Abstract

Resonators containing a focusing rod are thoroughly analyzed. It is shown that, as a function of the dioptric power of the rod, two stability zones of the same width exist and that the mode volume in the rod always presents a stationary point. At this point, the output power is insensitive to the focal length fluctuations, and the mode volume inside the rod is inversely proportional to the range of the input power for which the resonator is stable. The two zones are markedly different with respect to misalignment sensitivity, which is, in general, much greater in one zone than in the other. Two design procedures are presented for monomode solid-state laser resonators with large mode volume and low sensitivity both to focal length fluctuations and to misalignment.

© 1986 Optical Society of America

Full Article  |  PDF Article

Corrections

Vittorio Magni, "Resonators for solid-state lasers with large-volume fundamental mode and high alignment stability: errata," Appl. Opt. 25, 2039-2039 (1986)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-25-13-2039

References

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  1. C. M. Stickley, “Laser Brightness, Gain and Mode Control by Compensation for Thermal Distortion,” IEEE J. Quantum Electron. QE-2511 (1966).
    [CrossRef]
  2. L. M. Osternik, J. D. Foster, “Thermal Effects and Transverse Mode Control in a Nd:YAG Laser” Appl. Phys. Lett. 12, 128 (1968).
    [CrossRef]
  3. F. A. Levine, “TEM00 Enhancement in CW Nd:YAG by Thermal Lensing Compensation,” IEEE J. Quantum Electron. QE-7170 (1971).
    [CrossRef]
  4. R. B. Chesler, D. Maydan, “Convex-Concave Resonators for TEM00 Operation of Solid-State Ion Lasers,” J. Appl. Phys. 43, 2254 (1972).
    [CrossRef]
  5. J. Steffen, J. P. Lörtscher, G. Herziger, “Fundamental Mode Radiation with Solid-State Lasers,” IEEE J. Quantum Electron. QE-8239 (1972).
    [CrossRef]
  6. J. P. Lörtscher, J. Steffen, G. Herziger, “Dynamic Stable Resonators: a Design Procedure,” Opt. Quantum Electron. 7, 505 (1975).
    [CrossRef]
  7. G. Z. Baumann, “Resonator Optimization for Passively Mode-Locked Lasers,” Opt. Commun. 48, 421 (1984).
    [CrossRef]
  8. H. P. Kortz, R. Iffländer, H. Weber, “Stability and Beam Divergence of Multimode Lasers with Internal Variable Lenses,” Appl. Opt. 20, 4124 (1981).
    [CrossRef] [PubMed]
  9. P. H. Sarkies, “A Stable YAG Resonator Yielding a Beam of Very Low Divergence and High Output Energy,” Opt. Commun. 31, 189 (1979).
    [CrossRef]
  10. D. C. Hanna, C. G. Sawyers, M. A. Yuratich, “Large Volume TEM00 Mode Operation of Nd:YAG Lasers,” Opt. Commun. 37, 359 (1981).
    [CrossRef]
  11. A. J. Berry, D. C. Hanna, C. G. Sawyers, “High Power, Single Frequency Operation of a Q-switched TEM00 Mode Nd:YAG Laser,” Opt. Commun. 40, 54 (1981).
    [CrossRef]
  12. D. C. Hanna, C. G. Sawyers, M. A. Yuratich, “Telescopic Resonators for Large-Volume TEM00-Mode Operation,” Opt. Quantum Electron. 13, 493 (1981).
    [CrossRef]
  13. J. D. Foster, L. M. Osterink, “Thermal Effects in a Nd:YAG Laser,” J. Appl. Phys. 41, 3656 (1970).
    [CrossRef]
  14. W. Koechner, “Absorbed Pump Power, Thermal Profile and Stresses in a cw Pumped Nd:YAG Crystal,” Appl. Opt. 9, 1429 (1970).
    [CrossRef] [PubMed]
  15. W. Koechner, “Thermal Lensing in a Nd:YAG Laser Rod,” Appl. Opt. 9, 2548 (1970).
    [CrossRef] [PubMed]
  16. H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455 (1965).
  17. W. Koechner, Solid-State Laser Engineering (Springer-Verlag, New York, 1976), pp. 355 and 356.
  18. P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of a Light Beam of Gaussian Field Distribution in a Continuous and Periodic Lens-Like Media,” Proc. IEEE 53, 129 (1965).
    [CrossRef]
  19. G. Herziger, H. Weber, “Equivalent Optical Resonators,” Appl. Opt. 23, 1450 (1984).
    [CrossRef] [PubMed]
  20. Ref. 17, pp. 191–195.
  21. R. L. Sanderson, W. Streifer, “Laser Resonators with Tilted Reflectors,” Appl. Opt. 8, 2241 (1969).
    [CrossRef] [PubMed]
  22. N. K. Berger, N. A. Deryugin, Y. N. Lukyanov, Y. E. Studenikin, “Open Misaligned Spherical-Mirror Resonator,” Opt. Spectrosc. USSR 43, 176 (1977).
  23. R. Hauck, H. P. Kortz, H. Weber, “Misalignment Sensitivity of Optical Resonators,” Appl. Opt. 19, 598 (1980).
    [CrossRef] [PubMed]
  24. Ref. 17, Sec. 7.1.
  25. H. Kogelnik, T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]

