Abstract

We present a novel method for maximizing signal gain in continuously pumped silicon-waveguide Raman amplifiers made with silicon-on-insulator technology. Our method allows for pump-power depletion during Raman amplification and makes use of a variational technique. Its use leads to a system of four coupled nonlinear differential equations, whose numerical solution provides the optimal axial profile of the effective mode area along the waveguide length that maximizes the output signal power for a given amplifier length and a preset input (or output) cross-section area. In practice, the optimum profile can be realized by varying the cross-section area of a silicon waveguide along its length by tapering its width appropriately.

© 2009 Optical Society of America

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References

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  1. R. Claps, D. Dimitropoulos, and B. Jalali, "Stimulated Raman scattering in silicon waveguides," Electron. Lett. 38, 1352 (2002).
    [CrossRef]
  2. R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, "Observation of stimulated Raman amplification in silicon waveguides," Opt. Express 11, 1731 (2003).
    [CrossRef] [PubMed]
  3. B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, "Raman-based silicon photonics," IEEE J. Sel. Top. Quantum Electron. 12, 412 (2006).
    [CrossRef]
  4. C. Headley and G. P. Agrawal, Eds., Raman Amplification in Fiber-Optic Communication Systems, (Academic Press, San Diego, 2005).
  5. G. P. Agrawal, Nonlinear Fiber Optics, (Academic, Boston, 2007).
  6. Q. Lin, O. J. Painter, and G. P. Agrawal, "Nonlinear optical phenomena in silicon waveguides: Modeling and applications," Opt. Express 15, 16604 (2007).
    [CrossRef] [PubMed]
  7. X. Chen, N. C. Panoiu, and R. M. Osgood, "Theory of Raman-mediated pulsed amplification in silicon-wire waveguides," IEEE J. Quantum Electron. 42, 160 (2006).
    [CrossRef]
  8. M. Krause, H. Renner, and E. Brinkmeyer, "Analysis of Raman lasing characteristics in silicon-on-insulator waveguides," Opt. Express 12, 5703-5710 (2004).
    [CrossRef] [PubMed]
  9. H. Renner, M. Krause, and E. Brinkmeyer, "Maximal gain and optimal taper design for Raman amplifiers in silicon-on-insulator waveguides," in Integrated Photonics Research and Applications Topical Meetings (IPRA 2005), paper JWA3.
  10. M. Krause, H. Renner, and E. Brinkmeyer, "Efficiency increase of silicon-on-insulator Raman lasers by reduction of free-carrier absorption in tapered waveguides," in Conference on Lasers and Electro-Optics (CLEO 2005), paper CThB1.
  11. M. Krause, H. Renner, and E. Brinkmeyer, "Efficient Raman lasing in tapered silicon waveguides," Spectroscopy 21, 26-32 (2006).
  12. D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, "Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides," Appl. Phys. Lett. 86, 071115 (2005).
    [CrossRef]
  13. S. Fathpour, K. K. Tsia, and B. Jalali, "Energy harvesting in silicon Raman amplifiers," Appl. Phys. Lett. 89, 061109 (2006).
    [CrossRef]
  14. A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, "Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering," Opt. Express 12, 4261 (2004).
    [CrossRef] [PubMed]
  15. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, "Influence of nonlinear absorption on Raman amplification in Silicon waveguides," Opt. Express 12, 2774 (2004).
    [CrossRef] [PubMed]
  16. T. K. Liang and H. K. Tsang, "Role of free carriers from two-photon absorption in Raman amplification in silicon-on-insulator waveguides," Appl. Phys. Lett. 84, 2745 (2004).
    [CrossRef]
  17. T. K. Liang and H. K. Tsang, "Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides," IEEE J. Sel. Top. Quantum Electron. 10, 1149 (2004).
    [CrossRef]
  18. I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, "Nonlinear pulse evolution in silicon waveguides: An approximate analytic approach," J. Lightwave Technol. 28, in press (2009).
  19. I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, "Continuous-wave Raman amplification in silicon waveguides: beyond the undepleted pump approximation," Opt. Lett. 34, 536 (2009).
    [CrossRef] [PubMed]
  20. S. Roy, S. K. Bhadra, and G. P. Agrawal, "Raman amplification of optical pulses in silicon waveguides: effects of finite gain bandwidth, pulse width, and chirp," J. Opt. Soc. Am. B 26, 17 (2009).
    [CrossRef]
  21. B. Jalali, O. Boyraz, V. Raghunathan, D. Dimitropoulos, and P. Koonath, "Silicon Raman amplifiers, lasers and their applications," Proc. SPIE 6014, 21-26 (2005).
  22. C. Fox, An Introduction to the Calculus of Variations (Dover, New York, 1987).

