Abstract

The operation of a single-shot cross-correlator based on a pulse replicator is described. The correlator uses a discrete sequence of sampling pulses that are nonlinearly mixed with the pulse under test. The combination of a high reflector and partial reflector replicates an optical pulse by multiple internal reflections and generates a sequence of spatially displaced and temporally delayed sampling pulses. This principle is used in a cross-correlator characterizing optical pulses at 1053 nm. A dynamic range higher than 60 dB is obtained over a temporal range larger than 200 ps.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S.-W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovksy, "Generation and characterization of the highest laser intensities (1022 W/cm2)," Opt. Lett. 29, 2837�??2839 (2004).
    [CrossRef]
  2. J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).
  3. D. J. Kane and R. Trebino, "Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating," Opt. Lett. 18, 823�??825 (1993).
    [CrossRef] [PubMed]
  4. C. Dorrer, B. de Beauvoir, C. Le Blanc, S. Ranc, J.-P. Rousseau, P. Rousseau, and J.-P. Chambaret, "Single-shot real-time characterization of chirped-pulse amplification systems by spectral phase interferometry for direct electric-field reconstruction," Opt. Lett. 24, 1644�??1646 (1999).
    [CrossRef]
  5. J. Bromage, C. Dorrer, I. A. Begishev, N. G. Usechak, and J. D. Zuegel, "Highly sensitive, single-shot characterization for pulse widths from 0.4 to 85 ps using electro-optic shearing interferometry," Opt. Lett. 31, 3523�??3525 (2006).
    [CrossRef] [PubMed]
  6. S. Luan, M. H. R. Hutchinson, R. A. Smith, and F. Zhou, "High dynamic range third-order correlation measurement of picosecond laser pulse shapes," Meas. Sci. Technol. 4, 1426�??1429 (1993).
    [CrossRef]
  7. E. J. Divall and I. N. Ross, "High dynamic range contrast measurements by use of an optical parametric amplifier correlator," Opt. Lett. 29, 2273�??2275 (2004).
    [CrossRef] [PubMed]
  8. J. Collier, C. Hernandez-Gomez, R. Allott, C. Danson, and A. Hall, "A single-shot third-order autocorrelator for pulse contrast and pulse shape measurements," Laser Part. Beams 19, 231�??235 (2001).
    [CrossRef]
  9. I. Jovanovic, B. C. Haefner, M. Shverdin, M. Taranowski, and C. P. J. Barty, "High-dynamic-range, 200-ps window, single-shot cross-correlator for ultrahigh intensity laser characterization," presented at the CLEO/QELS Conference, Baltimore, MD, 6-11 May 2007 (Paper JThD137).
  10. C. Dufour, "Utilisation de l�??interféromètre Fabry-Perot pour la recherche des satellites faibles," Rev. Opt. 24, 11�??18 (1945).
  11. S. Xiao, J. D. McKinney, and A. M. Weiner, "Photonic microwave arbitrary waveform generation using a virtually imaged phased-array (VIPA) direct space-to-time pulse shaper," IEEE Photon. Technol. Lett. 16, 1936�??1938 (2004).
    [CrossRef]
  12. L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, Cambridge, England, 1995).

2006

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

J. Bromage, C. Dorrer, I. A. Begishev, N. G. Usechak, and J. D. Zuegel, "Highly sensitive, single-shot characterization for pulse widths from 0.4 to 85 ps using electro-optic shearing interferometry," Opt. Lett. 31, 3523�??3525 (2006).
[CrossRef] [PubMed]

2004

2001

J. Collier, C. Hernandez-Gomez, R. Allott, C. Danson, and A. Hall, "A single-shot third-order autocorrelator for pulse contrast and pulse shape measurements," Laser Part. Beams 19, 231�??235 (2001).
[CrossRef]

1999

1993

S. Luan, M. H. R. Hutchinson, R. A. Smith, and F. Zhou, "High dynamic range third-order correlation measurement of picosecond laser pulse shapes," Meas. Sci. Technol. 4, 1426�??1429 (1993).
[CrossRef]

D. J. Kane and R. Trebino, "Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating," Opt. Lett. 18, 823�??825 (1993).
[CrossRef] [PubMed]

1945

C. Dufour, "Utilisation de l�??interféromètre Fabry-Perot pour la recherche des satellites faibles," Rev. Opt. 24, 11�??18 (1945).

Allott, R.

J. Collier, C. Hernandez-Gomez, R. Allott, C. Danson, and A. Hall, "A single-shot third-order autocorrelator for pulse contrast and pulse shape measurements," Laser Part. Beams 19, 231�??235 (2001).
[CrossRef]

Bahk, S.-W.

Barty, C.

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

Begishev, I. A.

Blanchot, N.

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

Borneis, S.

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

Britten, J.

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

Bromage, J.

Chambaret, J.-P.

