Abstract

We present a theoretical study on the Z-scan characteristics of thin nonlinear optical media with simultaneous two- and three-photon absorption, a situation that exists, for example, in polydiacetylenes. With the introduction of a coupling function between two- and three-photon absorption, we find a quasi-analytic expression for open aperture Z-scan traces. We make a comparison of the analytic solutions with numerical solutions in detail, showing that they are in good agreement. This theoretical result allows us to easily identify and determine simultaneously the two-and three-photon absorption coefficients from the open aperture Z-scan traces.

© 2005 Optical Society of America

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References

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  1. R. L. Sutherland , Handbook of Nonlinear Optics, ( Marcel Dekker, New York , 1996 ).
  2. G. S. Maciel , N. Rakov , Cid B. de Araujo , A. A. Lipovskii , and D. K. Tagantsev , “ Optical limiting behavior of a glass-ceramic containing sodium niobate crystallites ,” Appl. Phys. Lett.   79 , 584 – 586 ( 2001 ).
    [CrossRef]
  3. G. S. He , P. P. Markowicz , T. C. Lin , and P. N. Prasad , “ Observation of stimulated emission by direct three-photon excitation ,” Nature   415 , 767 – 770 ( 2002 ).
    [CrossRef] [PubMed]
  4. F. Yoshino , S. Polyakov , M. Liu , and G. Stegeman , “ Observation of three-photon enhanced four-photon absorption ,” Phys. Rev. Lett.   91 , 063902 ( 2003 ).
    [CrossRef] [PubMed]
  5. S. Polyakov , F. Yoshino , M. Liu , and G. Stegeman , “ Nonlinear refraction and multiphoton absorption in polydiacetylenes from 1200 to 2200 nm ,” Phys. Rev. B.   69 , 115421 ( 2004 ).
    [CrossRef]
  6. K. S. Bindra , H. T. Bookey , A. K. Kar , B. S. Wherrett , X. Liu , and A. Jha , “ Nonlinear optical properties of chalcogenide glasses: observation of multiphoton absorption ,” Appl. Phys. Lett.   79 , 1939 – 1941 ( 2001 ).
    [CrossRef]
  7. R. A. Ganeev , A. I. Ryasnyansky , N. Ishizawa , M. Baba , M. Suzuki , M. Turu , S. Sakakibara , and H. Kuroda , “ Two- and three-photon absorption in CS 2 ,” Opt. Commun.   231 , 431 – 436 ( 2004 ).
    [CrossRef]
  8. 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 – 769 ( 1990 ).
    [CrossRef]
  9. Mathematica 4.0, ( Wolfram Research, Inc. , 1999 ).
  10. G. Boudebs , S. Cherukulappurath , M. Guignard , J. Troles , F. Smektala , and F. Sanchez , “ Experimental observation of higher order nonlinear absorption in tellurium based chalcogenide glasses ,” Opt. Commun.   232 , 417 – 423 ( 2004 ).
    [CrossRef]
  11. S. Cherukulappurath , J. L. Godet , and G. Boudebs , “ Higher order coefficient measurements in nonlinear absorption process ,” J. Nonlin. Opt. Phys. & Mater.   14 , 49 – 60 ( 2005 ).
    [CrossRef]

2005 (1)

S. Cherukulappurath , J. L. Godet , and G. Boudebs , “ Higher order coefficient measurements in nonlinear absorption process ,” J. Nonlin. Opt. Phys. & Mater.   14 , 49 – 60 ( 2005 ).
[CrossRef]

2004 (3)

S. Polyakov , F. Yoshino , M. Liu , and G. Stegeman , “ Nonlinear refraction and multiphoton absorption in polydiacetylenes from 1200 to 2200 nm ,” Phys. Rev. B.   69 , 115421 ( 2004 ).
[CrossRef]

R. A. Ganeev , A. I. Ryasnyansky , N. Ishizawa , M. Baba , M. Suzuki , M. Turu , S. Sakakibara , and H. Kuroda , “ Two- and three-photon absorption in CS 2 ,” Opt. Commun.   231 , 431 – 436 ( 2004 ).
[CrossRef]

