Abstract

The design of a coupling between a semiconductor laser and a single-mode fiber, or between any two optical or acoustical elements that support Gaussian modes, is presented as a trade-off among coupling efficiency Ta, offset misalignment tolerance de, and angular misalignment tolerance θe. We show that these three parameters are subject to a trade-off limitation which takes the form 0<Ta1/2θedeλ/π, and we show how to design a coupling so that the upper bound on the alignment product Ta1/2θede is achieved.

© 1984 Optical Society of America

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References

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  1. H. Kogelnik, “Coupling and Conversion Coefficients for Optical Modes in Quasi-Optics,” Microwave Research Institute Symposia Series 14 (Polytechnic Press, New York, 1964), pp. 333–347. Our expressions for de and θe differ from Kogelnik’s corresponding results.
  2. D. Marcuse, Light Trasnmission Optics (Van Nostrand, New York, 1972), Chap. 6.
  3. D. Marcuse, “Loss Analysis of Single-Mode Fiber Splices,” Bell Syst. Tech. J. 56, 703 (1977).
  4. D. D. Cook, F. R. Nash, “Gain-Induced Guiding and Astigmatic Output Beam of GaAs Lasers,” J. Appl. Phys. 46, 1660 (April1975).
    [Crossref]
  5. H. Kuwahara, M. Sasaki, N. Tokoyo, “Efficient Coupling from Semiconductor Lasers into Single-Mode Fibers with Tapered Hemispherical Ends,” Appl. Opt. 19, 2578 (1980).
    [Crossref] [PubMed]
  6. G. D. Khoe, J. Poulissen, H. M. de Vrieze, “Efficient Coupling of Laser Diodes to Tapered Monomode Fibers with High-Index End,” Electron. Lett. 19, 205 (1983).
    [Crossref]
  7. G. Wenke, Y. Zhu, “Comparison of Efficiency and Feedback Characteristics of Techniques for Coupling Semiconductor Lasers to Single-Mode Fiber,” Appl. Opt. 22, 3837 (1983).
    [Crossref] [PubMed]
  8. J. H. Rowen, unpublished.
  9. R. E. Wagner, W. J. Tomlinson, “Coupling Efficiency of Optics in Single-Mode Fiber Components,” Appl. Opt. 21, 2671 (1982).
    [Crossref] [PubMed]
  10. W. L. Emkey, “Optical Coupling between Single-Mode Semiconductor Lasers and Strip Waveguides,” IEEE/OSA J. Light-wave Technol. LT-1, 436 (June1983).
    [Crossref]
  11. There may also be Fresnel-reflection loss at the sink which should be accounted for separately. Often the normal-incidence reflectivity is a sufficiently accurate estimate. Light reflected back into the source may, of course, change the source-output power, but that is a separate question from the coupling efficiency. The aligned losses arising from aberrations and other causes of non-Gaussian beams are found in some cases as by-products when the Gaussian fit is determined by projection.
  12. All integrals in this paper are of the standard Gaussian form∫-∞∞exp(-ax2-2bx-c)dx=(π/a)1/2exp(b2/a-c)where the real part of a is positive.
  13. If one or both beams are astigmatic, the waist separation sx in the x-z plane may differ from the waist separation sy in the y-z plane; i.e., sy − sx = Δs. Nevertheless the forms of Eqs. (27), (28), and (30) are still valid, in the independent x-z and y-z planes approximation, if properly interpreted. From Eq. (12) it is apparent that, as the source-sink separation is varied, the maximum of Ta does not occur at the separations where τa,x and τa,y have their maxima. Equations (27) and (28) are still valid if s in Eq. (27) is replaced by S, the distance from the maximum point of Ta. From Eqs. (10) and (21) it follows that de,x(0) in Eq. (30) is measured at the maximum of τa,x, i.e., at that separation where de,x is a minimum and similarly de,y(0) is measured at that separation where de,y is a minimum.
  14. J. S. Cook, W. L. Mammel, R. J. Grow, “Effect of Misalignment on Coupling Efficiency of Single-Mode Optical Fiber Butt Joints,” Bell Syst. Tech. J. 52, 1439 (1973).
  15. w¯, w, κ¯, κ, and α are independent of n in the design sense that, for a given n, lenses and tapers are then chosen to yield the given values for the beam sizes and curvatures.
  16. W. B. Joyce, “Classical Particle Description of Photons and Phonons,” Phys. Rev. D 9, 3234 (1974).
    [Crossref]
  17. W. B. Joyce, R. Z. Bachrach, R. W. Dixon, D. A. Sealer, “Geoemtrical Properties of Random Particles and the Extraction of Photons from Electroluminescent Diodes,” J. Appl. Phys. 45, 2229 (1974).
    [Crossref]
  18. P. P. Deimel, “Integral Lens Design Considerations,” to be published.
  19. W. B. Joyce, B. C. DeLoach, “Alignment-Tolerant Optical-Fiber Tips for Laser Transmitters,” unpublished.
  20. Designing at s = 0 also has the advantage for nonastigmatic beams of facilitating the axial-direction alignment. That is, the design point (s = 0) corresponds to the easily recognized maximum in τa(s).

