Abstract

A numerical scheme is presented to solve the nonlinear Helmholtz (NLH) equation modeling second-harmonic generation (SHG) in photonic bandgap material doped with a nonlinear χ(2) effect and the NLH equation modeling wave propagation in Kerr type gratings with a nonlinear χ(3) effect in the one-dimensional case. Both of these nonlinear phenomena arise as a result of the combination of high electromagnetic mode density and nonlinear reaction from the medium. When the mode intensity of the incident wave is significantly strong, which makes the nonlinear effect non-negligible, numerical methods based on the linearization of the essentially nonlinear problem will become inadequate. In this work, a robust, stable numerical scheme is designed to simulate the NLH equations with strong nonlinearity.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. Yamamoto, K. Mizuuchi, Y. Kitaoka, and M. Kato, “Highly efficient quasi-phase-matched second-harmonic generation by frequency doubling of a high-frequency superimposed laser diode,” Opt. Lett. 20, 273–275 (1995).
    [CrossRef] [PubMed]
  2. D. Blanc, A. M. Bouchoux, C. Plumereau, A. Cachard, and J. F. Roux, “Phase-matched frequency doubling in an aluminum nitride waveguide with a tunable laser source,” Appl. Phys. Lett. 66, 659–661 (1995).
    [CrossRef]
  3. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–120 (1961).
    [CrossRef]
  4. M. M. Fejer, “Nonlinear optical frequency conversion,” Phys. Today 47, 25–32 (1994).
    [CrossRef]
  5. N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–486 (1970).
    [CrossRef]
  6. P. Yeh, Optical Waves in Layered Media (Wiley, 1998).
  7. J.-P. Fouque, J. Garnier, G. Papanicolaou, and K. Solna, Wave Propagation and Time Reversal in Randomly Layered Medium (Springer, 2007).
  8. G. Bao and D. Dobson, “Second harmonic generation in nonlinear optical films,” J. Math. Phys. 35, 1623–1633 (1994).
    [CrossRef]
  9. N. Bloembergen, Nonlinear Optics (Benjamin, 1965).
  10. G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. Bloemer, and C. Bowden, “Energy exchange properties during second-harmonic generation in finite one-dimensional photonic band-gap structure with deep gratings,” Phys. Rev. E 67, 016606 (2003).
    [CrossRef]
  11. A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, 1993).
  12. A. Suryanto, E. V. Groesen, M. Hammer, and H. J. W. M. Hoekstra, “A finite element scheme to study the nonlinear optical response of a finite grating without and with defect,” Opt. Quantum Electron. 35, 313–332 (2003).
    [CrossRef]
  13. P. Tran, “Optical limiting and switching of short pulses by use of a nonlinear photonic bandgap structure with a defect,” J. Opt. Soc. Am. B 14, 2589–2595 (1997).
    [CrossRef]
  14. A. Taflove and S. C. HagnessComputational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).
  15. G. Baruch, G. Fibich, and S. Tsynkov, “High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension,” J. Comput. Phys. 227, 820–850 (2007).
    [CrossRef]
  16. M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
    [CrossRef]

2007

J.-P. Fouque, J. Garnier, G. Papanicolaou, and K. Solna, Wave Propagation and Time Reversal in Randomly Layered Medium (Springer, 2007).

G. Baruch, G. Fibich, and S. Tsynkov, “High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension,” J. Comput. Phys. 227, 820–850 (2007).
[CrossRef]

2003

A. Suryanto, E. V. Groesen, M. Hammer, and H. J. W. M. Hoekstra, “A finite element scheme to study the nonlinear optical response of a finite grating without and with defect,” Opt. Quantum Electron. 35, 313–332 (2003).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. Bloemer, and C. Bowden, “Energy exchange properties during second-harmonic generation in finite one-dimensional photonic band-gap structure with deep gratings,” Phys. Rev. E 67, 016606 (2003).
[CrossRef]

2000

A. Taflove and S. C. HagnessComputational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

1998

P. Yeh, Optical Waves in Layered Media (Wiley, 1998).

1997

P. Tran, “Optical limiting and switching of short pulses by use of a nonlinear photonic bandgap structure with a defect,” J. Opt. Soc. Am. B 14, 2589–2595 (1997).
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

1995

K. Yamamoto, K. Mizuuchi, Y. Kitaoka, and M. Kato, “Highly efficient quasi-phase-matched second-harmonic generation by frequency doubling of a high-frequency superimposed laser diode,” Opt. Lett. 20, 273–275 (1995).
[CrossRef] [PubMed]

D. Blanc, A. M. Bouchoux, C. Plumereau, A. Cachard, and J. F. Roux, “Phase-matched frequency doubling in an aluminum nitride waveguide with a tunable laser source,” Appl. Phys. Lett. 66, 659–661 (1995).
[CrossRef]

1994

M. M. Fejer, “Nonlinear optical frequency conversion,” Phys. Today 47, 25–32 (1994).
[CrossRef]

G. Bao and D. Dobson, “Second harmonic generation in nonlinear optical films,” J. Math. Phys. 35, 1623–1633 (1994).
[CrossRef]

1993

A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, 1993).