G. Boudebs , S. Cherukulappurath , M. Guignard , J. Troles , F. Smektala , and F. Sanchez , “ Experimental observation of higher order nonlinear absorption in tellurium based chalcogenide glasses ,” Opt. Commun.   232 , 417 – 423 ( 2004 ).
[CrossRef]

2003 (1)

F. Yoshino , S. Polyakov , M. Liu , and G. Stegeman , “ Observation of three-photon enhanced four-photon absorption ,” Phys. Rev. Lett.   91 , 063902 ( 2003 ).
[CrossRef] [PubMed]

2002 (1)

G. S. He , P. P. Markowicz , T. C. Lin , and P. N. Prasad , “ Observation of stimulated emission by direct three-photon excitation ,” Nature   415 , 767 – 770 ( 2002 ).
[CrossRef] [PubMed]

2001 (2)

G. S. Maciel , N. Rakov , Cid B. de Araujo , A. A. Lipovskii , and D. K. Tagantsev , “ Optical limiting behavior of a glass-ceramic containing sodium niobate crystallites ,” Appl. Phys. Lett.   79 , 584 – 586 ( 2001 ).
[CrossRef]

K. S. Bindra , H. T. Bookey , A. K. Kar , B. S. Wherrett , X. Liu , and A. Jha , “ Nonlinear optical properties of chalcogenide glasses: observation of multiphoton absorption ,” Appl. Phys. Lett.   79 , 1939 – 1941 ( 2001 ).
[CrossRef]

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 – 769 ( 1990 ).
[CrossRef]

Araujo, Cid B. de

G. S. Maciel , N. Rakov , Cid B. de Araujo , A. A. Lipovskii , and D. K. Tagantsev , “ Optical limiting behavior of a glass-ceramic containing sodium niobate crystallites ,” Appl. Phys. Lett.   79 , 584 – 586 ( 2001 ).
[CrossRef]

Baba, M.

R. A. Ganeev , A. I. Ryasnyansky , N. Ishizawa , M. Baba , M. Suzuki , M. Turu , S. Sakakibara , and H. Kuroda , “ Two- and three-photon absorption in CS 2 ,” Opt. Commun.   231 , 431 – 436 ( 2004 ).
[CrossRef]

Bindra, K. S.

K. S. Bindra , H. T. Bookey , A. K. Kar , B. S. Wherrett , X. Liu , and A. Jha , “ Nonlinear optical properties of chalcogenide glasses: observation of multiphoton absorption ,” Appl. Phys. Lett.   79 , 1939 – 1941 ( 2001 ).
[CrossRef]

Bookey, H. T.

K. S. Bindra , H. T. Bookey , A. K. Kar , B. S. Wherrett , X. Liu , and A. Jha , “ Nonlinear optical properties of chalcogenide glasses: observation of multiphoton absorption ,” Appl. Phys. Lett.   79 , 1939 – 1941 ( 2001 ).
[CrossRef]

Boudebs, G.

S. Cherukulappurath , J. L. Godet , and G. Boudebs , “ Higher order coefficient measurements in nonlinear absorption process ,” J. Nonlin. Opt. Phys. & Mater.   14 , 49 – 60 ( 2005 ).
[CrossRef]

G. Boudebs , S. Cherukulappurath , M. Guignard , J. Troles , F. Smektala , and F. Sanchez , “ Experimental observation of higher order nonlinear absorption in tellurium based chalcogenide glasses ,” Opt. Commun.   232 , 417 – 423 ( 2004 ).
[CrossRef]

Cherukulappurath, S.

S. Cherukulappurath , J. L. Godet , and G. Boudebs , “ Higher order coefficient measurements in nonlinear absorption process ,” J. Nonlin. Opt. Phys. & Mater.   14 , 49 – 60 ( 2005 ).
[CrossRef]

G. Boudebs , S. Cherukulappurath , M. Guignard , J. Troles , F. Smektala , and F. Sanchez , “ Experimental observation of higher order nonlinear absorption in tellurium based chalcogenide glasses ,” Opt. Commun.   232 , 417 – 423 ( 2004 ).
[CrossRef]

Ganeev, R. A.

