Abstract

The past dozen years have seen several important publications on mode coupling losses in circular and rectangular waveguide lasers. It is frequently assumed that the laser mode is pure EH11 (quasi-TEM00). We note a flaw in the widely quoted Laguerre-Gaussian mode expansion method as it originally appeared and show how to reconcile it with later results. Also we summarize and try to remove several discrepancies in the published accounts of how the EH11 loss behaves for the popular near-Case I reflector (i.e., a plane mirror placed within a few guide widths of the guide aperture).

© 1985 Optical Society of America

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References

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  1. P. W. Smith, “A Waveguide Gas Laser,” Appl. Phys. Lett. 19, 132 (1971).
    [Crossref]
  2. J. J. Degnan, “The Waveguide Laser: A Review,” Appl. Phys. 11, 1 (1976).
    [Crossref]
  3. R. L. Abrams, “Waveguide Gas Lasers,” in Laser Handbook, Vol. 3 (North-Holland, Amsterdam, 1979).
  4. P. W. Smith, O. R. Wood, P. J. Maloney, C. R. Adams, “Transversely Excited Waveguide Gas Lasers,” IEEE J. Quantum Electron. QE-17, 1166 (1981).
    [Crossref]
  5. J. L. Lachambre, J. Macfarlane, G. Otis, P. Lavigne, “A Transversely RF Excited CO2 Waveguide Laser,” Appl. Phys. Lett. 32, 652 (1978).
    [Crossref]
  6. C. J. Baker, D. He, P. J. Wilson, D. R. Hall, “Discharge Scaling in rf-Excited Waveguide CO2 lasers,” in Technical Digest, Conference on Lasers and Electrooptics (Optical Society of America, Washington, D.C., 1984), paper THL3.
  7. 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 (1964).
  8. K. D. Laakmann, W. H. Steier, “Waveguides: Characteristic Modes of Hollow Rectangular Dielectric Waveguides,” Appl. Opt. 15, 1334 (1976).
    [Crossref] [PubMed]
  9. H. Krammer, “Field Configurations and Propagation Constants of Modes in Hollow Rectangular Dielectric Waveguides,” IEEE J. Quantum Electron. QE-12, 505 (1976).
    [Crossref]
  10. R. L. Abrams, “Coupling Losses in Hollow Waveguide Laser Resonators,” IEEE J. Quantum Electron. QE-8, 838 (1972).
    [Crossref]
  11. H. W. Kogelnik, T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550 (1966).
    [Crossref] [PubMed]
  12. D. M. Henderson, “Waveguide Lasers with Intracavity Electrooptic Modulators: Misalignment Loss,” Appl. Opt. 15, 1066 (1976).
    [Crossref] [PubMed]
  13. A. N. Chester, R. L. Abrams, “Mode Losses in Hollow-Waveguide Lasers,” Appl. Phys. Lett. 21, 576 (1972).
    [Crossref]
  14. G. V. Melekhin, G. P. Melekhina, “Matching Losses in Waveguide Cavities. Two-Dimensional Problem,” Opt. Spectrosc. 52, 527 (1982).
  15. J. J. Degnan, D. R. Hall, “Finite-Aperture Waveguide-Laser Resonators,” IEEE J. Quantum Electron. QE-9, 901 (1973).
    [Crossref]
  16. R. L. Abrams, A. N. Chester, “Resonator Theory for Hollow Waveguide Lasers,” Appl. Opt. 13, 2117 (1974).
    [Crossref] [PubMed]
  17. R. Gerlach, D. Wei, N. M. Amer, “Coupling Efficiency of Waveguide Laser Resonators Formed by Flat Mirrors: Analysis and Experiment,” IEEE J. Quantum Electron. QE-20, 948 (1984).
    [Crossref]
  18. C. A. Hill, D. R. Hall, “Waveguide Resonators with a Tilted Mirror,” in preparation.
  19. S. Avrillier, J. Verdonck, “Coupling Losses in Laser Resonators Containing a Hollow Rectangular Dielectric Waveguide,” J. Appl. Phys. 48, 4937 (1977).
    [Crossref]
  20. J.-L. Boulnois, G. P. Agrawal, “Mode Discrimination and Coupling Losses in Rectangular-Waveguide Resonators with Conventional and Phase-Conjugate Mirrors,” J. Opt. Soc. Am. 72, 853 (1982).
    [Crossref]
  21. A. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1983).
  22. The expression for what we have called z′ is misprinted in Ref. 10 (the −b2 term is missing).
  23. H. W. Kogelnik, in Quasi-Optics (PIB Symposium Proceedings, New York, 1964;J. Fox, Ed.).
  24. Henderson's result12 converts as follows:20(sb)2%=20(2zb)2=20[zka2·4(aω0)2]2,where ω0 = 0.7032a.This equals 20 × (8.089)2 (z/ka2)2 = 1310 (z/ka2)2 or 33.1N−2 %. The factor of 161.8 quoted in Ref. 19 equals 20 × 8.089.
  25. W. W. Rigrod, “Saturation Effects in High-Gain Lasers,” J. Appl. Phys. 36, 2487 (1965).
    [Crossref]
  26. D. He, D. R. Hall, “Influence of Xenon on Sealed-off Operation of RF-Excited CO2 Waveguide Lasers,” J. Appl. Phys. 56, 856 (1984).
    [Crossref]