1970

N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–486 (1970).
[CrossRef]

1965

N. Bloembergen, Nonlinear Optics (Benjamin, 1965).

1961

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–120 (1961).
[CrossRef]

Bao, G.

G. Bao and D. Dobson, “Second harmonic generation in nonlinear optical films,” J. Math. Phys. 35, 1623–1633 (1994).
[CrossRef]

Baruch, G.

G. Baruch, G. Fibich, and S. Tsynkov, “High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension,” J. Comput. Phys. 227, 820–850 (2007).
[CrossRef]

Bertolotti, M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. Bloemer, and C. Bowden, “Energy exchange properties during second-harmonic generation in finite one-dimensional photonic band-gap structure with deep gratings,” Phys. Rev. E 67, 016606 (2003).
[CrossRef]

Blanc, D.

D. Blanc, A. M. Bouchoux, C. Plumereau, A. Cachard, and J. F. Roux, “Phase-matched frequency doubling in an aluminum nitride waveguide with a tunable laser source,” Appl. Phys. Lett. 66, 659–661 (1995).
[CrossRef]

Bloembergen, N.

N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–486 (1970).
[CrossRef]

N. Bloembergen, Nonlinear Optics (Benjamin, 1965).

Bloemer, M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. Bloemer, and C. Bowden, “Energy exchange properties during second-harmonic generation in finite one-dimensional photonic band-gap structure with deep gratings,” Phys. Rev. E 67, 016606 (2003).
[CrossRef]

Bloemer, M. J.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Bouchoux, A. M.

D. Blanc, A. M. Bouchoux, C. Plumereau, A. Cachard, and J. F. Roux, “Phase-matched frequency doubling in an aluminum nitride waveguide with a tunable laser source,” Appl. Phys. Lett. 66, 659–661 (1995).
[CrossRef]

Bowden, C.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. Bloemer, and C. Bowden, “Energy exchange properties during second-harmonic generation in finite one-dimensional photonic band-gap structure with deep gratings,” Phys. Rev. E 67, 016606 (2003).
[CrossRef]

Bowden, C. M.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Cachard, A.

D. Blanc, A. M. Bouchoux, C. Plumereau, A. Cachard, and J. F. Roux, “Phase-matched frequency doubling in an aluminum nitride waveguide with a tunable laser source,” Appl. Phys. Lett. 66, 659–661 (1995).
[CrossRef]

Centini, M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. Bloemer, and C. Bowden, “Energy exchange properties during second-harmonic generation in finite one-dimensional photonic band-gap structure with deep gratings,” Phys. Rev. E 67, 016606 (2003).
[CrossRef]

D’Aguanno, G.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. Bloemer, and C. Bowden, “Energy exchange properties during second-harmonic generation in finite one-dimensional photonic band-gap structure with deep gratings,” Phys. Rev. E 67, 016606 (2003).
[CrossRef]

Dobson, D.

G. Bao and D. Dobson, “Second harmonic generation in nonlinear optical films,” J. Math. Phys. 35, 1623–1633 (1994).
[CrossRef]

Dowling, J. P.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Fejer, M. M.

M. M. Fejer, “Nonlinear optical frequency conversion,” Phys. Today 47, 25–32 (1994).
[CrossRef]

Fibich, G.

G. Baruch, G. Fibich, and S. Tsynkov, “High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension,” J. Comput. Phys. 227, 820–850 (2007).
[CrossRef]

Fouque, J.-P.

J.-P. Fouque, J. Garnier, G. Papanicolaou, and K. Solna, Wave Propagation and Time Reversal in Randomly Layered Medium (Springer, 2007).

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–120 (1961).
[CrossRef]

Garnier, J.

J.-P. Fouque, J. Garnier, G. Papanicolaou, and K. Solna, Wave Propagation and Time Reversal in Randomly Layered Medium (Springer, 2007).

Groesen, E. V.

A. Suryanto, E. V. Groesen, M. Hammer, and H. J. W. M. Hoekstra, “A finite element scheme to study the nonlinear optical response of a finite grating without and with defect,” Opt. Quantum Electron. 35, 313–332 (2003).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. HagnessComputational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

Hammer, M.

