Abstract

A unified formulation for the analysis of linear stable resonators containing a lens of variable focal length, which represents the rod of a solid-state laser, and other intracavity optical systems is presented. The stability, the mode spot sizes, the dynamical stability, and the misalignment sensitivity are investigated, and general properties that are valid for any resonator are derived. Some important practical consequences for resonator design are discussed.

© 1987 Optical Society of America

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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-2, 511–518 (1966).
    [CrossRef]
  2. L. M. Osterink, L. D. Foster, “Thermal effects and transverse mode control in a Nd:YAG laser,” Appl. Phys. Lett. 12, 128–131 (1968).
    [CrossRef]
  3. F. A. Levine, “TEM00enhancement in cw Nd:YAG by thermal lensing compensation,” IEEE J. Quantum Electron. QE-7, 170–172 (1971).
    [CrossRef]
  4. R. B. Chesler, D. Maydan, “Convex-concave resonators for TEM00operation of solid-state ion lasers,” J. Appl. Phys. 43, 2254–2257 (1972).
    [CrossRef]
  5. J. Steffen, J. P. Lörtscher, G. Herziger, “Fundamental mode radiation with solid-state lasers,” IEEE J. Quantum Electron. QE-8, 239–245 (1972).
    [CrossRef]
  6. J. P. Lörtscher, J. Steffen, G. Herziger, “Dynamic stable resonators: a design procedure,” Opt. Quantum Electron. 7, 505–514 (1975).
    [CrossRef]
  7. P. H. Sarkies, “A stable YAG resonator yielding a beam of very low divergence and high output energy,” Opt. Commun. 31, 189–192 (1979).
    [CrossRef]
  8. D. C. Hanna, C. G. Sawyers, M. A. Yuratich, “Telescopic resonators for large-volume TEM00-mode operation,” Opt. Quantum Electron. 13, 493–507 (1981).
    [CrossRef]
  9. A. J. Berry, D. C. Hanna, C. G. Sawyers, “High power, single frequency operation of a Q-switched TEM00mode Nd:YAG laser,” Opt. Commun. 40, 54–58 (1981).
    [CrossRef]
  10. R. Iffländer, H. P. Kortz, H. Weber, “Beam divergence and refractive power of directly coated solid-state lasers,” Opt. Commun. 29, 223–226 (1979).
    [CrossRef]
  11. H. P. Kortz, R. Iffländer, H. Weber, “Stability and beam divergence of multimode lasers with internal variable lenses,” Appl. Opt. 20, 4124–4134 (1981).
    [CrossRef] [PubMed]
  12. A. G. Fox, T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51, 80–89 (1963).
    [CrossRef]
  13. R. L. Sanderson, W. Streifer, “Laser resonators with tilted reflectors,” Appl. Opt. 8, 2241–2248 (1969).
    [CrossRef] [PubMed]
  14. J. A. Arnaud, “Degenerate optical cavities. II: Effect of misalignment,” Appl. Opt. 8, 1909–1917 (1969).
    [CrossRef] [PubMed]
  15. R. Hauck, H. P. Kortz, H. Weber, “Misalignment sensitivity of optical resonators,” Appl. Opt. 19, 598–601 (1980).
    [CrossRef] [PubMed]
  16. M. J. Konopnicki, M. E. Smithers, “Unstable resonator with multiple outputs,” Appl. Opt. 22, 947–951 (1983).
    [CrossRef] [PubMed]
  17. E. Sklar, “The advantages of a negative branch unstable resonator for use with free-electron lasers,” IEEE J. Quantum Electron. QE-22, 1088–1094 (1986).
    [CrossRef]
  18. W. F. Krupke, W. R. Sooy, “Properties of an unstable confocal resonator CO2laser system,” IEEE J. Quantum Electron. QE-5, 575–586 (1969).
    [CrossRef]
  19. K. E. Oughstun, “Intracavity adaptive optic compensation of phase aberrations. I: Analysis,”J. Opt. Soc. Am. 71, 862–872 (1981).
    [CrossRef]
  20. K. E. Oughstun, “Intracavity compensation of quadratic phase aberrations,”J. Opt. Soc. Am. 72, 1529–1537 (1982).
    [CrossRef]
  21. K. E. Oughstun, “Aberration sensitivity of unstable-cavity geometries,” J. Opt. Soc. Am. A 3, 1113–1141 (1986).
    [CrossRef]
  22. V. Magni, “Resonators for solid-state lasers with large-volume fundamental mode and high alignment stability,” Appl. Opt. 25, 107–117 (1986).
    [CrossRef] [PubMed]
  23. S. De Silvestri, P. Laporta, V. Magni, “Misalignment sensitivity of solid-state laser resonators with thermal lensing,” Opt. Commun. 59, 43–48 (1986).
    [CrossRef]
  24. S. De Silvestri, P. Laporta, V. Magni, “Novel stability diagrams for continuous wave solid-state lasers,” Opt. Lett. 11, 513–515 (1986).
    [CrossRef] [PubMed]
  25. S. De Silvestri, P. Laporta, V. Magni, “14-W continuous-wave mode-locked Nd:YAG laser,” Opt. Lett. 11, 785–787 (1986).
    [CrossRef] [PubMed]
  26. H. Kogelnik, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
  27. H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1965).
    [CrossRef]
  28. K. Halbach, “Matrix representation of Gaussian optics,” Am. J. Phys. 32, 90–108 (1964).
    [CrossRef]
  29. A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975), pp. 106–108, 286–291.
  30. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chap. 4.
  31. M. Nazarathy, A. Hardy, J. Shamir, “Misaligned first-order optics: canonical operator theory,” J. Opt. Soc. Am. A 3, 1360–1369 (1986).
    [CrossRef]
  32. P. Baues, “Huygens’ principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-Electronics 1, 37–44 (1969).
    [CrossRef]
  33. J. A. Arnaud, “Degenerate optical cavities,” Appl. Opt. 8, 189–195 (1969).
    [CrossRef] [PubMed]
  34. W. Koechner, “Thermal lensing in a Nd:YAG laser rod,” Appl. Opt. 9, 2548–2553 (1970).
    [CrossRef] [PubMed]
  35. W. Koechner, Solid-State Laser Engineering (Springer-Verlag, New York, 1976), p. 355.

