Abstract

The paraxial approximation is applied to calculate the propagation characteristics of cw beams possessing Gaussian spatial profiles in nonlinear optical waveguides. For powers below the critical power for self-focusing in a homogeneous medium, we find that the waveguide effect dominates, i.e., the beam becomes trapped and the spot size varies sinusoidally. Above the critical power, the propagation is dominated by nonlinearity; the beam becomes unstable and displays self-focusing analogous to that in a homogeneous medium. Mode mixing is defined in terms of the nonlinearity-induced mixing of the initially excited modes at the guide face, and explicit expressions are obtained for the mode mixing for special cases. This mixing vanishes only if the spot size is a constant, a condition that can be satisfied only for special values of the parameters in longitudinally homogeneous, lossless guides.

© 1981 Optical Society of America

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References

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  1. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142 (1973); “II. Normal dispersion,”  23, 171 (1973); M. Jain and N. Tzoar, “Propagation of nonlinear optical pulses in inhomogeneous media,” J. Appl. Phys. 49, 4649 (1978); “Nonlinear pulse propagation in optical fibers,” Opt. Lett. 3, 202 (1978); B. Bendow and P. D. Gianino, “Theory of nonlinear pulse propagation in inhomogeneous waveguides,” Opt. Lett. 4, 164 (1979); P. D. Gianino and B. Bendow, “Simplified formulae for solitons in media with slowly varying inhomogeneity,” J. Phys. Fluids 23, 220 (1980); B. Bendow and et al., “Theory of nonlinear pulse propagation in optical waveguides,” J. Opt. Soc. Am. 70, 539 (1980).
    [Crossref] [PubMed]
  2. See, e.g., R. H. Stolen and C. Lin, “Self-phase modulation in silica optical fibers,” Phys. Rev. A 17, 1448 (1978); N. Tzoar and M. Jain, “Self-phase modulation in optical fibers,” Phys. Rev. A (to be published), and references therein.
    [Crossref]
  3. See, e.g., K. O. Hill and et al., in Fiber Optics, Advances in Research and Development, B. Bendow and S. S. Mitra, eds. (Plenum, New York, 1979);R. Stolen, in Optical Fiber Telecommunications, S. Miller and A. Chynoweth, eds. (Academic, New York, 1979).
  4. M. Sodha and A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977), Secs. 3.1, 5.2, 8.2.
    [Crossref]
  5. S. Akhmanov, R. Khokhlov, and A. Sukhorukov, in Laser Handbook, Vol. 2, F. Arecchi and E. Schultz-Dubois, eds. (North-Holland, Amsterdam, 1972), and references therein.
  6. O. Svelto, “Self-focusing, self-trapping and self-phase modulation of laser beams,” in Progress in Optics XII, E. Wolf, ed. (North-Holland, Amsterdam, 1974).
    [Crossref]
  7. M. Sodha, A. Ghatak, and V. Tripathi, “Self focusing of laser beams in plasmas and semiconductors,” in Progress in Optics XIII, E. Wolf, ed. (North-Holland, Amsterdam, 1976).
    [Crossref]
  8. S. Akhmanov, A. Sukhorukov, and R. Khokhlov, “Self-focusing and self-trapping of intense light beams in a non-linear medium,” Zh. Eksp. Teor. Fiz. 50, 1537 (1966) [Sov. Phys. JETP 23, 1025 (1966)].
  9. N. Tzoar and J. I. Gersten, “Calculation of the self-focusing of electromagnetic radiation in semiconductors,” Phys. Rev. B 4, 3540 (1971).
    [Crossref]
  10. See, e.g., W. Gambling and H. Matsumura, “Pulse dispersion in a lenslike medium,” Opt. Electron. 5, 429 (1973).
    [Crossref]
  11. See, e.g., A. Yariv and et al., “Image phase compensation and real-time holography by four-wave mixing in optical fibers,” Appl. Phys. Lett. 32, 635–637 (1978).
    [Crossref]
  12. J. N. Fields and et al., in Physics of Fiber Optics, B. Bendow and S. S. Mitra, eds. (American Ceramic Society, Columbus, Ohio, 1981).
  13. R. H. Stolen, in Physics of Fiber Optics, B. Bendow and S. S. Mitra, eds. (American Ceramic Society, Columbus, Ohio, 1981).

