Abstract

Optical spectra broadening as a result self-phase modulation in a channel waveguide fabricated on a high quality tantalum pentoxide (Ta2O5) film by using RF sputtering is measured. The full-width at half maximum of the optical spectra for transverse electric (TE)/transverse magnetic (TM) polarizations of 42.5/31.7 nm is obtained using pulses of 10 nm at a wavelength of 800 nm with a peak-coupled power of 43.77 W. The nonlinear Kerr coefficients of 2.14 × 10−14 cm2/W and 1.92 × 10−14 cm2/W for TE and TM polarizations, respectively, are then extracted from the experiments using a theoretical model based on the method of moments. The obtained results on the nonlinearity further suggest that Ta2O5 is a promising material to develop nonlinear waveguide devices for integrated photonics.

© 2016 Optical Society of America

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References

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2015 (3)

2014 (2)

2013 (2)

N. Vukovic, N. Healy, F. H. Suhailin, P. Mehta, T. D. Day, J. V. Badding, and A. C. Peacock, “Ultrafast optical control using the Kerr nonlinearity in hydrogenated amorphous silicon microcylindrical resonators,” Sci. Rep. 3, 2885 (2013).
[Crossref] [PubMed]

H. Jung, C. Xiong, K. Y. Fong, X. Zhang, and H. X. Tang, “Optical frequency comb generation from aluminum nitride microring resonator,” Opt. Lett. 38(15), 2810–2813 (2013).
[Crossref] [PubMed]

2012 (1)

2011 (1)

2010 (3)

J. Y. Lee, L. Yin, G. P. Agrawal, and P. M. Fauchet, “Ultrafast optical switching based on nonlinear polarization rotation in silicon waveguides,” Opt. Express 18(11), 11514–11523 (2010).
[Crossref] [PubMed]

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

E. Y. M. Teraoka, D. H. Broaddus, T. Kita, A. Tsukazaki, M. Kawasaki, A. L. Gaeta, and H. Yamada, “Self-phase modulation at visible wavelengths in nonlinear ZnO channel waveguides,” Appl. Phys. Lett. 97(7), 071105 (2010).
[Crossref]

2008 (1)

2007 (2)

Q. Xu and M. Lipson, “All-optical logic based on silicon micro-ring resonators,” Opt. Express 15(3), 924–929 (2007).
[Crossref] [PubMed]

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007).
[Crossref]

2005 (1)

2004 (3)

1998 (1)

J. Linnros, “Carrier lifetime measurements using free carrier absorption transients. II. Lifetime mapping and effects of surface recombination,” J. Appl. Phys. 84(1), 284–291 (1998).
[Crossref]

1988 (1)

J. Waldmeyer, “A contactless method for determination of carrier lifetime, surface recombination velocity, and diffusion constant in semiconductors,” J. Appl. Phys. 63(6), 1977–1983 (1988).
[Crossref]

1987 (1)

P. N. Kean, K. Smith, and W. Sibbett, “Spectral and temporal investigation of self-phase modulation and stimulated Raman scattering in a single-mode optical fibre,” IEE Proc. 134(3), 163–170 (1987).

Agrawal, G. P.

Alic, N.

Badding, J. V.

N. Vukovic, N. Healy, F. H. Suhailin, P. Mehta, T. D. Day, J. V. Badding, and A. C. Peacock, “Ultrafast optical control using the Kerr nonlinearity in hydrogenated amorphous silicon microcylindrical resonators,” Sci. Rep. 3, 2885 (2013).
[Crossref] [PubMed]

Baets, R.

Baumberg, J.

Beausoleil, R. G.

Boyraz, O.

Bristow, A. D.

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007).
[Crossref]

Broaddus, D. H.

E. Y. M. Teraoka, D. H. Broaddus, T. Kita, A. Tsukazaki, M. Kawasaki, A. L. Gaeta, and H. Yamada, “Self-phase modulation at visible wavelengths in nonlinear ZnO channel waveguides,” Appl. Phys. Lett. 97(7), 071105 (2010).
[Crossref]

Cattaneo, F.

Charlton, M. D. B.

Chen, B.-T.

Chen, C.-L.

Chen, R. Y.

Chi, Y.-C.

C.-L. Wu, Y.-H. Lin, S.-P. Su, B.-J. Huang, C.-T. Tsai, H.-Y. Wang, Y.-C. Chi, C.-I. Wu, and G.-R. Lin, “Enhancing optical nonlinearity in a nonstoichiometric SiN waveguide for cross-wavelength all-optical data processing,” ACS Photonics 2(8), 1141–1154 (2015).
[Crossref]

Chiu, Y.-J.