1986 (7)

1983 (1)

1982 (1)

1981 (4)

K. E. Oughstun, “Intracavity adaptive optic compensation of phase aberrations. I: Analysis,”J. Opt. Soc. Am. 71, 862–872 (1981).
[CrossRef]

H. P. Kortz, R. Iffländer, H. Weber, “Stability and beam divergence of multimode lasers with internal variable lenses,” Appl. Opt. 20, 4124–4134 (1981).
[CrossRef] [PubMed]

D. C. Hanna, C. G. Sawyers, M. A. Yuratich, “Telescopic resonators for large-volume TEM00-mode operation,” Opt. Quantum Electron. 13, 493–507 (1981).
[CrossRef]

A. J. Berry, D. C. Hanna, C. G. Sawyers, “High power, single frequency operation of a Q-switched TEM00mode Nd:YAG laser,” Opt. Commun. 40, 54–58 (1981).
[CrossRef]

1980 (1)

1979 (2)

P. H. Sarkies, “A stable YAG resonator yielding a beam of very low divergence and high output energy,” Opt. Commun. 31, 189–192 (1979).
[CrossRef]

R. Iffländer, H. P. Kortz, H. Weber, “Beam divergence and refractive power of directly coated solid-state lasers,” Opt. Commun. 29, 223–226 (1979).
[CrossRef]

1975 (1)

J. P. Lörtscher, J. Steffen, G. Herziger, “Dynamic stable resonators: a design procedure,” Opt. Quantum Electron. 7, 505–514 (1975).
[CrossRef]