1983 (3)

G. D. Khoe, J. Poulissen, H. M. de Vrieze, “Efficient Coupling of Laser Diodes to Tapered Monomode Fibers with High-Index End,” Electron. Lett. 19, 205 (1983).
[Crossref]

W. L. Emkey, “Optical Coupling between Single-Mode Semiconductor Lasers and Strip Waveguides,” IEEE/OSA J. Light-wave Technol. LT-1, 436 (June1983).
[Crossref]

G. Wenke, Y. Zhu, “Comparison of Efficiency and Feedback Characteristics of Techniques for Coupling Semiconductor Lasers to Single-Mode Fiber,” Appl. Opt. 22, 3837 (1983).
[Crossref] [PubMed]

1982 (1)

1980 (1)

1977 (1)

D. Marcuse, “Loss Analysis of Single-Mode Fiber Splices,” Bell Syst. Tech. J. 56, 703 (1977).

1975 (1)

D. D. Cook, F. R. Nash, “Gain-Induced Guiding and Astigmatic Output Beam of GaAs Lasers,” J. Appl. Phys. 46, 1660 (April1975).
[Crossref]

1974 (2)

W. B. Joyce, “Classical Particle Description of Photons and Phonons,” Phys. Rev. D 9, 3234 (1974).
[Crossref]

W. B. Joyce, R. Z. Bachrach, R. W. Dixon, D. A. Sealer, “Geoemtrical Properties of Random Particles and the Extraction of Photons from Electroluminescent Diodes,” J. Appl. Phys. 45, 2229 (1974).
[Crossref]

1973 (1)

J. S. Cook, W. L. Mammel, R. J. Grow, “Effect of Misalignment on Coupling Efficiency of Single-Mode Optical Fiber Butt Joints,” Bell Syst. Tech. J. 52, 1439 (1973).

Bachrach, R. Z.

W. B. Joyce, R. Z. Bachrach, R. W. Dixon, D. A. Sealer, “Geoemtrical Properties of Random Particles and the Extraction of Photons from Electroluminescent Diodes,” J. Appl. Phys. 45, 2229 (1974).
[Crossref]

Cook, D. D.

D. D. Cook, F. R. Nash, “Gain-Induced Guiding and Astigmatic Output Beam of GaAs Lasers,” J. Appl. Phys. 46, 1660 (April1975).
[Crossref]

Cook, J. S.

J. S. Cook, W. L. Mammel, R. J. Grow, “Effect of Misalignment on Coupling Efficiency of Single-Mode Optical Fiber Butt Joints,” Bell Syst. Tech. J. 52, 1439 (1973).

de Vrieze, H. M.

G. D. Khoe, J. Poulissen, H. M. de Vrieze, “Efficient Coupling of Laser Diodes to Tapered Monomode Fibers with High-Index End,” Electron. Lett. 19, 205 (1983).
[Crossref]

Deimel, P. P.

P. P. Deimel, “Integral Lens Design Considerations,” to be published.

DeLoach, B. C.

W. B. Joyce, B. C. DeLoach, “Alignment-Tolerant Optical-Fiber Tips for Laser Transmitters,” unpublished.

Dixon, R. W.

W. B. Joyce, R. Z. Bachrach, R. W. Dixon, D. A. Sealer, “Geoemtrical Properties of Random Particles and the Extraction of Photons from Electroluminescent Diodes,” J. Appl. Phys. 45, 2229 (1974).
[Crossref]

Emkey, W. L.

W. L. Emkey, “Optical Coupling between Single-Mode Semiconductor Lasers and Strip Waveguides,” IEEE/OSA J. Light-wave Technol. LT-1, 436 (June1983).
[Crossref]

Grow, R. J.

