Abstract

For second-harmonic generation in two-dimensional waveguiding structures composed of segments that are invariant in the longitudinal direction, we develop an efficient numerical method based on the Dirichlet-to-Neumann (DtN) maps of the segments and a marching scheme using two operators and two functions. A Chebyshev collocation method is used to discretize the longitudinal variable for computing the DtN map and the locally generated second harmonic wave in each segment. The method rigorously solves the inhomogeneous Helmholtz equation of the second-harmonic wave without making any analytic approximations. Numerical examples are used to illustrate this new method.

© 2007 Optical Society of America

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References

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  1. M. M. Fejer, "Nonlinear optical frequency conversion," Phys. Today 47, 25-32 (1994).
    [CrossRef]
  2. M. Bertolotti, "Wave interactions in photonic band structures: an overview," J. Opt. A, Pure Appl. Opt. 8, S9-S32 (2006).
    [CrossRef]
  3. W. Nakagawa, R. C. Tyan, and Y. Fainman, "Analysis of enhanced second-harmonic generation in periodic nanostructures using modified rigorous coupled-wave analysis in the undepleted-pump approximation," J. Opt. Soc. Am. A 19, 1919-1928 (2002).
    [CrossRef]
  4. Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, "Second-harmonic generation in one-dimensional photonic edge waveguides," Phys. Rev. E 68, 066617 (2003).
    [CrossRef]
  5. A. Locatelli, D. Modotto, C. De Angelis, F. M. Pigozzo, and A. D. Capobianco, "Nonlinear bidirectional beam propagation method based on scattering operators for periodic microstructured waveguides," J. Opt. Soc. Am. B 20, 1724-1731 (2003).
    [CrossRef]
  6. B. Maes, P. Bienstman, and R. Baets, "Modeling second-harmonic generation by use of mode expansion," J. Opt. Soc. Am. B 22, 1378-1383 (2005).
    [CrossRef]
  7. Q. H. Liu and W. C. Chew, "Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method," IEEE Trans. Microwave Theory Tech. 39, 422-430 (1991).
    [CrossRef]
  8. G. Sztefka and H. P. Nolting, "Bidirectional eigenmode propagation for large refractive index steps," IEEE Photon. Technol. Lett. 5, 554-557 (1993).
    [CrossRef]
  9. J. Willems, J. Haes, and R. Baets, "The bidirectional mode expansion method for 2-dimensional waveguides--the TM case," Opt. Quantum Electron. 27, 995-1007 (1995).
    [CrossRef]
  10. S. F. Helfert and R. Pregla, "Efficient analysis of periodic structures," J. Lightwave Technol. 16, 1694-1702 (1998).
    [CrossRef]
  11. P. Bienstman and R. Baets, "Optical modeling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).
    [CrossRef]
  12. E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, "Use of grating theories in integrated optics," J. Opt. Soc. Am. A 18, 2865-2875 (2001).
    [CrossRef]
  13. J. Ctyroky, "A simple bidirectional mode expansion propagation algorithm based on modes of a parallel-plate waveguide," Opt. Quantum Electron. 38, 45-62 (2006).
    [CrossRef]
  14. H. Rao, R. Scarmozzino, and R. M. Osgood, "A bidirectional beam propagation method for multiple dielectric interfaces," IEEE Photon. Technol. Lett. 11, 830-832 (1999).
    [CrossRef]
  15. H. El-Refaei, D. Yevick, and I. Betty, "Stable and noniterative bidirectional beam propagation method," IEEE Photon. Technol. Lett. 12, 389-391 (2000).
    [CrossRef]
  16. P. L. Ho and Y. Y. Lu, "A stable bidirectional propagation method based on scattering operators," IEEE Photon. Technol. Lett. 13, 1316-1318 (2001).
    [CrossRef]
  17. P. L. Ho and Y. Y. Lu, "A bidirectional beam propagation method for periodic waveguides," IEEE Photon. Technol. Lett. 14, 325-327 (2002).
    [CrossRef]
  18. Y. Y. Lu and S. H. Wei, "A new iterative bidirectional beam propagation method," IEEE Photon. Technol. Lett. 14, 1533-1535 (2002).
    [CrossRef]
  19. Y. P. Chiou and H. C. Chang, "Analysis of optical waveguide discontinuities using the Padé approximations," IEEE Photon. Technol. Lett. 9, 946-966 (1997).
  20. C. Yu and D. Yevick, "Application of the bidirectional parabolic equation method to optical waveguide facets," J. Opt. Soc. Am. A 14, 1448-1450 (1997).
    [CrossRef]
  21. H. El-Refaei, I. Betty, and D. Yevick, "The application of complex Padé approximants to reflection a optical waveguide facets," IEEE Photon. Technol. Lett. 12, 168-170 (2000).
    [CrossRef]
  22. S. H. Wei and Y. Y. Lu, "Application of Bi-CGSTAB to waveguide discontinuity problems," IEEE Photon. Technol. Lett. 14, 645-647 (2002).
    [CrossRef]
  23. N. N. Feng and W. P. Huang, "A field-based numerical method for three-dimensional analysis of optical waveguide discontinuities," IEEE J. Quantum Electron. 39, 1661-1665 (2003).
    [CrossRef]
  24. J. Yamauchi, Propagating Beam Analysis of Optical Waveguides (Research Studies Press, 2003).
  25. L. Yuan and Y. Y. Lu, "An efficient bidirectional propagation method based on Dirichlet-to-Neumann maps," IEEE Photon. Technol. Lett. 18, 1967-1969 (2006).
    [CrossRef]
  26. L. N. Trefethen, Spectral Methods in MATLAB (SIAM, 2000).
    [CrossRef]