1978 (2)

See, e.g., R. H. Stolen and C. Lin, “Self-phase modulation in silica optical fibers,” Phys. Rev. A 17, 1448 (1978); N. Tzoar and M. Jain, “Self-phase modulation in optical fibers,” Phys. Rev. A (to be published), and references therein.
[Crossref]

See, e.g., A. Yariv and et al., “Image phase compensation and real-time holography by four-wave mixing in optical fibers,” Appl. Phys. Lett. 32, 635–637 (1978).
[Crossref]

1973 (2)

See, e.g., W. Gambling and H. Matsumura, “Pulse dispersion in a lenslike medium,” Opt. Electron. 5, 429 (1973).
[Crossref]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142 (1973); “II. Normal dispersion,”  23, 171 (1973); M. Jain and N. Tzoar, “Propagation of nonlinear optical pulses in inhomogeneous media,” J. Appl. Phys. 49, 4649 (1978); “Nonlinear pulse propagation in optical fibers,” Opt. Lett. 3, 202 (1978); B. Bendow and P. D. Gianino, “Theory of nonlinear pulse propagation in inhomogeneous waveguides,” Opt. Lett. 4, 164 (1979); P. D. Gianino and B. Bendow, “Simplified formulae for solitons in media with slowly varying inhomogeneity,” J. Phys. Fluids 23, 220 (1980); B. Bendow and et al., “Theory of nonlinear pulse propagation in optical waveguides,” J. Opt. Soc. Am. 70, 539 (1980).
[Crossref] [PubMed]

1971 (1)

N. Tzoar and J. I. Gersten, “Calculation of the self-focusing of electromagnetic radiation in semiconductors,” Phys. Rev. B 4, 3540 (1971).
[Crossref]

1966 (1)

S. Akhmanov, A. Sukhorukov, and R. Khokhlov, “Self-focusing and self-trapping of intense light beams in a non-linear medium,” Zh. Eksp. Teor. Fiz. 50, 1537 (1966) [Sov. Phys. JETP 23, 1025 (1966)].

Akhmanov, S.

S. Akhmanov, A. Sukhorukov, and R. Khokhlov, “Self-focusing and self-trapping of intense light beams in a non-linear medium,” Zh. Eksp. Teor. Fiz. 50, 1537 (1966) [Sov. Phys. JETP 23, 1025 (1966)].

S. Akhmanov, R. Khokhlov, and A. Sukhorukov, in Laser Handbook, Vol. 2, F. Arecchi and E. Schultz-Dubois, eds. (North-Holland, Amsterdam, 1972), and references therein.

Fields, J. N.

J. N. Fields and et al., in Physics of Fiber Optics, B. Bendow and S. S. Mitra, eds. (American Ceramic Society, Columbus, Ohio, 1981).

Gambling, W.

See, e.g., W. Gambling and H. Matsumura, “Pulse dispersion in a lenslike medium,” Opt. Electron. 5, 429 (1973).
[Crossref]

Gersten, J. I.

N. Tzoar and J. I. Gersten, “Calculation of the self-focusing of electromagnetic radiation in semiconductors,” Phys. Rev. B 4, 3540 (1971).
[Crossref]

Ghatak, A.

M. Sodha, A. Ghatak, and V. Tripathi, “Self focusing of laser beams in plasmas and semiconductors,” in Progress in Optics XIII, E. Wolf, ed. (North-Holland, Amsterdam, 1976).
[Crossref]

Ghatak, A. K.

M. Sodha and A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977), Secs. 3.1, 5.2, 8.2.
[Crossref]

Hasegawa, A.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142 (1973); “II. Normal dispersion,”  23, 171 (1973); M. Jain and N. Tzoar, “Propagation of nonlinear optical pulses in inhomogeneous media,” J. Appl. Phys. 49, 4649 (1978); “Nonlinear pulse propagation in optical fibers,” Opt. Lett. 3, 202 (1978); B. Bendow and P. D. Gianino, “Theory of nonlinear pulse propagation in inhomogeneous waveguides,” Opt. Lett. 4, 164 (1979); P. D. Gianino and B. Bendow, “Simplified formulae for solitons in media with slowly varying inhomogeneity,” J. Phys. Fluids 23, 220 (1980); B. Bendow and et al., “Theory of nonlinear pulse propagation in optical waveguides,” J. Opt. Soc. Am. 70, 539 (1980).
[Crossref] [PubMed]

Hill, K. O.

