Abstract

Simulations of laser-beam propagation through atmospheric turbulence and acoustic pulse propagation though ocean internal waves in the presence of a transverse waveguide are described. Both these problems are amenable to the parabolic wave equation for propagation in the forward direction. In the optical case, the question treated is the irradiance variance and spatial spectrum. In the ocean case, pulse propagation over long ranges is investigated. Determining the travel time of a pulse requires expanding the numerical simulation in a broadband, multifrequency calculation that takes even more time. Much effort has been expended in approximating the propagation by rays, so that the trajectory of the energy propagating from a given source to a given receiver can be tracked, and coherence calculations can be ray-based, using semi-classical formulas. This paper reviews the comparisons of several analytic ray-based approximations with numerical parabolic-equation simulations to determine their accuracies.

© 2002 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  5. R. Dashen and G.-Y. Wang, ??Intensity fluctuation for waves behind a phase screen: a new asymptotic scheme,?? J. Opt. Soc. Am. A 10, 1219??1225 (1993).
    [CrossRef]
  6. R. Dashen, G.-Y. Wang, S. M. Flatte, and C. Bracher, ??Moments of intensity and log intensity: new asymptotic results for waves in power-law media,?? J. Opt. Soc. Am. A 10, 1233??1242 (1993).
    [CrossRef]
  7. R. Dashen, G. Wang, S. M. Flatte, and C. Bracher, ??Moments of intensity and log-intensity: New asymptotic results for waves in power-law media,?? J. Opt. Soc. Am. A 10, 1233??1242 (1993).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. T. Duda, S. M. Flatte, J. Colosi, B. Cornuelle, J. Hildebrand, W. Hodgkiss, Jr., P. Worcester, B. Howe, J. Mercer, and R. Spindel, ??Measured wavefront fluctuations in 1000-km pulse propagation in the Pacific Ocean,?? J. Acoust. Soc. Am. 92, 939??955 (1992).
    [CrossRef]
  13. J. Colosi, S. M. Flatte, and C. Bracher, ??Internal-wave effects on 1000-km oceanic acoustic pulse propagation: Simulation and comparison with experiment,?? J. Acoust. Soc. Am. 96, 452??468 (1994).
    [CrossRef]
  14. S. Reynolds, S. Flatte, R. Dashen, B. Buehler, and P. Maciejewski, ??AFAR measurements of acoustic mutual coherence functions of time and frequency,?? J. Acoust. Soc. Am. 77, 1723??31 (1985).
    [CrossRef]
  15. R. Dashen, S. Flatte, and S. Reynolds, ??Path-integral treatment of acoustic mutual coherence functions for rays in a sound channel,?? J. Acoust. Soc. Am. 77, 1716??22 (1985).
    [CrossRef]
  16. S. Flatte, S. Reynolds, R. Dashen, B. Buehler, and P. Maciejewski, ??AFAR measurements of intensity and intensity moments,?? J. Acoust. Soc. Am. 82, 973??980 (1987).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. S. M. Flatte, C. Bracher, and G.-Y. Wang, ??Probability-density functions of irradiance for waves in atmospheric turbulence calculated by numerical simulation,?? J. Opt. Soc. Am. A 11, 2080??2092 (1994).
  23. K. Gochelashvili and V. Shishov, ??Saturated fluctuations in the laser radiation intensity in a turbulent medium,?? Sov. Phys. JETP 39, 605??609 (1974).
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    [CrossRef]
  25. R. Frehlich, ??Intensity covariance of a point source in a random medium with a Kolmogorov spectrum and an inner scale of turbulence,?? J. Opt. Soc. Am. A 4, 360??366 (1987).
    [CrossRef]
  26. L. Andrews, M. Al-Habash, C. Hopen, and R. Phillips, ??Theory of optical scintillation: Gaussianbeam wave model,?? Waves Random Media 11, 271??291 (2001).
    [CrossRef]
  27. R. Dashen, G.Y. Wang, Stanley M. Flatte, and C. Bracher, ??Moments of intensity and logintensity: new asymptotic results from waves in power-law random media,?? J. Opt. Soc. Am. A 10, 1233??1242 (1993)
    [CrossRef]
  28. S. Flatte and G. Rovner, ??Calculations of internal-wave-induced fluctuations in ocean-acoustic propagation,?? J. Acoust. Soc. Am. 108, 526??534 (2000).
    [CrossRef] [PubMed]
  29. R. Hardin and F. Tappert, ??Applications of the Split-Step Fourier Method to the Numerical Solution of Nonlinear and Variable CoefficientWave Equations,?? Society for Industrial and Applied Mathematics Review 15, 423 (1973).
  30. S. Flatte and F. Tappert, ??Calculation of the effect of internal waves on oceanic sound transmission,?? J. Acoust. Soc. Am. 58, 1151??1159 (1975).
    [CrossRef]
  31. W. H. Munk, ??Sound channel in an exponentially stratified ocean, with application to SOFAR,?? J. Acoust. Soc. Am. 55, 220??226 (1974).
    [CrossRef]
  32. C. Garrett and W. Munk, ??Space-time scales of ocean internal waves,?? Geophys. Fluid Dyn. 2, 225??264 (1972).
  33. C. Garrett and W. Munk, ??Space-time scales of internal waves: a progress report,?? J. Geophys. Res. 80, 291??297 (1975).
    [CrossRef]
  34. S. Flatte and R. Esswein, ??Calculation of the phase-structure function density from oceanic internal waves,?? J. Acoust. Soc. Am. 70, 1387??96 (1981).
    [CrossRef]
  35. J. Colosi and M. Brown, ??Efficient numerical simulation of stochastic internal-wave-induced soundspeed perturbation fields,?? J. Acoust. Soc. Am. 103, 2232??2235 (1998).
    [CrossRef]
  36. S. Flatte and G. Rovner, ??Path-integral expressions for fluctuations in acoustic transmission in the ocean waveguide,?? In Methods of The oretical Physics Applied to Oceanography, P. Muller, ed., pp. 167??174 (Proceedings of the Ninth ??Aha Huliko??a Hawaiian Winter Workshop, 1997).
  37. S. Flatte and R. Stoughton, ??Predictions of internal-wave effects on ocean acoustic coherence, travel-time variance, and intensity moments for very long-range propagation,?? J. Acoust. Soc. Am. 84, 1414??1424 (1988).
    [CrossRef]
  38. J. Colosi, Ph.D. thesis, University of California at Santa Cruz, 1993.
  39. J. Colosi et al., ??Comparisons of measured and predicted acoustic fluctuations for a 3250-km propagation experiment in the eastern North Pacific Ocean,?? J. Acoust. Soc. Am. 105, 3202??3218 (1999).
    [CrossRef]
  40. J. Colosi, F. Tappert, and M. Dzieciuch, ??Further analysis of intensity fluctuations from a 3252-km acoustic propagation experiment in the eastern North Pacific,?? J. Acoust. Soc. Am. 110, 163??169 (2001).
    [CrossRef]