2006 (3)

M. Bertolotti, "Wave interactions in photonic band structures: an overview," J. Opt. A, Pure Appl. Opt. 8, S9-S32 (2006).
[CrossRef]

J. Ctyroky, "A simple bidirectional mode expansion propagation algorithm based on modes of a parallel-plate waveguide," Opt. Quantum Electron. 38, 45-62 (2006).
[CrossRef]

L. Yuan and Y. Y. Lu, "An efficient bidirectional propagation method based on Dirichlet-to-Neumann maps," IEEE Photon. Technol. Lett. 18, 1967-1969 (2006).
[CrossRef]

2005 (1)

2003 (3)

A. Locatelli, D. Modotto, C. De Angelis, F. M. Pigozzo, and A. D. Capobianco, "Nonlinear bidirectional beam propagation method based on scattering operators for periodic microstructured waveguides," J. Opt. Soc. Am. B 20, 1724-1731 (2003).
[CrossRef]

N. N. Feng and W. P. Huang, "A field-based numerical method for three-dimensional analysis of optical waveguide discontinuities," IEEE J. Quantum Electron. 39, 1661-1665 (2003).
[CrossRef]

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, "Second-harmonic generation in one-dimensional photonic edge waveguides," Phys. Rev. E 68, 066617 (2003).
[CrossRef]

2002 (4)

S. H. Wei and Y. Y. Lu, "Application of Bi-CGSTAB to waveguide discontinuity problems," IEEE Photon. Technol. Lett. 14, 645-647 (2002).
[CrossRef]

P. L. Ho and Y. Y. Lu, "A bidirectional beam propagation method for periodic waveguides," IEEE Photon. Technol. Lett. 14, 325-327 (2002).
[CrossRef]

Y. Y. Lu and S. H. Wei, "A new iterative bidirectional beam propagation method," IEEE Photon. Technol. Lett. 14, 1533-1535 (2002).
[CrossRef]

W. Nakagawa, R. C. Tyan, and Y. Fainman, "Analysis of enhanced second-harmonic generation in periodic nanostructures using modified rigorous coupled-wave analysis in the undepleted-pump approximation," J. Opt. Soc. Am. A 19, 1919-1928 (2002).
[CrossRef]

2001 (3)

E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, "Use of grating theories in integrated optics," J. Opt. Soc. Am. A 18, 2865-2875 (2001).
[CrossRef]

P. L. Ho and Y. Y. Lu, "A stable bidirectional propagation method based on scattering operators," IEEE Photon. Technol. Lett. 13, 1316-1318 (2001).
[CrossRef]

P. Bienstman and R. Baets, "Optical modeling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).
[CrossRef]

2000 (2)

H. El-Refaei, I. Betty, and D. Yevick, "The application of complex Padé approximants to reflection a optical waveguide facets," IEEE Photon. Technol. Lett. 12, 168-170 (2000).
[CrossRef]

