measurements,” Plasma Phys. 15, 635–646 (1973).
[CrossRef]

Jahoda, F. C.

F. C. Jahoda, G. A. Sawyer, “Optical Refractivity of Plasmas,” in Plasma PhysicsR. H. Lovberg, H. R. Griem, Eds. (Academic, New York, 1971), Part B, Vol. 9, pp. 1–48.
[CrossRef]

Kapoor, H.

Kapteyn, H. C.

M. M. Murnane, H. C. Kapteyn, R. W. Falcone, “High Density Plasmas Produced by Ultrafast Laser Pulses,” Phys. Rev. Lett. 62, 155–158 (1989).
[CrossRef] [PubMed]

Keilmann, F.

F. Keilmann, “An Infrared Schlieren Interferometer for Measuring Electron Density Profiles,” Plasma Phys. 14, 111–122 (1972).
[CrossRef]

Kennedy, E. T.

P. K. Carroll, E. T. Kennedy, “Laser-Produced Plasmas,” Contemp. Phys. 22, 61–96 (1981).
[CrossRef]

Koren, G.

G. Koren, “Observation of Shock Waves and Cooling Waves in the Laser Ablation of Kapton Films in Air,” Appl. Phys. Lett. 51, 569–571 (1987).
[CrossRef]

Lacy, R. A.

Langridge-Smith, P. R. R.

J. B. Hopkins, P. R. R. Langridge-Smith, M. D. Morse, R. E. Smalley, “Supersonic Metal Cluster Beams of Refractory Metals: Spectral Investigations of Ultracold Mo2,” J. Chem. Phys. 78, 1627–1637 (1983).
[CrossRef]

Laptev, I. D.

Y. A. Bykovskii, V. G. Degtyarev, N. N. Degtyarenko, V. F. Elesin, I. D. Laptev, V. N. Nevolin, “Ion Energies in a Laser-Produced Plasma,” Sov. Phys. Tech. Phys. 17, 517–520 (1972).

Lundberg, H.

H. Bergstrom, G. W. Faris, H. Hallstadius, H. Lundberg, A. Persson, C.-G. Wahlstrom, “Radiative Lifetime and Hyperfine-Structure Studies on Laser-Evaporated Boron,” Z. Phys. D 8, 17–23 (1988).
[CrossRef]

H. Bergstrom, G. Faris, H. Hallstadius, H. Lundberg, A. Persson, C.-G. Wahlstrom, “Spectroscopy on Laser-Evaporated Boron and Carbon,” in Technical Digest, Eighth International Conference on Laser Spectroscopy, Are, Sweden (22–26 June 1987).

Ma, D.

Martin, G. A.

W. L. Wiese, G. A. Martin, “Atomic Transition Probabilities,” in Handbook of Chemistry and Physics, R. C. Weast, M. J. Astle, Eds. (CRC Press, West Palm Beach, FL, 1978), pp. 112–140.

Matthias, E.

S. Petzoldt, A. P. Elg, M. Reighling, J. Reif, E. Matthias, “Surface Laser Damage Thresholds Determined by Photo-acoustic Deflection,” Appl. Phys. Lett. 53, 2005–2007 (1988).
[CrossRef]

Maxon, S.

S. Maxon et al., “Calculation for Ni-Like Soft X-Ray Lasers: Optimization for W (43.1 Å),” Phys. Rev. Lett. 63, 236–239 (1989).
[CrossRef] [PubMed]

Mayne-Banton, V.

R. Srinivasan, V. Mayne-Banton, “Self-Developing Photoetching of Poly(ethylene Terephthalate) Films by Far-Ultraviolet Excimer Laser Radiation,” Appl. Phys. Lett. 41, 576–578 (1982); S. Lazare, V. Granier, “Excimer Laser Light Induced Ablation and Reactions at Polymer Surfaces as Measured with a Quartz-Crystal Microbalance,” J. Appl. Phys. 63, 2110–2115 (1988).
[CrossRef]

McIlrath, T. J.

Meachum, J. S.

