Abstract

The results of implanting silicon carbide with a 1-MeV proton beam at a dose of 1 × 1017 cm-2 are presented. Using high-resolution confocal Raman spectroscopy, we analyzed the depth profile of the implantation damage before and after thermal annealing. When it is applied to a high-refractive-index medium, such as SiC, this technique requires careful manipulation to ensure the correct interpretation of results. To this end we discuss a simple ray-tracking model that includes the effects of additional spherical aberration and of the Gaussian intensity profile of the excitation beam. In addition, infrared reflectance measurements show evidence of a well-defined step in the refractive-index profile at the expected implantation depth.

© 2004 Optical Society of America

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References

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  1. L. DiCioccio, Y. LeTiec, C. Jausaud, E. Hugonnard-Bruere, M. Bruel, “Silicon carbide on insulator formation by the smart cut process,” Mater. Sci. Forum 264–268, 765–770 (1998).
    [CrossRef]
  2. N. Sieber, B. F. Mantel, T. L. Seyller, J. Ristein, L. Ley, T. Heller, D. R. Batchelor, D. Schmeisser, “Electronic and chemical passivation of hexagonal 6H-SiC surfaces by hydrogen termination,” Appl. Phys. Lett. 78, 1216–1219 (2001).
    [CrossRef]
  3. J. Camassel, P. Vicente, L. Falkovski, “Optical characterization of SiC materials: bulk and implanted layers,” Mater. Sci. Forum 353–356, 335–340 (2001).
    [CrossRef]
  4. H. W. Kunert, T. P. Maurice, T. Hauser, J. B. Malherbe, L. C. Prinsloo, D. J. Brink, L. A. Falkovski, J. Camassel, “Effects of hydrogen implantation and annealing on the vibrational properties of 6H-SiC,” Mater. Sci. Forum 353–356, 353–355 (2001).
  5. N. Mestres, F. Alsina, F. J. Campos, J. Pascual, E. Morvan, P. Godignon, J. Millan, “Cofocal Raman microprobe of lattice damage in N+ implanted 6H-SiC,” Mater. Sci. Forum 338–342, 663–666 (2000).
    [CrossRef]
  6. J. F. Ziegler, J. P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, J. F. Ziegler, ed. (Pergamon, New York, 1985), Vol. 1.
  7. M. F. MacMillan, R. P. Devaty, W. J. Choyke, D. R. Goldstein, J. E. Spanier, A. D. Kurtz, “Infrared reflectance of thick p-type porous SiC layers,” J. Appl. Phys. 80, 2412–2419 (1996).
    [CrossRef]
  8. Ph. J. Rousel, L. Vanhellemont, H. E. Maes, “Numerical aspects of the effective medium approximation models in spectroscopic ellipsometry regression software,” Thin Solid Films 234, 423–427 (1993).
    [CrossRef]
  9. J. Campos, “Microscopia Raman confocal: aplicacio al 6H-SiC,” dissertation (Universitat Autonoma de Barcelona, Barcelona, 1998), pp. 26–29, 37–39.
  10. C. J. R. Shepard, T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858–859 (1981).
    [CrossRef]
  11. T. Wilson, Confocal Microscopy (Academic, San Diego, Calif., 1990), Chap. 1.
  12. T. Wilson, A. R. Carlini, “Size of the detector in confocal imaging systems,” Opt. Lett. 12, 227–229 (1987).
    [CrossRef] [PubMed]
  13. T. Wilson, “Optical sectioning in confocal fluorescent microscopes,” J. Microsc. 154, 143–156 (1989).
    [CrossRef]

2001 (3)

N. Sieber, B. F. Mantel, T. L. Seyller, J. Ristein, L. Ley, T. Heller, D. R. Batchelor, D. Schmeisser, “Electronic and chemical passivation of hexagonal 6H-SiC surfaces by hydrogen termination,” Appl. Phys. Lett. 78, 1216–1219 (2001).
[CrossRef]

J. Camassel, P. Vicente, L. Falkovski, “Optical characterization of SiC materials: bulk and implanted layers,” Mater. Sci. Forum 353–356, 335–340 (2001).
[CrossRef]

H. W. Kunert, T. P. Maurice, T. Hauser, J. B. Malherbe, L. C. Prinsloo, D. J. Brink, L. A. Falkovski, J. Camassel, “Effects of hydrogen implantation and annealing on the vibrational properties of 6H-SiC,” Mater. Sci. Forum 353–356, 353–355 (2001).