R. A. Ganeev , A. I. Ryasnyansky , N. Ishizawa , M. Baba , M. Suzuki , M. Turu , S. Sakakibara , and H. Kuroda , “ Two- and three-photon absorption in CS 2 ,” Opt. Commun.   231 , 431 – 436 ( 2004 ).
[CrossRef]

Godet, J. L.

S. Cherukulappurath , J. L. Godet , and G. Boudebs , “ Higher order coefficient measurements in nonlinear absorption process ,” J. Nonlin. Opt. Phys. & Mater.   14 , 49 – 60 ( 2005 ).
[CrossRef]

Guignard, M.

G. Boudebs , S. Cherukulappurath , M. Guignard , J. Troles , F. Smektala , and F. Sanchez , “ Experimental observation of higher order nonlinear absorption in tellurium based chalcogenide glasses ,” Opt. Commun.   232 , 417 – 423 ( 2004 ).
[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 – 769 ( 1990 ).
[CrossRef]

He, G. S.

G. S. He , P. P. Markowicz , T. C. Lin , and P. N. Prasad , “ Observation of stimulated emission by direct three-photon excitation ,” Nature   415 , 767 – 770 ( 2002 ).
[CrossRef] [PubMed]

Ishizawa, N.

R. A. Ganeev , A. I. Ryasnyansky , N. Ishizawa , M. Baba , M. Suzuki , M. Turu , S. Sakakibara , and H. Kuroda , “ Two- and three-photon absorption in CS 2 ,” Opt. Commun.   231 , 431 – 436 ( 2004 ).
[CrossRef]

Jha, A.

K. S. Bindra , H. T. Bookey , A. K. Kar , B. S. Wherrett , X. Liu , and A. Jha , “ Nonlinear optical properties of chalcogenide glasses: observation of multiphoton absorption ,” Appl. Phys. Lett.   79 , 1939 – 1941 ( 2001 ).
[CrossRef]

Kar, A. K.

K. S. Bindra , H. T. Bookey , A. K. Kar , B. S. Wherrett , X. Liu , and A. Jha , “ Nonlinear optical properties of chalcogenide glasses: observation of multiphoton absorption ,” Appl. Phys. Lett.   79 , 1939 – 1941 ( 2001 ).
[CrossRef]

Kuroda, H.

R. A. Ganeev , A. I. Ryasnyansky , N. Ishizawa , M. Baba , M. Suzuki , M. Turu , S. Sakakibara , and H. Kuroda , “ Two- and three-photon absorption in CS 2 ,” Opt. Commun.   231 , 431 – 436 ( 2004 ).
[CrossRef]

Lin, T. C.

G. S. He , P. P. Markowicz , T. C. Lin , and P. N. Prasad , “ Observation of stimulated emission by direct three-photon excitation ,” Nature   415 , 767 – 770 ( 2002 ).
[CrossRef] [PubMed]

Lipovskii, A. A.

G. S. Maciel , N. Rakov , Cid B. de Araujo , A. A. Lipovskii , and D. K. Tagantsev , “ Optical limiting behavior of a glass-ceramic containing sodium niobate crystallites ,” Appl. Phys. Lett.   79 , 584 – 586 ( 2001 ).
[CrossRef]

Liu, M.

S. Polyakov , F. Yoshino , M. Liu , and G. Stegeman , “ Nonlinear refraction and multiphoton absorption in polydiacetylenes from 1200 to 2200 nm ,” Phys. Rev. B.   69 , 115421 ( 2004 ).
[CrossRef]

F. Yoshino , S. Polyakov , M. Liu , and G. Stegeman , “ Observation of three-photon enhanced four-photon absorption ,” Phys. Rev. Lett.   91 , 063902 ( 2003 ).
[CrossRef] [PubMed]

Liu, X.