1984 (2)

R. Gerlach, D. Wei, N. M. Amer, “Coupling Efficiency of Waveguide Laser Resonators Formed by Flat Mirrors: Analysis and Experiment,” IEEE J. Quantum Electron. QE-20, 948 (1984).
[Crossref]

D. He, D. R. Hall, “Influence of Xenon on Sealed-off Operation of RF-Excited CO2 Waveguide Lasers,” J. Appl. Phys. 56, 856 (1984).
[Crossref]

1982 (2)

J.-L. Boulnois, G. P. Agrawal, “Mode Discrimination and Coupling Losses in Rectangular-Waveguide Resonators with Conventional and Phase-Conjugate Mirrors,” J. Opt. Soc. Am. 72, 853 (1982).
[Crossref]

G. V. Melekhin, G. P. Melekhina, “Matching Losses in Waveguide Cavities. Two-Dimensional Problem,” Opt. Spectrosc. 52, 527 (1982).

1981 (1)

P. W. Smith, O. R. Wood, P. J. Maloney, C. R. Adams, “Transversely Excited Waveguide Gas Lasers,” IEEE J. Quantum Electron. QE-17, 1166 (1981).
[Crossref]

1978 (1)

J. L. Lachambre, J. Macfarlane, G. Otis, P. Lavigne, “A Transversely RF Excited CO2 Waveguide Laser,” Appl. Phys. Lett. 32, 652 (1978).
[Crossref]

1977 (1)

S. Avrillier, J. Verdonck, “Coupling Losses in Laser Resonators Containing a Hollow Rectangular Dielectric Waveguide,” J. Appl. Phys. 48, 4937 (1977).
[Crossref]

1976 (4)

D. M. Henderson, “Waveguide Lasers with Intracavity Electrooptic Modulators: Misalignment Loss,” Appl. Opt. 15, 1066 (1976).
[Crossref] [PubMed]

J. J. Degnan, “The Waveguide Laser: A Review,” Appl. Phys. 11, 1 (1976).
[Crossref]

K. D. Laakmann, W. H. Steier, “Waveguides: Characteristic Modes of Hollow Rectangular Dielectric Waveguides,” Appl. Opt. 15, 1334 (1976).
[Crossref] [PubMed]

H. Krammer, “Field Configurations and Propagation Constants of Modes in Hollow Rectangular Dielectric Waveguides,” IEEE J. Quantum Electron. QE-12, 505 (1976).
[Crossref]

1974 (1)

1973 (1)

J. J. Degnan, D. R. Hall, “Finite-Aperture Waveguide-Laser Resonators,” IEEE J. Quantum Electron. QE-9, 901 (1973).
[Crossref]

1972 (2)

A. N. Chester, R. L. Abrams, “Mode Losses in Hollow-Waveguide Lasers,” Appl. Phys. Lett. 21, 576 (1972).
[Crossref]

R. L. Abrams, “Coupling Losses in Hollow Waveguide Laser Resonators,” IEEE J. Quantum Electron. QE-8, 838 (1972).
[Crossref]

1971 (1)

P. W. Smith, “A Waveguide Gas Laser,” Appl. Phys. Lett. 19, 132 (1971).
[Crossref]

1966 (1)

1965 (1)

W. W. Rigrod, “Saturation Effects in High-Gain Lasers,” J. Appl. Phys. 36, 2487 (1965).
[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 (1964).