A. Suryanto, E. V. Groesen, M. Hammer, and H. J. W. M. Hoekstra, “A finite element scheme to study the nonlinear optical response of a finite grating without and with defect,” Opt. Quantum Electron. 35, 313–332 (2003).
[CrossRef]

Haus, J. W.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–120 (1961).
[CrossRef]

Hoekstra, H. J. W. M.

A. Suryanto, E. V. Groesen, M. Hammer, and H. J. W. M. Hoekstra, “A finite element scheme to study the nonlinear optical response of a finite grating without and with defect,” Opt. Quantum Electron. 35, 313–332 (2003).
[CrossRef]

Kato, M.

Kitaoka, Y.

Manka, A. S.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Mizuuchi, K.

Nayfeh, A. H.

A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, 1993).

Papanicolaou, G.

J.-P. Fouque, J. Garnier, G. Papanicolaou, and K. Solna, Wave Propagation and Time Reversal in Randomly Layered Medium (Springer, 2007).

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–120 (1961).
[CrossRef]

Plumereau, C.

D. Blanc, A. M. Bouchoux, C. Plumereau, A. Cachard, and J. F. Roux, “Phase-matched frequency doubling in an aluminum nitride waveguide with a tunable laser source,” Appl. Phys. Lett. 66, 659–661 (1995).
[CrossRef]

Roux, J. F.

D. Blanc, A. M. Bouchoux, C. Plumereau, A. Cachard, and J. F. Roux, “Phase-matched frequency doubling in an aluminum nitride waveguide with a tunable laser source,” Appl. Phys. Lett. 66, 659–661 (1995).
[CrossRef]

Scalora, M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. Bloemer, and C. Bowden, “Energy exchange properties during second-harmonic generation in finite one-dimensional photonic band-gap structure with deep gratings,” Phys. Rev. E 67, 016606 (2003).
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Sibilia, C.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. Bloemer, and C. Bowden, “Energy exchange properties during second-harmonic generation in finite one-dimensional photonic band-gap structure with deep gratings,” Phys. Rev. E 67, 016606 (2003).
[CrossRef]

Sievers, A. J.

N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–486 (1970).
[CrossRef]

Solna, K.

J.-P. Fouque, J. Garnier, G. Papanicolaou, and K. Solna, Wave Propagation and Time Reversal in Randomly Layered Medium (Springer, 2007).

Suryanto, A.

A. Suryanto, E. V. Groesen, M. Hammer, and H. J. W. M. Hoekstra, “A finite element scheme to study the nonlinear optical response of a finite grating without and with defect,” Opt. Quantum Electron. 35, 313–332 (2003).
[CrossRef]

Taflove, A.

A. Taflove and S. C. HagnessComputational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

Tran, P.

Tsynkov, S.

G. Baruch, G. Fibich, and S. Tsynkov, “High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension,” J. Comput. Phys. 227, 820–850 (2007).
[CrossRef]

Viswanathan, R.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Weinreich, G.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–120 (1961).
[CrossRef]

Yamamoto, K.

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 1998).

Appl. Phys. Lett.

D. Blanc, A. M. Bouchoux, C. Plumereau, A. Cachard, and J. F. Roux, “Phase-matched frequency doubling in an aluminum nitride waveguide with a tunable laser source,” Appl. Phys. Lett. 66, 659–661 (1995).
[CrossRef]

N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–486 (1970).
[CrossRef]

J. Comput. Phys.

G. Baruch, G. Fibich, and S. Tsynkov, “High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension,” J. Comput. Phys. 227, 820–850 (2007).
[CrossRef]

J. Math. Phys.

G. Bao and D. Dobson, “Second harmonic generation in nonlinear optical films,” J. Math. Phys. 35, 1623–1633 (1994).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Opt. Quantum Electron.

A. Suryanto, E. V. Groesen, M. Hammer, and H. J. W. M. Hoekstra, “A finite element scheme to study the nonlinear optical response of a finite grating without and with defect,” Opt. Quantum Electron. 35, 313–332 (2003).
[CrossRef]

Phys. Rev. A

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997).
[CrossRef]

Phys. Rev. E

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. Bloemer, and C. Bowden, “Energy exchange properties during second-harmonic generation in finite one-dimensional photonic band-gap structure with deep gratings,” Phys. Rev. E 67, 016606 (2003).
[CrossRef]

Phys. Rev. Lett.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–120 (1961).
[CrossRef]

Phys. Today

M. M. Fejer, “Nonlinear optical frequency conversion,” Phys. Today 47, 25–32 (1994).
[CrossRef]

Other

P. Yeh, Optical Waves in Layered Media (Wiley, 1998).

J.-P. Fouque, J. Garnier, G. Papanicolaou, and K. Solna, Wave Propagation and Time Reversal in Randomly Layered Medium (Springer, 2007).

