Abstract

By using the extended Huygens-Fresnel principle, an expression has been obtained for the generalized two-position, two-frequency mutual-coherence function of an electromagnetic beam in a turbulent medium. That result has been examined for different states of source coherence and for different turbulence conditions.

© 1981 Optical Society of America

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  1. K. Yeh and C. Liu, "An investigation of temporal moments of stochastic waves," Radio Sci. 12, 671–677 (1977).
  2. H. Su and M. Plonus, "Optical-pulse propagation in a turbulent medium" J. Opt. Soc. Am. 61, 256–260 (1971).
  3. C. Gardner and M. Plonus, "Optical pulses in atmospheric turbulence," J. Opt. Soc. Am. 64, 63–77 (1974).
  4. C. Liu, A. Wernik, and K. Yeh, "Propagation of pulse trains through a random medium," IEEE Trans. Antennas Propag. AP-22, 624–627 (1974).
  5. S. Hong and A. Ishimaru, "Multiple scattering effects on coherent bandwidth and pulse distortion of a wave propagating in a random distribution of particles," Radio Sci. 10, 637–644 (1975).
  6. I. Sreenivasiah, A. Ishimaru, and S. Hong, "Two frequency mutual coherence function and pulse propagation in random medium: an analytic solution to the plane wave case," Radio Sci. 11, 775–778 (1976).
  7. R. Dashen, "Path integrals for waves in random media," J. Math. Phys. 20, 894–920 (1979).
  8. L. Lee and J. Jokipii, "Strong scintillations in astrophysics II: theory of temporal pulse broadening," Astrophys. J. 201, 532–543 (1975).
  9. C. Liu and K. Yeh, "Pulse spreading and wandering in random media," Radio Sci. 14, 925–931 (1979).
  10. C. Rino, V. Gonzalex, and A. Hessing, "Coherence bandwidth loss in transionospheric radio propagation," Radio Sci. 16, 245–255 (1981).
  11. I. Sreenivasiah and A. Ishimaru, "Beam wave two frequency mutual coherence function and pulse propagation in random media: an analytic solution," Appl. Opt. 18, 1613–1618 (1979).
  12. S. Wandzura, "Meaning of quadratic structure functions," J. Opt. Soc. Am. 70, 745–747 (1980).
  13. By narrow-band we mean that the temporal frequency spectrum is concentrated in a band Δω about ω = ±ω0 such that Δω/ω0 « 1.
  14. We need calculate 〈uu*〉 and not 〈uu〉 and 〈u*u*〉 because the last two terms are coefficients of exp(i2ω0t) and exp(-i0t) variations, which generally average to zero.
  15. V. Feizulin and Y. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 33–35 (1967).
  16. R. Lutomirski and H. Yura, "Propagation of a finite optical beam in an inhomogeneous medium," Appl. Opt. 10, 1652–1658 (1971).
  17. R. Fante, "Multiple-frequency mutual coherence functions for a beam in a random medium," IEEE Trans. Antennas Propag. AP-26, 621–623 (1978).
  18. R. Fante, "Electromagnetic beam propagation in turbulent media—an update," Proc. IEEE 68, 1424–1443 (1980).
  19. V. Tatarskii, The Effect of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971)
  20. A. Ishimaru, A., "Temporal frequency spectra of multifrequency waves in turbulent atmosphere," IEEE Trans. Antennas Propag. AP-20, 10–19 (1972).
  21. R. Fante and J. Poirier, "Mutual coherence function of a finite optical beam in a turbulent medium," Appl. Opt. 12, 2247–2248 (1973).
  22. The Kolmogorov spectrum is the limit of Eq. (10) when l0 → 0 and L0 → ∞.
  23. L. Mandel and E. Wolf, "Spectral coherence and the concept of cross spectral purity," J. Opt. Soc. Am. 66, 529–535 (1976).
  24. E. Wolf, "A new description of second-order coherence phenomena in the space frequency domain," in M. Machado and L. Nordicci, eds., Optics in Four Dimensions, Conference Proceedings No. 65 (American Institute of Physics, New York, 1981).
  25. L. Mandel and E. Wolf,Complete coherence in the space-frequency domain Opt. Commun. 36, 247–249 (1981).
  26. J. Leader, "Beam intensity fluctuations in atmospheric turbulence," J. Opt. Soc. Am. 71, 542–558 (1981).
  27. R. Fante, "Some physical insights into beam propagation in strong turbulence," Radio Sci. 15, 757–762 (1980).
  28. D. Knepp, "Multiple phase screen calculations of the temporal behavior of stochastic waves," presented at the URSI Symposium, Quebec, Canada.