K. S. Bindra , H. T. Bookey , A. K. Kar , B. S. Wherrett , X. Liu , and A. Jha , “ Nonlinear optical properties of chalcogenide glasses: observation of multiphoton absorption ,” Appl. Phys. Lett.   79 , 1939 – 1941 ( 2001 ).
[CrossRef]

Maciel, G. S.

G. S. Maciel , N. Rakov , Cid B. de Araujo , A. A. Lipovskii , and D. K. Tagantsev , “ Optical limiting behavior of a glass-ceramic containing sodium niobate crystallites ,” Appl. Phys. Lett.   79 , 584 – 586 ( 2001 ).
[CrossRef]

Markowicz, P. P.

G. S. He , P. P. Markowicz , T. C. Lin , and P. N. Prasad , “ Observation of stimulated emission by direct three-photon excitation ,” Nature   415 , 767 – 770 ( 2002 ).
[CrossRef] [PubMed]

Polyakov, S.

S. Polyakov , F. Yoshino , M. Liu , and G. Stegeman , “ Nonlinear refraction and multiphoton absorption in polydiacetylenes from 1200 to 2200 nm ,” Phys. Rev. B.   69 , 115421 ( 2004 ).
[CrossRef]

F. Yoshino , S. Polyakov , M. Liu , and G. Stegeman , “ Observation of three-photon enhanced four-photon absorption ,” Phys. Rev. Lett.   91 , 063902 ( 2003 ).
[CrossRef] [PubMed]

Prasad, P. N.

G. S. He , P. P. Markowicz , T. C. Lin , and P. N. Prasad , “ Observation of stimulated emission by direct three-photon excitation ,” Nature   415 , 767 – 770 ( 2002 ).
[CrossRef] [PubMed]

Rakov, N.

G. S. Maciel , N. Rakov , Cid B. de Araujo , A. A. Lipovskii , and D. K. Tagantsev , “ Optical limiting behavior of a glass-ceramic containing sodium niobate crystallites ,” Appl. Phys. Lett.   79 , 584 – 586 ( 2001 ).
[CrossRef]

Ryasnyansky, A. I.

R. A. Ganeev , A. I. Ryasnyansky , N. Ishizawa , M. Baba , M. Suzuki , M. Turu , S. Sakakibara , and H. Kuroda , “ Two- and three-photon absorption in CS 2 ,” Opt. Commun.   231 , 431 – 436 ( 2004 ).
[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 – 769 ( 1990 ).
[CrossRef]

Sakakibara, S.

R. A. Ganeev , A. I. Ryasnyansky , N. Ishizawa , M. Baba , M. Suzuki , M. Turu , S. Sakakibara , and H. Kuroda , “ Two- and three-photon absorption in CS 2 ,” Opt. Commun.   231 , 431 – 436 ( 2004 ).
[CrossRef]

Sanchez, F.

G. Boudebs , S. Cherukulappurath , M. Guignard , J. Troles , F. Smektala , and F. Sanchez , “ Experimental observation of higher order nonlinear absorption in tellurium based chalcogenide glasses ,” Opt. Commun.   232 , 417 – 423 ( 2004 ).
[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 – 769 ( 1990 ).
[CrossRef]

Smektala, F.

G. Boudebs , S. Cherukulappurath , M. Guignard , J. Troles , F. Smektala , and F. Sanchez , “ Experimental observation of higher order nonlinear absorption in tellurium based chalcogenide glasses ,” Opt. Commun.   232 , 417 – 423 ( 2004 ).
[CrossRef]

Stegeman, G.

S. Polyakov , F. Yoshino , M. Liu , and G. Stegeman , “ Nonlinear refraction and multiphoton absorption in polydiacetylenes from 1200 to 2200 nm ,” Phys. Rev. B.   69 , 115421 ( 2004 ).
[CrossRef]

F. Yoshino , S. Polyakov , M. Liu , and G. Stegeman , “ Observation of three-photon enhanced four-photon absorption ,” Phys. Rev. Lett.   91 , 063902 ( 2003 ).
[CrossRef] [PubMed]

Stryland, E. W. Van

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 – 769 ( 1990 ).
[CrossRef]

Sutherland, R. L.