A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, 1993).

N. Bloembergen, Nonlinear Optics (Benjamin, 1965).

A. Taflove and S. C. HagnessComputational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Periodic alternating structure with high–low index fluctuation.

Fig. 2
Fig. 2

Energy output of all the channels in SHG under extremely strong pump.

Fig. 3
Fig. 3

Energy output of all the channels in SHG.

Fig. 4
Fig. 4

Bistable solution through frequency tuning.

Fig. 5
Fig. 5

Bistable solution controlled by input intensity for fixed frequency.

Equations (28)

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

E ( r , t ) = E ( r ) e i ω t , H ( r , t ) = H ( r ) e i ω t ,
d 2 E 1 d z 2 + ( k 0 n 1 ) 2 E 1 = k 0 2 χ 1 ( 2 ) E ¯ 1 E 2 ,
d 2 E 2 d z 2 + ( 2 k 0 n 2 ) 2 E 2 = 2 k 0 2 χ 2 ( 2 ) E 1 2 ,
E 1 ( 0 ) = 2 i k 11 E I i k 11 E 1 ( 0 ) ,
E 2 ( 0 ) = i k 21 E 2 ( 0 ) ,
E 1 ( L ) = i k 12 E 1 ( L ) ,
E 2 ( L ) = i k 22 E 2 ( L ) .
d 2 E 1 d z 2 + ( k 0 n 1 ) 2 E 1 = 0 ,
d 2 E 2 d z 2 + ( 2 k 0 n 2 ) 2 E 2 = 2 k 0 2 χ 2 ( 2 ) E 1 2 ,
l = + , p ω ( + , ) d A ω ( l ) d z = i ω c ( k , l ) = ( + , ) Γ ( ω , + ) ( k , l ) A 2 ω ( k ) A ω ( l ) * ,
l = + , p ω ( , ) d A ω ( l ) d z = i ω c ( k , l ) = ( + , ) Γ ( ω , ) ( k , l ) A 2 ω ( k ) A ω ( l ) * ,
l = + , p 2 ω ( + , ) d A 2 ω ( l ) d z = i ω c ( k , l ) = ( + , ) Γ ( 2 ω , + ) ( k , l ) A ω ( k ) A ω ( l ) ,
l = + , p 2 ω ( , ) d A 2 ω ( l ) d z = i ω c ( k , l ) = ( + , ) Γ ( 2 ω , ) ( k , l ) A ω ( k ) A ω ( l ) ,
d 2 E 1 ( n + 1 ) d z 2 + ( k 0 n 1 ) 2 E 1 ( n + 1 ) = k 0 2 χ 1 ( 2 ) E ¯ 1 ( n ) E 2 ( n ) ,
d 2 E 2 ( n + 1 ) d z 2 + ( 2 k 0 n 2 ) 2 E 2 ( n + 1 ) = 2 k 0 2 χ 2 ( 2 ) ( E 1 ( n + 1 ) ) 2 .
i d E 1 d t = d 2 E 1 d z 2 ( k 0 n 1 ) 2 E 1 k 0 2 χ 1 ( 2 ) E ¯ 1 E 2 ,
i d E 2 d t = d 2 E 2 d z 2 ( 2 k 0 n 2 ) 2 E 2 2 k 0 2 χ 2 ( 2 ) E 1 2 ,
i d E 1 d t = d 2 E 1 d z 2 ( k 0 n 1 ) 2 E 1 ,
i d E 2 d t = d 2 E 2 d z 2 ( 2 k 0 n 2 ) 2 E 2 ,
i d E 1 d t = k 0 2 χ 1 ( 2 ) E ¯ 1 E 2 ,
i d E 2 d t = 2 k 0 2 χ 2 ( 2 ) E 1 2 .
d 2 E d z 2 + k 2 ( n 2 ( z ) + γ | E | 2 ) E = 0
E ( 0 ) = 2 i k 0 E I i k 0 E ( 0 ) ,
E ( L ) = i k 0 E ( L ) ,
E = E E inc 0 , γ = γ | E inc 0 | 2 .
i d E d t = d 2 E d z 2 ( k n ) 2 E γ k 2 | E | 2 E
i d E d t = d 2 E d z 2 ( k n ) 2 E ,
i d E d t = γ k 2 | E | 2 E .

Metrics