Abstract

The loss characteristics in multimode transmission were studied for hollow waveguides, and simple equations for optical power attenuation were derived. The equations that were derived by the electromagnetic wave theory and the geometrical optics theory coincide exactly. If the transmission loss is small, the optical power attenuates exponentially with the length of the waveguide. The transmission loss increases in proportion to the inverse of the bore radius and the square of the launch angle. The theoretical values that were calculated by the present equations were in good agreement with the data measured for the SiO2 and ZnSe-coated Ag hollow waveguides.

© 1993 Optical Society of America

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  1. D. R. Hall, E. K. Gorton, R. M. Jenkins, “10-μm propagation losses in hollow dielectric waveguides,” J. Appl. Phys 48, 1212–1216 (1977).
    [Crossref]
  2. E. Garmire, T. McMahon, M. Bass, “Flexible infrared waveguides for high-power transmission,” IEEE J. Quantum Electron. QE-16, 23–32 (1980).
    [Crossref]
  3. T. Hidaka, K. Kumada, J. Shimada, T. Morikawa, “GeO2–ZnO–K2O glass as the cladding material of 940 cm−1CO2laser light transmitting hollow-core waveguide,” J. Appl. Phys. 53, 5484–5490 (1982).
    [Crossref]
  4. M. Miyagi, A. Hongo, Y. Aizawa, S. Kawakami, “Fabrication of germanium-coated nickel hollow waveguides for infrared transmission,” Appl. Phys. Lett. 43, 430–432 (1983).
    [Crossref]
  5. J. A. Harrington, C. C. Gregory, “Hollow sapphire fibers for the delivery of CO2laser energy,” Opt. Lett. 15, 541–543 (1990).
    [Crossref] [PubMed]
  6. S. J. Saggese, J. A. Harrington, G. H. Siegel, “Attenuation of incoherent infrared radiation in hollow sapphire and silica waveguides,” Opt. Lett. 16, 27–29 (1991).
    [Crossref] [PubMed]
  7. S. J. Saggese, J. A. Harrington, G. H. Siegel, “Hollow sapphire waveguides for remote radiometric temperature measurement,” Electron. Lett. 27, 707–709 (1991).
    [Crossref]
  8. C. A. Worrell, I. P. Giles, N. A. Adatia, “Remote gas sensing with mid-infra-red hollow waveguide,” Electron. Lett. 28, 615–617 (1992).
    [Crossref]
  9. M. Saito, S. Sato, M. Miyagi, “Radiation thermometry for low temperature using an infrared hollow waveguide,” Opt. Eng. 31, 1793–1799 (1992).
    [Crossref]
  10. E. A. J. Marcatili, R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
  11. M. Miyagi, S. Kawakami, “Design theory of dielectriccoated circular metallic waveguides for infrared transmission,” IEEE J. Lightwave Technol. LT-2, 116–126 (1984).
    [Crossref]
  12. A. Hongo, M. Miyagi, K. Sakamoto, S. Karasawa, S. Nishida, “Excitation dependent losses and temperature increase in various hollow waveguides at 10.6 μm,” Opt Laser Technol. 19, 214–216 (1987).
    [Crossref]
  13. Y. Matsuura, M. Saito, M. Miyagi, A. Hongo, “Loss characteristics of circular hollow waveguides for incoherent infrared light,” J. Opt. Soc. Am. A 6, 423–427 (1989).
    [Crossref]
  14. K. Iga, K. Kokubun, M. Oikawa, Fundamentals of Microoptics (Academic, New York, 1984), Chaps. 5 and 6.
  15. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965), Chaps. 9 and 11.
  16. M. Saito, Y. Matsuura, M. Kawamura, M. Miyagi, “Bending losses of incoherent light in circular hollow waveguides,” J. Opt. Soc. Am. A 7, 2063–2068 (1990).
    [Crossref]
  17. N. Nagano, M. Saito, M. Miyagi, N. Baba, N. Sawanobori, “Prediction of optical losses in SiO2- and GeO2-based glass hollow waveguides for the infrared,” Appl. Phys. Lett. 58, 1807–1809 (1991).
    [Crossref]
  18. Y. Matsuura, M. Miyagi, A. Hongo, “Fabrication of low-loss zinc-selenide coated silver hollow waveguides for CO2laser light,” J. Appl. Phys. 68, 5463–5466 (1990).
    [Crossref]