See, e.g., K. O. Hill and et al., in Fiber Optics, Advances in Research and Development, B. Bendow and S. S. Mitra, eds. (Plenum, New York, 1979);R. Stolen, in Optical Fiber Telecommunications, S. Miller and A. Chynoweth, eds. (Academic, New York, 1979).

Khokhlov, R.

S. Akhmanov, A. Sukhorukov, and R. Khokhlov, “Self-focusing and self-trapping of intense light beams in a non-linear medium,” Zh. Eksp. Teor. Fiz. 50, 1537 (1966) [Sov. Phys. JETP 23, 1025 (1966)].

S. Akhmanov, R. Khokhlov, and A. Sukhorukov, in Laser Handbook, Vol. 2, F. Arecchi and E. Schultz-Dubois, eds. (North-Holland, Amsterdam, 1972), and references therein.

Lin, C.

See, e.g., R. H. Stolen and C. Lin, “Self-phase modulation in silica optical fibers,” Phys. Rev. A 17, 1448 (1978); N. Tzoar and M. Jain, “Self-phase modulation in optical fibers,” Phys. Rev. A (to be published), and references therein.
[Crossref]

Matsumura, H.

See, e.g., W. Gambling and H. Matsumura, “Pulse dispersion in a lenslike medium,” Opt. Electron. 5, 429 (1973).
[Crossref]

Sodha, M.

M. Sodha and A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977), Secs. 3.1, 5.2, 8.2.
[Crossref]

M. Sodha, A. Ghatak, and V. Tripathi, “Self focusing of laser beams in plasmas and semiconductors,” in Progress in Optics XIII, E. Wolf, ed. (North-Holland, Amsterdam, 1976).
[Crossref]

Stolen, R. H.

See, e.g., R. H. Stolen and C. Lin, “Self-phase modulation in silica optical fibers,” Phys. Rev. A 17, 1448 (1978); N. Tzoar and M. Jain, “Self-phase modulation in optical fibers,” Phys. Rev. A (to be published), and references therein.
[Crossref]

R. H. Stolen, in Physics of Fiber Optics, B. Bendow and S. S. Mitra, eds. (American Ceramic Society, Columbus, Ohio, 1981).

Sukhorukov, A.

S. Akhmanov, A. Sukhorukov, and R. Khokhlov, “Self-focusing and self-trapping of intense light beams in a non-linear medium,” Zh. Eksp. Teor. Fiz. 50, 1537 (1966) [Sov. Phys. JETP 23, 1025 (1966)].

S. Akhmanov, R. Khokhlov, and A. Sukhorukov, in Laser Handbook, Vol. 2, F. Arecchi and E. Schultz-Dubois, eds. (North-Holland, Amsterdam, 1972), and references therein.

Svelto, O.

O. Svelto, “Self-focusing, self-trapping and self-phase modulation of laser beams,” in Progress in Optics XII, E. Wolf, ed. (North-Holland, Amsterdam, 1974).
[Crossref]

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142 (1973); “II. Normal dispersion,”  23, 171 (1973); M. Jain and N. Tzoar, “Propagation of nonlinear optical pulses in inhomogeneous media,” J. Appl. Phys. 49, 4649 (1978); “Nonlinear pulse propagation in optical fibers,” Opt. Lett. 3, 202 (1978); B. Bendow and P. D. Gianino, “Theory of nonlinear pulse propagation in inhomogeneous waveguides,” Opt. Lett. 4, 164 (1979); P. D. Gianino and B. Bendow, “Simplified formulae for solitons in media with slowly varying inhomogeneity,” J. Phys. Fluids 23, 220 (1980); B. Bendow and et al., “Theory of nonlinear pulse propagation in optical waveguides,” J. Opt. Soc. Am. 70, 539 (1980).
[Crossref] [PubMed]

Tripathi, V.

