Abstract

An analytical method based on a renormalized q parameter for Gaussian-beam propagation and properly matched self-consistent complex q parameters is proposed to aid in the design of arbitrary-astigmatism-compensated Kerr-lens mode-locked ring cavity lasers. The q parameters throughout the ring cavity can be calculated by solution of an algebraically quadratic equation for an arbitrarily thick Kerr medium when the intracavity laser power is less than the self-trapping power. The Kerr-lens mode-locking strength at the curved mirror was calculated over the stable range. The results indicate that the astigmatism is best compensated so that the stable range of x and y directions have optimal overlapping. Although to maximize the hard-aperturing effect the curved mirror separation must be at the far edge of the stable range, pump and cavity field matching inside the Kerr medium is also necessary. In addition, the insertion of a vertical slit is more effective than insertion of a horizontal slit. A spiking phenomenon is found in the intracavity z scan curve, which can be observed only with thick Kerr materials and a high laser power.

© 1995 Optical Society of America

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References

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  1. D. E. Spence, P. N. Kean, and W. Sibbett, “60-fsec pulse generation from a self-mode-locked Ti:sapphire laser,” Opt. Lett. 16, 42 (1991).
    [CrossRef] [PubMed]
  2. M. T. Asaki, C.-P. Huang, D. Garvey, J. Zhou, H. C. Kapteyn, and M. M. Murnane, “Generation of 11-fs pulses from a self-mode-locked Ti:sapphire laser,” Opt. Lett. 18, 977 (1993).
    [CrossRef] [PubMed]
  3. P. F. Curley, Ch. Spielmann, T. Brabec, F. Krausz, E. Wintner, and A. J. Schmidt, “Operation of a femtosecond Ti:sapphire solitary laser in the vicinity of zero-group-delay dispersion,” Opt. Lett. 18, 54 (1993).
    [CrossRef] [PubMed]
  4. W. S. Pelouch, P. E. Powers, and C. L. Tang, “Self-starting mode-locked ring-cavity Ti:sapphire laser,” Opt. Lett. 17, 1581 (1992).
    [CrossRef] [PubMed]
  5. G. W. Pearson, C. Radzewicz, and J. S. Krasinski, “Analysis of self-focusing mode-locked lasers with additional highly nonlinear self-focusing elements,” Opt. Commun. 94, 221 (1992).
    [CrossRef]
  6. D. Georgiev, J. Herrmann, and U. Stamm, “Cavity design for optimum nonlinear absorption in Kerr-lens mode-locked solid-state lasers,” Opt. Commun. 92, 368 (1992).
    [CrossRef]
  7. T. Brabec, Ch. Spielmann, P. E. Curley, and F. Krausz, “Kerr lens mode locking,” Opt. Lett. 17, 1292 (1992).
    [CrossRef] [PubMed]
  8. D. R. Heatley, A. M. Dunlop, and W. J. Firth, “Kerr lens effects in a ring resonator with an aperture: mode locking and unidirectional operation,” Opt. Lett. 18, 170 (1993).
    [CrossRef] [PubMed]
  9. V. Magni, G. Cerullo, and S. D. Silvestri, “ABCD matrix analysis of propagation of Gaussian beams through Kerr media,” Opt. Commun. 96, 348 (1993).
    [CrossRef]
  10. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28, 2086 (1992).
    [CrossRef]
  11. K.-H. Lin and W.-F. Hsieh, “An analytical design of symmetrical Kerr-lens mode-locking laser cavities,” J. Opt. Soc. Am. B 11, 737 (1994).
    [CrossRef]
  12. D. Huang, M. Ulman, L. H. Acioli, H. A. Haus, and J. G. Fujimoto, “Self-focusing-induced saturable loss for laser mode locking,” Opt. Lett. 17, 511 (1992).
    [CrossRef] [PubMed]
  13. H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, “Astigmatically compensated cavities for cw dye lasers,” IEEE J. Quantum Electron. QE-8, 373 (1972).
    [CrossRef]
  14. B. E. Lemoff and C. P. J. Barty, “Generation of high-peak-power 20-fs pulses from a regeneratively initiated, self-mode-locked Ti:sapphire laser,” Opt. Lett. 17, 1367 (1992).
    [CrossRef] [PubMed]
  15. A. Stingl, C. Spielmann, F. Krausz, and R. Szipöcs, “Generation of 11-fs pulses from a Ti:sapphire laser without the use of prisms,” Opt. Lett. 19, 204 (1994).
    [CrossRef] [PubMed]
  16. M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
    [CrossRef]
  17. J.-M. Shieh, F. Ganikhanov, K.-H. Lin, W.-F. Hsieh, and C.-L. Pan, “Completely self-starting picosecond and femtosecond Kerr-lens mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B (to be published).