1992 (2)

C. A. Worrell, I. P. Giles, N. A. Adatia, “Remote gas sensing with mid-infra-red hollow waveguide,” Electron. Lett. 28, 615–617 (1992).
[Crossref]

M. Saito, S. Sato, M. Miyagi, “Radiation thermometry for low temperature using an infrared hollow waveguide,” Opt. Eng. 31, 1793–1799 (1992).
[Crossref]

1991 (3)

S. J. Saggese, J. A. Harrington, G. H. Siegel, “Attenuation of incoherent infrared radiation in hollow sapphire and silica waveguides,” Opt. Lett. 16, 27–29 (1991).
[Crossref] [PubMed]

S. J. Saggese, J. A. Harrington, G. H. Siegel, “Hollow sapphire waveguides for remote radiometric temperature measurement,” Electron. Lett. 27, 707–709 (1991).
[Crossref]

N. Nagano, M. Saito, M. Miyagi, N. Baba, N. Sawanobori, “Prediction of optical losses in SiO2- and GeO2-based glass hollow waveguides for the infrared,” Appl. Phys. Lett. 58, 1807–1809 (1991).
[Crossref]

1990 (3)

1989 (1)

1987 (1)

A. Hongo, M. Miyagi, K. Sakamoto, S. Karasawa, S. Nishida, “Excitation dependent losses and temperature increase in various hollow waveguides at 10.6 μm,” Opt Laser Technol. 19, 214–216 (1987).
[Crossref]

1984 (1)

M. Miyagi, S. Kawakami, “Design theory of dielectriccoated circular metallic waveguides for infrared transmission,” IEEE J. Lightwave Technol. LT-2, 116–126 (1984).
[Crossref]

1983 (1)

M. Miyagi, A. Hongo, Y. Aizawa, S. Kawakami, “Fabrication of germanium-coated nickel hollow waveguides for infrared transmission,” Appl. Phys. Lett. 43, 430–432 (1983).
[Crossref]

1982 (1)

T. Hidaka, K. Kumada, J. Shimada, T. Morikawa, “GeO2–ZnO–K2O glass as the cladding material of 940 cm−1CO2laser light transmitting hollow-core waveguide,” J. Appl. Phys. 53, 5484–5490 (1982).
[Crossref]

1980 (1)

E. Garmire, T. McMahon, M. Bass, “Flexible infrared waveguides for high-power transmission,” IEEE J. Quantum Electron. QE-16, 23–32 (1980).
[Crossref]

1977 (1)

D. R. Hall, E. K. Gorton, R. M. Jenkins, “10-μm propagation losses in hollow dielectric waveguides,” J. Appl. Phys 48, 1212–1216 (1977).
[Crossref]

1964 (1)

E. A. J. Marcatili, R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).

Adatia, N. A.

C. A. Worrell, I. P. Giles, N. A. Adatia, “Remote gas sensing with mid-infra-red hollow waveguide,” Electron. Lett. 28, 615–617 (1992).
[Crossref]

Aizawa, Y.

M. Miyagi, A. Hongo, Y. Aizawa, S. Kawakami, “Fabrication of germanium-coated nickel hollow waveguides for infrared transmission,” Appl. Phys. Lett. 43, 430–432 (1983).
[Crossref]

Baba, N.

N. Nagano, M. Saito, M. Miyagi, N. Baba, N. Sawanobori, “Prediction of optical losses in SiO2- and GeO2-based glass hollow waveguides for the infrared,” Appl. Phys. Lett. 58, 1807–1809 (1991).
[Crossref]

Bass, M.

E. Garmire, T. McMahon, M. Bass, “Flexible infrared waveguides for high-power transmission,” IEEE J. Quantum Electron. QE-16, 23–32 (1980).
[Crossref]

Garmire, E.

E. Garmire, T. McMahon, M. Bass, “Flexible infrared waveguides for high-power transmission,” IEEE J. Quantum Electron. QE-16, 23–32 (1980).
[Crossref]

Giles, I. P.

C. A. Worrell, I. P. Giles, N. A. Adatia, “Remote gas sensing with mid-infra-red hollow waveguide,” Electron. Lett. 28, 615–617 (1992).
[Crossref]

Gorton, E. K.

D. R. Hall, E. K. Gorton, R. M. Jenkins, “10-μm propagation losses in hollow dielectric waveguides,” J. Appl. Phys 48, 1212–1216 (1977).
[Crossref]

Gregory, C. C.