2000 (1)

N. Mestres, F. Alsina, F. J. Campos, J. Pascual, E. Morvan, P. Godignon, J. Millan, “Cofocal Raman microprobe of lattice damage in N+ implanted 6H-SiC,” Mater. Sci. Forum 338–342, 663–666 (2000).
[CrossRef]

1998 (1)

L. DiCioccio, Y. LeTiec, C. Jausaud, E. Hugonnard-Bruere, M. Bruel, “Silicon carbide on insulator formation by the smart cut process,” Mater. Sci. Forum 264–268, 765–770 (1998).
[CrossRef]

1996 (1)

M. F. MacMillan, R. P. Devaty, W. J. Choyke, D. R. Goldstein, J. E. Spanier, A. D. Kurtz, “Infrared reflectance of thick p-type porous SiC layers,” J. Appl. Phys. 80, 2412–2419 (1996).
[CrossRef]

1993 (1)

Ph. J. Rousel, L. Vanhellemont, H. E. Maes, “Numerical aspects of the effective medium approximation models in spectroscopic ellipsometry regression software,” Thin Solid Films 234, 423–427 (1993).
[CrossRef]

1989 (1)

T. Wilson, “Optical sectioning in confocal fluorescent microscopes,” J. Microsc. 154, 143–156 (1989).
[CrossRef]

1987 (1)

1981 (1)

C. J. R. Shepard, T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858–859 (1981).
[CrossRef]

Alsina, F.

N. Mestres, F. Alsina, F. J. Campos, J. Pascual, E. Morvan, P. Godignon, J. Millan, “Cofocal Raman microprobe of lattice damage in N+ implanted 6H-SiC,” Mater. Sci. Forum 338–342, 663–666 (2000).
[CrossRef]

Batchelor, D. R.

N. Sieber, B. F. Mantel, T. L. Seyller, J. Ristein, L. Ley, T. Heller, D. R. Batchelor, D. Schmeisser, “Electronic and chemical passivation of hexagonal 6H-SiC surfaces by hydrogen termination,” Appl. Phys. Lett. 78, 1216–1219 (2001).
[CrossRef]

Biersack, J. P.

J. F. Ziegler, J. P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, J. F. Ziegler, ed. (Pergamon, New York, 1985), Vol. 1.

Brink, D. J.

H. W. Kunert, T. P. Maurice, T. Hauser, J. B. Malherbe, L. C. Prinsloo, D. J. Brink, L. A. Falkovski, J. Camassel, “Effects of hydrogen implantation and annealing on the vibrational properties of 6H-SiC,” Mater. Sci. Forum 353–356, 353–355 (2001).

Bruel, M.

L. DiCioccio, Y. LeTiec, C. Jausaud, E. Hugonnard-Bruere, M. Bruel, “Silicon carbide on insulator formation by the smart cut process,” Mater. Sci. Forum 264–268, 765–770 (1998).
[CrossRef]

Camassel, J.

J. Camassel, P. Vicente, L. Falkovski, “Optical characterization of SiC materials: bulk and implanted layers,” Mater. Sci. Forum 353–356, 335–340 (2001).
[CrossRef]

H. W. Kunert, T. P. Maurice, T. Hauser, J. B. Malherbe, L. C. Prinsloo, D. J. Brink, L. A. Falkovski, J. Camassel, “Effects of hydrogen implantation and annealing on the vibrational properties of 6H-SiC,” Mater. Sci. Forum 353–356, 353–355 (2001).

Campos, F. J.

N. Mestres, F. Alsina, F. J. Campos, J. Pascual, E. Morvan, P. Godignon, J. Millan, “Cofocal Raman microprobe of lattice damage in N+ implanted 6H-SiC,” Mater. Sci. Forum 338–342, 663–666 (2000).
[CrossRef]

Campos, J.