J. S. Cook, W. L. Mammel, R. J. Grow, “Effect of Misalignment on Coupling Efficiency of Single-Mode Optical Fiber Butt Joints,” Bell Syst. Tech. J. 52, 1439 (1973).

Joyce, W. B.

W. B. Joyce, “Classical Particle Description of Photons and Phonons,” Phys. Rev. D 9, 3234 (1974).
[Crossref]

W. B. Joyce, R. Z. Bachrach, R. W. Dixon, D. A. Sealer, “Geoemtrical Properties of Random Particles and the Extraction of Photons from Electroluminescent Diodes,” J. Appl. Phys. 45, 2229 (1974).
[Crossref]

W. B. Joyce, B. C. DeLoach, “Alignment-Tolerant Optical-Fiber Tips for Laser Transmitters,” unpublished.

Khoe, G. D.

G. D. Khoe, J. Poulissen, H. M. de Vrieze, “Efficient Coupling of Laser Diodes to Tapered Monomode Fibers with High-Index End,” Electron. Lett. 19, 205 (1983).
[Crossref]

Kogelnik, H.

H. Kogelnik, “Coupling and Conversion Coefficients for Optical Modes in Quasi-Optics,” Microwave Research Institute Symposia Series 14 (Polytechnic Press, New York, 1964), pp. 333–347. Our expressions for de and θe differ from Kogelnik’s corresponding results.

Kuwahara, H.

Mammel, W. L.

J. S. Cook, W. L. Mammel, R. J. Grow, “Effect of Misalignment on Coupling Efficiency of Single-Mode Optical Fiber Butt Joints,” Bell Syst. Tech. J. 52, 1439 (1973).

Marcuse, D.

D. Marcuse, “Loss Analysis of Single-Mode Fiber Splices,” Bell Syst. Tech. J. 56, 703 (1977).

D. Marcuse, Light Trasnmission Optics (Van Nostrand, New York, 1972), Chap. 6.

Nash, F. R.

D. D. Cook, F. R. Nash, “Gain-Induced Guiding and Astigmatic Output Beam of GaAs Lasers,” J. Appl. Phys. 46, 1660 (April1975).
[Crossref]

Poulissen, J.

G. D. Khoe, J. Poulissen, H. M. de Vrieze, “Efficient Coupling of Laser Diodes to Tapered Monomode Fibers with High-Index End,” Electron. Lett. 19, 205 (1983).
[Crossref]

Rowen, J. H.

J. H. Rowen, unpublished.

Sasaki, M.

Sealer, D. A.

W. B. Joyce, R. Z. Bachrach, R. W. Dixon, D. A. Sealer, “Geoemtrical Properties of Random Particles and the Extraction of Photons from Electroluminescent Diodes,” J. Appl. Phys. 45, 2229 (1974).
[Crossref]

Tokoyo, N.

Tomlinson, W. J.

Wagner, R. E.

Wenke, G.

Zhu, Y.

Appl. Opt. (3)

Bell Syst. Tech. J. (2)

D. Marcuse, “Loss Analysis of Single-Mode Fiber Splices,” Bell Syst. Tech. J. 56, 703 (1977).

J. S. Cook, W. L. Mammel, R. J. Grow, “Effect of Misalignment on Coupling Efficiency of Single-Mode Optical Fiber Butt Joints,” Bell Syst. Tech. J. 52, 1439 (1973).

Electron. Lett. (1)

G. D. Khoe, J. Poulissen, H. M. de Vrieze, “Efficient Coupling of Laser Diodes to Tapered Monomode Fibers with High-Index End,” Electron. Lett. 19, 205 (1983).
[Crossref]

IEEE/OSA J. Light-wave Technol. (1)

W. L. Emkey, “Optical Coupling between Single-Mode Semiconductor Lasers and Strip Waveguides,” IEEE/OSA J. Light-wave Technol. LT-1, 436 (June1983).
[Crossref]

J. Appl. Phys. (2)

W. B. Joyce, R. Z. Bachrach, R. W. Dixon, D. A. Sealer, “Geoemtrical Properties of Random Particles and the Extraction of Photons from Electroluminescent Diodes,” J. Appl. Phys. 45, 2229 (1974).
[Crossref]

D. D. Cook, F. R. Nash, “Gain-Induced Guiding and Astigmatic Output Beam of GaAs Lasers,” J. Appl. Phys. 46, 1660 (April1975).
[Crossref]

Phys. Rev. D (1)

W. B. Joyce, “Classical Particle Description of Photons and Phonons,” Phys. Rev. D 9, 3234 (1974).
[Crossref]

Other (10)

w¯, w, κ¯, κ, and α are independent of n in the design sense that, for a given n, lenses and tapers are then chosen to yield the given values for the beam sizes and curvatures.