Hall, D. R.

D. R. Hall, E. K. Gorton, R. M. Jenkins, “10-μm propagation losses in hollow dielectric waveguides,” J. Appl. Phys 48, 1212–1216 (1977).
[Crossref]

Harrington, J. A.

Hidaka, T.

T. Hidaka, K. Kumada, J. Shimada, T. Morikawa, “GeO2–ZnO–K2O glass as the cladding material of 940 cm−1CO2laser light transmitting hollow-core waveguide,” J. Appl. Phys. 53, 5484–5490 (1982).
[Crossref]

Hongo, A.

Y. Matsuura, M. Miyagi, A. Hongo, “Fabrication of low-loss zinc-selenide coated silver hollow waveguides for CO2laser light,” J. Appl. Phys. 68, 5463–5466 (1990).
[Crossref]

Y. Matsuura, M. Saito, M. Miyagi, A. Hongo, “Loss characteristics of circular hollow waveguides for incoherent infrared light,” J. Opt. Soc. Am. A 6, 423–427 (1989).
[Crossref]

A. Hongo, M. Miyagi, K. Sakamoto, S. Karasawa, S. Nishida, “Excitation dependent losses and temperature increase in various hollow waveguides at 10.6 μm,” Opt Laser Technol. 19, 214–216 (1987).
[Crossref]

M. Miyagi, A. Hongo, Y. Aizawa, S. Kawakami, “Fabrication of germanium-coated nickel hollow waveguides for infrared transmission,” Appl. Phys. Lett. 43, 430–432 (1983).
[Crossref]

Iga, K.

K. Iga, K. Kokubun, M. Oikawa, Fundamentals of Microoptics (Academic, New York, 1984), Chaps. 5 and 6.

Jenkins, R. M.

D. R. Hall, E. K. Gorton, R. M. Jenkins, “10-μm propagation losses in hollow dielectric waveguides,” J. Appl. Phys 48, 1212–1216 (1977).
[Crossref]

Karasawa, S.

A. Hongo, M. Miyagi, K. Sakamoto, S. Karasawa, S. Nishida, “Excitation dependent losses and temperature increase in various hollow waveguides at 10.6 μm,” Opt Laser Technol. 19, 214–216 (1987).
[Crossref]

Kawakami, S.

M. Miyagi, S. Kawakami, “Design theory of dielectriccoated circular metallic waveguides for infrared transmission,” IEEE J. Lightwave Technol. LT-2, 116–126 (1984).
[Crossref]

M. Miyagi, A. Hongo, Y. Aizawa, S. Kawakami, “Fabrication of germanium-coated nickel hollow waveguides for infrared transmission,” Appl. Phys. Lett. 43, 430–432 (1983).
[Crossref]

Kawamura, M.

Kokubun, K.

K. Iga, K. Kokubun, M. Oikawa, Fundamentals of Microoptics (Academic, New York, 1984), Chaps. 5 and 6.

Kumada, K.

T. Hidaka, K. Kumada, J. Shimada, T. Morikawa, “GeO2–ZnO–K2O glass as the cladding material of 940 cm−1CO2laser light transmitting hollow-core waveguide,” J. Appl. Phys. 53, 5484–5490 (1982).
[Crossref]

Marcatili, E. A. J.

E. A. J. Marcatili, R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).

Matsuura, Y.

McMahon, T.

E. Garmire, T. McMahon, M. Bass, “Flexible infrared waveguides for high-power transmission,” IEEE J. Quantum Electron. QE-16, 23–32 (1980).
[Crossref]

Miyagi, M.

M. Saito, S. Sato, M. Miyagi, “Radiation thermometry for low temperature using an infrared hollow waveguide,” Opt. Eng. 31, 1793–1799 (1992).
[Crossref]

N. Nagano, M. Saito, M. Miyagi, N. Baba, N. Sawanobori, “Prediction of optical losses in SiO2- and GeO2-based glass hollow waveguides for the infrared,” Appl. Phys. Lett. 58, 1807–1809 (1991).
[Crossref]

Y. Matsuura, M. Miyagi, A. Hongo, “Fabrication of low-loss zinc-selenide coated silver hollow waveguides for CO2laser light,” J. Appl. Phys. 68, 5463–5466 (1990).
[Crossref]