1984 (2)

G. Z. Baumann, “Resonator Optimization for Passively Mode-Locked Lasers,” Opt. Commun. 48, 421 (1984).
[CrossRef]

G. Herziger, H. Weber, “Equivalent Optical Resonators,” Appl. Opt. 23, 1450 (1984).
[CrossRef] [PubMed]

1981 (4)

H. P. Kortz, R. Iffländer, H. Weber, “Stability and Beam Divergence of Multimode Lasers with Internal Variable Lenses,” Appl. Opt. 20, 4124 (1981).
[CrossRef] [PubMed]

D. C. Hanna, C. G. Sawyers, M. A. Yuratich, “Large Volume TEM00 Mode Operation of Nd:YAG Lasers,” Opt. Commun. 37, 359 (1981).
[CrossRef]

A. J. Berry, D. C. Hanna, C. G. Sawyers, “High Power, Single Frequency Operation of a Q-switched TEM00 Mode Nd:YAG Laser,” Opt. Commun. 40, 54 (1981).
[CrossRef]

D. C. Hanna, C. G. Sawyers, M. A. Yuratich, “Telescopic Resonators for Large-Volume TEM00-Mode Operation,” Opt. Quantum Electron. 13, 493 (1981).
[CrossRef]

1980 (1)

1979 (1)

P. H. Sarkies, “A Stable YAG Resonator Yielding a Beam of Very Low Divergence and High Output Energy,” Opt. Commun. 31, 189 (1979).
[CrossRef]

1977 (1)

N. K. Berger, N. A. Deryugin, Y. N. Lukyanov, Y. E. Studenikin, “Open Misaligned Spherical-Mirror Resonator,” Opt. Spectrosc. USSR 43, 176 (1977).

1975 (1)

J. P. Lörtscher, J. Steffen, G. Herziger, “Dynamic Stable Resonators: a Design Procedure,” Opt. Quantum Electron. 7, 505 (1975).
[CrossRef]

1972 (2)

R. B. Chesler, D. Maydan, “Convex-Concave Resonators for TEM00 Operation of Solid-State Ion Lasers,” J. Appl. Phys. 43, 2254 (1972).
[CrossRef]

J. Steffen, J. P. Lörtscher, G. Herziger, “Fundamental Mode Radiation with Solid-State Lasers,” IEEE J. Quantum Electron. QE-8239 (1972).
[CrossRef]

1971 (1)

F. A. Levine, “TEM00 Enhancement in CW Nd:YAG by Thermal Lensing Compensation,” IEEE J. Quantum Electron. QE-7170 (1971).
[CrossRef]

1970 (3)

1969 (1)

1968 (1)

L. M. Osternik, J. D. Foster, “Thermal Effects and Transverse Mode Control in a Nd:YAG Laser” Appl. Phys. Lett. 12, 128 (1968).
[CrossRef]

1966 (2)

C. M. Stickley, “Laser Brightness, Gain and Mode Control by Compensation for Thermal Distortion,” IEEE J. Quantum Electron. QE-2511 (1966).
[CrossRef]

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

1965 (2)

H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455 (1965).