Chu, A.-K.

Day, T. D.

N. Vukovic, N. Healy, F. H. Suhailin, P. Mehta, T. D. Day, J. V. Badding, and A. C. Peacock, “Ultrafast optical control using the Kerr nonlinearity in hydrogenated amorphous silicon microcylindrical resonators,” Sci. Rep. 3, 2885 (2013).
[Crossref] [PubMed]

Emplit, P.

Fainman, Y.

Fattal, D. A.

Fauchet, P. M.

Finlayson, C.

Fong, K. Y.

Foster, M. A.

R. Halir, Y. Okawachi, J. S. Levy, M. A. Foster, M. Lipson, and A. L. Gaeta, “Ultrabroadband supercontinuum generation in a CMOS-compatible platform,” Opt. Lett. 37(10), 1685–1687 (2012).
[Crossref] [PubMed]

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

Fukuda, H.

Gaeta, A. L.

R. Halir, Y. Okawachi, J. S. Levy, M. A. Foster, M. Lipson, and A. L. Gaeta, “Ultrabroadband supercontinuum generation in a CMOS-compatible platform,” Opt. Lett. 37(10), 1685–1687 (2012).
[Crossref] [PubMed]

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

E. Y. M. Teraoka, D. H. Broaddus, T. Kita, A. Tsukazaki, M. Kawasaki, A. L. Gaeta, and H. Yamada, “Self-phase modulation at visible wavelengths in nonlinear ZnO channel waveguides,” Appl. Phys. Lett. 97(7), 071105 (2010).
[Crossref]

Gondarenko, A.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

Gorza, S.-P.

Halir, R.

Healy, N.

N. Vukovic, N. Healy, F. H. Suhailin, P. Mehta, T. D. Day, J. V. Badding, and A. C. Peacock, “Ultrafast optical control using the Kerr nonlinearity in hydrogenated amorphous silicon microcylindrical resonators,” Sci. Rep. 3, 2885 (2013).
[Crossref] [PubMed]

Huang, B.-J.

C.-L. Wu, Y.-H. Lin, S.-P. Su, B.-J. Huang, C.-T. Tsai, H.-Y. Wang, Y.-C. Chi, C.-I. Wu, and G.-R. Lin, “Enhancing optical nonlinearity in a nonstoichiometric SiN waveguide for cross-wavelength all-optical data processing,” ACS Photonics 2(8), 1141–1154 (2015).
[Crossref]

Hung, Y.-J.

Ikeda, K.

Indukuri, T.

Itabashi, S.

Jalali, B.

Jung, H.

Kawasaki, M.

E. Y. M. Teraoka, D. H. Broaddus, T. Kita, A. Tsukazaki, M. Kawasaki, A. L. Gaeta, and H. Yamada, “Self-phase modulation at visible wavelengths in nonlinear ZnO channel waveguides,” Appl. Phys. Lett. 97(7), 071105 (2010).
[Crossref]

Kean, P. N.

P. N. Kean, K. Smith, and W. Sibbett, “Spectral and temporal investigation of self-phase modulation and stimulated Raman scattering in a single-mode optical fibre,” IEE Proc. 134(3), 163–170 (1987).

Kita, T.

E. Y. M. Teraoka, D. H. Broaddus, T. Kita, A. Tsukazaki, M. Kawasaki, A. L. Gaeta, and H. Yamada, “Self-phase modulation at visible wavelengths in nonlinear ZnO channel waveguides,” Appl. Phys. Lett. 97(7), 071105 (2010).
[Crossref]

Koonath, P.

Kuyken, B.

Lagoudakis, P. G.

Lee, C.-K.

Lee, J. Y.

Leo, F.

Levy, J. S.

R. Halir, Y. Okawachi, J. S. Levy, M. A. Foster, M. Lipson, and A. L. Gaeta, “Ultrabroadband supercontinuum generation in a CMOS-compatible platform,” Opt. Lett. 37(10), 1685–1687 (2012).
[Crossref] [PubMed]

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

Lin, G.-R.

C.-L. Wu, Y.-H. Lin, S.-P. Su, B.-J. Huang, C.-T. Tsai, H.-Y. Wang, Y.-C. Chi, C.-I. Wu, and G.-R. Lin, “Enhancing optical nonlinearity in a nonstoichiometric SiN waveguide for cross-wavelength all-optical data processing,” ACS Photonics 2(8), 1141–1154 (2015).
[Crossref]

C.-L. Wu, B.-T. Chen, Y.-Y. Lin, W.-C. Tien, G.-R. Lin, Y.-J. Chiu, Y.-J. Hung, A.-K. Chu, and C.-K. Lee, “Low-loss and high-Q Ta2O5 based micro-ring resonator with inverse taper structure,” Opt. Express 23(20), 26268–26275 (2015).
[Crossref] [PubMed]

Lin, Y.-H.