H. Kogelnik, “Coupling and Conversion Coefficients for Optical Modes in Quasi-Optics,” Microwave Research Institute Symposia Series 14 (Polytechnic Press, New York, 1964), pp. 333–347. Our expressions for de and θe differ from Kogelnik’s corresponding results.

D. Marcuse, Light Trasnmission Optics (Van Nostrand, New York, 1972), Chap. 6.

J. H. Rowen, unpublished.

P. P. Deimel, “Integral Lens Design Considerations,” to be published.

W. B. Joyce, B. C. DeLoach, “Alignment-Tolerant Optical-Fiber Tips for Laser Transmitters,” unpublished.

Designing at s = 0 also has the advantage for nonastigmatic beams of facilitating the axial-direction alignment. That is, the design point (s = 0) corresponds to the easily recognized maximum in τa(s).

There may also be Fresnel-reflection loss at the sink which should be accounted for separately. Often the normal-incidence reflectivity is a sufficiently accurate estimate. Light reflected back into the source may, of course, change the source-output power, but that is a separate question from the coupling efficiency. The aligned losses arising from aberrations and other causes of non-Gaussian beams are found in some cases as by-products when the Gaussian fit is determined by projection.

All integrals in this paper are of the standard Gaussian form∫-∞∞exp(-ax2-2bx-c)dx=(π/a)1/2exp(b2/a-c)where the real part of a is positive.

If one or both beams are astigmatic, the waist separation sx in the x-z plane may differ from the waist separation sy in the y-z plane; i.e., sy − sx = Δs. Nevertheless the forms of Eqs. (27), (28), and (30) are still valid, in the independent x-z and y-z planes approximation, if properly interpreted. From Eq. (12) it is apparent that, as the source-sink separation is varied, the maximum of Ta does not occur at the separations where τa,x and τa,y have their maxima. Equations (27) and (28) are still valid if s in Eq. (27) is replaced by S, the distance from the maximum point of Ta. From Eqs. (10) and (21) it follows that de,x(0) in Eq. (30) is measured at the maximum of τa,x, i.e., at that separation where de,x is a minimum and similarly de,y(0) is measured at that separation where de,y is a minimum.

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Figures (5)

Fig. 1
Fig. 1

Parametrization of a Gaussian beam.

Fig. 2
Fig. 2

Mutual projection (inner product) of the modes of a circuit and a fiber. The alignment plane contains the intersection of the two optical axes and applies as shown to the case d = 0.

Fig. 3
Fig. 3

Possible combinations of the circular-beam coupling efficiency T a (or the elliptical-beam one-plane coupling efficiency τ a ), the 1-dB angular alignment tolerance θ1, and the 1-dB offset tolerance d1 when the alignment product T a 1 / 2 θ 1 d 1 (or τ a θ1d1) is maximized.

Fig. 4
Fig. 4

Three cases where the source and sink waists (2w+ and 2w) of two round beams fall on the alignment plane. Each case takes a low value for one of the three parameters T a , θ e , and d e to achieve high values for the other two. Case (a) has a low misalignment tilt tolerance θ e . The large (compared to λ) waists mean that a large distance and hence a large phase shift open up between the waists with a small tilt angle. Case (b) clearly has a low offset tolerance d e for vertical displacement of either waist. Case (c) has a low coupling efficiency T a because of the disparity in waist size.

Fig. 5
Fig. 5

That realization of Fig. 3 for which the beam waists, 2w+ and 2w, fall on the alignment plane. The corresponding tolerance for axial separation of the waists is given by Eq. (29) or (30) using d e (0) = d1/0.48. The tilt tolerance for rotation about an axis outside of the plane of the waists is given approxmately by Eq. (51), where θ e ,max is the value of θ e from Fig. 5. To use Fig. 5 pick any two of five quantities and read off the other three. The five quantities are the two coplanar waists, 2w+ and 2w, the 1-dB offset distance d1, the 1-dB tilt angle θ1, and the round-beam coupling efficiency T a or the elliptical-beam one-plane coupling efficiency τ a [Eq. (12)]. λ is the wavelength in the medium between the optical source and the sink.