2009 (3)

2007 (1)

2006 (4)

X. Chen, N. C. Panoiu, and R. M. Osgood, "Theory of Raman-mediated pulsed amplification in silicon-wire waveguides," IEEE J. Quantum Electron. 42, 160 (2006).
[CrossRef]

S. Fathpour, K. K. Tsia, and B. Jalali, "Energy harvesting in silicon Raman amplifiers," Appl. Phys. Lett. 89, 061109 (2006).
[CrossRef]

M. Krause, H. Renner, and E. Brinkmeyer, "Efficient Raman lasing in tapered silicon waveguides," Spectroscopy 21, 26-32 (2006).

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, "Raman-based silicon photonics," IEEE J. Sel. Top. Quantum Electron. 12, 412 (2006).
[CrossRef]

2005 (2)

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, "Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides," Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

B. Jalali, O. Boyraz, V. Raghunathan, D. Dimitropoulos, and P. Koonath, "Silicon Raman amplifiers, lasers and their applications," Proc. SPIE 6014, 21-26 (2005).

2004 (5)

2003 (1)

2002 (1)

R. Claps, D. Dimitropoulos, and B. Jalali, "Stimulated Raman scattering in silicon waveguides," Electron. Lett. 38, 1352 (2002).
[CrossRef]

Agrawal, G. P.

Bhadra, S. K.

Boyraz, O.

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, "Raman-based silicon photonics," IEEE J. Sel. Top. Quantum Electron. 12, 412 (2006).
[CrossRef]

B. Jalali, O. Boyraz, V. Raghunathan, D. Dimitropoulos, and P. Koonath, "Silicon Raman amplifiers, lasers and their applications," Proc. SPIE 6014, 21-26 (2005).

Brinkmeyer, E.

M. Krause, H. Renner, and E. Brinkmeyer, "Efficient Raman lasing in tapered silicon waveguides," Spectroscopy 21, 26-32 (2006).

M. Krause, H. Renner, and E. Brinkmeyer, "Analysis of Raman lasing characteristics in silicon-on-insulator waveguides," Opt. Express 12, 5703-5710 (2004).
[CrossRef] [PubMed]

Chen, X.

X. Chen, N. C. Panoiu, and R. M. Osgood, "Theory of Raman-mediated pulsed amplification in silicon-wire waveguides," IEEE J. Quantum Electron. 42, 160 (2006).
[CrossRef]

Claps, R.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, "Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides," Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, "Influence of nonlinear absorption on Raman amplification in Silicon waveguides," Opt. Express 12, 2774 (2004).
[CrossRef] [PubMed]

R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, "Observation of stimulated Raman amplification in silicon waveguides," Opt. Express 11, 1731 (2003).
[CrossRef] [PubMed]

R. Claps, D. Dimitropoulos, and B. Jalali, "Stimulated Raman scattering in silicon waveguides," Electron. Lett. 38, 1352 (2002).
[CrossRef]

Cohen, O.

Dimitropoulos, D.

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, "Raman-based silicon photonics," IEEE J. Sel. Top. Quantum Electron. 12, 412 (2006).
[CrossRef]

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, "Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides," Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

B. Jalali, O. Boyraz, V. Raghunathan, D. Dimitropoulos, and P. Koonath, "Silicon Raman amplifiers, lasers and their applications," Proc. SPIE 6014, 21-26 (2005).

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, "Influence of nonlinear absorption on Raman amplification in Silicon waveguides," Opt. Express 12, 2774 (2004).
[CrossRef] [PubMed]

R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, "Observation of stimulated Raman amplification in silicon waveguides," Opt. Express 11, 1731 (2003).
[CrossRef] [PubMed]

R. Claps, D. Dimitropoulos, and B. Jalali, "Stimulated Raman scattering in silicon waveguides," Electron. Lett. 38, 1352 (2002).
[CrossRef]

Dissanayake, C.

I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, "Continuous-wave Raman amplification in silicon waveguides: beyond the undepleted pump approximation," Opt. Lett. 34, 536 (2009).
[CrossRef] [PubMed]

I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, "Nonlinear pulse evolution in silicon waveguides: An approximate analytic approach," J. Lightwave Technol. 28, in press (2009).