1972 (2)

R. B. Chesler, D. Maydan, “Convex-concave resonators for TEM00operation of solid-state ion lasers,” J. Appl. Phys. 43, 2254–2257 (1972).
[CrossRef]

J. Steffen, J. P. Lörtscher, G. Herziger, “Fundamental mode radiation with solid-state lasers,” IEEE J. Quantum Electron. QE-8, 239–245 (1972).
[CrossRef]

1971 (1)

F. A. Levine, “TEM00enhancement in cw Nd:YAG by thermal lensing compensation,” IEEE J. Quantum Electron. QE-7, 170–172 (1971).
[CrossRef]

1970 (1)

1969 (5)

P. Baues, “Huygens’ principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-Electronics 1, 37–44 (1969).
[CrossRef]

J. A. Arnaud, “Degenerate optical cavities,” Appl. Opt. 8, 189–195 (1969).
[CrossRef] [PubMed]

W. F. Krupke, W. R. Sooy, “Properties of an unstable confocal resonator CO2laser system,” IEEE J. Quantum Electron. QE-5, 575–586 (1969).
[CrossRef]

R. L. Sanderson, W. Streifer, “Laser resonators with tilted reflectors,” Appl. Opt. 8, 2241–2248 (1969).
[CrossRef] [PubMed]

J. A. Arnaud, “Degenerate optical cavities. II: Effect of misalignment,” Appl. Opt. 8, 1909–1917 (1969).
[CrossRef] [PubMed]

1968 (1)

L. M. Osterink, L. D. Foster, “Thermal effects and transverse mode control in a Nd:YAG laser,” Appl. Phys. Lett. 12, 128–131 (1968).
[CrossRef]

1966 (1)

C. M. Stickley, “Laser brightness gain and mode control by compensation for thermal distortion,” IEEE J. Quantum Electron. QE-2, 511–518 (1966).
[CrossRef]

1965 (2)

H. Kogelnik, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1965).
[CrossRef]

1964 (1)

K. Halbach, “Matrix representation of Gaussian optics,” Am. J. Phys. 32, 90–108 (1964).
[CrossRef]

1963 (1)

A. G. Fox, T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51, 80–89 (1963).
[CrossRef]

Arnaud, J. A.

Baues, P.

P. Baues, “Huygens’ principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-Electronics 1, 37–44 (1969).
[CrossRef]

Berry, A. J.

A. J. Berry, D. C. Hanna, C. G. Sawyers, “High power, single frequency operation of a Q-switched TEM00mode Nd:YAG laser,” Opt. Commun. 40, 54–58 (1981).
[CrossRef]

Burch, J. M.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975), pp. 106–108, 286–291.

Chesler, R. B.

R. B. Chesler, D. Maydan, “Convex-concave resonators for TEM00operation of solid-state ion lasers,” J. Appl. Phys. 43, 2254–2257 (1972).
[CrossRef]

De Silvestri, S.

Foster, L. D.

L. M. Osterink, L. D. Foster, “Thermal effects and transverse mode control in a Nd:YAG laser,” Appl. Phys. Lett. 12, 128–131 (1968).
[CrossRef]

Fox, A. G.

A. G. Fox, T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51, 80–89 (1963).
[CrossRef]

Gerrard, A.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975), pp. 106–108, 286–291.

Halbach, K.

K. Halbach, “Matrix representation of Gaussian optics,” Am. J. Phys. 32, 90–108 (1964).
[CrossRef]

Hanna, D. C.

A. J. Berry, D. C. Hanna, C. G. Sawyers, “High power, single frequency operation of a Q-switched TEM00mode Nd:YAG laser,” Opt. Commun. 40, 54–58 (1981).
[CrossRef]

D. C. Hanna, C. G. Sawyers, M. A. Yuratich, “Telescopic resonators for large-volume TEM00-mode operation,” Opt. Quantum Electron. 13, 493–507 (1981).
[CrossRef]

Hardy, A.