Equations (75)

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R = 1 / κ = z [ 1 + ( k w 0 2 / 2 z ) 2 ]
w = w 0 [ 1 + ( 2 z / k w 0 2 ) 2 ] 1 / 2 ,
1 / w 0 2 = 1 / w 2 + ( ½ k w κ ) 2 ,
w = a ( 0.65 + 1.619 V - 3 / 2 + 2.879 V - 6 )
V 2 = ( n 1 2 - n 2 2 ) k 2 a 2 ,
Ψ ( x , y ) = ψ ( x , w x , κ x ) ψ ( y , w y , κ y ) ,
w , κ > = ψ ( w , κ ; x ) = ( 2 / π ) 1 / 4 w - 1 / 2 exp [ - ( x / w ) 2 + ½ i k κ x 2 ] .
τ a = w ¯ , κ ¯ w , κ 2 = | - ψ * ( w ¯ , κ ¯ ; x ) ψ ( w , κ ; x ) d x | 2
= 2 [ ( w ¯ / w + w / w ¯ ) 2 + ( ½ k w ¯ w ) 2 ( κ ¯ - κ ) 2 ] 1 / 2 .
τ a = 2 [ ( w ¯ 0 / w 0 + w 0 / w ¯ 0 ) 2 + ( 2 / k w ¯ 0 w 0 ) 2 s 2 ] 1 / 2 .
τ a ( s ) τ a ( 0 ) = 2 w ¯ 0 / w 0 + w 0 / w ¯ 0 .
T a = τ a , x τ a , y ,
- 10 log T a = - 10 log τ a , x - 10 log τ a , y .
T a = τ a 2 .
w , κ ; θ > = ( 2 / π ) 1 / 4 w - 1 / 2 exp { - ( x / w ) 2 + i [ k θ x + ½ k κ x 2 ] } .
τ = w ¯ , κ ¯ , 0 w , κ , θ 2 = τ a exp [ - ( θ / θ e ) 2 ] ,
θ e = 2 3 / 2 k τ a ( w ¯ 2 + w 2 ) 1 / 2 = 2 1 / 2 π τ a [ ( w ¯ / λ ) 2 + ( w / λ ) 2 ] 1 / 2
T = T a exp [ - ( θ / θ e ) 2 ] .
τ = τ a exp [ - ( d / d e ) 2 ] ,
T = T a exp [ - ( d / d e ) 2 ] ,
d e = 2 1 / 2 τ a ( 1 / w ¯ 0 2 + 1 / w 0 2 ) 1 / 2 ,
d e ( s ) d e ( 0 ) = 2 - 1 / 2 ( w ¯ 0 2 + w 0 2 ) 1 / 2 .
d e = 2 1 / 2 τ a [ 1 / w ¯ 2 + 1 / w 2 + ( ½ w κ ¯ ) 2 + ( ½ k w κ ) 2 ] 1 / 2 ,
τ a ( s ) = τ a ( 0 ) exp [ - ( s / s e ) 2 ] .
s e = ( - ½ τ a - 1 2 τ a / s 2 ) s = 0 - 1 / 2 = 2 1 / 2 k w ¯ 0 w 0 / τ a ( 0 ) = 2 - 1 / 2 k ( w ¯ 0 2 + w 0 2 ) .
s e / λ = 2 1 / 2 π [ ( w ¯ 0 / λ ) 2 + ( w 0 / λ ) 2 ]
T a ( s ) = T a ( 0 ) exp [ - ( s / S e ) 2 ] ,
1 / S e 2 = 1 / s e , x 2 + 1 / s e , y 2 ,
s e / λ = 2 3 / 2 π [ d e ( 0 ) / λ ] 2 ,
8 π ( S e / λ ) - 2 = [ d e , x ( 0 ) / λ ] - 4 + [ d e , y ( 0 ) / λ ] - 4 .
A = τ a θ e d e
A = T a 1 / 2 θ e d e
α = ½ k A = ½ k τ a θ e d e .
α = 2 τ a ( w ¯ 2 + w 2 ) 1 / 2 ( 1 / w ¯ 0 2 + 1 / w 0 2 ) 1 / 2
α 2 = ( ½ k τ a θ e d e ) 2 = ( w ¯ / w + w / w ¯ ) 2 + ( ½ k w ¯ w ) 2 ( κ ¯ - κ ) 2 ( w ¯ / w + w / w ¯ ) 2 + ( w ¯ 2 + w 2 ) [ ( ½ k w ¯ κ ¯ ) 2 + ( ½ k w κ ) 2 ]