Geophys. Fluid Dyn. (1)

C. Garrett and W. Munk, ??Space-time scales of ocean internal waves,?? Geophys. Fluid Dyn. 2, 225??264 (1972).

J. Acoust. Soc. Am. (16)

S. Flatte and F. Tappert, ??Calculation of the effect of internal waves on oceanic sound transmission,?? J. Acoust. Soc. Am. 58, 1151??1159 (1975).
[CrossRef]

W. H. Munk, ??Sound channel in an exponentially stratified ocean, with application to SOFAR,?? J. Acoust. Soc. Am. 55, 220??226 (1974).
[CrossRef]

S. Flatte and R. Esswein, ??Calculation of the phase-structure function density from oceanic internal waves,?? J. Acoust. Soc. Am. 70, 1387??96 (1981).
[CrossRef]

J. Colosi and M. Brown, ??Efficient numerical simulation of stochastic internal-wave-induced soundspeed perturbation fields,?? J. Acoust. Soc. Am. 103, 2232??2235 (1998).
[CrossRef]

S. Flatte and R. Stoughton, ??Predictions of internal-wave effects on ocean acoustic coherence, travel-time variance, and intensity moments for very long-range propagation,?? J. Acoust. Soc. Am. 84, 1414??1424 (1988).
[CrossRef]