P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of a Light Beam of Gaussian Field Distribution in a Continuous and Periodic Lens-Like Media,” Proc. IEEE 53, 129 (1965).
[CrossRef]

Baumann, G. Z.

G. Z. Baumann, “Resonator Optimization for Passively Mode-Locked Lasers,” Opt. Commun. 48, 421 (1984).
[CrossRef]

Berger, N. K.

N. K. Berger, N. A. Deryugin, Y. N. Lukyanov, Y. E. Studenikin, “Open Misaligned Spherical-Mirror Resonator,” Opt. Spectrosc. USSR 43, 176 (1977).

Berry, A. J.

A. J. Berry, D. C. Hanna, C. G. Sawyers, “High Power, Single Frequency Operation of a Q-switched TEM00 Mode Nd:YAG Laser,” Opt. Commun. 40, 54 (1981).
[CrossRef]

Chesler, R. B.

R. B. Chesler, D. Maydan, “Convex-Concave Resonators for TEM00 Operation of Solid-State Ion Lasers,” J. Appl. Phys. 43, 2254 (1972).
[CrossRef]

Deryugin, N. A.

N. K. Berger, N. A. Deryugin, Y. N. Lukyanov, Y. E. Studenikin, “Open Misaligned Spherical-Mirror Resonator,” Opt. Spectrosc. USSR 43, 176 (1977).

Foster, J. D.

J. D. Foster, L. M. Osterink, “Thermal Effects in a Nd:YAG Laser,” J. Appl. Phys. 41, 3656 (1970).
[CrossRef]

L. M. Osternik, J. D. Foster, “Thermal Effects and Transverse Mode Control in a Nd:YAG Laser” Appl. Phys. Lett. 12, 128 (1968).
[CrossRef]

Gordon, J. P.

P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of a Light Beam of Gaussian Field Distribution in a Continuous and Periodic Lens-Like Media,” Proc. IEEE 53, 129 (1965).
[CrossRef]

Hanna, D. C.

A. J. Berry, D. C. Hanna, C. G. Sawyers, “High Power, Single Frequency Operation of a Q-switched TEM00 Mode Nd:YAG Laser,” Opt. Commun. 40, 54 (1981).
[CrossRef]

D. C. Hanna, C. G. Sawyers, M. A. Yuratich, “Telescopic Resonators for Large-Volume TEM00-Mode Operation,” Opt. Quantum Electron. 13, 493 (1981).
[CrossRef]

D. C. Hanna, C. G. Sawyers, M. A. Yuratich, “Large Volume TEM00 Mode Operation of Nd:YAG Lasers,” Opt. Commun. 37, 359 (1981).
[CrossRef]

Hauck, R.

Herziger, G.

G. Herziger, H. Weber, “Equivalent Optical Resonators,” Appl. Opt. 23, 1450 (1984).
[CrossRef] [PubMed]

J. P. Lörtscher, J. Steffen, G. Herziger, “Dynamic Stable Resonators: a Design Procedure,” Opt. Quantum Electron. 7, 505 (1975).
[CrossRef]

J. Steffen, J. P. Lörtscher, G. Herziger, “Fundamental Mode Radiation with Solid-State Lasers,” IEEE J. Quantum Electron. QE-8239 (1972).
[CrossRef]

Iffländer, R.

Koechner, W.

Kogelnik, H.

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

H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455 (1965).

Kortz, H. P.

Levine, F. A.