1981

C. Rino, V. Gonzalex, and A. Hessing, "Coherence bandwidth loss in transionospheric radio propagation," Radio Sci. 16, 245–255 (1981).

L. Mandel and E. Wolf,Complete coherence in the space-frequency domain Opt. Commun. 36, 247–249 (1981).

J. Leader, "Beam intensity fluctuations in atmospheric turbulence," J. Opt. Soc. Am. 71, 542–558 (1981).

1980

S. Wandzura, "Meaning of quadratic structure functions," J. Opt. Soc. Am. 70, 745–747 (1980).

R. Fante, "Some physical insights into beam propagation in strong turbulence," Radio Sci. 15, 757–762 (1980).

1979

I. Sreenivasiah and A. Ishimaru, "Beam wave two frequency mutual coherence function and pulse propagation in random media: an analytic solution," Appl. Opt. 18, 1613–1618 (1979).

C. Liu and K. Yeh, "Pulse spreading and wandering in random media," Radio Sci. 14, 925–931 (1979).

R. Dashen, "Path integrals for waves in random media," J. Math. Phys. 20, 894–920 (1979).

1978

R. Fante, "Multiple-frequency mutual coherence functions for a beam in a random medium," IEEE Trans. Antennas Propag. AP-26, 621–623 (1978).

1977

K. Yeh and C. Liu, "An investigation of temporal moments of stochastic waves," Radio Sci. 12, 671–677 (1977).

1976

I. Sreenivasiah, A. Ishimaru, and S. Hong, "Two frequency mutual coherence function and pulse propagation in random medium: an analytic solution to the plane wave case," Radio Sci. 11, 775–778 (1976).

L. Mandel and E. Wolf, "Spectral coherence and the concept of cross spectral purity," J. Opt. Soc. Am. 66, 529–535 (1976).

1975

S. Hong and A. Ishimaru, "Multiple scattering effects on coherent bandwidth and pulse distortion of a wave propagating in a random distribution of particles," Radio Sci. 10, 637–644 (1975).

L. Lee and J. Jokipii, "Strong scintillations in astrophysics II: theory of temporal pulse broadening," Astrophys. J. 201, 532–543 (1975).

1974

C. Gardner and M. Plonus, "Optical pulses in atmospheric turbulence," J. Opt. Soc. Am. 64, 63–77 (1974).

C. Liu, A. Wernik, and K. Yeh, "Propagation of pulse trains through a random medium," IEEE Trans. Antennas Propag. AP-22, 624–627 (1974).

1973

1972

A. Ishimaru, A., "Temporal frequency spectra of multifrequency waves in turbulent atmosphere," IEEE Trans. Antennas Propag. AP-20, 10–19 (1972).

1971

1967

V. Feizulin and Y. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 33–35 (1967).

Dashen, R.

R. Dashen, "Path integrals for waves in random media," J. Math. Phys. 20, 894–920 (1979).

Fante, R.

R. Fante, "Some physical insights into beam propagation in strong turbulence," Radio Sci. 15, 757–762 (1980).

R. Fante, "Multiple-frequency mutual coherence functions for a beam in a random medium," IEEE Trans. Antennas Propag. AP-26, 621–623 (1978).

R. Fante and J. Poirier, "Mutual coherence function of a finite optical beam in a turbulent medium," Appl. Opt. 12, 2247–2248 (1973).

R. Fante, "Electromagnetic beam propagation in turbulent media—an update," Proc. IEEE 68, 1424–1443 (1980).

Feizulin, V.

V. Feizulin and Y. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 33–35 (1967).

Gardner, C.

C. Gardner and M. Plonus, "Optical pulses in atmospheric turbulence," J. Opt. Soc. Am. 64, 63–77 (1974).

Gonzalex, V.