J. Campos, “Microscopia Raman confocal: aplicacio al 6H-SiC,” dissertation (Universitat Autonoma de Barcelona, Barcelona, 1998), pp. 26–29, 37–39.

Carlini, A. R.

Choyke, W. J.

M. F. MacMillan, R. P. Devaty, W. J. Choyke, D. R. Goldstein, J. E. Spanier, A. D. Kurtz, “Infrared reflectance of thick p-type porous SiC layers,” J. Appl. Phys. 80, 2412–2419 (1996).
[CrossRef]

Devaty, R. P.

M. F. MacMillan, R. P. Devaty, W. J. Choyke, D. R. Goldstein, J. E. Spanier, A. D. Kurtz, “Infrared reflectance of thick p-type porous SiC layers,” J. Appl. Phys. 80, 2412–2419 (1996).
[CrossRef]

DiCioccio, L.

L. DiCioccio, Y. LeTiec, C. Jausaud, E. Hugonnard-Bruere, M. Bruel, “Silicon carbide on insulator formation by the smart cut process,” Mater. Sci. Forum 264–268, 765–770 (1998).
[CrossRef]

Falkovski, L.

J. Camassel, P. Vicente, L. Falkovski, “Optical characterization of SiC materials: bulk and implanted layers,” Mater. Sci. Forum 353–356, 335–340 (2001).
[CrossRef]

Falkovski, L. A.

H. W. Kunert, T. P. Maurice, T. Hauser, J. B. Malherbe, L. C. Prinsloo, D. J. Brink, L. A. Falkovski, J. Camassel, “Effects of hydrogen implantation and annealing on the vibrational properties of 6H-SiC,” Mater. Sci. Forum 353–356, 353–355 (2001).

Godignon, P.

N. Mestres, F. Alsina, F. J. Campos, J. Pascual, E. Morvan, P. Godignon, J. Millan, “Cofocal Raman microprobe of lattice damage in N+ implanted 6H-SiC,” Mater. Sci. Forum 338–342, 663–666 (2000).
[CrossRef]

Goldstein, D. R.

M. F. MacMillan, R. P. Devaty, W. J. Choyke, D. R. Goldstein, J. E. Spanier, A. D. Kurtz, “Infrared reflectance of thick p-type porous SiC layers,” J. Appl. Phys. 80, 2412–2419 (1996).
[CrossRef]

Hauser, T.

H. W. Kunert, T. P. Maurice, T. Hauser, J. B. Malherbe, L. C. Prinsloo, D. J. Brink, L. A. Falkovski, J. Camassel, “Effects of hydrogen implantation and annealing on the vibrational properties of 6H-SiC,” Mater. Sci. Forum 353–356, 353–355 (2001).

Heller, T.

N. Sieber, B. F. Mantel, T. L. Seyller, J. Ristein, L. Ley, T. Heller, D. R. Batchelor, D. Schmeisser, “Electronic and chemical passivation of hexagonal 6H-SiC surfaces by hydrogen termination,” Appl. Phys. Lett. 78, 1216–1219 (2001).
[CrossRef]

Hugonnard-Bruere, E.

L. DiCioccio, Y. LeTiec, C. Jausaud, E. Hugonnard-Bruere, M. Bruel, “Silicon carbide on insulator formation by the smart cut process,” Mater. Sci. Forum 264–268, 765–770 (1998).
[CrossRef]

Jausaud, C.

L. DiCioccio, Y. LeTiec, C. Jausaud, E. Hugonnard-Bruere, M. Bruel, “Silicon carbide on insulator formation by the smart cut process,” Mater. Sci. Forum 264–268, 765–770 (1998).
[CrossRef]

Kunert, H. W.

H. W. Kunert, T. P. Maurice, T. Hauser, J. B. Malherbe, L. C. Prinsloo, D. J. Brink, L. A. Falkovski, J. Camassel, “Effects of hydrogen implantation and annealing on the vibrational properties of 6H-SiC,” Mater. Sci. Forum 353–356, 353–355 (2001).

Kurtz, A. D.

M. F. MacMillan, R. P. Devaty, W. J. Choyke, D. R. Goldstein, J. E. Spanier, A. D. Kurtz, “Infrared reflectance of thick p-type porous SiC layers,” J. Appl. Phys. 80, 2412–2419 (1996).
[CrossRef]

LeTiec, Y.