C. L. Enloe, R. M. Gilgenbach, J. S. Meachum, “Fast, Sensitive Laser Deflection System Suitable for Transient Plasma Analysis,” Rev. Sci. Instrum. 58, 1597–1600 (1987).
[CrossRef]

Morse, M. D.

J. B. Hopkins, P. R. R. Langridge-Smith, M. D. Morse, R. E. Smalley, “Supersonic Metal Cluster Beams of Refractory Metals: Spectral Investigations of Ultracold Mo2,” J. Chem. Phys. 78, 1627–1637 (1983).
[CrossRef]

Muhling, I.

A. Richter, H.-J. Scheibe, W. Pompe, K.-W. Brzezinka, I. Muhling, “About the Structure and Bonding of Laser Generated Carbon Films by Raman and Electron Energy Loss Spectroscopy,” J. Non-Cryst. Solids 88, 131–144 (1986); C. B. Collins, F. Davanloo, E. M. Juengerman, W. R. Osborn, D. R. Jander, “Laser Plasma Source of Amorphic Diamond,” Appl. Phys. Lett. 54, 216–218 (1989).
[CrossRef]

Murnane, M. M.

M. M. Murnane, H. C. Kapteyn, R. W. Falcone, “High Density Plasmas Produced by Ultrafast Laser Pulses,” Phys. Rev. Lett. 62, 155–158 (1989).
[CrossRef] [PubMed]

Nevolin, V. N.

Y. A. Bykovskii, V. G. Degtyarev, N. N. Degtyarenko, V. F. Elesin, I. D. Laptev, V. N. Nevolin, “Ion Energies in a Laser-Produced Plasma,” Sov. Phys. Tech. Phys. 17, 517–520 (1972).

Nilsson, A. C.

Pariiskaya, L. V.

A. V. Gurevich, L. V. Pariiskaya, L. P. Pitaevskii, “Self-Similar Motion of a Rarefied Plasma,” Sov. Phys. JETP 22, 449–454 (1966); P. Mora, “Self-Similar Expansion of a Plasma into a Vacuum,” Phys. Fluids 22, 2300–2304 (1979).
[CrossRef]

Paul, J. W. M.

P. T. Rumsby, J. W. M. Paul, “Temperature and Density of an Expanding Laser-Produced Plasma,” Plasma Phys. 16, 247–260 (1974).
[CrossRef]

Pawliszyn, J.

J. Pawliszyn, “LEDs and Laser Diodes in Schlieren Optics Methods,” Rev. Sci. Instrum. 58, 245–248 (1987).
[CrossRef]

Pekerar, M. C.

Persson, A.

H. Bergstrom, G. W. Faris, H. Hallstadius, H. Lundberg, A. Persson, C.-G. Wahlstrom, “Radiative Lifetime and Hyperfine-Structure Studies on Laser-Evaporated Boron,” Z. Phys. D 8, 17–23 (1988).
[CrossRef]

H. Bergstrom, G. Faris, H. Hallstadius, H. Lundberg, A. Persson, C.-G. Wahlstrom, “Spectroscopy on Laser-Evaporated Boron and Carbon,” in Technical Digest, Eighth International Conference on Laser Spectroscopy, Are, Sweden (22–26 June 1987).

Petzoldt, S.

S. Petzoldt, A. P. Elg, M. Reighling, J. Reif, E. Matthias, “Surface Laser Damage Thresholds Determined by Photo-acoustic Deflection,” Appl. Phys. Lett. 53, 2005–2007 (1988).
[CrossRef]

Pitaevskii, L. P.

A. V. Gurevich, L. V. Pariiskaya, L. P. Pitaevskii, “Self-Similar Motion of a Rarefied Plasma,” Sov. Phys. JETP 22, 449–454 (1966); P. Mora, “Self-Similar Expansion of a Plasma into a Vacuum,” Phys. Fluids 22, 2300–2304 (1979).
[CrossRef]

Pompe, W.