S. Flatte and G. Rovner, ??Calculations of internal-wave-induced fluctuations in ocean-acoustic propagation,?? J. Acoust. Soc. Am. 108, 526??534 (2000).
[CrossRef] [PubMed]

T. Duda, S. M. Flatte, J. Colosi, B. Cornuelle, J. Hildebrand, W. Hodgkiss, Jr., P. Worcester, B. Howe, J. Mercer, and R. Spindel, ??Measured wavefront fluctuations in 1000-km pulse propagation in the Pacific Ocean,?? J. Acoust. Soc. Am. 92, 939??955 (1992).
[CrossRef]

J. Colosi, S. M. Flatte, and C. Bracher, ??Internal-wave effects on 1000-km oceanic acoustic pulse propagation: Simulation and comparison with experiment,?? J. Acoust. Soc. Am. 96, 452??468 (1994).
[CrossRef]

S. Reynolds, S. Flatte, R. Dashen, B. Buehler, and P. Maciejewski, ??AFAR measurements of acoustic mutual coherence functions of time and frequency,?? J. Acoust. Soc. Am. 77, 1723??31 (1985).
[CrossRef]

R. Dashen, S. Flatte, and S. Reynolds, ??Path-integral treatment of acoustic mutual coherence functions for rays in a sound channel,?? J. Acoust. Soc. Am. 77, 1716??22 (1985).
[CrossRef]

S. Flatte, S. Reynolds, R. Dashen, B. Buehler, and P. Maciejewski, ??AFAR measurements of intensity and intensity moments,?? J. Acoust. Soc. Am. 82, 973??980 (1987).
[CrossRef]

S. Flatte, S. Reynolds, and R. Dashen, ??Path-integral treatment of intensity behavior for rays in a sound channel,?? J. Acoust. Soc. Am. 82, 967??972 (1987).
[CrossRef]

S. Flatte and M. Vera, ??Comparison between ocean-acoustic fluctuations in parabolic-equation simulations and estimates from integral approximations,?? J. Acoust. Soc. Am. in review (2002).

J. Simmen, S. M. Flatte, and G.-Y. Wang, ??Wavefront folding, chaos, and diffraction for sound propagation through ocean internal waves,?? J. Acoust. Soc. Am. 102, 239??255 (1997).
[CrossRef]

J. Colosi et al., ??Comparisons of measured and predicted acoustic fluctuations for a 3250-km propagation experiment in the eastern North Pacific Ocean,?? J. Acoust. Soc. Am. 105, 3202??3218 (1999).
[CrossRef]

J. Colosi, F. Tappert, and M. Dzieciuch, ??Further analysis of intensity fluctuations from a 3252-km acoustic propagation experiment in the eastern North Pacific,?? J. Acoust. Soc. Am. 110, 163??169 (2001).
[CrossRef]

J. Geophys. Res. (1)

C. Garrett and W. Munk, ??Space-time scales of internal waves: a progress report,?? J. Geophys. Res. 80, 291??297 (1975).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (10)

S. Flatte and J. Gerber, ??Irradiance variance behavior for plane- and spherical-wave optical propagation through strong turbulence,?? J. Opt. Soc. Am. A 17, 1092??1097 (2000).
[CrossRef]

R. Dashen and G.-Y. Wang, ??Intensity fluctuation for waves behind a phase screen: a new asymptotic scheme,?? J. Opt. Soc. Am. A 10, 1219??1225 (1993).
[CrossRef]

R. Dashen, G.-Y. Wang, S. M. Flatte, and C. Bracher, ??Moments of intensity and log intensity: new asymptotic results for waves in power-law media,?? J. Opt. Soc. Am. A 10, 1233??1242 (1993).
[CrossRef]

R. Dashen, G. Wang, S. M. Flatte, and C. Bracher, ??Moments of intensity and log-intensity: New asymptotic results for waves in power-law media,?? J. Opt. Soc. Am. A 10, 1233??1242 (1993).
[CrossRef]

A. Consortini, F. Cochetti, J. H. Churnside, and R. J. Hill, ??Inner-scale effect on intensity variance measured for weak to strong atmospheric scintillation,?? J. Opt. Soc. Am. A 10, 2354??2362 (1993).
[CrossRef]