1994 (2)

1993 (4)

1992 (7)

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28, 2086 (1992).
[CrossRef]

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, “Analysis of self-focusing mode-locked lasers with additional highly nonlinear self-focusing elements,” Opt. Commun. 94, 221 (1992).
[CrossRef]

D. Georgiev, J. Herrmann, and U. Stamm, “Cavity design for optimum nonlinear absorption in Kerr-lens mode-locked solid-state lasers,” Opt. Commun. 92, 368 (1992).
[CrossRef]

D. Huang, M. Ulman, L. H. Acioli, H. A. Haus, and J. G. Fujimoto, “Self-focusing-induced saturable loss for laser mode locking,” Opt. Lett. 17, 511 (1992).
[CrossRef] [PubMed]

T. Brabec, Ch. Spielmann, P. E. Curley, and F. Krausz, “Kerr lens mode locking,” Opt. Lett. 17, 1292 (1992).
[CrossRef] [PubMed]

B. E. Lemoff and C. P. J. Barty, “Generation of high-peak-power 20-fs pulses from a regeneratively initiated, self-mode-locked Ti:sapphire laser,” Opt. Lett. 17, 1367 (1992).
[CrossRef] [PubMed]

W. S. Pelouch, P. E. Powers, and C. L. Tang, “Self-starting mode-locked ring-cavity Ti:sapphire laser,” Opt. Lett. 17, 1581 (1992).
[CrossRef] [PubMed]

1991 (1)

1990 (1)

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

1972 (1)

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, “Astigmatically compensated cavities for cw dye lasers,” IEEE J. Quantum Electron. QE-8, 373 (1972).
[CrossRef]

Acioli, L. H.

Asaki, M. T.

Barty, C. P. J.

Brabec, T.

Cerullo, G.

V. Magni, G. Cerullo, and S. D. Silvestri, “ABCD matrix analysis of propagation of Gaussian beams through Kerr media,” Opt. Commun. 96, 348 (1993).
[CrossRef]

Curley, P. E.

Curley, P. F.

Dienes, A.

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, “Astigmatically compensated cavities for cw dye lasers,” IEEE J. Quantum Electron. QE-8, 373 (1972).
[CrossRef]

Dunlop, A. M.

Firth, W. J.

Fujimoto, J. G.

D. Huang, M. Ulman, L. H. Acioli, H. A. Haus, and J. G. Fujimoto, “Self-focusing-induced saturable loss for laser mode locking,” Opt. Lett. 17, 511 (1992).
[CrossRef] [PubMed]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28, 2086 (1992).
[CrossRef]

Ganikhanov, F.

J.-M. Shieh, F. Ganikhanov, K.-H. Lin, W.-F. Hsieh, and C.-L. Pan, “Completely self-starting picosecond and femtosecond Kerr-lens mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B (to be published).

Garvey, D.

Georgiev, D.

D. Georgiev, J. Herrmann, and U. Stamm, “Cavity design for optimum nonlinear absorption in Kerr-lens mode-locked solid-state lasers,” Opt. Commun. 92, 368 (1992).
[CrossRef]

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Haus, H. A.

D. Huang, M. Ulman, L. H. Acioli, H. A. Haus, and J. G. Fujimoto, “Self-focusing-induced saturable loss for laser mode locking,” Opt. Lett. 17, 511 (1992).
[CrossRef] [PubMed]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28, 2086 (1992).
[CrossRef]

Heatley, D. R.

Herrmann, J.

D. Georgiev, J. Herrmann, and U. Stamm, “Cavity design for optimum nonlinear absorption in Kerr-lens mode-locked solid-state lasers,” Opt. Commun. 92, 368 (1992).
[CrossRef]

Hsieh, W.-F.