C.-L. Wu, Y.-H. Lin, S.-P. Su, B.-J. Huang, C.-T. Tsai, H.-Y. Wang, Y.-C. Chi, C.-I. Wu, and G.-R. Lin, “Enhancing optical nonlinearity in a nonstoichiometric SiN waveguide for cross-wavelength all-optical data processing,” ACS Photonics 2(8), 1141–1154 (2015).
[Crossref]

Lin, Y.-Y.

Linnros, J.

J. Linnros, “Carrier lifetime measurements using free carrier absorption transients. II. Lifetime mapping and effects of surface recombination,” J. Appl. Phys. 84(1), 284–291 (1998).
[Crossref]

Lipson, M.

Massar, S.

Mehta, P.

N. Vukovic, N. Healy, F. H. Suhailin, P. Mehta, T. D. Day, J. V. Badding, and A. C. Peacock, “Ultrafast optical control using the Kerr nonlinearity in hydrogenated amorphous silicon microcylindrical resonators,” Sci. Rep. 3, 2885 (2013).
[Crossref] [PubMed]

Netti, M.

Okawachi, Y.

Peacock, A. C.

N. Vukovic, N. Healy, F. H. Suhailin, P. Mehta, T. D. Day, J. V. Badding, and A. C. Peacock, “Ultrafast optical control using the Kerr nonlinearity in hydrogenated amorphous silicon microcylindrical resonators,” Sci. Rep. 3, 2885 (2013).
[Crossref] [PubMed]

Pelc, J. S.

Perney, N.

Raghunathan, V.

Rivoire, K.

Roelkens, G.

Rotenberg, N.

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007).
[Crossref]

Safioui, J.

Santori, C.

Saperstein, R.-E.

Selvaraja, S. K.

Shoji, T.

Sibbett, W.

P. N. Kean, K. Smith, and W. Sibbett, “Spectral and temporal investigation of self-phase modulation and stimulated Raman scattering in a single-mode optical fibre,” IEE Proc. 134(3), 163–170 (1987).

Smith, K.

P. N. Kean, K. Smith, and W. Sibbett, “Spectral and temporal investigation of self-phase modulation and stimulated Raman scattering in a single-mode optical fibre,” IEE Proc. 134(3), 163–170 (1987).

Su, S.-P.

C.-L. Wu, Y.-H. Lin, S.-P. Su, B.-J. Huang, C.-T. Tsai, H.-Y. Wang, Y.-C. Chi, C.-I. Wu, and G.-R. Lin, “Enhancing optical nonlinearity in a nonstoichiometric SiN waveguide for cross-wavelength all-optical data processing,” ACS Photonics 2(8), 1141–1154 (2015).
[Crossref]

Suhailin, F. H.

N. Vukovic, N. Healy, F. H. Suhailin, P. Mehta, T. D. Day, J. V. Badding, and A. C. Peacock, “Ultrafast optical control using the Kerr nonlinearity in hydrogenated amorphous silicon microcylindrical resonators,” Sci. Rep. 3, 2885 (2013).
[Crossref] [PubMed]

Tai, C.-Y.

Takahashi, J.

Takahashi, M.

Tang, H. X.

Teraoka, E. Y. M.

E. Y. M. Teraoka, D. H. Broaddus, T. Kita, A. Tsukazaki, M. Kawasaki, A. L. Gaeta, and H. Yamada, “Self-phase modulation at visible wavelengths in nonlinear ZnO channel waveguides,” Appl. Phys. Lett. 97(7), 071105 (2010).
[Crossref]

Tien, W.-C.

Tsai, C.-T.

C.-L. Wu, Y.-H. Lin, S.-P. Su, B.-J. Huang, C.-T. Tsai, H.-Y. Wang, Y.-C. Chi, C.-I. Wu, and G.-R. Lin, “Enhancing optical nonlinearity in a nonstoichiometric SiN waveguide for cross-wavelength all-optical data processing,” ACS Photonics 2(8), 1141–1154 (2015).
[Crossref]

Tsuchizawa, T.

Tsukazaki, A.