Hauck, R.

Herziger, G.

J. P. Lörtscher, J. Steffen, G. Herziger, “Dynamic stable resonators: a design procedure,” Opt. Quantum Electron. 7, 505–514 (1975).
[CrossRef]

J. Steffen, J. P. Lörtscher, G. Herziger, “Fundamental mode radiation with solid-state lasers,” IEEE J. Quantum Electron. QE-8, 239–245 (1972).
[CrossRef]

Iffländer, R.

H. P. Kortz, R. Iffländer, H. Weber, “Stability and beam divergence of multimode lasers with internal variable lenses,” Appl. Opt. 20, 4124–4134 (1981).
[CrossRef] [PubMed]

R. Iffländer, H. P. Kortz, H. Weber, “Beam divergence and refractive power of directly coated solid-state lasers,” Opt. Commun. 29, 223–226 (1979).
[CrossRef]

Koechner, W.

W. Koechner, “Thermal lensing in a Nd:YAG laser rod,” Appl. Opt. 9, 2548–2553 (1970).
[CrossRef] [PubMed]

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

Kogelnik, H.

H. Kogelnik, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1965).
[CrossRef]

Konopnicki, M. J.

Kortz, H. P.

Krupke, W. F.

W. F. Krupke, W. R. Sooy, “Properties of an unstable confocal resonator CO2laser system,” IEEE J. Quantum Electron. QE-5, 575–586 (1969).
[CrossRef]

Laporta, P.

Levine, F. A.

F. A. Levine, “TEM00enhancement in cw Nd:YAG by thermal lensing compensation,” IEEE J. Quantum Electron. QE-7, 170–172 (1971).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1965).
[CrossRef]

A. G. Fox, T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51, 80–89 (1963).
[CrossRef]

Lörtscher, J. P.

J. P. Lörtscher, J. Steffen, G. Herziger, “Dynamic stable resonators: a design procedure,” Opt. Quantum Electron. 7, 505–514 (1975).
[CrossRef]

J. Steffen, J. P. Lörtscher, G. Herziger, “Fundamental mode radiation with solid-state lasers,” IEEE J. Quantum Electron. QE-8, 239–245 (1972).
[CrossRef]

Magni, V.

Maydan, D.

R. B. Chesler, D. Maydan, “Convex-concave resonators for TEM00operation of solid-state ion lasers,” J. Appl. Phys. 43, 2254–2257 (1972).
[CrossRef]

Nazarathy, M.

Osterink, L. M.

L. M. Osterink, L. D. Foster, “Thermal effects and transverse mode control in a Nd:YAG laser,” Appl. Phys. Lett. 12, 128–131 (1968).
[CrossRef]

Oughstun, K. E.

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, 189–192 (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, 493–507 (1981).
[CrossRef]

A. J. Berry, D. C. Hanna, C. G. Sawyers, “High power, single frequency operation of a Q-switched TEM00mode Nd:YAG laser,” Opt. Commun. 40, 54–58 (1981).
[CrossRef]

Shamir, J.

Sklar, E.

E. Sklar, “The advantages of a negative branch unstable resonator for use with free-electron lasers,” IEEE J. Quantum Electron. QE-22, 1088–1094 (1986).
[CrossRef]

Smithers, M. E.

Sooy, W. R.

W. F. Krupke, W. R. Sooy, “Properties of an unstable confocal resonator CO2laser system,” IEEE J. Quantum Electron. QE-5, 575–586 (1969).
[CrossRef]

Steffen, J.

J. P. Lörtscher, J. Steffen, G. Herziger, “Dynamic stable resonators: a design procedure,” Opt. Quantum Electron. 7, 505–514 (1975).
[CrossRef]

J. Steffen, J. P. Lörtscher, G. Herziger, “Fundamental mode radiation with solid-state lasers,” IEEE J. Quantum Electron. QE-8, 239–245 (1972).
[CrossRef]

Stickley, C. M.