K.-H. Lin and W.-F. Hsieh, “An analytical design of symmetrical Kerr-lens mode-locking laser cavities,” J. Opt. Soc. Am. B 11, 737 (1994).
[CrossRef]

J.-M. Shieh, F. Ganikhanov, K.-H. Lin, W.-F. Hsieh, and C.-L. Pan, “Completely self-starting picosecond and femtosecond Kerr-lens mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B (to be published).

Huang, C.-P.

Huang, D.

Ippen, E. P.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28, 2086 (1992).
[CrossRef]

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, “Astigmatically compensated cavities for cw dye lasers,” IEEE J. Quantum Electron. QE-8, 373 (1972).
[CrossRef]

Kapteyn, H. C.

Kean, P. N.

Kogelnik, H. W.

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, “Astigmatically compensated cavities for cw dye lasers,” IEEE J. Quantum Electron. QE-8, 373 (1972).
[CrossRef]

Krasinski, J. S.

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, “Analysis of self-focusing mode-locked lasers with additional highly nonlinear self-focusing elements,” Opt. Commun. 94, 221 (1992).
[CrossRef]

Krausz, F.

Lemoff, B. E.

Lin, K.-H.

K.-H. Lin and W.-F. Hsieh, “An analytical design of symmetrical Kerr-lens mode-locking laser cavities,” J. Opt. Soc. Am. B 11, 737 (1994).
[CrossRef]

J.-M. Shieh, F. Ganikhanov, K.-H. Lin, W.-F. Hsieh, and C.-L. Pan, “Completely self-starting picosecond and femtosecond Kerr-lens mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B (to be published).

Magni, V.

V. Magni, G. Cerullo, and S. D. Silvestri, “ABCD matrix analysis of propagation of Gaussian beams through Kerr media,” Opt. Commun. 96, 348 (1993).
[CrossRef]

Murnane, M. M.

Pan, C.-L.

J.-M. Shieh, F. Ganikhanov, K.-H. Lin, W.-F. Hsieh, and C.-L. Pan, “Completely self-starting picosecond and femtosecond Kerr-lens mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B (to be published).

Pearson, G. W.

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, “Analysis of self-focusing mode-locked lasers with additional highly nonlinear self-focusing elements,” Opt. Commun. 94, 221 (1992).
[CrossRef]

Pelouch, W. S.

Powers, P. E.

Radzewicz, C.

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, “Analysis of self-focusing mode-locked lasers with additional highly nonlinear self-focusing elements,” Opt. Commun. 94, 221 (1992).
[CrossRef]

Said, A. A.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Schmidt, A. J.

Shank, C. V.

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, “Astigmatically compensated cavities for cw dye lasers,” IEEE J. Quantum Electron. QE-8, 373 (1972).
[CrossRef]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Shieh, J.-M.

J.-M. Shieh, F. Ganikhanov, K.-H. Lin, W.-F. Hsieh, and C.-L. Pan, “Completely self-starting picosecond and femtosecond Kerr-lens mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B (to be published).

Sibbett, W.

Silvestri, S. D.

V. Magni, G. Cerullo, and S. D. Silvestri, “ABCD matrix analysis of propagation of Gaussian beams through Kerr media,” Opt. Commun. 96, 348 (1993).
[CrossRef]

Spence, D. E.

Spielmann, C.

Spielmann, Ch.

Stamm, U.

D. Georgiev, J. Herrmann, and U. Stamm, “Cavity design for optimum nonlinear absorption in Kerr-lens mode-locked solid-state lasers,” Opt. Commun. 92, 368 (1992).
[CrossRef]

Stingl, A.

Szipöcs, R.

Tang, C. L.

Ulman, M.

Van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Wei, T.-H.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Wintner, E.

Zhou, J.