Fathpour, S.

S. Fathpour, K. K. Tsia, and B. Jalali, "Energy harvesting in silicon Raman amplifiers," Appl. Phys. Lett. 89, 061109 (2006).
[CrossRef]

Hak, D.

Han, Y.

Jalali, B.

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, "Raman-based silicon photonics," IEEE J. Sel. Top. Quantum Electron. 12, 412 (2006).
[CrossRef]

S. Fathpour, K. K. Tsia, and B. Jalali, "Energy harvesting in silicon Raman amplifiers," Appl. Phys. Lett. 89, 061109 (2006).
[CrossRef]

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, "Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides," Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

B. Jalali, O. Boyraz, V. Raghunathan, D. Dimitropoulos, and P. Koonath, "Silicon Raman amplifiers, lasers and their applications," Proc. SPIE 6014, 21-26 (2005).

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, "Influence of nonlinear absorption on Raman amplification in Silicon waveguides," Opt. Express 12, 2774 (2004).
[CrossRef] [PubMed]

R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, "Observation of stimulated Raman amplification in silicon waveguides," Opt. Express 11, 1731 (2003).
[CrossRef] [PubMed]

R. Claps, D. Dimitropoulos, and B. Jalali, "Stimulated Raman scattering in silicon waveguides," Electron. Lett. 38, 1352 (2002).
[CrossRef]

Jhaveri, R.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, "Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides," Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

Koonath, P.

B. Jalali, O. Boyraz, V. Raghunathan, D. Dimitropoulos, and P. Koonath, "Silicon Raman amplifiers, lasers and their applications," Proc. SPIE 6014, 21-26 (2005).

Krause, M.

M. Krause, H. Renner, and E. Brinkmeyer, "Efficient Raman lasing in tapered silicon waveguides," Spectroscopy 21, 26-32 (2006).

M. Krause, H. Renner, and E. Brinkmeyer, "Analysis of Raman lasing characteristics in silicon-on-insulator waveguides," Opt. Express 12, 5703-5710 (2004).
[CrossRef] [PubMed]

Liang, T. K.

T. K. Liang and H. K. Tsang, "Role of free carriers from two-photon absorption in Raman amplification in silicon-on-insulator waveguides," Appl. Phys. Lett. 84, 2745 (2004).
[CrossRef]

T. K. Liang and H. K. Tsang, "Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides," IEEE J. Sel. Top. Quantum Electron. 10, 1149 (2004).
[CrossRef]

Lin, Q.

Liu, A.

Osgood, R. M.

X. Chen, N. C. Panoiu, and R. M. Osgood, "Theory of Raman-mediated pulsed amplification in silicon-wire waveguides," IEEE J. Quantum Electron. 42, 160 (2006).
[CrossRef]

Painter, O. J.

Paniccia, M.

Panoiu, N. C.

X. Chen, N. C. Panoiu, and R. M. Osgood, "Theory of Raman-mediated pulsed amplification in silicon-wire waveguides," IEEE J. Quantum Electron. 42, 160 (2006).
[CrossRef]

Premaratne, M.

I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, "Nonlinear pulse evolution in silicon waveguides: An approximate analytic approach," J. Lightwave Technol. 28, in press (2009).

I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, "Continuous-wave Raman amplification in silicon waveguides: beyond the undepleted pump approximation," Opt. Lett. 34, 536 (2009).
[CrossRef] [PubMed]

Raghunathan, V.

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, "Raman-based silicon photonics," IEEE J. Sel. Top. Quantum Electron. 12, 412 (2006).
[CrossRef]

B. Jalali, O. Boyraz, V. Raghunathan, D. Dimitropoulos, and P. Koonath, "Silicon Raman amplifiers, lasers and their applications," Proc. SPIE 6014, 21-26 (2005).

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, "Influence of nonlinear absorption on Raman amplification in Silicon waveguides," Opt. Express 12, 2774 (2004).
[CrossRef] [PubMed]

R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, "Observation of stimulated Raman amplification in silicon waveguides," Opt. Express 11, 1731 (2003).
[CrossRef] [PubMed]

Renner, H.

M. Krause, H. Renner, and E. Brinkmeyer, "Efficient Raman lasing in tapered silicon waveguides," Spectroscopy 21, 26-32 (2006).