S. M. Flatte, G. Y. Wang, and J. Martin, ??Irradiance variance of optical waves through atmospheric turbulence by numerical simulation and comparison with experiment,?? J. Opt. Soc. Am. A 10, 2363??2370 (1993).
[CrossRef]

L. Andrews, R. Phillips, C. Hopen, and M. Al-Habash, ??Theory of optical scintillation,?? J. Opt. Soc. Am. A 16, 1417??1429 (1999).
[CrossRef]

R. Frehlich, ??Intensity covariance of a point source in a random medium with a Kolmogorov spectrum and an inner scale of turbulence,?? J. Opt. Soc. Am. A 4, 360??366 (1987).
[CrossRef]

S. M. Flatte, C. Bracher, and G.-Y. Wang, ??Probability-density functions of irradiance for waves in atmospheric turbulence calculated by numerical simulation,?? J. Opt. Soc. Am. A 11, 2080??2092 (1994).

R. Dashen, G.Y. Wang, Stanley M. Flatte, and C. Bracher, ??Moments of intensity and logintensity: new asymptotic results from waves in power-law random media,?? J. Opt. Soc. Am. A 10, 1233??1242 (1993)
[CrossRef]

Proc. of the IEEE (1)

S. Flatte, ??Wave propagation through random media: Contributions from ocean acoustics,?? Proc. of the IEEE 71, 1267??1294 (1983).
[CrossRef]

Soc. Industrial Appl. Math. Rev. (1)

R. Hardin and F. Tappert, ??Applications of the Split-Step Fourier Method to the Numerical Solution of Nonlinear and Variable CoefficientWave Equations,?? Society for Industrial and Applied Mathematics Review 15, 423 (1973).

Sov. Phys. JETP (1)

K. Gochelashvili and V. Shishov, ??Saturated fluctuations in the laser radiation intensity in a turbulent medium,?? Sov. Phys. JETP 39, 605??609 (1974).

Waves Random Media (1)

L. Andrews, M. Al-Habash, C. Hopen, and R. Phillips, ??Theory of optical scintillation: Gaussianbeam wave model,?? Waves Random Media 11, 271??291 (2001).
[CrossRef]

Other (6)

J. Colosi, Ph.D. thesis, University of California at Santa Cruz, 1993.

S. Flatte and G. Rovner, ??Path-integral expressions for fluctuations in acoustic transmission in the ocean waveguide,?? In Methods of The oretical Physics Applied to Oceanography, P. Muller, ed., pp. 167??174 (Proceedings of the Ninth ??Aha Huliko??a Hawaiian Winter Workshop, 1997).

S. Flatte, R. Dashen, W. Munk, K. Watson, and F. Zachariasen, Sound Transmission Through a Fluctuating Ocean (A 300 page monograph published by the Cambridge University Press in their series on Mechanics and Applied Mathematics, 1979).

L. Andrews and R. Phillips, Laser beam propagation through random media (SPIE Publishing Services, Bellingham, WA, 1999).

F. Jensen, W. Kuperman, M. Porter, and H. Schmidt, Computational Ocean Acoustics (American Institute of Physics Press, 1994).

F. Tappert, ??The parabolic approximation method,?? In Wave Propagation and Underwater Acoustics, J. Keller and J. Papadakis, eds., pp. 224??287 (Springer-Verlag, 1977).
[CrossRef]

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Figures (22)

Fig. 1.
Fig. 1.

Variance preserving spectra for plane and spherical-wave simulations. The integral under each curve is the value of irradiance variance σI2 for the corresponding turbulence parameters. In this paper we are studying the difference of this variance from unity.

Fig. 2.
Fig. 2.

Spectra of irradiance from plane-wave simulations for different values of turbulence strength and inner scale. The values of β02 are 0.01, 30, 40, 50, 60, 70, 80, 90, and 100. The order is from top to bottom at κRf = 1. Each spectrum has been divided by its respective β02. For zero inner scale, it is expected that the high-wavenumber asymptote of all the curves will be the same.

Fig. 3.
Fig. 3.