C. M. Stickley, “Laser brightness gain and mode control by compensation for thermal distortion,” IEEE J. Quantum Electron. QE-2, 511–518 (1966).
[CrossRef]

Streifer, W.

Weber, H.

Yuratich, M. A.

D. C. Hanna, C. G. Sawyers, M. A. Yuratich, “Telescopic resonators for large-volume TEM00-mode operation,” Opt. Quantum Electron. 13, 493–507 (1981).
[CrossRef]

Am. J. Phys. (1)

K. Halbach, “Matrix representation of Gaussian optics,” Am. J. Phys. 32, 90–108 (1964).
[CrossRef]

Appl. Opt. (9)

Appl. Phys. Lett. (1)

L. M. Osterink, L. D. Foster, “Thermal effects and transverse mode control in a Nd:YAG laser,” Appl. Phys. Lett. 12, 128–131 (1968).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

IEEE J. Quantum Electron. (5)

C. M. Stickley, “Laser brightness gain and mode control by compensation for thermal distortion,” IEEE J. Quantum Electron. QE-2, 511–518 (1966).
[CrossRef]

F. A. Levine, “TEM00enhancement in cw Nd:YAG by thermal lensing compensation,” IEEE J. Quantum Electron. QE-7, 170–172 (1971).
[CrossRef]

J. Steffen, J. P. Lörtscher, G. Herziger, “Fundamental mode radiation with solid-state lasers,” IEEE J. Quantum Electron. QE-8, 239–245 (1972).
[CrossRef]

E. Sklar, “The advantages of a negative branch unstable resonator for use with free-electron lasers,” IEEE J. Quantum Electron. QE-22, 1088–1094 (1986).
[CrossRef]

W. F. Krupke, W. R. Sooy, “Properties of an unstable confocal resonator CO2laser system,” IEEE J. Quantum Electron. QE-5, 575–586 (1969).
[CrossRef]

J. Appl. Phys. (1)

R. B. Chesler, D. Maydan, “Convex-concave resonators for TEM00operation of solid-state ion lasers,” J. Appl. Phys. 43, 2254–2257 (1972).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Commun. (4)

S. De Silvestri, P. Laporta, V. Magni, “Misalignment sensitivity of solid-state laser resonators with thermal lensing,” Opt. Commun. 59, 43–48 (1986).
[CrossRef]

P. H. Sarkies, “A stable YAG resonator yielding a beam of very low divergence and high output energy,” Opt. Commun. 31, 189–192 (1979).
[CrossRef]

A. J. Berry, D. C. Hanna, C. G. Sawyers, “High power, single frequency operation of a Q-switched TEM00mode Nd:YAG laser,” Opt. Commun. 40, 54–58 (1981).
[CrossRef]

R. Iffländer, H. P. Kortz, H. Weber, “Beam divergence and refractive power of directly coated solid-state lasers,” Opt. Commun. 29, 223–226 (1979).
[CrossRef]

Opt. Lett. (2)

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, 493–507 (1981).
[CrossRef]

J. P. Lörtscher, J. Steffen, G. Herziger, “Dynamic stable resonators: a design procedure,” Opt. Quantum Electron. 7, 505–514 (1975).
[CrossRef]

Opto-Electronics (1)

P. Baues, “Huygens’ principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-Electronics 1, 37–44 (1969).
[CrossRef]

Proc. IEEE (1)

A. G. Fox, T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51, 80–89 (1963).
[CrossRef]

Other (3)

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975), pp. 106–108, 286–291.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chap. 4.

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

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Figures (5)

Fig. 1
Fig. 1

Misaligned optical system. (a) The matrix and the vector relate the position and the slope of the rays at the output plane to those at the input plane. (b) Interpretation of the elements of the misalignment vector for an input ray coincident with the reference axis.