H. El-Refaei, D. Yevick, and I. Betty, "Stable and noniterative bidirectional beam propagation method," IEEE Photon. Technol. Lett. 12, 389-391 (2000).
[CrossRef]

1999 (1)

H. Rao, R. Scarmozzino, and R. M. Osgood, "A bidirectional beam propagation method for multiple dielectric interfaces," IEEE Photon. Technol. Lett. 11, 830-832 (1999).
[CrossRef]

1998 (1)

1997 (2)

Y. P. Chiou and H. C. Chang, "Analysis of optical waveguide discontinuities using the Padé approximations," IEEE Photon. Technol. Lett. 9, 946-966 (1997).

C. Yu and D. Yevick, "Application of the bidirectional parabolic equation method to optical waveguide facets," J. Opt. Soc. Am. A 14, 1448-1450 (1997).
[CrossRef]

1995 (1)

J. Willems, J. Haes, and R. Baets, "The bidirectional mode expansion method for 2-dimensional waveguides--the TM case," Opt. Quantum Electron. 27, 995-1007 (1995).
[CrossRef]

1994 (1)

M. M. Fejer, "Nonlinear optical frequency conversion," Phys. Today 47, 25-32 (1994).
[CrossRef]

1993 (1)

G. Sztefka and H. P. Nolting, "Bidirectional eigenmode propagation for large refractive index steps," IEEE Photon. Technol. Lett. 5, 554-557 (1993).
[CrossRef]

1991 (1)

Q. H. Liu and W. C. Chew, "Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method," IEEE Trans. Microwave Theory Tech. 39, 422-430 (1991).
[CrossRef]

IEEE J. Quantum Electron. (1)

N. N. Feng and W. P. Huang, "A field-based numerical method for three-dimensional analysis of optical waveguide discontinuities," IEEE J. Quantum Electron. 39, 1661-1665 (2003).
[CrossRef]

IEEE Photon. Technol. Lett. (10)

G. Sztefka and H. P. Nolting, "Bidirectional eigenmode propagation for large refractive index steps," IEEE Photon. Technol. Lett. 5, 554-557 (1993).
[CrossRef]

H. Rao, R. Scarmozzino, and R. M. Osgood, "A bidirectional beam propagation method for multiple dielectric interfaces," IEEE Photon. Technol. Lett. 11, 830-832 (1999).
[CrossRef]

H. El-Refaei, D. Yevick, and I. Betty, "Stable and noniterative bidirectional beam propagation method," IEEE Photon. Technol. Lett. 12, 389-391 (2000).
[CrossRef]

P. L. Ho and Y. Y. Lu, "A stable bidirectional propagation method based on scattering operators," IEEE Photon. Technol. Lett. 13, 1316-1318 (2001).
[CrossRef]

P. L. Ho and Y. Y. Lu, "A bidirectional beam propagation method for periodic waveguides," IEEE Photon. Technol. Lett. 14, 325-327 (2002).
[CrossRef]

Y. Y. Lu and S. H. Wei, "A new iterative bidirectional beam propagation method," IEEE Photon. Technol. Lett. 14, 1533-1535 (2002).
[CrossRef]

Y. P. Chiou and H. C. Chang, "Analysis of optical waveguide discontinuities using the Padé approximations," IEEE Photon. Technol. Lett. 9, 946-966 (1997).

H. El-Refaei, I. Betty, and D. Yevick, "The application of complex Padé approximants to reflection a optical waveguide facets," IEEE Photon. Technol. Lett. 12, 168-170 (2000).
[CrossRef]

S. H. Wei and Y. Y. Lu, "Application of Bi-CGSTAB to waveguide discontinuity problems," IEEE Photon. Technol. Lett. 14, 645-647 (2002).
[CrossRef]

L. Yuan and Y. Y. Lu, "An efficient bidirectional propagation method based on Dirichlet-to-Neumann maps," IEEE Photon. Technol. Lett. 18, 1967-1969 (2006).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

Q. H. Liu and W. C. Chew, "Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method," IEEE Trans. Microwave Theory Tech. 39, 422-430 (1991).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. A, Pure Appl. Opt. (1)

M. Bertolotti, "Wave interactions in photonic band structures: an overview," J. Opt. A, Pure Appl. Opt. 8, S9-S32 (2006).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (2)