IEEE J. Quantum Electron. (3)

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28, 2086 (1992).
[CrossRef]

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, “Astigmatically compensated cavities for cw dye lasers,” IEEE J. Quantum Electron. QE-8, 373 (1972).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (3)

V. Magni, G. Cerullo, and S. D. Silvestri, “ABCD matrix analysis of propagation of Gaussian beams through Kerr media,” Opt. Commun. 96, 348 (1993).
[CrossRef]

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, “Analysis of self-focusing mode-locked lasers with additional highly nonlinear self-focusing elements,” Opt. Commun. 94, 221 (1992).
[CrossRef]

D. Georgiev, J. Herrmann, and U. Stamm, “Cavity design for optimum nonlinear absorption in Kerr-lens mode-locked solid-state lasers,” Opt. Commun. 92, 368 (1992).
[CrossRef]

Opt. Lett. (9)

T. Brabec, Ch. Spielmann, P. E. Curley, and F. Krausz, “Kerr lens mode locking,” Opt. Lett. 17, 1292 (1992).
[CrossRef] [PubMed]

D. R. Heatley, A. M. Dunlop, and W. J. Firth, “Kerr lens effects in a ring resonator with an aperture: mode locking and unidirectional operation,” Opt. Lett. 18, 170 (1993).
[CrossRef] [PubMed]

D. E. Spence, P. N. Kean, and W. Sibbett, “60-fsec pulse generation from a self-mode-locked Ti:sapphire laser,” Opt. Lett. 16, 42 (1991).
[CrossRef] [PubMed]

M. T. Asaki, C.-P. Huang, D. Garvey, J. Zhou, H. C. Kapteyn, and M. M. Murnane, “Generation of 11-fs pulses from a self-mode-locked Ti:sapphire laser,” Opt. Lett. 18, 977 (1993).
[CrossRef] [PubMed]

P. F. Curley, Ch. Spielmann, T. Brabec, F. Krausz, E. Wintner, and A. J. Schmidt, “Operation of a femtosecond Ti:sapphire solitary laser in the vicinity of zero-group-delay dispersion,” Opt. Lett. 18, 54 (1993).
[CrossRef] [PubMed]

W. S. Pelouch, P. E. Powers, and C. L. Tang, “Self-starting mode-locked ring-cavity Ti:sapphire laser,” Opt. Lett. 17, 1581 (1992).
[CrossRef] [PubMed]

D. Huang, M. Ulman, L. H. Acioli, H. A. Haus, and J. G. Fujimoto, “Self-focusing-induced saturable loss for laser mode locking,” Opt. Lett. 17, 511 (1992).
[CrossRef] [PubMed]

B. E. Lemoff and C. P. J. Barty, “Generation of high-peak-power 20-fs pulses from a regeneratively initiated, self-mode-locked Ti:sapphire laser,” Opt. Lett. 17, 1367 (1992).
[CrossRef] [PubMed]

A. Stingl, C. Spielmann, F. Krausz, and R. Szipöcs, “Generation of 11-fs pulses from a Ti:sapphire laser without the use of prisms,” Opt. Lett. 19, 204 (1994).
[CrossRef] [PubMed]

Other (1)

J.-M. Shieh, F. Ganikhanov, K.-H. Lin, W.-F. Hsieh, and C.-L. Pan, “Completely self-starting picosecond and femtosecond Kerr-lens mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B (to be published).

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Figures (9)

Fig. 1
Fig. 1

Four-mirror figure-8 ring cavity laser with a Kerr medium of length t placed between two curved mirrors. M1 and M2 are curved mirrors with focal length f. M3 and M4 are flat mirrors. The coordinate conventions are chosen such that the z axis is along the propagation direction of the laser beam.

Fig. 2
Fig. 2

Astigmatic CW spot sizes 2wx and 2wy of the two transverse modes at the output coupler as functions of the distance between the curved mirrors. The cavity parameters are f = 5 cm, t = 2 cm, and d = 170 cm.

Fig. 3
Fig. 3

Astigmatism-compensated spot sizes 2wx and 2wy of the two transverse modes at the output coupler as functions of the distance between the curved mirrors with astigmatism-compensation angle θ = 15.8°. (a) CW operation; (b) KLM operation.

Fig. 4
Fig. 4

KLM strength at M1 versus the curved-mirror separation for the x and the y directions.

Fig. 5
Fig. 5

Calculated spot size throughout the laser cavity for (a) the x direction and (b) the y direction as functions of the distance from endface I under different cavity powers K = 0 and K = 0.5. The distance between the curved mirrors is 115 mm.

Fig. 6
Fig. 6

Calculated spot sizes inside the Ti:sapphire crystal for (a) the x direction and (b) the y direction as functions of the distance from endface I under different cavity powers K = 0 and K = 0.5. The curved-mirror separation is 115 mm. The x beam waist is located 1.8 and 1.6 mm from the center of Kerr medium toward M2 for K = 0 and K = 0.5 respectively, whereas the y beam waist is located 5.4 and 5.0 mm from the center of Kerr medium for K = 0 and K = 0.5.