E. Y. M. Teraoka, D. H. Broaddus, T. Kita, A. Tsukazaki, M. Kawasaki, A. L. Gaeta, and H. Yamada, “Self-phase modulation at visible wavelengths in nonlinear ZnO channel waveguides,” Appl. Phys. Lett. 97(7), 071105 (2010).
[Crossref]

Turner-Foster, A. C.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

van Driel, H. M.

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007).
[Crossref]

Vo, S.

Vukovic, N.

N. Vukovic, N. Healy, F. H. Suhailin, P. Mehta, T. D. Day, J. V. Badding, and A. C. Peacock, “Ultrafast optical control using the Kerr nonlinearity in hydrogenated amorphous silicon microcylindrical resonators,” Sci. Rep. 3, 2885 (2013).
[Crossref] [PubMed]

Waldmeyer, J.

J. Waldmeyer, “A contactless method for determination of carrier lifetime, surface recombination velocity, and diffusion constant in semiconductors,” J. Appl. Phys. 63(6), 1977–1983 (1988).
[Crossref]

Wang, H.-Y.

C.-L. Wu, Y.-H. Lin, S.-P. Su, B.-J. Huang, C.-T. Tsai, H.-Y. Wang, Y.-C. Chi, C.-I. Wu, and G.-R. Lin, “Enhancing optical nonlinearity in a nonstoichiometric SiN waveguide for cross-wavelength all-optical data processing,” ACS Photonics 2(8), 1141–1154 (2015).
[Crossref]

Watanabe, T.

Wilkinson, J.

Wu, C.-I.

C.-L. Wu, Y.-H. Lin, S.-P. Su, B.-J. Huang, C.-T. Tsai, H.-Y. Wang, Y.-C. Chi, C.-I. Wu, and G.-R. Lin, “Enhancing optical nonlinearity in a nonstoichiometric SiN waveguide for cross-wavelength all-optical data processing,” ACS Photonics 2(8), 1141–1154 (2015).
[Crossref]

Wu, C.-L.

Xiong, C.

Xu, Q.

Yamada, H.

E. Y. M. Teraoka, D. H. Broaddus, T. Kita, A. Tsukazaki, M. Kawasaki, A. L. Gaeta, and H. Yamada, “Self-phase modulation at visible wavelengths in nonlinear ZnO channel waveguides,” Appl. Phys. Lett. 97(7), 071105 (2010).
[Crossref]

Yamada, K.

Yin, L.

Zhang, X.

ACS Photonics (1)

C.-L. Wu, Y.-H. Lin, S.-P. Su, B.-J. Huang, C.-T. Tsai, H.-Y. Wang, Y.-C. Chi, C.-I. Wu, and G.-R. Lin, “Enhancing optical nonlinearity in a nonstoichiometric SiN waveguide for cross-wavelength all-optical data processing,” ACS Photonics 2(8), 1141–1154 (2015).
[Crossref]

Appl. Phys. Lett. (2)

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007).
[Crossref]

E. Y. M. Teraoka, D. H. Broaddus, T. Kita, A. Tsukazaki, M. Kawasaki, A. L. Gaeta, and H. Yamada, “Self-phase modulation at visible wavelengths in nonlinear ZnO channel waveguides,” Appl. Phys. Lett. 97(7), 071105 (2010).
[Crossref]

IEE Proc. (1)

P. N. Kean, K. Smith, and W. Sibbett, “Spectral and temporal investigation of self-phase modulation and stimulated Raman scattering in a single-mode optical fibre,” IEE Proc. 134(3), 163–170 (1987).

J. Appl. Phys. (2)

J. Waldmeyer, “A contactless method for determination of carrier lifetime, surface recombination velocity, and diffusion constant in semiconductors,” J. Appl. Phys. 63(6), 1977–1983 (1988).
[Crossref]

J. Linnros, “Carrier lifetime measurements using free carrier absorption transients. II. Lifetime mapping and effects of surface recombination,” J. Appl. Phys. 84(1), 284–291 (1998).
[Crossref]

Nat. Photonics (1)

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

Opt. Express (11)

J. Safioui, F. Leo, B. Kuyken, S.-P. Gorza, S. K. Selvaraja, R. Baets, P. Emplit, G. Roelkens, and S. Massar, “Supercontinuum generation in hydrogenated amorphous silicon waveguides at telecommunication wavelengths,” Opt. Express 22(3), 3089–3097 (2014).
[Crossref] [PubMed]

J. S. Pelc, K. Rivoire, S. Vo, C. Santori, D. A. Fattal, and R. G. Beausoleil, “Picosecond all-optical switching in hydrogenated amorphous silicon microring resonators,” Opt. Express 22(4), 3797–3810 (2014).
[Crossref] [PubMed]