C. Rino, V. Gonzalex, and A. Hessing, "Coherence bandwidth loss in transionospheric radio propagation," Radio Sci. 16, 245–255 (1981).

Hessing, A.

C. Rino, V. Gonzalex, and A. Hessing, "Coherence bandwidth loss in transionospheric radio propagation," Radio Sci. 16, 245–255 (1981).

Hong, S.

I. Sreenivasiah, A. Ishimaru, and S. Hong, "Two frequency mutual coherence function and pulse propagation in random medium: an analytic solution to the plane wave case," Radio Sci. 11, 775–778 (1976).

S. Hong and A. Ishimaru, "Multiple scattering effects on coherent bandwidth and pulse distortion of a wave propagating in a random distribution of particles," Radio Sci. 10, 637–644 (1975).

Ishimaru, A.

I. Sreenivasiah and A. Ishimaru, "Beam wave two frequency mutual coherence function and pulse propagation in random media: an analytic solution," Appl. Opt. 18, 1613–1618 (1979).

I. Sreenivasiah, A. Ishimaru, and S. Hong, "Two frequency mutual coherence function and pulse propagation in random medium: an analytic solution to the plane wave case," Radio Sci. 11, 775–778 (1976).

S. Hong and A. Ishimaru, "Multiple scattering effects on coherent bandwidth and pulse distortion of a wave propagating in a random distribution of particles," Radio Sci. 10, 637–644 (1975).

A. Ishimaru, A., "Temporal frequency spectra of multifrequency waves in turbulent atmosphere," IEEE Trans. Antennas Propag. AP-20, 10–19 (1972).

Jokipii, J.

L. Lee and J. Jokipii, "Strong scintillations in astrophysics II: theory of temporal pulse broadening," Astrophys. J. 201, 532–543 (1975).

Knepp, D.

D. Knepp, "Multiple phase screen calculations of the temporal behavior of stochastic waves," presented at the URSI Symposium, Quebec, Canada.

Kravtsov, Y.

V. Feizulin and Y. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 33–35 (1967).

Leader, J.

Lee, L.

L. Lee and J. Jokipii, "Strong scintillations in astrophysics II: theory of temporal pulse broadening," Astrophys. J. 201, 532–543 (1975).

Liu, C.

C. Liu and K. Yeh, "Pulse spreading and wandering in random media," Radio Sci. 14, 925–931 (1979).

K. Yeh and C. Liu, "An investigation of temporal moments of stochastic waves," Radio Sci. 12, 671–677 (1977).

C. Liu, A. Wernik, and K. Yeh, "Propagation of pulse trains through a random medium," IEEE Trans. Antennas Propag. AP-22, 624–627 (1974).

Lutomirski, R.

Mandel, L.

L. Mandel and E. Wolf,Complete coherence in the space-frequency domain Opt. Commun. 36, 247–249 (1981).

L. Mandel and E. Wolf, "Spectral coherence and the concept of cross spectral purity," J. Opt. Soc. Am. 66, 529–535 (1976).

Plonus, M.

C. Gardner and M. Plonus, "Optical pulses in atmospheric turbulence," J. Opt. Soc. Am. 64, 63–77 (1974).

H. Su and M. Plonus, "Optical-pulse propagation in a turbulent medium" J. Opt. Soc. Am. 61, 256–260 (1971).

Poirier, J.

Rino, C.

C. Rino, V. Gonzalex, and A. Hessing, "Coherence bandwidth loss in transionospheric radio propagation," Radio Sci. 16, 245–255 (1981).

Sreenivasiah, I.

I. Sreenivasiah and A. Ishimaru, "Beam wave two frequency mutual coherence function and pulse propagation in random media: an analytic solution," Appl. Opt. 18, 1613–1618 (1979).

I. Sreenivasiah, A. Ishimaru, and S. Hong, "Two frequency mutual coherence function and pulse propagation in random medium: an analytic solution to the plane wave case," Radio Sci. 11, 775–778 (1976).

Su, H.

Tatarskii, V.

V. Tatarskii, The Effect of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971)

Wandzura, S.

Wernik, A.

C. Liu, A. Wernik, and K. Yeh, "Propagation of pulse trains through a random medium," IEEE Trans. Antennas Propag. AP-22, 624–627 (1974).