Abrams, R. L.

R. L. Abrams, A. N. Chester, “Resonator Theory for Hollow Waveguide Lasers,” Appl. Opt. 13, 2117 (1974).
[Crossref] [PubMed]

A. N. Chester, R. L. Abrams, “Mode Losses in Hollow-Waveguide Lasers,” Appl. Phys. Lett. 21, 576 (1972).
[Crossref]

R. L. Abrams, “Coupling Losses in Hollow Waveguide Laser Resonators,” IEEE J. Quantum Electron. QE-8, 838 (1972).
[Crossref]

R. L. Abrams, “Waveguide Gas Lasers,” in Laser Handbook, Vol. 3 (North-Holland, Amsterdam, 1979).

Adams, C. R.

P. W. Smith, O. R. Wood, P. J. Maloney, C. R. Adams, “Transversely Excited Waveguide Gas Lasers,” IEEE J. Quantum Electron. QE-17, 1166 (1981).
[Crossref]

Agrawal, G. P.

Amer, N. M.

R. Gerlach, D. Wei, N. M. Amer, “Coupling Efficiency of Waveguide Laser Resonators Formed by Flat Mirrors: Analysis and Experiment,” IEEE J. Quantum Electron. QE-20, 948 (1984).
[Crossref]

Avrillier, S.

S. Avrillier, J. Verdonck, “Coupling Losses in Laser Resonators Containing a Hollow Rectangular Dielectric Waveguide,” J. Appl. Phys. 48, 4937 (1977).
[Crossref]

Baker, C. J.

C. J. Baker, D. He, P. J. Wilson, D. R. Hall, “Discharge Scaling in rf-Excited Waveguide CO2 lasers,” in Technical Digest, Conference on Lasers and Electrooptics (Optical Society of America, Washington, D.C., 1984), paper THL3.

Boulnois, J.-L.

Chester, A. N.

R. L. Abrams, A. N. Chester, “Resonator Theory for Hollow Waveguide Lasers,” Appl. Opt. 13, 2117 (1974).
[Crossref] [PubMed]

A. N. Chester, R. L. Abrams, “Mode Losses in Hollow-Waveguide Lasers,” Appl. Phys. Lett. 21, 576 (1972).
[Crossref]

Degnan, J. J.

J. J. Degnan, “The Waveguide Laser: A Review,” Appl. Phys. 11, 1 (1976).
[Crossref]

J. J. Degnan, D. R. Hall, “Finite-Aperture Waveguide-Laser Resonators,” IEEE J. Quantum Electron. QE-9, 901 (1973).
[Crossref]

Gerlach, R.

R. Gerlach, D. Wei, N. M. Amer, “Coupling Efficiency of Waveguide Laser Resonators Formed by Flat Mirrors: Analysis and Experiment,” IEEE J. Quantum Electron. QE-20, 948 (1984).
[Crossref]

Hall, D. R.

D. He, D. R. Hall, “Influence of Xenon on Sealed-off Operation of RF-Excited CO2 Waveguide Lasers,” J. Appl. Phys. 56, 856 (1984).
[Crossref]

J. J. Degnan, D. R. Hall, “Finite-Aperture Waveguide-Laser Resonators,” IEEE J. Quantum Electron. QE-9, 901 (1973).
[Crossref]

C. A. Hill, D. R. Hall, “Waveguide Resonators with a Tilted Mirror,” in preparation.

C. J. Baker, D. He, P. J. Wilson, D. R. Hall, “Discharge Scaling in rf-Excited Waveguide CO2 lasers,” in Technical Digest, Conference on Lasers and Electrooptics (Optical Society of America, Washington, D.C., 1984), paper THL3.

He, D.

D. He, D. R. Hall, “Influence of Xenon on Sealed-off Operation of RF-Excited CO2 Waveguide Lasers,” J. Appl. Phys. 56, 856 (1984).
[Crossref]

C. J. Baker, D. He, P. J. Wilson, D. R. Hall, “Discharge Scaling in rf-Excited Waveguide CO2 lasers,” in Technical Digest, Conference on Lasers and Electrooptics (Optical Society of America, Washington, D.C., 1984), paper THL3.

Henderson, D. M.

Hill, C. A.

C. A. Hill, D. R. Hall, “Waveguide Resonators with a Tilted Mirror,” in preparation.

Kogelnik, H. W.