A. Richter, H.-J. Scheibe, W. Pompe, K.-W. Brzezinka, I. Muhling, “About the Structure and Bonding of Laser Generated Carbon Films by Raman and Electron Energy Loss Spectroscopy,” J. Non-Cryst. Solids 88, 131–144 (1986); C. B. Collins, F. Davanloo, E. M. Juengerman, W. R. Osborn, D. R. Jander, “Laser Plasma Source of Amorphic Diamond,” Appl. Phys. Lett. 54, 216–218 (1989).
[CrossRef]

Potter, I. C.

I. C. Potter, I. S. Falconer, W. I. B. Smith, “Measurement of the Electron Density Distribution in Plasmas from the Bending of a Gas Laser Beam,” J. Phys. E 5, 910–914 (1972).
[CrossRef]

Reddy, K. V.

M. A. Greenspan, K. V. Reddy, “A Laser Deflection Technique for Sensitive Measurements of a Reduced-Density Channel in Neutral Gas,” Appl. Phys. Lett. 40, 576–578 (1982).
[CrossRef]

Reif, J.

S. Petzoldt, A. P. Elg, M. Reighling, J. Reif, E. Matthias, “Surface Laser Damage Thresholds Determined by Photo-acoustic Deflection,” Appl. Phys. Lett. 53, 2005–2007 (1988).
[CrossRef]

Reighling, M.

S. Petzoldt, A. P. Elg, M. Reighling, J. Reif, E. Matthias, “Surface Laser Damage Thresholds Determined by Photo-acoustic Deflection,” Appl. Phys. Lett. 53, 2005–2007 (1988).
[CrossRef]

Richter, A.

A. Richter, H.-J. Scheibe, W. Pompe, K.-W. Brzezinka, I. Muhling, “About the Structure and Bonding of Laser Generated Carbon Films by Raman and Electron Energy Loss Spectroscopy,” J. Non-Cryst. Solids 88, 131–144 (1986); C. B. Collins, F. Davanloo, E. M. Juengerman, W. R. Osborn, D. R. Jander, “Laser Plasma Source of Amorphic Diamond,” Appl. Phys. Lett. 54, 216–218 (1989).
[CrossRef]

Ruckle, B.

Rumsby, P. T.

P. T. Rumsby, J. W. M. Paul, “Temperature and Density of an Expanding Laser-Produced Plasma,” Plasma Phys. 16, 247–260 (1974).
[CrossRef]

Sawyer, G. A.

F. C. Jahoda, G. A. Sawyer, “Optical Refractivity of Plasmas,” in Plasma PhysicsR. H. Lovberg, H. R. Griem, Eds. (Academic, New York, 1971), Part B, Vol. 9, pp. 1–48.
[CrossRef]

Scheibe, H.-J.

A. Richter, H.-J. Scheibe, W. Pompe, K.-W. Brzezinka, I. Muhling, “About the Structure and Bonding of Laser Generated Carbon Films by Raman and Electron Energy Loss Spectroscopy,” J. Non-Cryst. Solids 88, 131–144 (1986); C. B. Collins, F. Davanloo, E. M. Juengerman, W. R. Osborn, D. R. Jander, “Laser Plasma Source of Amorphic Diamond,” Appl. Phys. Lett. 54, 216–218 (1989).
[CrossRef]

Schmidt, H.

Schreiber, P. W.

P. W. Schreiber, A. M. Hunter, D. R. Smith, “The Determination of Plasma Electron Density from Refraction Measurements,” Plasma Phys. 15, 635–646 (1973).
[CrossRef]

Sell, J. A.

J. A. Sell, D. M. Heffelfinger, P. Vantzek, R. M. Gilgenbach, “Laser Beam Deflection as a Probe of Laser Ablation of Materials,” Appl. Phys. Lett. 55, 2435–2437 (1989).
[CrossRef]

Silfvast, W. T.

Smalley, R. E.

J. B. Hopkins, P. R. R. Langridge-Smith, M. D. Morse, R. E. Smalley, “Supersonic Metal Cluster Beams of Refractory Metals: Spectral Investigations of Ultracold Mo2,” J. Chem. Phys. 78, 1627–1637 (1983).
[CrossRef]

Smith, D. R.