R. L. Sutherland , Handbook of Nonlinear Optics, ( Marcel Dekker, New York , 1996 ).

Suzuki, M.

R. A. Ganeev , A. I. Ryasnyansky , N. Ishizawa , M. Baba , M. Suzuki , M. Turu , S. Sakakibara , and H. Kuroda , “ Two- and three-photon absorption in CS 2 ,” Opt. Commun.   231 , 431 – 436 ( 2004 ).
[CrossRef]

Tagantsev, D. K.

G. S. Maciel , N. Rakov , Cid B. de Araujo , A. A. Lipovskii , and D. K. Tagantsev , “ Optical limiting behavior of a glass-ceramic containing sodium niobate crystallites ,” Appl. Phys. Lett.   79 , 584 – 586 ( 2001 ).
[CrossRef]

Troles, J.

G. Boudebs , S. Cherukulappurath , M. Guignard , J. Troles , F. Smektala , and F. Sanchez , “ Experimental observation of higher order nonlinear absorption in tellurium based chalcogenide glasses ,” Opt. Commun.   232 , 417 – 423 ( 2004 ).
[CrossRef]

Turu, M.

R. A. Ganeev , A. I. Ryasnyansky , N. Ishizawa , M. Baba , M. Suzuki , M. Turu , S. Sakakibara , and H. Kuroda , “ Two- and three-photon absorption in CS 2 ,” Opt. Commun.   231 , 431 – 436 ( 2004 ).
[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 – 769 ( 1990 ).
[CrossRef]

Wherrett, B. S.

K. S. Bindra , H. T. Bookey , A. K. Kar , B. S. Wherrett , X. Liu , and A. Jha , “ Nonlinear optical properties of chalcogenide glasses: observation of multiphoton absorption ,” Appl. Phys. Lett.   79 , 1939 – 1941 ( 2001 ).
[CrossRef]

Yoshino, F.

S. Polyakov , F. Yoshino , M. Liu , and G. Stegeman , “ Nonlinear refraction and multiphoton absorption in polydiacetylenes from 1200 to 2200 nm ,” Phys. Rev. B.   69 , 115421 ( 2004 ).
[CrossRef]

F. Yoshino , S. Polyakov , M. Liu , and G. Stegeman , “ Observation of three-photon enhanced four-photon absorption ,” Phys. Rev. Lett.   91 , 063902 ( 2003 ).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

G. S. Maciel , N. Rakov , Cid B. de Araujo , A. A. Lipovskii , and D. K. Tagantsev , “ Optical limiting behavior of a glass-ceramic containing sodium niobate crystallites ,” Appl. Phys. Lett.   79 , 584 – 586 ( 2001 ).
[CrossRef]

K. S. Bindra , H. T. Bookey , A. K. Kar , B. S. Wherrett , X. Liu , and A. Jha , “ Nonlinear optical properties of chalcogenide glasses: observation of multiphoton absorption ,” Appl. Phys. Lett.   79 , 1939 – 1941 ( 2001 ).
[CrossRef]

IEEE J. Quantum Electron. (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 – 769 ( 1990 ).
[CrossRef]

J. Nonlin. Opt. Phys. & Mater. (1)

S. Cherukulappurath , J. L. Godet , and G. Boudebs , “ Higher order coefficient measurements in nonlinear absorption process ,” J. Nonlin. Opt. Phys. & Mater.   14 , 49 – 60 ( 2005 ).
[CrossRef]

Nature (1)

G. S. He , P. P. Markowicz , T. C. Lin , and P. N. Prasad , “ Observation of stimulated emission by direct three-photon excitation ,” Nature   415 , 767 – 770 ( 2002 ).
[CrossRef] [PubMed]

Opt. Commun. (2)