H. W. Kogelnik, T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550 (1966).
[Crossref] [PubMed]

H. W. Kogelnik, in Quasi-Optics (PIB Symposium Proceedings, New York, 1964;J. Fox, Ed.).

Krammer, H.

H. Krammer, “Field Configurations and Propagation Constants of Modes in Hollow Rectangular Dielectric Waveguides,” IEEE J. Quantum Electron. QE-12, 505 (1976).
[Crossref]

Laakmann, K. D.

Lachambre, J. L.

J. L. Lachambre, J. Macfarlane, G. Otis, P. Lavigne, “A Transversely RF Excited CO2 Waveguide Laser,” Appl. Phys. Lett. 32, 652 (1978).
[Crossref]

Lavigne, P.

J. L. Lachambre, J. Macfarlane, G. Otis, P. Lavigne, “A Transversely RF Excited CO2 Waveguide Laser,” Appl. Phys. Lett. 32, 652 (1978).
[Crossref]

Li, T.

Macfarlane, J.

J. L. Lachambre, J. Macfarlane, G. Otis, P. Lavigne, “A Transversely RF Excited CO2 Waveguide Laser,” Appl. Phys. Lett. 32, 652 (1978).
[Crossref]

Maloney, P. J.

P. W. Smith, O. R. Wood, P. J. Maloney, C. R. Adams, “Transversely Excited Waveguide Gas Lasers,” IEEE J. Quantum Electron. QE-17, 1166 (1981).
[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 (1964).

Melekhin, G. V.

G. V. Melekhin, G. P. Melekhina, “Matching Losses in Waveguide Cavities. Two-Dimensional Problem,” Opt. Spectrosc. 52, 527 (1982).

Melekhina, G. P.

G. V. Melekhin, G. P. Melekhina, “Matching Losses in Waveguide Cavities. Two-Dimensional Problem,” Opt. Spectrosc. 52, 527 (1982).

Otis, G.

J. L. Lachambre, J. Macfarlane, G. Otis, P. Lavigne, “A Transversely RF Excited CO2 Waveguide Laser,” Appl. Phys. Lett. 32, 652 (1978).
[Crossref]

Rigrod, W. W.

W. W. Rigrod, “Saturation Effects in High-Gain Lasers,” J. Appl. Phys. 36, 2487 (1965).
[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 (1964).

Siegman, A.

A. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1983).

Smith, P. W.

P. W. Smith, O. R. Wood, P. J. Maloney, C. R. Adams, “Transversely Excited Waveguide Gas Lasers,” IEEE J. Quantum Electron. QE-17, 1166 (1981).
[Crossref]

P. W. Smith, “A Waveguide Gas Laser,” Appl. Phys. Lett. 19, 132 (1971).
[Crossref]

Steier, W. H.

Verdonck, J.

S. Avrillier, J. Verdonck, “Coupling Losses in Laser Resonators Containing a Hollow Rectangular Dielectric Waveguide,” J. Appl. Phys. 48, 4937 (1977).
[Crossref]

Wei, D.

R. Gerlach, D. Wei, N. M. Amer, “Coupling Efficiency of Waveguide Laser Resonators Formed by Flat Mirrors: Analysis and Experiment,” IEEE J. Quantum Electron. QE-20, 948 (1984).
[Crossref]

Wilson, P. J.

C. J. Baker, D. He, P. J. Wilson, D. R. Hall, “Discharge Scaling in rf-Excited Waveguide CO2 lasers,” in Technical Digest, Conference on Lasers and Electrooptics (Optical Society of America, Washington, D.C., 1984), paper THL3.

Wood, O. R.

P. W. Smith, O. R. Wood, P. J. Maloney, C. R. Adams, “Transversely Excited Waveguide Gas Lasers,” IEEE J. Quantum Electron. QE-17, 1166 (1981).
[Crossref]

Appl. Opt. (4)

Appl. Phys. (1)

J. J. Degnan, “The Waveguide Laser: A Review,” Appl. Phys. 11, 1 (1976).
[Crossref]

Appl. Phys. Lett. (3)

P. W. Smith, “A Waveguide Gas Laser,” Appl. Phys. Lett. 19, 132 (1971).
[Crossref]

J. L. Lachambre, J. Macfarlane, G. Otis, P. Lavigne, “A Transversely RF Excited CO2 Waveguide Laser,” Appl. Phys. Lett. 32, 652 (1978).
[Crossref]

A. N. Chester, R. L. Abrams, “Mode Losses in Hollow-Waveguide Lasers,” Appl. Phys. Lett. 21, 576 (1972).
[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 (1964).