P. W. Schreiber, A. M. Hunter, D. R. Smith, “The Determination of Plasma Electron Density from Refraction Measurements,” Plasma Phys. 15, 635–646 (1973).
[CrossRef]

Smith, W. I. B.

I. C. Potter, I. S. Falconer, W. I. B. Smith, “Measurement of the Electron Density Distribution in Plasmas from the Bending of a Gas Laser Beam,” J. Phys. E 5, 910–914 (1972).
[CrossRef]

Srinivasan, R.

R. Srinivasan, V. Mayne-Banton, “Self-Developing Photoetching of Poly(ethylene Terephthalate) Films by Far-Ultraviolet Excimer Laser Radiation,” Appl. Phys. Lett. 41, 576–578 (1982); S. Lazare, V. Granier, “Excimer Laser Light Induced Ablation and Reactions at Polymer Surfaces as Measured with a Quartz-Crystal Microbalance,” J. Appl. Phys. 63, 2110–2115 (1988).
[CrossRef]

Svanberg, S.

Tallents, G. J.

G. J. Tallents, “On the Fitting of Displaced Maxwellians to Laser-Produced Plasma Ion Velocity Distributions,” Opt. Commun. 37, 108–112 (1981).
[CrossRef]

Trail, J. A.

Vantzek, P.

J. A. Sell, D. M. Heffelfinger, P. Vantzek, R. M. Gilgenbach, “Laser Beam Deflection as a Probe of Laser Ablation of Materials,” Appl. Phys. Lett. 55, 2435–2437 (1989).
[CrossRef]

Wahlstrom, C.-G.

H. Bergstrom, G. W. Faris, H. Hallstadius, H. Lundberg, A. Persson, C.-G. Wahlstrom, “Radiative Lifetime and Hyperfine-Structure Studies on Laser-Evaporated Boron,” Z. Phys. D 8, 17–23 (1988).
[CrossRef]

H. Bergstrom, G. Faris, H. Hallstadius, H. Lundberg, A. Persson, C.-G. Wahlstrom, “Spectroscopy on Laser-Evaporated Boron and Carbon,” in Technical Digest, Eighth International Conference on Laser Spectroscopy, Are, Sweden (22–26 June 1987).

Wiese, W. L.

W. L. Wiese, G. A. Martin, “Atomic Transition Probabilities,” in Handbook of Chemistry and Physics, R. C. Weast, M. J. Astle, Eds. (CRC Press, West Palm Beach, FL, 1978), pp. 112–140.

Wood, O. R.

Appl. Opt. (4)

Appl. Phys. Lett. (7)

M. L. Bortz, R. H. French, “Optical Reflectivity Measurements Using a Laser Plasma Light Source,” Appl. Phys. Lett. 55, 1955–1957 (1989).
[CrossRef]

D. Dijkkamp et al., “Preparation of Y–Ba–Cu Oxide Superconductor Thin Films Using Pulsed Laser Evaporation from High Tc Bulk Material,” Appl. Phys. Lett. 51, 619–621 (1987); M. J. Ferrari et al., “Low Magnetic Flux Noise Observed in Laser-Deposited In Situ Films of YB2Cu3O and Implications for High-Tc SQUIDs,” Nature (London) 341, 723–725 (1989).
[CrossRef]

R. Srinivasan, V. Mayne-Banton, “Self-Developing Photoetching of Poly(ethylene Terephthalate) Films by Far-Ultraviolet Excimer Laser Radiation,” Appl. Phys. Lett. 41, 576–578 (1982); S. Lazare, V. Granier, “Excimer Laser Light Induced Ablation and Reactions at Polymer Surfaces as Measured with a Quartz-Crystal Microbalance,” J. Appl. Phys. 63, 2110–2115 (1988).
[CrossRef]

M. A. Greenspan, K. V. Reddy, “A Laser Deflection Technique for Sensitive Measurements of a Reduced-Density Channel in Neutral Gas,” Appl. Phys. Lett. 40, 576–578 (1982).
[CrossRef]

G. Koren, “Observation of Shock Waves and Cooling Waves in the Laser Ablation of Kapton Films in Air,” Appl. Phys. Lett. 51, 569–571 (1987).
[CrossRef]