Fig. 7
Fig. 7

Output spot sizes in the intracavity z-scan analysis for various intracavity laser powers. The optical distance between the curved mirrors is 114.6 mm. (a) 1-mm Ti:sapphire crystal; (b) 2-cm Ti:sapphire crystal.

Fig. 8
Fig. 8

Calculated spot size throughout the laser cavity for (a) r = 43.3 mm and (b) r = 51.3 mm as functions of the distance from endface I under various cavity power conditions. The Ti:sapphire rod is 2 cm long.

Fig. 9
Fig. 9

Calculated spot sizes inside the Ti:sapphire crystal for (a) r = 43.3 mm and (b) r = 51.3 mm as functions of distance from endface I under various cavity power conditions.

Equations (27)

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U = A 0 exp ( j φ ) exp ( j 2 2 π n 0 λ r 2 q ) .
1 q = 1 R j λ π n 0 w 2 .
Δ Φ = 2 π λ n 2 A 0 2 exp ( 2 r 2 w 2 ) d z .
2 π λ n 2 A 0 2 ( 1 2 r 2 w 2 ) d z ,
d d z ( 1 q ) = 1 q 2 + K Im 2 ( 1 q ) ,
K = 8 P π ( π λ ) 2 n 0 n 2 .
1 q = Re ( 1 q ) + j Im ( 1 q ) 1 K ,
d d z ( 1 q ) = 1 q 2 .
[ A B C D ] ;
1 q IIM = [ y I 2 ( A R I + B ) ( C R I + D ) + B D R I 2 ] j y I R I 2 y I 2 ( A R I + B ) 2 + B 2 R I 2 .
1 q IIL = [ y I 2 R I L y I 2 L R I 2 ( 1 K ) ] j y I R I 2 y I 2 R I 2 2 L y I 2 R I + L 2 y I 2 + L R I 2 ( 1 K ) ,
[ L ( 1 K ) + B D ] [ ( A R I + B ) 2 ( R I L ) 2 ] + [ L 2 ( 1 K ) B 2 ] × [ ( A R I + B ) ( C R I + D ) R I + L ] = 0 ,
y I 2 = R I 2 [ L 2 ( 1 K ) B 2 ] ( A R I + B ) 2 ( R I L ) 2 .
F = 1 w d w d P P = 0 .
A = f 2 2 s f d f + s d f 2 ,
B = r f 2 + s f 2 + d f 2 rdf sdf + rsd 2 rsf f 2 ,
C = d 2 f f 2 ,
D = f 2 2 r f d f + r d f 2 ,
L x = L ,
L y = L / n 0 2 .
f x = f / cos θ ,
f y = f cos θ .
1 q II = [ y I 2 R I L y I 2 L R I 2 ( 1 K ) ] j y I R I 2 y I 2 R I 2 2 L y I 2 R I + L 2 y I 2 + L R I 2 ( 1 K ) .
1 q II = [ y I 2 R I L y I 2 L R I 2 ( 1 K ) ] j y I R I 2 1 K y I 2 R I 2 2 L y I 2 R I + L 2 y I 2 + L R I 2 ( 1 K ) .
[ 1 L 0 1 ] ,
q I = 1 / q II 1 + L ( 1 / q II ) = [ y I 2 R I L y I 2 L R I 2 ( 1 K ) ] j y I R I 2 1 K y I 2 R I 2 2 L y I 2 R I + L 2 y I 2 + L R I 2 ( 1 K ) 2 . 1 + L [ y I 2 R I L y I 2 L R I 2 ( 1 K ) ] j y I R I 2 1 K y I 2 R I 2 2 L y I 2 R I + L 2 y I 2 + L R I 2 ( 1 K ) .
q I = 1 R I [ 1 2 L R I + L 2 R I 2 + L 2 y I 2 ( 1 K ) ] j 1 K y I [ 1 2 L R I + L 2 R I 2 + L 2 y I 2 ( 1 K ) ] 1 2 L R I + L 2 R I 2 + L 2 y I 2 ( 1 K ) = 1 R I j 1 K y I .

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