M. Saito, Y. Matsuura, M. Kawamura, M. Miyagi, “Bending losses of incoherent light in circular hollow waveguides,” J. Opt. Soc. Am. A 7, 2063–2068 (1990).
[Crossref]

Y. Matsuura, M. Saito, M. Miyagi, A. Hongo, “Loss characteristics of circular hollow waveguides for incoherent infrared light,” J. Opt. Soc. Am. A 6, 423–427 (1989).
[Crossref]

A. Hongo, M. Miyagi, K. Sakamoto, S. Karasawa, S. Nishida, “Excitation dependent losses and temperature increase in various hollow waveguides at 10.6 μm,” Opt Laser Technol. 19, 214–216 (1987).
[Crossref]

M. Miyagi, S. Kawakami, “Design theory of dielectriccoated circular metallic waveguides for infrared transmission,” IEEE J. Lightwave Technol. LT-2, 116–126 (1984).
[Crossref]

M. Miyagi, A. Hongo, Y. Aizawa, S. Kawakami, “Fabrication of germanium-coated nickel hollow waveguides for infrared transmission,” Appl. Phys. Lett. 43, 430–432 (1983).
[Crossref]

Morikawa, T.

T. Hidaka, K. Kumada, J. Shimada, T. Morikawa, “GeO2–ZnO–K2O glass as the cladding material of 940 cm−1CO2laser light transmitting hollow-core waveguide,” J. Appl. Phys. 53, 5484–5490 (1982).
[Crossref]

Nagano, N.

N. Nagano, M. Saito, M. Miyagi, N. Baba, N. Sawanobori, “Prediction of optical losses in SiO2- and GeO2-based glass hollow waveguides for the infrared,” Appl. Phys. Lett. 58, 1807–1809 (1991).
[Crossref]

Nishida, S.

A. Hongo, M. Miyagi, K. Sakamoto, S. Karasawa, S. Nishida, “Excitation dependent losses and temperature increase in various hollow waveguides at 10.6 μm,” Opt Laser Technol. 19, 214–216 (1987).
[Crossref]

Oikawa, M.

K. Iga, K. Kokubun, M. Oikawa, Fundamentals of Microoptics (Academic, New York, 1984), Chaps. 5 and 6.

Saggese, S. J.

S. J. Saggese, J. A. Harrington, G. H. Siegel, “Attenuation of incoherent infrared radiation in hollow sapphire and silica waveguides,” Opt. Lett. 16, 27–29 (1991).
[Crossref] [PubMed]

S. J. Saggese, J. A. Harrington, G. H. Siegel, “Hollow sapphire waveguides for remote radiometric temperature measurement,” Electron. Lett. 27, 707–709 (1991).
[Crossref]

Saito, M.

M. Saito, S. Sato, M. Miyagi, “Radiation thermometry for low temperature using an infrared hollow waveguide,” Opt. Eng. 31, 1793–1799 (1992).
[Crossref]

N. Nagano, M. Saito, M. Miyagi, N. Baba, N. Sawanobori, “Prediction of optical losses in SiO2- and GeO2-based glass hollow waveguides for the infrared,” Appl. Phys. Lett. 58, 1807–1809 (1991).
[Crossref]

M. Saito, Y. Matsuura, M. Kawamura, M. Miyagi, “Bending losses of incoherent light in circular hollow waveguides,” J. Opt. Soc. Am. A 7, 2063–2068 (1990).
[Crossref]

Y. Matsuura, M. Saito, M. Miyagi, A. Hongo, “Loss characteristics of circular hollow waveguides for incoherent infrared light,” J. Opt. Soc. Am. A 6, 423–427 (1989).
[Crossref]

Sakamoto, K.

A. Hongo, M. Miyagi, K. Sakamoto, S. Karasawa, S. Nishida, “Excitation dependent losses and temperature increase in various hollow waveguides at 10.6 μm,” Opt Laser Technol. 19, 214–216 (1987).
[Crossref]

Sato, S.

M. Saito, S. Sato, M. Miyagi, “Radiation thermometry for low temperature using an infrared hollow waveguide,” Opt. Eng. 31, 1793–1799 (1992).
[Crossref]

Sawanobori, N.

N. Nagano, M. Saito, M. Miyagi, N. Baba, N. Sawanobori, “Prediction of optical losses in SiO2- and GeO2-based glass hollow waveguides for the infrared,” Appl. Phys. Lett. 58, 1807–1809 (1991).
[Crossref]

Schmeltzer, R. A.