M. Sodha, A. Ghatak, and V. Tripathi, “Self focusing of laser beams in plasmas and semiconductors,” in Progress in Optics XIII, E. Wolf, ed. (North-Holland, Amsterdam, 1976).
[Crossref]

Tzoar, N.

N. Tzoar and J. I. Gersten, “Calculation of the self-focusing of electromagnetic radiation in semiconductors,” Phys. Rev. B 4, 3540 (1971).
[Crossref]

Yariv, A.

See, e.g., A. Yariv and et al., “Image phase compensation and real-time holography by four-wave mixing in optical fibers,” Appl. Phys. Lett. 32, 635–637 (1978).
[Crossref]

Appl. Phys. Lett. (2)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142 (1973); “II. Normal dispersion,”  23, 171 (1973); M. Jain and N. Tzoar, “Propagation of nonlinear optical pulses in inhomogeneous media,” J. Appl. Phys. 49, 4649 (1978); “Nonlinear pulse propagation in optical fibers,” Opt. Lett. 3, 202 (1978); B. Bendow and P. D. Gianino, “Theory of nonlinear pulse propagation in inhomogeneous waveguides,” Opt. Lett. 4, 164 (1979); P. D. Gianino and B. Bendow, “Simplified formulae for solitons in media with slowly varying inhomogeneity,” J. Phys. Fluids 23, 220 (1980); B. Bendow and et al., “Theory of nonlinear pulse propagation in optical waveguides,” J. Opt. Soc. Am. 70, 539 (1980).
[Crossref] [PubMed]

See, e.g., A. Yariv and et al., “Image phase compensation and real-time holography by four-wave mixing in optical fibers,” Appl. Phys. Lett. 32, 635–637 (1978).
[Crossref]

Opt. Electron. (1)

See, e.g., W. Gambling and H. Matsumura, “Pulse dispersion in a lenslike medium,” Opt. Electron. 5, 429 (1973).
[Crossref]

Phys. Rev. A (1)

See, e.g., R. H. Stolen and C. Lin, “Self-phase modulation in silica optical fibers,” Phys. Rev. A 17, 1448 (1978); N. Tzoar and M. Jain, “Self-phase modulation in optical fibers,” Phys. Rev. A (to be published), and references therein.
[Crossref]

Phys. Rev. B (1)

N. Tzoar and J. I. Gersten, “Calculation of the self-focusing of electromagnetic radiation in semiconductors,” Phys. Rev. B 4, 3540 (1971).
[Crossref]

Zh. Eksp. Teor. Fiz. (1)

S. Akhmanov, A. Sukhorukov, and R. Khokhlov, “Self-focusing and self-trapping of intense light beams in a non-linear medium,” Zh. Eksp. Teor. Fiz. 50, 1537 (1966) [Sov. Phys. JETP 23, 1025 (1966)].

Other (7)

J. N. Fields and et al., in Physics of Fiber Optics, B. Bendow and S. S. Mitra, eds. (American Ceramic Society, Columbus, Ohio, 1981).

R. H. Stolen, in Physics of Fiber Optics, B. Bendow and S. S. Mitra, eds. (American Ceramic Society, Columbus, Ohio, 1981).

See, e.g., K. O. Hill and et al., in Fiber Optics, Advances in Research and Development, B. Bendow and S. S. Mitra, eds. (Plenum, New York, 1979);R. Stolen, in Optical Fiber Telecommunications, S. Miller and A. Chynoweth, eds. (Academic, New York, 1979).

M. Sodha and A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977), Secs. 3.1, 5.2, 8.2.
[Crossref]

S. Akhmanov, R. Khokhlov, and A. Sukhorukov, in Laser Handbook, Vol. 2, F. Arecchi and E. Schultz-Dubois, eds. (North-Holland, Amsterdam, 1972), and references therein.