M. Krause, H. Renner, and E. Brinkmeyer, "Analysis of Raman lasing characteristics in silicon-on-insulator waveguides," Opt. Express 12, 5703-5710 (2004).
[CrossRef] [PubMed]

Rong, H.

Roy, S.

Rukhlenko, I. D.

I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, "Continuous-wave Raman amplification in silicon waveguides: beyond the undepleted pump approximation," Opt. Lett. 34, 536 (2009).
[CrossRef] [PubMed]

I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, "Nonlinear pulse evolution in silicon waveguides: An approximate analytic approach," J. Lightwave Technol. 28, in press (2009).

Tsang, H. K.

T. K. Liang and H. K. Tsang, "Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides," IEEE J. Sel. Top. Quantum Electron. 10, 1149 (2004).
[CrossRef]

T. K. Liang and H. K. Tsang, "Role of free carriers from two-photon absorption in Raman amplification in silicon-on-insulator waveguides," Appl. Phys. Lett. 84, 2745 (2004).
[CrossRef]

Tsia, K. K.

S. Fathpour, K. K. Tsia, and B. Jalali, "Energy harvesting in silicon Raman amplifiers," Appl. Phys. Lett. 89, 061109 (2006).
[CrossRef]

Woo, J. C. S.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, "Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides," Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

J. Lightwave Technol. (1)

I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, "Nonlinear pulse evolution in silicon waveguides: An approximate analytic approach," J. Lightwave Technol. 28, in press (2009).

Appl. Phys. Lett. (3)

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, "Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides," Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

S. Fathpour, K. K. Tsia, and B. Jalali, "Energy harvesting in silicon Raman amplifiers," Appl. Phys. Lett. 89, 061109 (2006).
[CrossRef]

T. K. Liang and H. K. Tsang, "Role of free carriers from two-photon absorption in Raman amplification in silicon-on-insulator waveguides," Appl. Phys. Lett. 84, 2745 (2004).
[CrossRef]

Electron. Lett. (1)

R. Claps, D. Dimitropoulos, and B. Jalali, "Stimulated Raman scattering in silicon waveguides," Electron. Lett. 38, 1352 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

X. Chen, N. C. Panoiu, and R. M. Osgood, "Theory of Raman-mediated pulsed amplification in silicon-wire waveguides," IEEE J. Quantum Electron. 42, 160 (2006).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, "Raman-based silicon photonics," IEEE J. Sel. Top. Quantum Electron. 12, 412 (2006).
[CrossRef]

T. K. Liang and H. K. Tsang, "Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides," IEEE J. Sel. Top. Quantum Electron. 10, 1149 (2004).
[CrossRef]

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

Opt. Express (5)

Opt. Lett. (1)

Proc. SPIE (1)

B. Jalali, O. Boyraz, V. Raghunathan, D. Dimitropoulos, and P. Koonath, "Silicon Raman amplifiers, lasers and their applications," Proc. SPIE 6014, 21-26 (2005).

Spectroscopy (1)

M. Krause, H. Renner, and E. Brinkmeyer, "Efficient Raman lasing in tapered silicon waveguides," Spectroscopy 21, 26-32 (2006).

Other (5)

H. Renner, M. Krause, and E. Brinkmeyer, "Maximal gain and optimal taper design for Raman amplifiers in silicon-on-insulator waveguides," in Integrated Photonics Research and Applications Topical Meetings (IPRA 2005), paper JWA3.

M. Krause, H. Renner, and E. Brinkmeyer, "Efficiency increase of silicon-on-insulator Raman lasers by reduction of free-carrier absorption in tapered waveguides," in Conference on Lasers and Electro-Optics (CLEO 2005), paper CThB1.

C. Headley and G. P. Agrawal, Eds., Raman Amplification in Fiber-Optic Communication Systems, (Academic Press, San Diego, 2005).

G. P. Agrawal, Nonlinear Fiber Optics, (Academic, Boston, 2007).

C. Fox, An Introduction to the Calculus of Variations (Dover, New York, 1987).

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Figures (3)

Fig. 1.
Fig. 1.

Left panel: Evolution of the pump and signal powers along SOI waveguides of three different length (L = 8, 10, 12 mm) with optimized EMA profiles. Right panel: Optimized EMA axial profiles for five waveguide lengths ranging from 8 to 12 mm. In all cases, P p0 = 1 W, P s0 = 0.01 W, and A 0 = 1μm2. See the text for other device parameters.

Fig. 2.
Fig. 2.