IEEE J. Quantum Electron. (5)

J. J. Degnan, D. R. Hall, “Finite-Aperture Waveguide-Laser Resonators,” IEEE J. Quantum Electron. QE-9, 901 (1973).
[Crossref]

R. Gerlach, D. Wei, N. M. Amer, “Coupling Efficiency of Waveguide Laser Resonators Formed by Flat Mirrors: Analysis and Experiment,” IEEE J. Quantum Electron. QE-20, 948 (1984).
[Crossref]

H. Krammer, “Field Configurations and Propagation Constants of Modes in Hollow Rectangular Dielectric Waveguides,” IEEE J. Quantum Electron. QE-12, 505 (1976).
[Crossref]

R. L. Abrams, “Coupling Losses in Hollow Waveguide Laser Resonators,” IEEE J. Quantum Electron. QE-8, 838 (1972).
[Crossref]

P. W. Smith, O. R. Wood, P. J. Maloney, C. R. Adams, “Transversely Excited Waveguide Gas Lasers,” IEEE J. Quantum Electron. QE-17, 1166 (1981).
[Crossref]

J. Appl. Phys. (3)

S. Avrillier, J. Verdonck, “Coupling Losses in Laser Resonators Containing a Hollow Rectangular Dielectric Waveguide,” J. Appl. Phys. 48, 4937 (1977).
[Crossref]

W. W. Rigrod, “Saturation Effects in High-Gain Lasers,” J. Appl. Phys. 36, 2487 (1965).
[Crossref]

D. He, D. R. Hall, “Influence of Xenon on Sealed-off Operation of RF-Excited CO2 Waveguide Lasers,” J. Appl. Phys. 56, 856 (1984).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Spectrosc. (1)

G. V. Melekhin, G. P. Melekhina, “Matching Losses in Waveguide Cavities. Two-Dimensional Problem,” Opt. Spectrosc. 52, 527 (1982).

Other (7)

C. A. Hill, D. R. Hall, “Waveguide Resonators with a Tilted Mirror,” in preparation.

R. L. Abrams, “Waveguide Gas Lasers,” in Laser Handbook, Vol. 3 (North-Holland, Amsterdam, 1979).

C. J. Baker, D. He, P. J. Wilson, D. R. Hall, “Discharge Scaling in rf-Excited Waveguide CO2 lasers,” in Technical Digest, Conference on Lasers and Electrooptics (Optical Society of America, Washington, D.C., 1984), paper THL3.

A. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1983).

The expression for what we have called z′ is misprinted in Ref. 10 (the −b2 term is missing).

H. W. Kogelnik, in Quasi-Optics (PIB Symposium Proceedings, New York, 1964;J. Fox, Ed.).

Henderson's result12 converts as follows:20(sb)2%=20(2zb)2=20[zka2·4(aω0)2]2,where ω0 = 0.7032a.This equals 20 × (8.089)2 (z/ka2)2 = 1310 (z/ka2)2 or 33.1N−2 %. The factor of 161.8 quoted in Ref. 19 equals 20 × 8.089.

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Figures (10)

Fig. 1
Fig. 1

General waveguide reflector.

Fig. 2
Fig. 2

Coupling efficiencies for the EH11 guide mode (reproduced from Degnan and Hall15 with permission).

Fig. 3
Fig. 3

EH11 coupling efficiencies [after Abrams10 with ϕp according to Eq. (6)].

Fig. 4
Fig. 4

Beam parameters of approximating Gaussian reflected from a spherical mirror.

Fig. 5
Fig. 5

Beam parameters and phase angles for reflection from a plane mirror.

Fig. 6
Fig. 6

EH11 coupling efficiencies [after Abrams10 with ϕp according to Eq. (10)].

Fig. 7
Fig. 7

Plane-mirror EH11 coupling loss.

Fig. 8
Fig. 8

Plane-mirror near-field EH11 coupling losses: A, normalized so that Γ11 (z = 0) = 0; B, normalized so that p = 0 5 | A p | 2 = 1 (truncation error remains).