O. Svelto, “Self-focusing, self-trapping and self-phase modulation of laser beams,” in Progress in Optics XII, E. Wolf, ed. (North-Holland, Amsterdam, 1974).
[Crossref]

M. Sodha, A. Ghatak, and V. Tripathi, “Self focusing of laser beams in plasmas and semiconductors,” in Progress in Optics XIII, E. Wolf, ed. (North-Holland, Amsterdam, 1976).
[Crossref]

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Figures (3)

Fig. 1
Fig. 1

Mode-mixing coefficients g0 and g1 versus 2ζ for σ1 = 0.1, σ2 = 5 for modes N = 0, 1, and 4. Results for g1 (dotted curves) are nearly coincident with those for g0 (solid curves) for this case.

Fig. 2
Fig. 2

Mode-mixing coefficients g0 (solid lines) and gI (dotted lines) versus 2ζ for σ1 = 0.1, N = 0 for various values for σ2.

Fig. 3
Fig. 3

Same quantities as in Fig. 2 but for σ2 = 1, N = 0 for various values of σ1.

Equations (45)

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( ρ , z , E ) = 0 ( z ) - R ( z ) ρ 2 + NL E 2 - i I .
[ 2 + k 0 2 ( ρ , z , E ) ] E = 0.
E ( ρ , z ) = A ( ρ , z ) [ 0 ( 0 ) 0 ( z ) ] 1 / 4 exp [ - i k ( z ) d z ] ,
( 1 k d k d z + 2 i k ) A z = 1 ρ ρ ( ρ A ρ ) - k 2 0 ( z ) [ R ( z ) ρ 2 - NL E 2 + i I ] A .
A = A 0 ( ρ , z ) exp [ - i k S ( ρ , z ) ]
2 S k d k d z + 2 S z + ( S ρ ) 2 = 1 k 2 A 0 ( 2 A 0 ρ 2 + 1 ρ A 0 ρ ) + NL 0 ( z ) E 2 - R ( z ) 0 ( z ) ρ 2 ,
A 0 2 z + S ρ + A 0 2 ( 2 S ρ 2 + 1 ρ S ρ ) + k I 0 A 0 2 = 0.
A 2 A 0 2 + A 2 ρ 2 | 0 ρ 2 ,
A 0 = E 0 f ( z ) exp ( - k I z ) exp [ - ρ 2 2 ρ 0 2 f 2 ( z ) ] ,
S = 1 2 f ( z ) d f ( z ) d z ρ 2 + ϕ ( z ) ,
d 2 f ( z ) d z 2 + 1 2 0 ( z ) d 0 ( z ) d z d f d z = ( 1 R d 2 - 1 R NL 2 e - 2 k I z ) 1 f 3 ( z ) - R ( z ) 0 ( z ) f ( z ) ,
d ϕ ( z ) d z + 1 2 0 z d 0 z d z ϕ ( z ) = ρ 0 2 f 2 ( z ) ( e - 2 k I z 2 R NL 2 - 1 R d 2 ) ,
R d 2 = k 2 ρ 0 4 = ω 2 0 ( z ) ρ 0 4 / c 4 , R NL 2 = ρ 0 2 0 ( z ) E 0 2 NL [ 0 ( z ) 0 ( 0 ) ] 1 / 2 .
d f d z | z = 0 = 0 ,             f ( 0 ) = 1 ,             ϕ ( 0 ) = ϕ 0 .
E = E 0 f ( z ) exp ( - k I z ) exp [ - ρ 2 2 ρ 0 2 f 2 ( z ) ] [ 0 ( 0 ) 0 ( z ) ] 1 / 4 × exp { - i k [ ρ 2 1 2 f ( z ) d f ( z ) d z + ϕ ( z ) ] } × exp ( i ω t ) exp [ - i k ( z ) d z ] ,