Spectra of irradiance from spherical-wave simulations for different values of turbulence strength and inner scale. The values of β02 are 0.01, 30, 40, 50, 60, 70, 80, 90, and 100. The order is from top to bottom at κRf = 1. Each spectrum has been divided by its respective β02. For zero inner scale, it is expected that the high-wavenumber asymptote of all the curves will be the same.

Fig. 4.
Fig. 4.

Irradiance variance from plane-wave simulations for different values of turbulence strength and inner scale. The dashed lines are from asymptotic theory (as β02 becomes large). It is seen that the simulations yield power-law behavior, but with power-law indices that are dramatically different from expectations.

Fig. 5.
Fig. 5.

Irradiance variance from spherical-wave simulations for different values of turbulence strength and inner scale. The dashed lines are from asymptotic theory (as β02 becomes large). It is seen that the simulations yield power-law behavior, but with power-law indices that are dramatically different from expectations.

Fig. 6.
Fig. 6.

Irradiance variance as a function of inner scale, compared with analytic asymptotic theory. It is seen that the simulations fall roughly, but not exactly, on power-law behavior, and that the power-law indices disagree with analytic predictions.

Fig. 7.
Fig. 7.

The sound-speed profiles, c(z), used in this paper. The Canonical profile is the solid curve; the dashed curve is from the Slice89 experiment.

Fig. 8.
Fig. 8.

Buoyancy frequency profiles. The Canonical profile is the solid curve; the dashed curve is from the Slice89 experiment.

Fig. 9.
Fig. 9.

The effect of internal waves on a timefront ID segment: ID-36 from a sample 250 Hz, Canonical timefront. The internal-wave strengths are (from left to right) 0, 0.5, 1 and 2 GM.

Fig. 10.
Fig. 10.

The effect of internal waves on a timefront ID segment: ID-37 from a sample 250 Hz, Slice89 timefront. The internal-wave strengths are (from left to right) 0, 0.5, 1, 2 GM.

Fig. 11.
Fig. 11.

A sample 250 Hz timefront in the Canonical environment with 2 GM internal waves. Significant front distortion is evident.

Fig. 12.
Fig. 12.

A sample 250 Hz timefront in the Slice89 environment with 2 GM internal waves. Significant distortion is evident, especially in the transmission finalé.

Fig. 13.
Fig. 13.

Analytic integral approximation (line) and simulation results (points) for the rms variation in travel time, τ, for a 250 Hz acoustic signal.

Fig. 14.
Fig. 14.

The rms travel-time variability, τ, for a 250 Hz frequency as a function of timefront ID number at 1000 km. The squares are the analytic estimates and the circles result from the PE simulations. Shaded (unshaded) points correspond to positive (negative) ID numbers.

Fig. 15.
Fig. 15.

The internal-wave bias τ 1, for a 250 Hz acoustic signal (the average difference from the travel time in the absence of internal waves). Results from integral approximations (line) and simulation results (points) differ substantially at long ranges, particularly for high internal-wave strengths.

Fig. 16.
Fig. 16.

The coherence function, Cz (δz), from integral approximations (circles) and the parabolic-equation simulations (line) at 1000 km for 250 Hz. Integral techniques yield a substantially shorter depth-coherence length scale.

Fig. 17.
Fig. 17.

Mean pulse shapes in the Canonical profile for a 250 Hz acoustic signal at 1000 km. The dashed curve is the pulse without internal waves present; the solid curve is affected by 1 GM internal waves. τ 0 is the average integral approximation for the induced spreading in ms.

Fig. 18.
Fig. 18.

Mean pulse shapes in the Slice89 profile for a 250 Hz acoustic signal at 1000 km. The dashed curve is the pulse without internal waves present; the solid curve is affected by 1 GM internal waves. τ 0 is the average integral approximation for the induced spreading in ms.

Fig. 19.
Fig. 19.

Timefronts for 1000-kilometer propagation through an ocean described by the SLICE89 average sound-speed profile with and without internal waves present. The figure in the upper left is the result for the case of no internal waves, and the remaining three figures are the results for three different realizations of the internal-wave field. Each point on the timefronts represents the arrival of one ray, and there are 4096 ray arrivals having uniformly incremented launch angles.

Fig. 20.
Fig. 20.