Fig. 9
Fig. 9

Approximate EH11 coupling losses (square guide): A, Boulnois and Agrawal,20 Γ11 = 64.5 (z/b)3/2%; B, Henderson,12 Γ11 = 80 (z/b)2%.

Fig. 10
Fig. 10

Approximate EH11 coupling losses (circular guide): A, Degnan and Hall,15 Γ11 = 57 (z/b)3/2%; B, Hill and Hall, Γ11 = 158 (z/b)2%; C, actual loss.

Tables (3)

Tables Icon

Table I Waveguide Resonator Notation

Tables Icon

Table II Published Expressions for Plane–Mirror EH11 Coupling Loss

Tables Icon

Table III Conversion Factors for Coupling Losses

Equations (47)

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E ( r ) = [ π a 2 J 1 2 ( u 11 ) ] 1 J 0 ( u 11 r / a ) p A p ψ p ( r ) ,
0 a E 2 ( r ) 2 π rdr = 1 .
ψ p ( r ) = 2 π 1 ω 0 L p ( 2 r 2 / ω 0 2 ) exp ( r 2 / ω 0 2 ) ,
0 ψ p 2 ( r ) 2 π rdr = 1 .
E ( r ) = 2 π 1 ω p A p L p ( 2 r 2 / ω 2 ) exp [ r 2 ω 2 ( 1 + i β ρ ) ] exp ( i ϕ p ) ,
ϕ p = ( 2 p + 1 ) [ tan 1 ( z / b ) + tan 1 ( z / β ) ] .
Γ 11 ( z , R M ) = 1 | 0 a E ( r ) E ( r ) 2 π rdr | 2 .
1 q 1 ρ i β ,
q = z + f z ( f z ) b 2 ( f z ) 2 + b 2 + ibf 2 ( f z ) 2 + b 2 = ( z + z ) + i b .
ϕ p = ( 2 p + 1 ) [ tan 1 ( z b ) + tan 1 ( z + z b ) tan 1 ( z b ) ] .
ω = ω 0 [ 1 + ( z + z b ) 2 ] 1 / 2
β ρ = z + z b .
b M = b [ 1 + ( z b ) 2 ] 1 / 2 ,
β = b [ 1 + ( 2 z b ) 2 ] 1 / 2 ,
tan 1 ( z b ) + tan 1 ( z β ) = Â + B ̂ ,
tan 1 ( 2 z b ) = Ĉ Â + B ̂ .
R M = z + ( b 2 / z ) ,
z = z ,
b = b ,
ϕ p = 2 ( 2 p + 1 ) tan 1 ( z / b ) .
Γ 11 605 ( z k a 2 ) 3 / 2 % for z k a 2 < 0.1 .
Γ 11 = 161.8 ( z k a 2 ) 2 .
Γ 11 N 3 / 2 524 ( z k a 2 ) 3 / 2 %
Γ 11 ( z , R M ) = 1 | 0 a E ( r ) E ( r ) 2 π rdr | 2 .
c = κ π 2 α ω 0 b b p A p exp ( i ϕ p ) α p ( z , R M ) ,
κ = [ π α 2 J 1 2 ( u 11 ) ] 1 / 2 ,
α = [ 1 + ( z + z b ) 2 ] 1 / 2 ,
α p ( z , R M ) = 0 2 a 2 / ω 0 2 J 0 ( u 11 w 0 a x 2 ) L p ( α 2 x b b ) × exp [ α 2 x b 2 b ( 1 + i β ρ ) ] dx .
κ π 2 ω 0 = 1 2 ( ω 0 a ) 1 J 1 ( u 11 ) ,
p = 0 | A p | 2 = 1 .
p = 0 5 | A p | 2 = 0.9986 .
0 a E 2 ( r ) 2 π rdr = 1 ,
1 ( p = 0 5 | A p | 2 ) 2 0.28 % ,
100 ( z b ) 2
70 ( z b ) 3 / 2
z / b 0.4 ( from graph )
605 ( z k a 2 ) 3 / 2 or
z / ( k a 2 ) 0.1 , z / b 0.48
z / b 0.2
z / ( k a 2 ) 0.0125
z k a 2
N = a 2 λ z
( z k a 2 ) 2 ( a ω 0 ) 2
N 1 π ( a ω 0 ) 2
= 4.83 ( z k a 2 )
= 4.04 ( z k a 2 )
20(sb)2%=20(2zb)2=20[zka2·4(aω0)2]2,

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