Chvykov, V.

Collier, J.

J. Collier, C. Hernandez-Gomez, R. Allott, C. Danson, and A. Hall, "A single-shot third-order autocorrelator for pulse contrast and pulse shape measurements," Laser Part. Beams 19, 231�??235 (2001).
[CrossRef]

Danson, C.

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

J. Collier, C. Hernandez-Gomez, R. Allott, C. Danson, and A. Hall, "A single-shot third-order autocorrelator for pulse contrast and pulse shape measurements," Laser Part. Beams 19, 231�??235 (2001).
[CrossRef]

Dawson, J.

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

de Beauvoir, B.

Divall, E. J.

Dorrer, C.

Dufour, C.

C. Dufour, "Utilisation de l�??interféromètre Fabry-Perot pour la recherche des satellites faibles," Rev. Opt. 24, 11�??18 (1945).

Hall, A.

J. Collier, C. Hernandez-Gomez, R. Allott, C. Danson, and A. Hall, "A single-shot third-order autocorrelator for pulse contrast and pulse shape measurements," Laser Part. Beams 19, 231�??235 (2001).
[CrossRef]

Hernandez-Gomez, C.

J. Collier, C. Hernandez-Gomez, R. Allott, C. Danson, and A. Hall, "A single-shot third-order autocorrelator for pulse contrast and pulse shape measurements," Laser Part. Beams 19, 231�??235 (2001).
[CrossRef]

Hutchinson, M. H. R.

S. Luan, M. H. R. Hutchinson, R. A. Smith, and F. Zhou, "High dynamic range third-order correlation measurement of picosecond laser pulse shapes," Meas. Sci. Technol. 4, 1426�??1429 (1993).
[CrossRef]

Jovanovic, I.

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

Jungquist, R.

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

Kalintchenko, G.

Kane, D. J.

Kelly, J. H.

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

Kessler, T. J.

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

Kruschwitz, B.

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

Le Blanc, C.

LeBlanc, C.

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

LeGarrec, B.

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

Luan, S.

S. Luan, M. H. R. Hutchinson, R. A. Smith, and F. Zhou, "High dynamic range third-order correlation measurement of picosecond laser pulse shapes," Meas. Sci. Technol. 4, 1426�??1429 (1993).
[CrossRef]

Maksimchuk, A.

McKinney, J. D.

S. Xiao, J. D. McKinney, and A. M. Weiner, "Photonic microwave arbitrary waveform generation using a virtually imaged phased-array (VIPA) direct space-to-time pulse shaper," IEEE Photon. Technol. Lett. 16, 1936�??1938 (2004).
[CrossRef]

Miyanaga, N.

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

Moses, E.

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

Mourou, G. A.

Planchon, T. A.

Rambo, P. K.

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

Ranc, S.

Ross, I. N.

Rousseau, J.-P.

Rousseau, P.

Schmid, A. W.

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

Smith, R. A.

S. Luan, M. H. R. Hutchinson, R. A. Smith, and F. Zhou, "High dynamic range third-order correlation measurement of picosecond laser pulse shapes," Meas. Sci. Technol. 4, 1426�??1429 (1993).
[CrossRef]

Trebino, R.

Usechak, N. G.

Waxer, L. J.

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

Weiner, A. M.

S. Xiao, J. D. McKinney, and A. M. Weiner, "Photonic microwave arbitrary waveform generation using a virtually imaged phased-array (VIPA) direct space-to-time pulse shaper," IEEE Photon. Technol. Lett. 16, 1936�??1938 (2004).
[CrossRef]

Xiao, S.

S. Xiao, J. D. McKinney, and A. M. Weiner, "Photonic microwave arbitrary waveform generation using a virtually imaged phased-array (VIPA) direct space-to-time pulse shaper," IEEE Photon. Technol. Lett. 16, 1936�??1938 (2004).
[CrossRef]

Yanovksy, V.

Zhou, F.

S. Luan, M. H. R. Hutchinson, R. A. Smith, and F. Zhou, "High dynamic range third-order correlation measurement of picosecond laser pulse shapes," Meas. Sci. Technol. 4, 1426�??1429 (1993).
[CrossRef]

Zuegel, J. D.

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

J. Bromage, C. Dorrer, I. A. Begishev, N. G. Usechak, and J. D. Zuegel, "Highly sensitive, single-shot characterization for pulse widths from 0.4 to 85 ps using electro-optic shearing interferometry," Opt. Lett. 31, 3523�??3525 (2006).
[CrossRef] [PubMed]

Fusion Sci. Technol.

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I.  Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453�??482 (2006).

IEEE Photon. Technol. Lett.