L. DiCioccio, Y. LeTiec, C. Jausaud, E. Hugonnard-Bruere, M. Bruel, “Silicon carbide on insulator formation by the smart cut process,” Mater. Sci. Forum 264–268, 765–770 (1998).
[CrossRef]

Ley, L.

N. Sieber, B. F. Mantel, T. L. Seyller, J. Ristein, L. Ley, T. Heller, D. R. Batchelor, D. Schmeisser, “Electronic and chemical passivation of hexagonal 6H-SiC surfaces by hydrogen termination,” Appl. Phys. Lett. 78, 1216–1219 (2001).
[CrossRef]

Littmark, U.

J. F. Ziegler, J. P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, J. F. Ziegler, ed. (Pergamon, New York, 1985), Vol. 1.

MacMillan, M. F.

M. F. MacMillan, R. P. Devaty, W. J. Choyke, D. R. Goldstein, J. E. Spanier, A. D. Kurtz, “Infrared reflectance of thick p-type porous SiC layers,” J. Appl. Phys. 80, 2412–2419 (1996).
[CrossRef]

Maes, H. E.

Ph. J. Rousel, L. Vanhellemont, H. E. Maes, “Numerical aspects of the effective medium approximation models in spectroscopic ellipsometry regression software,” Thin Solid Films 234, 423–427 (1993).
[CrossRef]

Malherbe, J. B.

H. W. Kunert, T. P. Maurice, T. Hauser, J. B. Malherbe, L. C. Prinsloo, D. J. Brink, L. A. Falkovski, J. Camassel, “Effects of hydrogen implantation and annealing on the vibrational properties of 6H-SiC,” Mater. Sci. Forum 353–356, 353–355 (2001).

Mantel, B. F.

N. Sieber, B. F. Mantel, T. L. Seyller, J. Ristein, L. Ley, T. Heller, D. R. Batchelor, D. Schmeisser, “Electronic and chemical passivation of hexagonal 6H-SiC surfaces by hydrogen termination,” Appl. Phys. Lett. 78, 1216–1219 (2001).
[CrossRef]

Maurice, T. P.

H. W. Kunert, T. P. Maurice, T. Hauser, J. B. Malherbe, L. C. Prinsloo, D. J. Brink, L. A. Falkovski, J. Camassel, “Effects of hydrogen implantation and annealing on the vibrational properties of 6H-SiC,” Mater. Sci. Forum 353–356, 353–355 (2001).

Mestres, N.

N. Mestres, F. Alsina, F. J. Campos, J. Pascual, E. Morvan, P. Godignon, J. Millan, “Cofocal Raman microprobe of lattice damage in N+ implanted 6H-SiC,” Mater. Sci. Forum 338–342, 663–666 (2000).
[CrossRef]

Millan, J.

N. Mestres, F. Alsina, F. J. Campos, J. Pascual, E. Morvan, P. Godignon, J. Millan, “Cofocal Raman microprobe of lattice damage in N+ implanted 6H-SiC,” Mater. Sci. Forum 338–342, 663–666 (2000).
[CrossRef]

Morvan, E.

N. Mestres, F. Alsina, F. J. Campos, J. Pascual, E. Morvan, P. Godignon, J. Millan, “Cofocal Raman microprobe of lattice damage in N+ implanted 6H-SiC,” Mater. Sci. Forum 338–342, 663–666 (2000).
[CrossRef]

Pascual, J.

N. Mestres, F. Alsina, F. J. Campos, J. Pascual, E. Morvan, P. Godignon, J. Millan, “Cofocal Raman microprobe of lattice damage in N+ implanted 6H-SiC,” Mater. Sci. Forum 338–342, 663–666 (2000).
[CrossRef]

Prinsloo, L. C.

H. W. Kunert, T. P. Maurice, T. Hauser, J. B. Malherbe, L. C. Prinsloo, D. J. Brink, L. A. Falkovski, J. Camassel, “Effects of hydrogen implantation and annealing on the vibrational properties of 6H-SiC,” Mater. Sci. Forum 353–356, 353–355 (2001).

Ristein, J.