Left panel: Solution of Eqs. (1) and (8) for a 3-mm-long waveguide with input values P p0 = 1 W, P s0 = 0.5 W, and A 0 = 1 μm2. Solid curves show the pump and signal powers; dashed and dotted curves represent the two auxiliary functions. Right panel: Optimized EMA profile (solid curve, right scale) and the pump and signal intensities (dashed curves, left scale) corresponding to the pump and signal powers shown on the left panel.

Fig. 3.
Fig. 3.

Output signal power as a function of output EMA for linearly tapered waveguides of four different lengths. The incident powers correspond to those used in Figs. 1 and 2, and A 0 = 1 μm2. The central dotted line represents a waveguide with constant EMA. The crosses show signal power and modal area at the end of the optimized waveguides in Figs. 1 and 2.

Equations (24)

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d P p d z = α p P p β p P p 2 A eff ζ ps P p P s A eff σ p τ c ( ρ p P p 2 + ρ s P s 2 + ρ ps P p P s ) P p A eff 2 ,
d P s d z = α s P s β s P s 2 A eff ζ sp P s P p A eff σ s τ c ( ρ p P p 2 + ρ s P s 2 + ρ ps P p P s ) P s A eff 2 ,
ζ ps = 2 β ps + 4 g R γ R 2 Ω R Ω ps ( Ω R 2 Ω ps 2 ) 2 + 4 γ R 2 Ω ps 2 ,
ζ sp = ω s ω p ( 2 β ps 4 g R γ R 2 Ω R Ω ps ( Ω R 2 Ω ps 2 ) 2 + 4 γ R 2 Ω ps 2 ) .
G ( A eff ) = 0 L 1 P s d P s d z d z = 0 L g ( z , A eff ) d z ,
g ( z , A eff ) = α s β s P s ( z ) A eff ( z ) ζ ps P p ( z ) A eff ( z ) σ s τ c ρ p P p 2 ( z ) + ρ s P s 2 ( z ) + ρ ps P p ( z ) P s ( z ) A eff 2 ( z ) .
0 L ( 𝒜 δ P p + 𝓑 δ P s ) d z = 0 L 𝒞 δ A eff d z ,
𝒜 = 1 A eff ( ζ sp + σ s τ c 2 ρ p P p + ρ ps P s A eff ) ,
𝓑 = 1 A eff ( β s + σ s τ c 2 ρ s P s + ρ ps P p A eff ) ,
𝒞 = β s P s + ζ sp P p A eff 2 + 2 σ s τ c ρ p P p 2 + ρ s P s 2 + ρ ps P p P s A eff 3 .
d ( δ P p d z ) = a p δ P p b s δ P s c p δ A eff ,
d ( δ P s d z ) = a s δ P s b p δ P p c s δ A eff ,
a p = α p + 2 β p P p A eff + ζ ps P s A eff + σ p τ c 3 ρ p P p 2 + ρ s P s 2 + 2 ρ ps P p P s A eff 2 ,
b p = 𝒜 P s , c p = 𝒞 P p .
0 L φ d ( δ P p ) d z d z = φδ P p | 0 L 0 L φ δ P p d z = 0 L φ ( a p δ P p + b s δ P s c p δ A eff ) d z ,
0 L ψ d ( δ P s ) d z d z = ψδ P s | 0 L 0 L ψ δ P s d z = 0 L ψ ( a s δ P s + b p δ P p c s δ A eff ) d z ,
φ ( L ) = 0 .
0 L [ ( φ a p φ ) δ P p + φ b s δ P s ] d z = 0 L φ c p δ A eff d z ,
0 L [ ψ b p δ P p + ( ψ a s ψ ) δ P s ] d z = 0 L ψ c p δ A eff d z .
0 L [ ( φ a p + ψ b p φ ) δ P p + ( ψ a s + φ b s ψ ) δ P s ] d z = 0 L ( φ c p + ψ c s ) δ A eff d z .
φ c p + ψ c s = 𝒞 ,
φ = a p φ + b p ψ 𝒜 ,
ψ = a s ψ + b s ψ 𝓑 .
A eff ( z ) = 2 τ c [ σ s ( 1 ψ P s ) σ p φ P p ] ( ρ p P p 2 + ρ s P s 2 + ρ ps P p P s ) φ P p ( β p P p + ζ ps P s ) ( 1 ψ P s ) ( β s P s + ζ sp P p ) .

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