F. A. Levine, “TEM00 Enhancement in CW Nd:YAG by Thermal Lensing Compensation,” IEEE J. Quantum Electron. QE-7170 (1971).
[CrossRef]

Li, T.

Lörtscher, J. P.

J. P. Lörtscher, J. Steffen, G. Herziger, “Dynamic Stable Resonators: a Design Procedure,” Opt. Quantum Electron. 7, 505 (1975).
[CrossRef]

J. Steffen, J. P. Lörtscher, G. Herziger, “Fundamental Mode Radiation with Solid-State Lasers,” IEEE J. Quantum Electron. QE-8239 (1972).
[CrossRef]

Lukyanov, Y. N.

N. K. Berger, N. A. Deryugin, Y. N. Lukyanov, Y. E. Studenikin, “Open Misaligned Spherical-Mirror Resonator,” Opt. Spectrosc. USSR 43, 176 (1977).

Maydan, D.

R. B. Chesler, D. Maydan, “Convex-Concave Resonators for TEM00 Operation of Solid-State Ion Lasers,” J. Appl. Phys. 43, 2254 (1972).
[CrossRef]

Osterink, L. M.

J. D. Foster, L. M. Osterink, “Thermal Effects in a Nd:YAG Laser,” J. Appl. Phys. 41, 3656 (1970).
[CrossRef]

Osternik, L. M.

L. M. Osternik, J. D. Foster, “Thermal Effects and Transverse Mode Control in a Nd:YAG Laser” Appl. Phys. Lett. 12, 128 (1968).
[CrossRef]

Sanderson, R. L.

Sarkies, P. H.

P. H. Sarkies, “A Stable YAG Resonator Yielding a Beam of Very Low Divergence and High Output Energy,” Opt. Commun. 31, 189 (1979).
[CrossRef]

Sawyers, C. G.

D. C. Hanna, C. G. Sawyers, M. A. Yuratich, “Telescopic Resonators for Large-Volume TEM00-Mode Operation,” Opt. Quantum Electron. 13, 493 (1981).
[CrossRef]

A. J. Berry, D. C. Hanna, C. G. Sawyers, “High Power, Single Frequency Operation of a Q-switched TEM00 Mode Nd:YAG Laser,” Opt. Commun. 40, 54 (1981).
[CrossRef]

D. C. Hanna, C. G. Sawyers, M. A. Yuratich, “Large Volume TEM00 Mode Operation of Nd:YAG Lasers,” Opt. Commun. 37, 359 (1981).
[CrossRef]

Steffen, J.

J. P. Lörtscher, J. Steffen, G. Herziger, “Dynamic Stable Resonators: a Design Procedure,” Opt. Quantum Electron. 7, 505 (1975).
[CrossRef]

J. Steffen, J. P. Lörtscher, G. Herziger, “Fundamental Mode Radiation with Solid-State Lasers,” IEEE J. Quantum Electron. QE-8239 (1972).
[CrossRef]

Stickley, C. M.

C. M. Stickley, “Laser Brightness, Gain and Mode Control by Compensation for Thermal Distortion,” IEEE J. Quantum Electron. QE-2511 (1966).
[CrossRef]

Streifer, W.

Studenikin, Y. E.

N. K. Berger, N. A. Deryugin, Y. N. Lukyanov, Y. E. Studenikin, “Open Misaligned Spherical-Mirror Resonator,” Opt. Spectrosc. USSR 43, 176 (1977).

Tien, P. K.

P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of a Light Beam of Gaussian Field Distribution in a Continuous and Periodic Lens-Like Media,” Proc. IEEE 53, 129 (1965).
[CrossRef]

Weber, H.

Whinnery, J. R.

P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of a Light Beam of Gaussian Field Distribution in a Continuous and Periodic Lens-Like Media,” Proc. IEEE 53, 129 (1965).
[CrossRef]

Yuratich, M. A.