S. Petzoldt, A. P. Elg, M. Reighling, J. Reif, E. Matthias, “Surface Laser Damage Thresholds Determined by Photo-acoustic Deflection,” Appl. Phys. Lett. 53, 2005–2007 (1988).
[CrossRef]

J. A. Sell, D. M. Heffelfinger, P. Vantzek, R. M. Gilgenbach, “Laser Beam Deflection as a Probe of Laser Ablation of Materials,” Appl. Phys. Lett. 55, 2435–2437 (1989).
[CrossRef]

Contemp. Phys. (1)

P. K. Carroll, E. T. Kennedy, “Laser-Produced Plasmas,” Contemp. Phys. 22, 61–96 (1981).
[CrossRef]

J. Appl. Phys. (2)

J. M. Green, W. T. Silfvast, O. R. Wood, “Evolution of a CO2-Laser-Produced Cadmium Plasma,” J. Appl. Phys. 48, 2753–2761 (1977).
[CrossRef]

F. J. Allen, “Production of High-Energy Ions in Laser-Produced Plasmas,” J. Appl. Phys. 43, 2169–2175 (1972).
[CrossRef]

J. Chem. Phys. (1)

J. B. Hopkins, P. R. R. Langridge-Smith, M. D. Morse, R. E. Smalley, “Supersonic Metal Cluster Beams of Refractory Metals: Spectral Investigations of Ultracold Mo2,” J. Chem. Phys. 78, 1627–1637 (1983).
[CrossRef]

J. Non-Cryst. Solids (1)

A. Richter, H.-J. Scheibe, W. Pompe, K.-W. Brzezinka, I. Muhling, “About the Structure and Bonding of Laser Generated Carbon Films by Raman and Electron Energy Loss Spectroscopy,” J. Non-Cryst. Solids 88, 131–144 (1986); C. B. Collins, F. Davanloo, E. M. Juengerman, W. R. Osborn, D. R. Jander, “Laser Plasma Source of Amorphic Diamond,” Appl. Phys. Lett. 54, 216–218 (1989).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. E (1)

I. C. Potter, I. S. Falconer, W. I. B. Smith, “Measurement of the Electron Density Distribution in Plasmas from the Bending of a Gas Laser Beam,” J. Phys. E 5, 910–914 (1972).
[CrossRef]

Opt. Commun. (1)

G. J. Tallents, “On the Fitting of Displaced Maxwellians to Laser-Produced Plasma Ion Velocity Distributions,” Opt. Commun. 37, 108–112 (1981).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (2)

M. M. Murnane, H. C. Kapteyn, R. W. Falcone, “High Density Plasmas Produced by Ultrafast Laser Pulses,” Phys. Rev. Lett. 62, 155–158 (1989).
[CrossRef] [PubMed]

S. Maxon et al., “Calculation for Ni-Like Soft X-Ray Lasers: Optimization for W (43.1 Å),” Phys. Rev. Lett. 63, 236–239 (1989).
[CrossRef] [PubMed]

Plasma Phys. (3)

F. Keilmann, “An Infrared Schlieren Interferometer for Measuring Electron Density Profiles,” Plasma Phys. 14, 111–122 (1972).
[CrossRef]

P. W. Schreiber, A. M. Hunter, D. R. Smith, “The Determination of Plasma Electron Density from Refraction Measurements,” Plasma Phys. 15, 635–646 (1973).
[CrossRef]

P. T. Rumsby, J. W. M. Paul, “Temperature and Density of an Expanding Laser-Produced Plasma,” Plasma Phys. 16, 247–260 (1974).
[CrossRef]

Rev. Sci. Instrum. (2)

J. Pawliszyn, “LEDs and Laser Diodes in Schlieren Optics Methods,” Rev. Sci. Instrum. 58, 245–248 (1987).
[CrossRef]

C. L. Enloe, R. M. Gilgenbach, J. S. Meachum, “Fast, Sensitive Laser Deflection System Suitable for Transient Plasma Analysis,” Rev. Sci. Instrum. 58, 1597–1600 (1987).
[CrossRef]