θ e d e = 2 / k = λ / π .
α 2 = F + G F + H
F = ( w ¯ w + w w ¯ ) 2 + ( w ¯ w ) 2 [ ( ½ k κ ¯ ) 2 + ( ½ k κ ) 2 ] > 0 G - H = - 2 ( ½ k w ¯ 2 κ ¯ ) ( ½ k w 2 κ ) - ( ½ k w ¯ 2 κ ¯ ) 2 - ( ½ k w 2 κ ) 2 = - ( ½ k w ¯ 2 κ ¯ + ½ k w 2 κ ) 2 0.
w ¯ 2 κ ¯ + w 2 κ = 0.
w ¯ / w ¯ 0 = w / w 0 .
w ¯ 0 2 / z ¯ + w 0 2 / z = 0.
θ e 2 - 1 / 2 ( 1 / w ¯ 0 2 + 1 / w 0 2 ) 1 / 2 λ / π
κ ¯ = κ ,
κ ¯ = κ = 0 ,
A = τ a θ e d e = 1 π λ α = 1 π λ 0 n α 1 π λ 0 n ,
0 < τ a θ e d e / λ = α / π
1 / π ,
0 < T a 1 / 2 θ e d e / λ = α / π
1 / π .
A ( z ) / A max = α ( z ) / α max = θ e ( z ) / θ e , max = exp [ - ( z - z ) 2 / ( δ z ) 2 ]
δ z / λ = ( - ½ θ e - 1 d 2 θ e / d z 2 ) z = z - 1 / 2 = 2 1 / 2 π τ a ( d e / λ ) 2 = 2 1 / 2 ( d e / λ ) / θ e , max = 2 1 / 2 / π τ a θ e , max 2 ,
0 < τ a 1 ,
0 < θ e < ,
0 < d e < ,
α = π τ a θ e d e / λ = 1 ,
α = 2 - 3 / 4 π 1 / 2 τ a θ e ( s e / λ ) 1 / 2 = 1 ,
τ a θ t d s / λ = 1 π ln 10 10 ( s t ) 1 / 2
= 7.33 × 10 - 2 ( s t ) 1 / 2
= 7.33 × 10 - 2 , s = t = 1 ,
d e = ( 10 / s ln 10 ) 1 / 2 d s = d s / 0.48 s 1 / 2 ,
θ e = ( 10 / t ln 10 ) 1 / 2 θ t = θ t / 0.48 t 1 / 2 .
r + = w + / w - ,             r - = w - / w + = 1 / r +
r ± = 1 / τ a ± ( 1 / τ a 2 - 1 ) 1 / 2 .
( w ¯ 0 w 0 ) 1 / 2 = ( w + w - ) 1 / 2 = τ a 1 / 2 d e = 2.08 λ τ a 1 / 2 ( d 1 / λ ) .
( w ¯ 0 w 0 ) 1 / 2 = ( w + w - ) 1 / 2 = λ / π τ a 1 / 2 θ e = 0.153 λ / τ a 1 / 2 θ 1 .
w ± = r ± 1 / 2 ( w + w - ) 1 / 2 = r + ± 1 / 2 ( w + w - ) 1 / 2 .
= d e [ 1 ± ( 1 - τ a 2 ) 1 / 2 ] 1 / 2 .
w ± = [ 1 ± ( 1 - τ a 2 ) 1 / 2 ] 1 / 2 λ / π τ a θ e .
( 10 / ln 10 ) 1 / 2 θ 1 = θ e 2 - 1 / 2 π λ w - .
( 10 / ln 10 ) 1 / 2 d 1 = d e 2 - 1 / 2 w + ,
( 10 / ln 10 ) 1 / 2 θ 1 = θ e = 1 π λ w ± ,
( 10 / ln 10 ) 1 / 2 d 1 = d e = w ± .
T a = τ a 2 = 1 - [ 1 - ( w - / d e ) 2 ] 2 ,             d e w - ,
( 10 / ln 10 ) 1 / 2 d 1 = d e < 2 1 / 2 w - / T a 1 / 2
- exp ( - a x 2 - 2 b x - c ) d x = ( π / a ) 1 / 2 exp ( b 2 / a - c )

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