D. C. Hanna, C. G. Sawyers, M. A. Yuratich, “Telescopic Resonators for Large-Volume TEM00-Mode Operation,” Opt. Quantum Electron. 13, 493 (1981).
[CrossRef]

D. C. Hanna, C. G. Sawyers, M. A. Yuratich, “Large Volume TEM00 Mode Operation of Nd:YAG Lasers,” Opt. Commun. 37, 359 (1981).
[CrossRef]

Appl. Opt. (7)

Appl. Phys. Lett. (1)

L. M. Osternik, J. D. Foster, “Thermal Effects and Transverse Mode Control in a Nd:YAG Laser” Appl. Phys. Lett. 12, 128 (1968).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455 (1965).

IEEE J. Quantum Electron. (3)

C. M. Stickley, “Laser Brightness, Gain and Mode Control by Compensation for Thermal Distortion,” IEEE J. Quantum Electron. QE-2511 (1966).
[CrossRef]

F. A. Levine, “TEM00 Enhancement in CW Nd:YAG by Thermal Lensing Compensation,” IEEE J. Quantum Electron. QE-7170 (1971).
[CrossRef]

J. Steffen, J. P. Lörtscher, G. Herziger, “Fundamental Mode Radiation with Solid-State Lasers,” IEEE J. Quantum Electron. QE-8239 (1972).
[CrossRef]

J. Appl. Phys. (2)

R. B. Chesler, D. Maydan, “Convex-Concave Resonators for TEM00 Operation of Solid-State Ion Lasers,” J. Appl. Phys. 43, 2254 (1972).
[CrossRef]

J. D. Foster, L. M. Osterink, “Thermal Effects in a Nd:YAG Laser,” J. Appl. Phys. 41, 3656 (1970).
[CrossRef]

Opt. Commun. (4)

G. Z. Baumann, “Resonator Optimization for Passively Mode-Locked Lasers,” Opt. Commun. 48, 421 (1984).
[CrossRef]

P. H. Sarkies, “A Stable YAG Resonator Yielding a Beam of Very Low Divergence and High Output Energy,” Opt. Commun. 31, 189 (1979).
[CrossRef]

D. C. Hanna, C. G. Sawyers, M. A. Yuratich, “Large Volume TEM00 Mode Operation of Nd:YAG Lasers,” Opt. Commun. 37, 359 (1981).
[CrossRef]

A. J. Berry, D. C. Hanna, C. G. Sawyers, “High Power, Single Frequency Operation of a Q-switched TEM00 Mode Nd:YAG Laser,” Opt. Commun. 40, 54 (1981).
[CrossRef]

Opt. Quantum Electron. (2)

D. C. Hanna, C. G. Sawyers, M. A. Yuratich, “Telescopic Resonators for Large-Volume TEM00-Mode Operation,” Opt. Quantum Electron. 13, 493 (1981).
[CrossRef]

J. P. Lörtscher, J. Steffen, G. Herziger, “Dynamic Stable Resonators: a Design Procedure,” Opt. Quantum Electron. 7, 505 (1975).
[CrossRef]

Opt. Spectrosc. USSR (1)

N. K. Berger, N. A. Deryugin, Y. N. Lukyanov, Y. E. Studenikin, “Open Misaligned Spherical-Mirror Resonator,” Opt. Spectrosc. USSR 43, 176 (1977).

Proc. IEEE (1)

P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of a Light Beam of Gaussian Field Distribution in a Continuous and Periodic Lens-Like Media,” Proc. IEEE 53, 129 (1965).
[CrossRef]

Other (3)

W. Koechner, Solid-State Laser Engineering (Springer-Verlag, New York, 1976), pp. 355 and 356.

Ref. 17, pp. 191–195.

Ref. 17, Sec. 7.1.

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Figures (8)

Fig. 1
Fig. 1

Equivalent resonators with (a) a lenslike rod; (b) a thin lens between two pieces of homogeneous material [exactly equivalent to (a) outside the rod and approximately inside]; (c) a thin lens [equivalent to (a) and (b) outside the rod]. The spot sizes on the lenses are the same in (b) and in (c), and approximately equal to that in the middle of the rod in (a); p.p. = principal plane.