N. Sieber, B. F. Mantel, T. L. Seyller, J. Ristein, L. Ley, T. Heller, D. R. Batchelor, D. Schmeisser, “Electronic and chemical passivation of hexagonal 6H-SiC surfaces by hydrogen termination,” Appl. Phys. Lett. 78, 1216–1219 (2001).
[CrossRef]

Rousel, Ph. J.

Ph. J. Rousel, L. Vanhellemont, H. E. Maes, “Numerical aspects of the effective medium approximation models in spectroscopic ellipsometry regression software,” Thin Solid Films 234, 423–427 (1993).
[CrossRef]

Schmeisser, D.

N. Sieber, B. F. Mantel, T. L. Seyller, J. Ristein, L. Ley, T. Heller, D. R. Batchelor, D. Schmeisser, “Electronic and chemical passivation of hexagonal 6H-SiC surfaces by hydrogen termination,” Appl. Phys. Lett. 78, 1216–1219 (2001).
[CrossRef]

Seyller, T. L.

N. Sieber, B. F. Mantel, T. L. Seyller, J. Ristein, L. Ley, T. Heller, D. R. Batchelor, D. Schmeisser, “Electronic and chemical passivation of hexagonal 6H-SiC surfaces by hydrogen termination,” Appl. Phys. Lett. 78, 1216–1219 (2001).
[CrossRef]

Shepard, C. J. R.

C. J. R. Shepard, T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858–859 (1981).
[CrossRef]

Sieber, N.

N. Sieber, B. F. Mantel, T. L. Seyller, J. Ristein, L. Ley, T. Heller, D. R. Batchelor, D. Schmeisser, “Electronic and chemical passivation of hexagonal 6H-SiC surfaces by hydrogen termination,” Appl. Phys. Lett. 78, 1216–1219 (2001).
[CrossRef]

Spanier, J. E.

M. F. MacMillan, R. P. Devaty, W. J. Choyke, D. R. Goldstein, J. E. Spanier, A. D. Kurtz, “Infrared reflectance of thick p-type porous SiC layers,” J. Appl. Phys. 80, 2412–2419 (1996).
[CrossRef]

Vanhellemont, L.

Ph. J. Rousel, L. Vanhellemont, H. E. Maes, “Numerical aspects of the effective medium approximation models in spectroscopic ellipsometry regression software,” Thin Solid Films 234, 423–427 (1993).
[CrossRef]

Vicente, P.

J. Camassel, P. Vicente, L. Falkovski, “Optical characterization of SiC materials: bulk and implanted layers,” Mater. Sci. Forum 353–356, 335–340 (2001).
[CrossRef]

Wilson, T.

T. Wilson, “Optical sectioning in confocal fluorescent microscopes,” J. Microsc. 154, 143–156 (1989).
[CrossRef]

T. Wilson, A. R. Carlini, “Size of the detector in confocal imaging systems,” Opt. Lett. 12, 227–229 (1987).
[CrossRef] [PubMed]

C. J. R. Shepard, T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858–859 (1981).
[CrossRef]

T. Wilson, Confocal Microscopy (Academic, San Diego, Calif., 1990), Chap. 1.

Ziegler, J. F.

J. F. Ziegler, J. P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, J. F. Ziegler, ed. (Pergamon, New York, 1985), Vol. 1.

Appl. Phys. Lett. (2)

N. Sieber, B. F. Mantel, T. L. Seyller, J. Ristein, L. Ley, T. Heller, D. R. Batchelor, D. Schmeisser, “Electronic and chemical passivation of hexagonal 6H-SiC surfaces by hydrogen termination,” Appl. Phys. Lett. 78, 1216–1219 (2001).
[CrossRef]

C. J. R. Shepard, T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858–859 (1981).
[CrossRef]

J. Appl. Phys. (1)

M. F. MacMillan, R. P. Devaty, W. J. Choyke, D. R. Goldstein, J. E. Spanier, A. D. Kurtz, “Infrared reflectance of thick p-type porous SiC layers,” J. Appl. Phys. 80, 2412–2419 (1996).
[CrossRef]