S. Xiao, J. D. McKinney, and A. M. Weiner, "Photonic microwave arbitrary waveform generation using a virtually imaged phased-array (VIPA) direct space-to-time pulse shaper," IEEE Photon. Technol. Lett. 16, 1936�??1938 (2004).
[CrossRef]

Laser Part. Beams

J. Collier, C. Hernandez-Gomez, R. Allott, C. Danson, and A. Hall, "A single-shot third-order autocorrelator for pulse contrast and pulse shape measurements," Laser Part. Beams 19, 231�??235 (2001).
[CrossRef]

Meas. Sci. Technol.

S. Luan, M. H. R. Hutchinson, R. A. Smith, and F. Zhou, "High dynamic range third-order correlation measurement of picosecond laser pulse shapes," Meas. Sci. Technol. 4, 1426�??1429 (1993).
[CrossRef]

Opt. Lett.

Rev. Opt.

C. Dufour, "Utilisation de l�??interféromètre Fabry-Perot pour la recherche des satellites faibles," Rev. Opt. 24, 11�??18 (1945).

Other

I. Jovanovic, B. C. Haefner, M. Shverdin, M. Taranowski, and C. P. J. Barty, "High-dynamic-range, 200-ps window, single-shot cross-correlator for ultrahigh intensity laser characterization," presented at the CLEO/QELS Conference, Baltimore, MD, 6-11 May 2007 (Paper JThD137).

L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, Cambridge, England, 1995).

Supplementary Material (1)

» Media 1: MOV (3951 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

(a) Design of a single-shot cross-correlator based on the nonlinear interaction of the pulse under test with a sequence of sampling pulses from an optical pulse replicator. (b) Layout for the calculation of the spatial and temporal properties of the sequence of pulses from the optical pulse replicator. HR: high reflector; PR: partial reflector.

Fig. 2.
Fig. 2.

(a) Relative position of 50 replicas from the pulse replicator versus relative angle of the two reflectors of the OPR. The value of α minimizing the spread of the relative positions is indicated by a white dashed line. (b) Relative position of 50 replicas from the pulse replicator for the relative angle minimizing the standard deviation of the relative positions.

Fig. 3.
Fig. 3.

(a) Nonlinear component of the delay for 50 replicas from the pulse replicator versus the relative angle of the two reflectors of the OPR. The value of α minimizing the spread of the relative positions is indicated by a white dashed line. (b) Nonlinear component of the delay for 50 replicas from the pulse replicator for the relative angle minimizing the standard deviation of the relative positions.

Fig. 4.
Fig. 4.

Measured 3ω signal versus 1ω signal without neutral-density filters on the sampling pulse (blue circles) and with a 30-dB attenuation on the sampling pulse after signal correction (red squares). The solid line corresponds to the expected linear relation between the input and output signals.

Fig. 5.
Fig. 5.

Intensity histogram of the output of a Fabry–Perot etalon on a (a) linear scale and (b) logarithmic scale. In each case, the average value of the intensity at each time slot is plotted with white circles.

Fig. 6.
Fig. 6.

Intensity histogram of 10,000 successive shots of the DPRA on a (a) linear and (b) logarithmic scale. In each time slot, brighter intensity corresponds to a larger proportion of samples in a given intensity bin. The average intensity is plotted with white circles. A prepulse can be observed approximately 100 ps before the main pulse.

Fig. 7.
Fig. 7.

(Media 1) Animated plot of the intensity of 100 successive shots of the DPRA on a logarithmic scale.

Fig. 8.
Fig. 8.

Intensity histogram of 10,000 realizations of ASE on (a) a linear and (b) a logarithmic scale. In each case, the average value of the intensity at each time slot is plotted with white circles.

Fig. 9.
Fig. 9.

Intensity probability density distribution for amplified spontaneous emission. Plots (a) and (c) correspond to an overlap of the distributions obtained at each of the 36 time slots characterized by the diagnostic on a linear and logarithmic scale, respectively. Plots (b) and (d) correspond to the distribution calculated with all the measured samples on a linear and logarithmic scale, respectively. Experimental distributions are plotted with blue lines and the theoretical prediction of Eq. (5) is plotted with black markers.

Fig. 10.
Fig. 10.

Normalized intensity standard deviation as a function of time for amplified spontaneous emission (in relation to Fig. 8) and for a 200-ps window including the main laser pulse (in relation to Fig. 6).

Equations (5)

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

θ n = θ n 1 2 α
d n = d n 1 1 tan ( α ) tan ( θ n 1 2 α ) 1 + tan ( θ n 1 ) tan ( α )
x n = x n 1 + d n 1 tan ( θ n 1 ) + tan ( θ n 1 2 α ) 1 + tan ( α ) tan ( θ n 1 )
L n = L n 1 + d n 1 1 + tan ( α ) tan ( θ n 1 ) [ 1 cos ( θ n 1 ) + 1 cos ( θ n 1 2 α ) ] + z cos ( θ n ) .
P ( I ) = 1 I exp ( I I ) ,

Metrics