Fig. 2
Fig. 2

Stability diagrams and mode profiles at the edges of the stability zones (marked I and II). The circles indicate the curvature centers of the mirrors. The point (g1,g2) representing the resonator moves linearly with the dioptric power 1/f of the lens along the straight line: (a) positive slope, i.e., u1u2 > 0; (b) negative slope, i.e., u1u2 < 0.

Fig. 3
Fig. 3

Spot size w3 on the lens as a function of the dioptric power 1/f of the lens (x = 1/f − 1/L1 − 1/L2).

Fig. 4
Fig. 4

Mode axis in a misaligned resonator (mirrors in a solid line). The axis passes through the curvature center C1 and C2 of the mirrors and through the images C1 and C2 of those points produced by the lens.

Fig. 5
Fig. 5

Misalignment sensitivity S as a function of the dioptric power 1/f of the lens (x = 1/f − 1/L1 − 1/L2) for | u1 | > | u2| and u1 < 0: (a) | u1 | ≫ | u2|; (b) | u1 | ≃ | u2|. The shaded areas correspond to the stability zones.

Fig. 6
Fig. 6

Mirror curvatures l/R1 and 1/R2 and misalignment sensitivity S0 vs L1 for determination of the optimized resonator configuration using design procedure A. The parameter on the curve is f0 (cm); w30 = 1 mm; L = 150 cm; λ = 1.064 μm: (a) u1 < 0; (b) u1 > 0. Note that the curvature of mirror 1 is independent of f0.

Fig. 7
Fig. 7

Same as Fig. 6 but for w30 = 2 mm.

Fig. 8
Fig. 8

Distance L1, mirror curvatures 1/R1 and 1/R2, and misalignment sensitivity S1/2 vs the dioptric power 1/f1/2 for resonators in zone I optimized for minimum misalignment sensitivity (design procedure B). The parameter on the curves is w30 (mm); L = 150 cm; λ = 1.064 μm. The plus sign in Eq. (47) and u1 > 0 have been assumed.

Tables (1)

Tables Icon

Table I Spot Sizes Near Instability

Equations (58)