E. A. J. Marcatili, R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).

Shimada, J.

T. Hidaka, K. Kumada, J. Shimada, T. Morikawa, “GeO2–ZnO–K2O glass as the cladding material of 940 cm−1CO2laser light transmitting hollow-core waveguide,” J. Appl. Phys. 53, 5484–5490 (1982).
[Crossref]

Siegel, G. H.

S. J. Saggese, J. A. Harrington, G. H. Siegel, “Hollow sapphire waveguides for remote radiometric temperature measurement,” Electron. Lett. 27, 707–709 (1991).
[Crossref]

S. J. Saggese, J. A. Harrington, G. H. Siegel, “Attenuation of incoherent infrared radiation in hollow sapphire and silica waveguides,” Opt. Lett. 16, 27–29 (1991).
[Crossref] [PubMed]

Worrell, C. A.

C. A. Worrell, I. P. Giles, N. A. Adatia, “Remote gas sensing with mid-infra-red hollow waveguide,” Electron. Lett. 28, 615–617 (1992).
[Crossref]

Appl. Phys. Lett. (2)

M. Miyagi, A. Hongo, Y. Aizawa, S. Kawakami, “Fabrication of germanium-coated nickel hollow waveguides for infrared transmission,” Appl. Phys. Lett. 43, 430–432 (1983).
[Crossref]

N. Nagano, M. Saito, M. Miyagi, N. Baba, N. Sawanobori, “Prediction of optical losses in SiO2- and GeO2-based glass hollow waveguides for the infrared,” Appl. Phys. Lett. 58, 1807–1809 (1991).
[Crossref]

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).

Electron. Lett. (2)

S. J. Saggese, J. A. Harrington, G. H. Siegel, “Hollow sapphire waveguides for remote radiometric temperature measurement,” Electron. Lett. 27, 707–709 (1991).
[Crossref]

C. A. Worrell, I. P. Giles, N. A. Adatia, “Remote gas sensing with mid-infra-red hollow waveguide,” Electron. Lett. 28, 615–617 (1992).
[Crossref]

IEEE J. Lightwave Technol. (1)

M. Miyagi, S. Kawakami, “Design theory of dielectriccoated circular metallic waveguides for infrared transmission,” IEEE J. Lightwave Technol. LT-2, 116–126 (1984).
[Crossref]

IEEE J. Quantum Electron. (1)

E. Garmire, T. McMahon, M. Bass, “Flexible infrared waveguides for high-power transmission,” IEEE J. Quantum Electron. QE-16, 23–32 (1980).
[Crossref]

J. Appl. Phys (1)

D. R. Hall, E. K. Gorton, R. M. Jenkins, “10-μm propagation losses in hollow dielectric waveguides,” J. Appl. Phys 48, 1212–1216 (1977).
[Crossref]

J. Appl. Phys. (2)

Y. Matsuura, M. Miyagi, A. Hongo, “Fabrication of low-loss zinc-selenide coated silver hollow waveguides for CO2laser light,” J. Appl. Phys. 68, 5463–5466 (1990).
[Crossref]

T. Hidaka, K. Kumada, J. Shimada, T. Morikawa, “GeO2–ZnO–K2O glass as the cladding material of 940 cm−1CO2laser light transmitting hollow-core waveguide,” J. Appl. Phys. 53, 5484–5490 (1982).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt Laser Technol. (1)

A. Hongo, M. Miyagi, K. Sakamoto, S. Karasawa, S. Nishida, “Excitation dependent losses and temperature increase in various hollow waveguides at 10.6 μm,” Opt Laser Technol. 19, 214–216 (1987).
[Crossref]

Opt. Eng. (1)

M. Saito, S. Sato, M. Miyagi, “Radiation thermometry for low temperature using an infrared hollow waveguide,” Opt. Eng. 31, 1793–1799 (1992).
[Crossref]

Opt. Lett. (2)

Other (2)

K. Iga, K. Kokubun, M. Oikawa, Fundamentals of Microoptics (Academic, New York, 1984), Chaps. 5 and 6.

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965), Chaps. 9 and 11.

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Figures (7)

Fig. 1
Fig. 1

Optical system for launching a beam into a hollow waveguide. T, bore radius; ws, minimum spot size.