Wolf, E.

L. Mandel and E. Wolf,Complete coherence in the space-frequency domain Opt. Commun. 36, 247–249 (1981).

L. Mandel and E. Wolf, "Spectral coherence and the concept of cross spectral purity," J. Opt. Soc. Am. 66, 529–535 (1976).

E. Wolf, "A new description of second-order coherence phenomena in the space frequency domain," in M. Machado and L. Nordicci, eds., Optics in Four Dimensions, Conference Proceedings No. 65 (American Institute of Physics, New York, 1981).

Yeh, K.

C. Liu and K. Yeh, "Pulse spreading and wandering in random media," Radio Sci. 14, 925–931 (1979).

K. Yeh and C. Liu, "An investigation of temporal moments of stochastic waves," Radio Sci. 12, 671–677 (1977).

C. Liu, A. Wernik, and K. Yeh, "Propagation of pulse trains through a random medium," IEEE Trans. Antennas Propag. AP-22, 624–627 (1974).

Yura, H.

Appl. Opt.

Astrophys. J.

L. Lee and J. Jokipii, "Strong scintillations in astrophysics II: theory of temporal pulse broadening," Astrophys. J. 201, 532–543 (1975).

IEEE Trans. Antennas Propag.

C. Liu, A. Wernik, and K. Yeh, "Propagation of pulse trains through a random medium," IEEE Trans. Antennas Propag. AP-22, 624–627 (1974).

A. Ishimaru, A., "Temporal frequency spectra of multifrequency waves in turbulent atmosphere," IEEE Trans. Antennas Propag. AP-20, 10–19 (1972).

R. Fante, "Multiple-frequency mutual coherence functions for a beam in a random medium," IEEE Trans. Antennas Propag. AP-26, 621–623 (1978).

J. Math. Phys.

R. Dashen, "Path integrals for waves in random media," J. Math. Phys. 20, 894–920 (1979).

J. Opt. Soc. Am.

Opt. Commun.

L. Mandel and E. Wolf,Complete coherence in the space-frequency domain Opt. Commun. 36, 247–249 (1981).

Radio Sci.

R. Fante, "Some physical insights into beam propagation in strong turbulence," Radio Sci. 15, 757–762 (1980).

K. Yeh and C. Liu, "An investigation of temporal moments of stochastic waves," Radio Sci. 12, 671–677 (1977).

S. Hong and A. Ishimaru, "Multiple scattering effects on coherent bandwidth and pulse distortion of a wave propagating in a random distribution of particles," Radio Sci. 10, 637–644 (1975).

I. Sreenivasiah, A. Ishimaru, and S. Hong, "Two frequency mutual coherence function and pulse propagation in random medium: an analytic solution to the plane wave case," Radio Sci. 11, 775–778 (1976).

C. Liu and K. Yeh, "Pulse spreading and wandering in random media," Radio Sci. 14, 925–931 (1979).

C. Rino, V. Gonzalex, and A. Hessing, "Coherence bandwidth loss in transionospheric radio propagation," Radio Sci. 16, 245–255 (1981).

Radiophys. Quantum Electron.

V. Feizulin and Y. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 33–35 (1967).

Other

By narrow-band we mean that the temporal frequency spectrum is concentrated in a band Δω about ω = ±ω0 such that Δω/ω0 « 1.

We need calculate 〈uu*〉 and not 〈uu〉 and 〈u*u*〉 because the last two terms are coefficients of exp(i2ω0t) and exp(-i0t) variations, which generally average to zero.

The Kolmogorov spectrum is the limit of Eq. (10) when l0 → 0 and L0 → ∞.

E. Wolf, "A new description of second-order coherence phenomena in the space frequency domain," in M. Machado and L. Nordicci, eds., Optics in Four Dimensions, Conference Proceedings No. 65 (American Institute of Physics, New York, 1981).

D. Knepp, "Multiple phase screen calculations of the temporal behavior of stochastic waves," presented at the URSI Symposium, Quebec, Canada.

R. Fante, "Electromagnetic beam propagation in turbulent media—an update," Proc. IEEE 68, 1424–1443 (1980).

V. Tatarskii, The Effect of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971)

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