Fig. 2
Fig. 2

Optical system closed at one side by a mirror. (a) Generic system: the transfer matrix and the misalignment vector are associated with the ray path from the entrance plane to the mirror and also include the possible mirror curvature and misalignment. (b) Equivalent system made by a misaligned spherical mirror.

Fig. 3
Fig. 3

(a) Linear resonator with curved mirrors (M1 and M2) and an OS. (b) Description of the resonator by plane mirrors (P1 and P2) and by a transfer matrix and a misalignment vector representing the ray path from P1 and P2.

Fig. 4
Fig. 4

Linear resonator with an internal lens of variable focal length f and other intracavity optical systems. The arrows indicate that the matrices and the vectors represent the paths from the lens to the mirrors. The dashed lines are reference planes.

Fig. 5
Fig. 5

Spot sizes and misalignment sensitivity of a linear resonator with an internal variable lens for uv > 0 and |u| < |v| as a function of η (dioptric power of the lens shifted by a constant amount). (a) Spot size on mirror 1. (b) Spot size on mirror 2. (c) Spot size on the lens. (d) Absolute value of the focal length of the optics between mirrors (including the mirrors’ power), which determines the misalignment sensitivity. The dashed vertical lines correspond to the dynamical stability.

Tables (3)

Tables Icon

Table 1 Ray-Transfer Matrices and Misalignment Vectors

Tables Icon

Table 2 Resonators at the Stability Limits

Tables Icon

Table 3 Stability Limits of a Resonator Containing a Variable Lens

Equations (28)

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

( x 0 θ 0 ) = [ A B C D ] ( x i θ i ) + ( s σ ) .
T = [ A B C D ]
m = ( s σ ) ,
[ D B C A ] ,             [ D B C A ] ( - s σ ) .
T r = [ 2 A D - 1 2 B D 2 A C 2 A D - 1 ]
m r = 2 σ ( B A ) .
A B C D < 0.
w 1 4 = - ( λ π ) 2 B D A C ,
w 2 4 = - ( λ π ) 2 A B C D ,
( x 1 θ 1 ) = [ 2 A D - 1 2 B D 2 A C 2 A D - 1 ] ( x 1 θ 1 ) + 2 σ ( B A ) .
( x 1 θ 1 ) = ( - σ / C 0 ) .
η = 1 f - 1 2 ( A 1 B 1 + C 1 D 1 + A 2 B 2 + C 2 D 2 ) ,
u = 1 2 B 1 D 1 - 1 2 B 2 D 2 ,
v = - 1 2 B 1 D 1 - 1 2 B 2 D 2 .
T = - [ D 1 B 2 ( η + u ) B 1 B 2 ( η + v ) D 1 D 2 ( η - v ) B 1 D 2 ( η - u ) ] .
Δ η = Δ 1 f = min ( u + v , u - v )
= min ( | 1 B 1 D 1 | , | 1 B 2 D 2 | ) .
w 3 4 = - ( 2 λ π ) 2 η 2 ( η 2 - u 2 ) ( η 2 - v 2 ) .
d w 3 4 d η 2 = 0 ,
η = ± ( u v ) 1 / 2 .
w 30 2 = 2 λ π 1 Δ 1 f ,
1 f = k π r 2 P in ,
Δ P in = 2 λ k ( r w 30 ) 2 .
w 1 4 = - ( λ B 1 π D 1 ) 2 ( η - u ) ( η + v ) ( η + u ) ( η - v ) ,
w 2 4 = - ( λ B 2 π D 2 ) ( η + u ) ( η + v ) ( η - u ) ( η - v ) .
w 10 2 = λ π | B 1 D 1 | .
w 20 2 = λ π 1 D 2 2 [ B 1 D 1 ( B 1 D 1 ) 2 - ( B 2 D 2 ) 2 ] ,
( x 3 θ 3 ) = - 1 C [ D 2 σ 1 + D 1 σ 2 - C 2 σ 1 + ( C 1 - D 1 / f ) σ 2 ] ,

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