Fig. 2
Fig. 2

Optical power ratio ηm of the HE1m mode excited in a hollow waveguide by (a) a Gaussian beam or (b) a focused plane-wave beam (theoretical values). The numbers beside the curves indicate the launch angles θ0 or θL. The values were calculated for T = 1 mm and λ= 10 μm.

Fig. 3
Fig. 3

Experimental setup for evaluating the waveguide loss with the beam of uniform angular distribution.

Fig. 4
Fig. 4

Angular distributions of the launching beams. Light intensity was measured through a pinhole of 1-mm diameter placed 50 mm from the coupling waveguide (Gaussian distribution, ○) or the aperture (uniform distribution, ●). The edges of the profile become dull because of the limited spatial resolution (∼1 deg) of the pinhole.

Fig. 5
Fig. 5

Optical attenuation spectra measured for (a) SiO2 hollow waveguides and (b) a ZnSe-coated Ag hollow waveguide. All the waveguides are 1 m in length. The bore radii T of the SiO2 waveguides are 0.9, 1.0, and 1.5 mm. The bore radius of the ZnSe/Ag waveguide is 0.75 mm, and the thickness of the ZnSe film is 0.75 μm. The beams of uniform intensity were launched into the waveguides with launch angles θL of (a) 2.8 and (b) 1.8, 2.8, and 3.3 deg.

Fig. 6
Fig. 6

Optical losses of SiO2 hollow waveguides at 8- and 9-μm wavelengths as a function of the bore radius T. The waveguides measured were 1 m in length. ○ and △ show the data measured with a beam of the Gaussian distribution of θ0 = 2.2 deg; ● and ▴ show the data measured with a beam of the uniform distribution of θL = 3.3 deg. The solid lines show the corresponding theoretical values calculated by using Eqs. (15) and (21).

Fig. 7
Fig. 7

Optical losses of hollow waveguides as a function of the launch angle θ0 or θL. The circles and triangles show the data measured at 8-μm wavelength for SiO2 hollow waveguides with a length of 1 m and bore radii of 0.75 and 1.5 mm. The squares show the data measured at 5-μm wavelength for a ZnSe-coated Ag hollow waveguide with a length of 1 m and a bore radius of 0.75 mm. ○, △, and □ correspond to the beams of the Gaussian distribution, and ●, ▴, and ▪ correspond to the beams of the uniform distribution. The solid lines show the theoretical values calculated by using Eqs. (15) and (21).

Equations (36)