Opt. Quantum Electron. (3)

P. Bienstman and R. Baets, "Optical modeling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).
[CrossRef]

J. Willems, J. Haes, and R. Baets, "The bidirectional mode expansion method for 2-dimensional waveguides--the TM case," Opt. Quantum Electron. 27, 995-1007 (1995).
[CrossRef]

J. Ctyroky, "A simple bidirectional mode expansion propagation algorithm based on modes of a parallel-plate waveguide," Opt. Quantum Electron. 38, 45-62 (2006).
[CrossRef]

Phys. Rev. E (1)

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, "Second-harmonic generation in one-dimensional photonic edge waveguides," Phys. Rev. E 68, 066617 (2003).
[CrossRef]

Phys. Today (1)

M. M. Fejer, "Nonlinear optical frequency conversion," Phys. Today 47, 25-32 (1994).
[CrossRef]

Other (2)

J. Yamauchi, Propagating Beam Analysis of Optical Waveguides (Research Studies Press, 2003).

L. N. Trefethen, Spectral Methods in MATLAB (SIAM, 2000).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Linear transmission and reflection properties of a piecewise uniform waveguide (example 1) for two wavelength intervals. The solid and dashed curves represent T 2 and R 2 , respectively.

Fig. 2
Fig. 2

Maximum amplitude of the generated second harmonic wave at z = a for example 1.

Fig. 3
Fig. 3

Magnitude of the fundamental frequency wave for example 1.

Fig. 4
Fig. 4

Magnitude of the second harmonic wave for example 1.

Fig. 5
Fig. 5

Schematic view of the waveguiding structure for example 2.

Fig. 6
Fig. 6

Linear transmission and reflection properties of the piecewise uniform structure in example 2. The solid and dashed curves represent T 2 and R 2 , respectively.

Fig. 7
Fig. 7

Maximum amplitude of the generated second harmonic wave at z = a versus the free space wavelength of the fundamental frequency for example 2.

Equations (51)