Science (1)

G. W. Faris, R. L. Byer, “Quantitative Three-Dimensional Optical Toiographic Imaging of Supersonic Flows,” Science 238, 1700–1702 (1987).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

A. V. Gurevich, L. V. Pariiskaya, L. P. Pitaevskii, “Self-Similar Motion of a Rarefied Plasma,” Sov. Phys. JETP 22, 449–454 (1966); P. Mora, “Self-Similar Expansion of a Plasma into a Vacuum,” Phys. Fluids 22, 2300–2304 (1979).
[CrossRef]

Sov. Phys. Tech. Phys. (1)

Y. A. Bykovskii, V. G. Degtyarev, N. N. Degtyarenko, V. F. Elesin, I. D. Laptev, V. N. Nevolin, “Ion Energies in a Laser-Produced Plasma,” Sov. Phys. Tech. Phys. 17, 517–520 (1972).

Z. Phys. D (1)

H. Bergstrom, G. W. Faris, H. Hallstadius, H. Lundberg, A. Persson, C.-G. Wahlstrom, “Radiative Lifetime and Hyperfine-Structure Studies on Laser-Evaporated Boron,” Z. Phys. D 8, 17–23 (1988).
[CrossRef]

Other (4)

T. P. Hughes, Plasmas and Laser Light (Hilger, Bristol, 1975).

H. Bergstrom, G. Faris, H. Hallstadius, H. Lundberg, A. Persson, C.-G. Wahlstrom, “Spectroscopy on Laser-Evaporated Boron and Carbon,” in Technical Digest, Eighth International Conference on Laser Spectroscopy, Are, Sweden (22–26 June 1987).

F. C. Jahoda, G. A. Sawyer, “Optical Refractivity of Plasmas,” in Plasma PhysicsR. H. Lovberg, H. R. Griem, Eds. (Academic, New York, 1971), Part B, Vol. 9, pp. 1–48.
[CrossRef]

W. L. Wiese, G. A. Martin, “Atomic Transition Probabilities,” in Handbook of Chemistry and Physics, R. C. Weast, M. J. Astle, Eds. (CRC Press, West Palm Beach, FL, 1978), pp. 112–140.

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Figures (6)

Fig. 1
Fig. 1

Experimental arrangement for two-wavelength measurement of beam deflection angles in a laser-produced plasma.

Fig. 2
Fig. 2

Example of temporal beam deflection profile in a laser-produced plasma measured at 0.5 mm from a silicon target. The solid line corresponds to the He–Ne measurement, the dashed line is for He–Cd.

Fig. 3
Fig. 3

Electron density profiles for a laser-produced plasma with a graphite target, reconstructed from a composite of many temporal beam deflection curves.

Fig. 4
Fig. 4

Electron density profiles for a laser-produced plasma with a graphite target, reconstructed from single temporal beam deflection curves. The 10-ns profile is calculated from a measurement made at 0.5 mm from the target. The correspondences for the other times are 50 ns at 1.4 mm, 100 ns at 3.7 mm, 200 ns at 5.0 mm, and 300 ns at 5.5 mm.

Fig. 5
Fig. 5

Electron density profiles for a laser-produced plasma with a graphite target, calculated from peak deflection angles assuming a Gaussian density dependence.

Fig. 6
Fig. 6

Distance from the target vs time for the peak deflection angle for a number of targets.

Equations (11)

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

α ( y ) = 1 n o d n d y d x ,
n = 1 - 1 2 n e n c .
n c = π m e c 2 e 2 λ 2 ,
w o = λ l 2 π ,
α max ~ 0.39 2 π λ w o .
α min = 1 2 2 π V min V max λ w o .
n e ( r , R ) = ( n e R 3 ) o R 3 n ^ e ( r R ) ,
α ( y , R ) = ( α R 3 ) o R 3 α ^ ( y R ) .
n e = e 2 2 π α * n c exp ( - 2 r 2 / y * 2 ) .
y * = ( 0.915 ± 0.06 ) R ,
α * = ( 0.683 ± 0.26 ) n e ( r = 0 ) n c ,

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