O. Boyraz, T. Indukuri, and B. Jalali, “Self-phase-modulation induced spectral broadening in silicon waveguides,” Opt. Express 12(5), 829–834 (2004).
[Crossref] [PubMed]

O. Boyraz, P. Koonath, V. Raghunathan, and B. Jalali, “All optical switching and continuum generation in silicon waveguides,” Opt. Express 12(17), 4094–4102 (2004).
[Crossref] [PubMed]

C.-Y. Tai, J. Wilkinson, N. Perney, M. Netti, F. Cattaneo, C. Finlayson, and J. Baumberg, “Determination of nonlinear refractive index in a Ta2O5 rib waveguide using self-phase modulation,” Opt. Express 12(21), 5110–5116 (2004).
[Crossref] [PubMed]

H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J. Takahashi, and S. Itabashi, “Four-wave mixing in silicon wire waveguides,” Opt. Express 13(12), 4629–4637 (2005).
[Crossref] [PubMed]

Q. Xu and M. Lipson, “All-optical logic based on silicon micro-ring resonators,” Opt. Express 15(3), 924–929 (2007).
[Crossref] [PubMed]

K. Ikeda, R.-E. Saperstein, N. Alic, and Y. Fainman, “Thermal and Kerr nonlinear properties of plasma-deposited silicon nitride/ silicon dioxide waveguides,” Opt. Express 16(17), 12987–12994 (2008).
[Crossref] [PubMed]

J. Y. Lee, L. Yin, G. P. Agrawal, and P. M. Fauchet, “Ultrafast optical switching based on nonlinear polarization rotation in silicon waveguides,” Opt. Express 18(11), 11514–11523 (2010).
[Crossref] [PubMed]

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Opt. Lett. (3)

Sci. Rep. (1)

N. Vukovic, N. Healy, F. H. Suhailin, P. Mehta, T. D. Day, J. V. Badding, and A. C. Peacock, “Ultrafast optical control using the Kerr nonlinearity in hydrogenated amorphous silicon microcylindrical resonators,” Sci. Rep. 3, 2885 (2013).
[Crossref] [PubMed]

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1995).

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Figures (4)

Fig. 1
Fig. 1 (a) Illustration of Ta2O5 channel waveguide. (b) SEM cross-sectional image of polished Ta2O5 channel waveguide. (c) Experimental setup of SPM measurement.
Fig. 2
Fig. 2 (a) Optical spectra of the Ti:sapphire laser passing through the 5-mm long Ta2O5 channel waveguide with TE polarization. (b) Power-dependent spectral linewidth for TE/TM polarizations in the Ta2O5 channel waveguide. (c) Simulated transverse mode profiles of the Ta2O5 channel waveguide at 800 nm. (d) Intensity-dependent spectral linewidth of TE/TM polarizations in the Ta2O5 channel waveguide.
Fig. 3
Fig. 3 Calculated chromatic dispersion (group velocity dispersion) of the Ta2O5 channel waveguide with a dimension 700 nm × 400 nm2.
Fig. 4
Fig. 4 Experimental results are fitted to the model without (in red) and with (in blue) GVD for optimized γLeff for (a) TE and (b) TM modes.

Equations (11)

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Δ λ = Δ λ i + 4 2 ln 2 e λ n 2 L e f f c A e f f P 0 t p
i U z = β 2 2 2 U t 2 + γ P 0 | U | 2 U
U ( z , T ) = T p ( 0 ) T p exp [ ( 1 + i C p ) T 2 2 T p 2 ]
d T p d Z = sgn ( β 2 ) C p T p ,
d C p d Z = ( 1 + C p 2 ) 1 T p 2 + γ P L 2 T p ( 0 ) T p
δ ω = ( 2 2 ln 2 ) 1 + C p 2 T p 2
d 2 T p d Z 2 = 1 T p 3 + γ P L 2 T p ( 0 ) T p 2
d 2 δ T p d Z 2 = Γ Ξ δ T p
δ T p ( Z ) = Ξ C p ( 0 ) T p ( 0 ) sin ( Ξ Z ) Γ cos ( Ξ Z ) + Γ Ξ
C p ( Z ) = [ T p ( 0 ) + Ξ C p ( 0 ) T p ( 0 ) sin ( Ξ Z ) Γ cos ( Ξ Z ) + Γ Ξ ] × [ Ξ C p ( 0 ) T p ( 0 ) cos ( Ξ Z ) + Γ sin ( Ξ Z ) Ξ ]
Γ [ 1 cos ( Ξ ) ] T p ( 0 ) Ξ

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