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z 2 u + x 2 u + [ k 0 n ( 1 ) ( x , z ) ] 2 u = k 0 2 χ ( 2 ) ( x , z ) u ¯ v ,
z 2 v + x 2 v + [ 2 k 0 n ( 2 ) ( x , z ) ] 2 v = 2 k 0 2 χ ( 2 ) ( x , z ) u 2 ,
χ ( 2 ) ( x , z ) = χ j ( 2 ) ( x ) , n ( l ) ( x , z ) = n j ( l ) ( x ) , l = 1 , 2 .
B 0 = x 2 + [ k 0 n 0 ( 1 ) ( x ) ] 2 .
z u ( i ) = i L 0 ( 1 ) u ( i ) , z u ( r ) = i L 0 ( 1 ) u ( r ) ,
z u + i L 0 ( 1 ) u = 2 i L 0 ( 1 ) u ( i ) ( x , 0 ) at z = 0 .
L j ( l ) = x 2 + [ l k 0 n j ( l ) ( x ) ] 2 , l = 1 , 2 .
z u i L m + 1 ( 1 ) u = 0 at z = a ,
z v + i L 0 ( 2 ) v = 0 at z = 0 ,
z v i L m + 1 ( 2 ) v = 0 at z = a .
z 2 u + x 2 u + [ k 0 n ( 1 ) ( x , z ) ] 2 u = 0 .
Q j v j = z v j g j ,
Y j v j = v m + f j ,
Q m = i L m + 1 ( 2 ) , g m = 0 , Y m = I , f m = 0 ,
[ Q 0 + i L 0 ( 2 ) ] v 0 = g 0 ,
v m = Y 0 v 0 f 0 .
M [ s ( x , z j 1 ) s ( x , z j ) ] = [ M 11 M 12 M 21 M 22 ] [ s ( x , z j 1 ) s ( x , z j ) ] = [ z s ( x , z j 1 ) z s ( x , z j ) ] ,
z 2 s + x 2 s + [ 2 k 0 n j ( 2 ) ( x ) ] 2 s = 0 , z j 1 < z < z j .
z 2 w + x 2 w + [ 2 k 0 n j ( 2 ) ( x ) ] 2 w = 2 k 0 2 χ j ( 2 ) ( x ) u 2 , z j 1 < z < z j ,
Z = ( Q j M 22 ) 1 M 21 ,
h = ( Q j M 22 ) 1 ( g j z w z j ) ,
Q j 1 = M 11 + M 12 Z ,
Y j 1 = Y j Z ,
g j 1 = z w z j 1 M 12 h ,
f j 1 = Y j h + f j .
( M 11 Q j 1 ) v j 1 + M 12 v j = g j 1 z w z j 1 ,
M 21 v j 1 + ( M 22 Q j ) v j = g j z w z j .
v j = Z v j 1 h ,
( M 11 Q j 1 + M 12 Z ) v j 1 = g j 1 z w z j 1 + M 12 h .
( Y j 1 Y j Z ) v j 1 = f j 1 f j Y j h .
ξ k = z j 1 + z j z j 1 2 [ 1 cos ( k π q ) ] , k = 0 , 1 , , q .
[ F ( ξ 0 ) F ( ξ 1 ) F ( ξ q ) ] C [ F ( ξ 0 ) F ( ξ 1 ) F ( ξ q ) ] .
c k l = 2 z j z j 1 × { ( 2 q 2 + 1 ) 6 if k = l = 0 ( 2 q 2 + 1 ) 6 if k = l = q 0.5 τ k ( 1 τ k 2 ) if 0 < k = l < q ( 1 ) k + l σ k σ l 1 ( τ k τ l ) otherwise ,
τ k = cos ( k π q ) , σ k = { 2 if k = 0 , q 1 if 0 < k < q .
C = [ c 00 c ̃ 0 c 0 q c q 0 c ̃ q c q q ] , C 2 = [ d 00 d 0 q d ̂ 0 D ̂ d ̂ q d q 0 d q q ] ,
D ̂ = R [ μ 1 μ 2 μ q 1 ] R 1 .
α = R 1 d ̂ 0 , β = R 1 d ̂ q , γ = c ̃ 0 R , δ = c ̃ q R .
d ̂ 0 s 0 + D ̂ [ s 1 s 2 s q 1 ] + d ̂ q s q + [ B s 1 B s 2 B s q 1 ] = 0 ,
d 2 p k d x 2 + { [ 2 k 0 2 n j ( 2 ) ] 2 + μ k } p k = α k s ( x , z j 1 ) β k s ( x , z j ) ,
1 k < q ,
[ p 1 p 2 p q 1 ] = R 1 [ s 1 s 2 s q 1 ] .
z s 0 = c 00 s 0 + c ̃ 0 [ s 1 s 2 s q 1 ] + c 0 q s q .
z s ( x , z j 1 ) = c 00 s ( x , z j 1 ) + k = 1 q 1 γ k p k ( x ) + c 0 q s ( x , z j ) .
z s ( x , z j ) = c q 0 s ( x , z j 1 ) + k = 1 q 1 δ k p k ( x ) + c q q s ( x , z j ) .
D ̂ [ w 1 w 2 w q 1 ] + [ B w 1 B w 2 B w q 1 ] = 2 k 0 2 χ j ( 2 ) ( x ) [ u 2 ( x , ξ 1 ) u 2 ( x , ξ 2 ) u 2 ( x , ξ q 1 ) ] ,
d 2 r k d x 2 + { [ 2 k 0 2 n j ( 2 ) ] 2 + μ k } r k = 2 k 0 2 χ j ( 2 ) ( x ) ρ k ( x ) , 1 k < q ,
[ r 1 r 2 r q 1 ] = R 1 [ w 1 w 2 w q 1 ] , [ ρ 1 ( x ) ρ 2 ( x ) ρ q 1 ( x ) ] = R 1 [ u 2 ( x , ξ 1 ) u 2 ( x , ξ 2 ) u 2 ( x , ξ q 1 ) ] .
z w ( x , z j 1 ) = k = 1 q 1 γ k r k ( x ) , z w ( x , z j ) = k = 1 q 1 δ k r k ( x ) .
z 2 k = 1.037 k ( μ m ) , z 2 k + 1 = 1.037 k + 0.2 ( μ m ) for k = 0 , 1 , , 7 .
z 2 k = 0.726 k ( μ m ) , z 2 k + 1 = 0.726 k + 0.126 ( μ m ) ,
k = 0 , 1 , , 10 .

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