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

E ( θ ) = E 0 exp ( θ 2 / θ 0 2 ) ,
w s λ / ( π θ 0 ) ,
E ( r ) = E ( 0 ) exp ( r 2 / w s 2 ) E ( 0 ) exp ( π 2 θ 0 2 r 2 / λ 2 ) .
η m = | 0 T E ( r ) J 0 ( u m r / T ) r d r | 2 0 | E ( r ) | 2 r d r 0 T J 0 2 ( u m r / T ) r d r ,
η m = | 0 T exp ( π 2 θ 0 2 r 2 / λ 2 ) J 0 ( u m r / T ) r d r | 2 0 exp ( 2 π 2 θ 0 2 r 2 / λ 2 ) r d r 0 T J 0 2 ( u m r / T ) r d r .
θ 0 0.02 ( 1 deg ) , T 0.3 mm , λ 10 μ m ,
exp ( π 2 θ 0 2 r 2 / λ 2 ) < exp ( π 2 θ 0 2 T 2 / λ 2 ) 10 2 .
η m | 0 exp ( π 2 θ 0 2 r 2 / λ 2 ) J 0 ( u m r / T ) r d r | 2 0 exp ( 2 π 2 θ 0 2 r 2 / λ 2 ) r d r 0 T J 0 2 ( u m r / T ) r d r = 2 λ 2 π 2 θ 0 2 T 2 J 1 2 ( u m ) exp ( u m 2 λ 2 2 π 2 θ 0 2 T 2 ) ( m 1 / 4 ) λ 2 θ 0 2 T 2 exp [ ( m 1 / 4 ) 2 λ 2 2 θ 0 2 T 2 ] ,
u m ( m 1 / 4 ) π , J 1 ( u m ) ( 1 ) m 1 [ 2 / ( m 1 / 4 ) ] 1 / 2 / π
2 α m = u m C 2 λ 2 / ( 8 π 2 T 3 ) ( m 1 / 4 ) 2 C λ 2 / ( 8 T 3 ) ,
C = 2 Re { [ ( n j κ ) 2 + 1 ] / [ ( n j κ ) 2 1 ] 1 / 2 } = 2 { 1 + 2 / [ ( n 2 κ 2 1 ) 2 + 4 n 2 κ 2 ] 1 / 2 } × { ( n 2 κ 2 1 ) + [ ( n 2 κ 2 1 ) 2 + 4 n 2 κ 2 ] 1 / 2 } 1 / 2 .
P ( z ) = m = 1 η m exp ( 2 α m z ) m = 1 ( m 1 / 4 ) λ 2 θ 0 2 T 2 exp [ ( m 1 / 4 ) 2 λ 2 2 θ 0 2 T 2 ( 1 + C θ 0 2 4 T z ) ] λ 2 θ 0 2 T 2 0 m exp [ λ 2 2 θ 0 2 T 2 ( 1 + C θ 0 2 4 T z ) m 2 ] d m
= [ 1 + C θ 0 2 z / ( 4 T ) ] 1 .
P ( z ) exp { [ C θ 0 2 / ( 4 T ) ] z } .
2 α = C θ 0 2 / ( 4 T )
E ( r ) = E ( 0 ) J 1 ( 2 π θ L r / λ ) / r ,
η m = | 0 T J 1 ( 2 π θ L r / λ ) J 0 ( u m r / T ) d r | 2 0 J 1 2 ( 2 π θ L r / λ ) / r d r 0 T J 0 2 ( u m r / T ) r d r | 0 J 1 ( 2 π θ L r / λ ) J 0 ( u m r / T ) d r | 2 0 J 1 2 ( 2 π θ L r / λ ) / r d r 0 T J 0 2 ( u m r / T ) r d r = { λ 2 / [ π θ L T J 1 ( u m ) ] 2 ( u m / T < 2 π θ L / λ ) 0 ( u m / T > 2 π θ L / λ ) .
η m = { ( m 1 / 4 ) λ 2 / ( 2 θ L 2 T 2 ) ( m < 2 θ L T / λ + 1 / 4 ) 0 ( m > 2 θ L T / λ + 1 / 4 ) .
P ( z ) = m = 1 2 θ L T / λ + 1 / 4 ( m 1 / 4 ) λ 2 2 θ L 2 T 2 exp [ ( m 1 / 4 ) 2 C λ 2 8 T 3 z ] λ 2 2 θ L 2 T 2 0 2 θ L T / λ m exp ( C λ 2 z 8 T 3 m 2 ) d m = [ 2 T / ( C θ L 2 z ) ] { 1 exp [ C θ L 2 z / ( 2 T ) ] } .
P ( z ) exp { [ C θ L 2 / ( 4 T ) ] z } .
2 α = C θ L 2 / ( 4 T )
I ( θ ) = | E ( θ ) | 2 = E 0 2 exp ( 2 θ 2 / θ 0 2 ) .
2 α ( θ ) = [ 1 R ( θ ) ] / ( 2 T cot θ ) ,
R ( θ ) 1 C θ .
2 α ( θ ) C θ 2 / ( 2 T )
P ( z ) = 0 π / 2 I ( θ ) exp [ 2 α ( θ ) z ] sin θ d θ / 0 π / 2 I ( θ ) sin θ d θ
0 exp [ 2 θ 2 θ 0 2 ( 1 + C θ 0 2 4 T z ) ] θ d θ / 0 exp ( 2 θ 2 θ 0 2 ) θ d θ = [ 1 + C θ 0 2 z / ( 4 T ) ] 1 .
I ( θ ) = { const. ( θ < θ L ) 0 ( θ > θ L ) ,
P ( z ) = 0 θ L exp ( C θ 2 2 T z ) sin θ d θ / 0 θ L sin θ d θ [ 2 T / ( C θ L 2 z ) ] { 1 exp [ C θ L 2 z / ( 2 T ) ] } .
I ( θ ) = ξ i I i ( θ ) ,
P ( z ) = ξ i exp ( 2 α i z ) ξ i ( 1 2 α i z ) = ξ i 2 α i ξ i z = 1 ( C 4 T ξ i θ i 2 ) z .
2 α 1 / T 3 .
2 α θ 0 2 / T
u m = ( 2 π / λ ) T sin θ 0 2 π T θ 0 / λ .
θ 0 1 / T .
2 α θ 0 2 / T 1 / T 3 ,

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