d 2 f d z 2 = 1 R e 2 1 f 3 = R 0 f ,             1 R e 2 = 1 R d 2 - 1 R NL 2 .
d 2 f d z 2 = 1 R d 2 1 f 3 - R 0 f .
f 2 = 1 2 [ ( 1 + C d ) + ( 1 - C d ) cos 2 ζ ] , C d = 0 R 1 R d 2 ,             ζ ( R 0 ) 1 / 2 z .
d 2 f d z 2 = 1 R e 2 f 3 .
f 2 = 1 + z 2 R e 2 .
f 2 = 1 - z 2 z F 2 ,             0 < z < z F ,
z F 2 = - R e 2 .
f 2 = 1 - ( z - 2 z F z F ) 2 .
f 2 = ½ [ ( 1 + C ) + ( 1 - C ) cos 2 ζ ] ,             C > 0 , C 0 R 1 R e 2 .
1 ρ 0 4 = R k 2 0 + k 2 NL E 0 2 0 ρ 0 2 1 w 1 4 .
f 2 = ½ [ ( 1 - C ) + ( 1 + C ) cos 2 ζ ] .
ζ F = ½ cos - 1 ( C - 1 C + 1 ) ,             z F = ( 0 R ) 1 / 2 ζ F .
f 2 = ½ [ ( 1 - C ) + ( 1 + C ) cos 2 ( ζ - 2 ζ F ) ] .
f 2 = ½ [ ( 1 - C ) + ( 1 + C ) cos 2 ( ζ - j ζ F ) ] ,             j ζ F ζ ( j + 2 ) ζ F
E 0 2 = 8 P c 0 1 / 2 ρ 0 2 .
P c = c λ 0 2 0 1 / 2 8 ( 2 π ) 2 NL ,
{ 1 ρ ρ ρ ρ - k 2 0 ( 0 ) [ R ( 0 ) + NL E 0 2 ρ 0 2 ] ρ 2 + k 2 NL E 0 2 0 ( 0 ) + k 2 - k n 2 } u n ( ρ ) = 0 ,
u n ( ρ ) = 1 w 1 π 1 / 2 exp ( - ρ 2 2 w 1 2 ) L n ( ρ 2 w 1 2 ) , 1 w 1 4 = 1 w 0 4 + k 2 NL E 0 2 0 ρ 0 2 ,             1 w 0 4 R 0 k 2 ,
n = ¼ α w 1 2 - ½ , k n 2 = k 2 ( 1 + NL 0 E 0 2 ) - α .
E ( ρ , z , t ) E 0 = n = 0 B n ( z , t ) u n ( ρ ) ,
B n ( z , t ) = 2 π 1 / 2 w 1 f ( z ) ( V 2 - 1 ) n ( V 2 + 1 ) n + 1 × exp { i [ ω t - k z - k ϕ ( z ) ] - k I z } , V 2 w 1 2 ( 1 ρ 0 2 f 2 + i k f d f d z ) .
B n ( z ) 2 = 4 π w 1 2 f 2 ( z ) V 2 - 1 2 n V 2 + 1 2 n + 2 e - 2 k I z , V 2 ± 1 2 = ( w 1 2 ρ 0 2 f 2 ± 1 ) 2 + k 2 w 1 4 f 2 ( f z ) 2 .
B n ( z ) 2 = 4 π ρ 0 2 σ 3 D - n / D + n + 1 , D ± = f 2 ( σ 3 2 - σ 1 - σ 2 ) + σ 1 + σ 2 + 1 ± 2 σ 3 + σ 1 ( f - f - 1 ) 2 , σ 1 k 2 ρ 0 4 R NL 2 = P P c ,             σ 2 k 2 ρ 0 4 R 0 ,             σ 3 ρ 0 2 w 1 2 ,
g ( n , z ) = B n ( z ) 2 / 4 π ρ 0 2 ,
g 1 ( n , z ) = ( σ 1 + σ 2 ) 1 / 2 × { [ ( σ 1 + σ 2 ) 1 / 2 - 1 ] 2 + σ 1 ( f - f - 1 ) 2 } n { [ ( σ 1 + σ 2 ) 1 / 2 + 1 ] 2 + σ 1 ( f - f - 1 ) 2 } n + 1 .
g 0 ( n , z ) = σ 2 1 / 2 [ ( σ 2 1 / 2 - 1 ) 2 - σ 1 ( 1 - f - 2 ) ] n [ ( σ 2 1 / 2 + 1 ) 2 - σ 1 ( 1 - f - 2 ) ] n + 1 .
C = σ 2 - 1 ( 1 - σ 1 ) ,
g = g ( n , σ 1 , σ 2 , ζ ) .
σ 1 ~ ( 10 - 4 - 10 - 6 ) × P ,
z F z p ~ P c g λ ( P P c ) 1 / 2 .