J. Microsc. (1)

T. Wilson, “Optical sectioning in confocal fluorescent microscopes,” J. Microsc. 154, 143–156 (1989).
[CrossRef]

Mater. Sci. Forum (4)

L. DiCioccio, Y. LeTiec, C. Jausaud, E. Hugonnard-Bruere, M. Bruel, “Silicon carbide on insulator formation by the smart cut process,” Mater. Sci. Forum 264–268, 765–770 (1998).
[CrossRef]

J. Camassel, P. Vicente, L. Falkovski, “Optical characterization of SiC materials: bulk and implanted layers,” Mater. Sci. Forum 353–356, 335–340 (2001).
[CrossRef]

H. W. Kunert, T. P. Maurice, T. Hauser, J. B. Malherbe, L. C. Prinsloo, D. J. Brink, L. A. Falkovski, J. Camassel, “Effects of hydrogen implantation and annealing on the vibrational properties of 6H-SiC,” Mater. Sci. Forum 353–356, 353–355 (2001).

N. Mestres, F. Alsina, F. J. Campos, J. Pascual, E. Morvan, P. Godignon, J. Millan, “Cofocal Raman microprobe of lattice damage in N+ implanted 6H-SiC,” Mater. Sci. Forum 338–342, 663–666 (2000).
[CrossRef]

Opt. Lett. (1)

Thin Solid Films (1)

Ph. J. Rousel, L. Vanhellemont, H. E. Maes, “Numerical aspects of the effective medium approximation models in spectroscopic ellipsometry regression software,” Thin Solid Films 234, 423–427 (1993).
[CrossRef]

Other (3)

J. Campos, “Microscopia Raman confocal: aplicacio al 6H-SiC,” dissertation (Universitat Autonoma de Barcelona, Barcelona, 1998), pp. 26–29, 37–39.

T. Wilson, Confocal Microscopy (Academic, San Diego, Calif., 1990), Chap. 1.

J. F. Ziegler, J. P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, J. F. Ziegler, ed. (Pergamon, New York, 1985), Vol. 1.

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Figures (7)

Fig. 1
Fig. 1

TRIM simulation of H+ implantation into SiC.

Fig. 2
Fig. 2

Calculated and measured IR reflectance spectra. The two lower curves represent measured results for the implanted (large modulations) and the implanted and annealed (small modulations) samples. The upper curve is calculated assuming a 5% volume fraction of hydrogen bubbles.

Fig. 3
Fig. 3

Raman spectra of untreated, H+-implanted (1017 cm-2) and annealed (950 °C) 6H-SiC. The spectrum for the implanted sample is magnified four times. Base lines were shifted vertically to clarify the spectra.

Fig. 4
Fig. 4

Detected Raman signal after the conjugate aperture. Here and in the following figures, SPH. AB. means spherical aberration.

Fig. 5
Fig. 5

Convolution of the focal-depth curves with the damage profile produced by the TRIM simulation program. Experimental points represent a measured depth profile in implanted and annealed SiC (D1 line). We obtained the focal position by multiplying the nominal focus by the index of refraction. Experimental points for the D2 line are almost identical.

Fig. 6
Fig. 6

Convolutions of the focal-depth curves with the hydrogen-ion profile from TRIM. Experimental points are the same as in Fig. 5.

Fig. 7
Fig. 7

Depth profile of the 789 cm-1 line in implanted and annealed SiC (circles). Solid curves are convolutions of the focal-depth functions with (upper curve) and without (lower curve) spherical aberration. The positions of implanted H+ ions according to the simulation TRIM and IR reflectance are shown.

Equations (9)

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

ε SiC = ε ω 2 - ω L 2 + i γ ω ω 2 - ω T 2 + i γ ω ,
E = A   sin u / 4 / u / 4 ,
I R = B | E | 2 | E | 2     D B | E | 4 ,
FWHM = 2 Δ z = u H λ 4 π   sin 2 α 0 / 2 ,
FWHM = u H λ 4 π n   sin 2 α 1 / 2 ,
u H = 0.76 V P + 0.80 ,
z F = z 0   tan   β 0 / tan   β 1 = f C z 0 nz 0 ,
E r = A   exp - r / w 2 sin ku / 4 / ku / 4 ,
I R = r   | E r | 4 .

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