Eigenrays that contribute to a single peak in intensity. The top figure shows the eigenray traces from the source to the receiving point 1000 kilometers away at a depth of 1000 meters. The middle figure magnifies a portion of the eigenray traces. The bottom figure describes the extent of the raytube as a function of range in two ways: the asterisks and the crosses give the maximum horizontal distance and vertical distance, respectively, between corresponding ray upper turning points within the raytube. The units of the vertical axis are km and m/100 for the asterisks and crosses, respectively. In this instance, horizontal resolution increases (i.e., raytube horizontal extent decreases) while vertical resolution decreases (i.e., vertical extent increases), as the raytube nears the receiver.

Fig. 21.
Fig. 21.

Comparison of semi-classical and PE solutions for the case of an internal-wave field when the simulated source signal has 1000-Hz center frequency. The top figure shows intensity versus time for the semi-classical solution (dashed), the incoherent semi-classical sum (dotted), and the PE solution (solid). The horizontal line at z=-1000 m in the bottom figure intersects the local timefront structure at the many ray arrival times indicated by crosses in the upper figure.

Fig. 22.
Fig. 22.

Comparison of semi-classical and PE solutions for another case of an internal-wave field when the simulated source signal has 1000-Hz center frequency. The top figure shows intensity versus time for both the semi-classical solution (dashed) and the PE solution (solid). The intensity calculated from an incoherent sum of ray arrivals (dotted) shows the error incurred if phase information is excluded from the semi-classical sum. In this case, multi-paths interfere to produce two peaks rather than one.

Tables (3)

Tables Icon

Table 1. Results of fits to irradiance variances from plane-wave simulations to Equation 4. The intercept a is also the value of σI2 – 1 at βc2 = 60.6.

Tables Icon

Table 2. Results of fits to irradiance variances from spherical-wave simulations to Equation 4. The intercept a is also the value of σI2 – 1 at βc2 = 60.6.

Tables Icon

Table 3. Results of common-slope fits to all non-zero inner scales for a given initial condition, by use of Equation 5. Fits to straight lines were made to values of σI2 – 1 for βc2 = 60.6 and for non-zero inner scale (from Tables 1 and 2) to obtain values of a and c. These fits are plotted in Figure 6. The common slope b for each initial condition was obtained by a method described in the reference.

Equations (17)

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

2 i q 0 ψ x + 2 ψ z 2 + 2 ψ y 2 2 [ U 0 + μ x y z ] q 0 2 ψ = 0
β 0 2 = 1.23 C n 2 k 7 6 L 11 6
β 0 2 = 0.50 C n 2 k 7 6 L 11 6
σ 1 2 1 = a ( β c 2 β 0 2 ) b
σ I 2 1 = a ( β c 2 β 0 2 ) b ( l 0 R f ) c
c ( x , t ) = c 0 [ 1 + U 0 ( z ) + μ ( x , t ) ]
μ ( x , t ) = [ z U 0 ( z ) ] p ζ ( x , t )
S ζ ( j , k ) = 2 B 2 E π M N 0 N 1 ( j 2 + j * 2 ) k j k 2 ( k 2 + k j 2 ) 2
{ k υ 2 } = N ( z ) 2 [ l o N 0 ] 2
ψ * ( δ z ) ψ ( 0 ) exp [ 1 2 ( δ z z 0 ) 2 ]
ψ * ( δσ ) ψ ( 0 ) exp [ 1 2 ( δστ ) 2 + iδσ τ 1 1 2 ( δσ τ 0 ) 2 ]
τ 2 = c 0 2 0 R dx μ 2 ( z ) L p ( θ , z )
ξ ( x ) = z ( x ) z ray ( x )
xx ξ ( x ) + U 0 ξ ( x ) = 0
z 0 2 = q 0 2 ln ( στ ) 0 R dx μ 2 L p { k υ 2 } [ ξ 1 ( x ) ] 2
τ 1 = ln ( στ ) 2 c 0 0 R dx μ 2 L p { k υ 2 } g ( x , x )
τ 0 2 = 1 2 ( ln ( στ ) c 0 ) 2 0 R dx μ 2 L p { k υ 2 } 0 R dx μ 2 L p { k υ 2 } [ g ( x , x ) ] 2

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