R. A. Ganeev , A. I. Ryasnyansky , N. Ishizawa , M. Baba , M. Suzuki , M. Turu , S. Sakakibara , and H. Kuroda , “ Two- and three-photon absorption in CS 2 ,” Opt. Commun.   231 , 431 – 436 ( 2004 ).
[CrossRef]

G. Boudebs , S. Cherukulappurath , M. Guignard , J. Troles , F. Smektala , and F. Sanchez , “ Experimental observation of higher order nonlinear absorption in tellurium based chalcogenide glasses ,” Opt. Commun.   232 , 417 – 423 ( 2004 ).
[CrossRef]

Phys. Rev. B. (1)

S. Polyakov , F. Yoshino , M. Liu , and G. Stegeman , “ Nonlinear refraction and multiphoton absorption in polydiacetylenes from 1200 to 2200 nm ,” Phys. Rev. B.   69 , 115421 ( 2004 ).
[CrossRef]

Phys. Rev. Lett. (1)

F. Yoshino , S. Polyakov , M. Liu , and G. Stegeman , “ Observation of three-photon enhanced four-photon absorption ,” Phys. Rev. Lett.   91 , 063902 ( 2003 ).
[CrossRef] [PubMed]

Other (2)

R. L. Sutherland , Handbook of Nonlinear Optics, ( Marcel Dekker, New York , 1996 ).

Mathematica 4.0, ( Wolfram Research, Inc. , 1999 ).

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Figures (5)

Fig. 1.
Fig. 1.

Analytical Z-scan traces (solid lines) compared with the numerical simulations (discrete symbols) for different intensities at I 0= 15 GW/cm2 (squares), 20 GW/cm2 (circles) and 25 GW/cm2 (triangles), respectively.

Fig. 2.
Fig. 2.

The normalized transmittance at the valley TV as a function of the linear absorption coefficient α 0 obtained by numerical simulation (circles) and our analytic solution (solid lines), respectively. The parameters are inside the figures.

Fig. 3.
Fig. 3.

The normalized transmittance at the valley TV as a function of the 2PA coefficient β1 obtained by numerical simulation (circles) and our analytic solution (solid lines), respectively. The parameters are inside the figures.

Fig. 4.
Fig. 4.

The normalized transmittance at the valley TV as a function of 3PA coefficient β2 obtained by numerical simulation (circles) and our analytic solution (solid lines), respectively. The parameters are inside the figures.

Fig. 5.
Fig. 5.

The normalized transmittance at the valley TV as a function of the on-axis peak intensity I 0 obtained by numerical simulation (circles) and our analytic solution (solid lines), respectively. The parameters are inside the figures.

Equations (9)

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

I ( r , x ) = I 0 1 1 + x 2 exp [ 2 r 2 ω 2 ( x ) ] ,
dI / dz′ = ( α 0 + β n I n ) I .
I ( L , x ) = I ( r , x ) exp ( α 0 L ) / ( 1 + n β n I n ( r , x ) L eff ( n ) ) 1 / n .
T n + 1 ( x , Ψ n ) = 0 2 π I ( L , x ) rdr exp ( α 0 L ) 0 2 π I ( r , x ) rdr = 2 F 1 [ 1 n , 1 n , 1 + n n , ψ n n ]
T 2 ( x , Ψ 1 ) = ln ( 1 + ψ 1 ) / ψ 1 .
T 3 ( x , Ψ 2 ) = sin h 1 ( ψ 2 ) / ψ 2 .
dI / dz′ = ( α 0 + β 1 I + β 2 I 2 ) I .
T ( x , Ψ 1 , Ψ 2 ) = T 2 ( x , Ψ 1 ) T 3 ( x , Ψ 2 ) f ( x , Ψ 1 , Ψ 2 ) ,
f ( x , Ψ 1 , Ψ 2 ) = 1 + ψ 1 [ 0.339 sin ( 0.498 ψ 2 ) 0.029 ] 1 + 0.966 ψ 1 ψ 2 0.718 .

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