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

1 f = 2 n 0 b sin 2 1 b ,
h = b 2 n 0 tan 1 b ,
1 f = 4 n 0 l b 2 ,
h = 1 2 n 0 .
L = L 1 + L 2 ,
L = L 1 + L 2 L 1 L 2 / f .
g 1 = 1 L 2 f L R 1 ,
g 2 = 1 L 1 f L R 2 .
u 1 = L 1 ( 1 L 1 R 1 ) ,
u 2 = L 2 ( 1 L 2 R 2 ) ,
x = 1 f 1 L 1 1 L 2 .
g 1 = L 2 L 1 ( 1 + x u 1 ) ,
g 2 = L 1 L 2 ( 1 + x u 2 ) ,
L = L 1 L 2 x .
0 < ( 1 + x u 1 ) ( 1 + x u 2 ) < 1 .
g 2 = ( L 1 L 2 ) 2 u 2 u 1 g 1 + L 1 L 2 ( u 2 u 1 1 ) .
1 u 1 for g 1 = 0 ,
x = 1 u 2 for g 2 = 0 ,
0 for g 1 g 2 = 1 .
1 u 1 1 u 2
| Δ 1 f | = | Δ x | = min ( | 1 u 1 | , | 1 u 2 | ) .
1 f = { 1 L 2 + 1 L 1 R 1 for g 1 = 0 , 1 L 1 + 1 L 2 R 2 for g 2 = 0 , 1 L 1 + 1 L 2 1 L 1 R 1 + 1 L 2 R 2 for g 1 g 2 = 1 .
w 1 2 = λ | L | π [ g 2 g 1 ( 1 g 1 g 2 ) ] 1 / 2 ,
w 2 2 = λ | L | π [ g 1 g 2 ( 1 g 1 g 2 ) ] 1 / 2 ,
w 3 2 = λ π [ 4 u 1 u 2 g 1 g 2 + ( u 1 u 2 ) 2 ( 1 g 1 g 2 ) g 1 g 2 ] 1 / 2
= λ π | 2 x u 1 u 2 + u 1 + u 2 | [ ( 1 g 1 g 2 ) g 1 g 2 ] 1 / 2 .
d w 3 d ( 1 / f ) = d w 3 d x = 0
g 1 g 2 = 1 2 ( 1 u 2 u 1 ) , | u 1 | > | u 2 | .
w 30 2 = 2 λ π | u 1 | , | u 1 | > | u 2 | ,
w 30 2 = 2 λ π max ( | u 1 | , | u 2 | ) .
w 30 2 = 2 λ π 1 | Δ 1 f | .
S 1 = d 1 w 30 α 1 ,
S 2 = d 2 w 30 α 2 ,
S = ( S 1 2 + S 2 2 ) 1 / 2 .
S 1 = 1 w 30 L 1 u 1 1 1 u 1 + 1 u 2 + x ,
S 2 = 1 w 30 L 2 u 2 1 1 u 1 + 1 u 2 + x .
S 0 = 1 w 30 [ L 1 2 + ( u 1 u 2 L 2 ) 2 ] 1 / 2 | 1 ( u 1 u 2 u 1 + u 2 ) 1 / 2 | ,
S 1 / 2 = { 2 w 30 [ L 1 2 + ( u 1 u 2 L 2 ) 2 ] 1 / 2 | u 2 2 u 1 + u 2 | zone I , 2 w 30 [ L 1 2 + ( u 1 u 2 L 2 ) 2 ] 1 / 2 zone II ,
u 1 = ± π 2 λ w 30 2 .
u 2 = u 1 2 x 0 u 1 + 1 2 x 0 2 u 1 2 + 2 x 0 u 1 + 1 ,
1 R 1 = 1 L 1 ( 1 u 1 L 1 ) ,
1 R 2 = 1 L 2 ( 1 u 2 L 2 ) .
S 0 = S 0 ( L 1 ) .
1 f 1 / 2 1 L 1 1 L 2 = 1 2 u 1 .
L 1 = L 2 { 1 ± [ 1 8 u 1 f 1 / 2 L ( 2 u 1 + f 1 / 2 ) ] 1 / 2 } .
u 2 = 1 2 ( L 2 L 1 ) 2 u 1 ,
S 1 / 2 = 1 w 30 2 L 1 L 2 ( 4 L 1 2 + L 2 2 ) 1 / 2 .
u 2 = u 1 ,
S 1 / 2 = 2 3 w 30 ( L 1 2 + L 2 2 ) 1 / 2 .
x 0 = 1 2 u 2 { 1 + u 2 u 1 [ 1 ( u 2 u 1 ) 2 ] 1 / 2 } ,
w 20 2 = λ π ( L 2 u 2 ) 2 [ | u 1 | ( u 1 2 u 2 2 ) 1 / 2 ] ,
w 10 2 = λ π L 1 2 | u 1 | .
R 1 w 30 w 30 R 1 = 1 2 ( L 1 u 1 1 ) ,
R 2 w 30 w 30 R 2 = 0 ,
L w 30 w 30 L 1 = L L 1 L 2 u 1 ,
R 1 f 1 / 2 f 1 / 2 R 1 = f 1 / 2 2 u 1 ( 1 L 1 u 1 ) ,
R 2 f 1 / 2 f 1 / 2 R 2 = { 0 zone I , f 1 / 2 u 2 ( 1 L 2 u 2 ) zone II ,
L f 1 / 2 f 1 / 2 L 1 = { L f 1 / 2 ( 1 L 1 2 1 L 2 2 + 1 2 u 1 2 1 L 1 u 1 ) zone I , L f 1 / 2 [ 1 L 1 2 + 1 2 u 1 2 1 L 1 u 1 ( 1 L 2 1 u 2 ) 2 ] zone II .

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