Abstract

An electromagnetic method based on rigorous diffraction theory of gratings is applied to the analysis of fields in semiconductor laser cavities. The method is based on the Fourier modal method; it is fully rigorous for infinitely periodic resonators and highly accurate for single resonators when absorbing boundary conditions are applied. Fundamental-mode intracavity and near-field distributions are evaluated for some selected geometries, and resonance frequencies are predicted.

© 2006 Optical Society of America

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    [CrossRef]
  4. P. Vahimaa and J. Turunen, "Electromagnetic analysis of waveguide Bragg reflectors," in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, 1998), pp. 69-71.
  5. P. Lalanne and E. Silberstein, "Fourier-modal method applied to waveguide computational problems," Opt. Lett. 25, 1092-1094 (2000).
    [CrossRef]
  6. J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, "Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer's star product," Opt. Commun. 198, 265-272 (2001).
    [CrossRef]
  7. P. Dansaṡ and N. Paraire, "Fast modeling of photonic bandgap structures by use of a diffraction-grating approach," J. Opt. Soc. Am. A 15, 1586-1598 (1998).
    [CrossRef]
  8. P. Lalanne and H. Benistry, "Out-of-plane losses of two-dimensional photonic crystal waveguides: electromagnetic analysis," J. Appl. Phys. 89, 1512-1524 (2001).
    [CrossRef]
  9. M. G. Moharam and A. Greenwell, "Integrated output grating coupler in semiconductor lasers," in 2004 ICO International Conference on Optics and Photonics in Technology Frontier (International Commission for Optics, 2004), pp. 543-544.
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    [CrossRef] [PubMed]
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    [CrossRef]
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  18. R. Redheffer, "Difference equations and functional equations in transmission-line theory," in Modern Mathematics for the Engineer, E.F.Beckenbach, ed. (McGraw-Hill, 1961), Chap. 12, pp. 282-337.
  19. R. J. Nelson, R. B. Wilson, P. D. Wright, P. A. Barnes, and N. K. Dutta, "CW electrooptical properties of InGaAsP buried-heterostructure lasers," IEEE J. Quantum Electron. QE-17, 202-207 (1981).
    [CrossRef]
  20. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), Chap. 3.
    [CrossRef]
  21. E. Noponen and J. Turunen, "Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles," J. Opt. Soc. Am. A 11, 2494-2502 (1994).
    [CrossRef]
  22. L. Li, "Mathematical reflections on the Fourier modal method in grating theory," in Mathematical Modeling in Optical Science, G.Bao, L.Cowsar, and W.Masters, eds. (Society for Industrial and Applied Mathematics, 2001), pp. 111-139.
    [CrossRef]

2005 (1)

2003 (1)

2001 (2)

P. Lalanne and H. Benistry, "Out-of-plane losses of two-dimensional photonic crystal waveguides: electromagnetic analysis," J. Appl. Phys. 89, 1512-1524 (2001).
[CrossRef]

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, "Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer's star product," Opt. Commun. 198, 265-272 (2001).
[CrossRef]

2000 (1)

1998 (2)

P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, "Guided-mode propagation through an ion-exchanged graded-index boundary," Opt. Commun. 14, 247-253 (1998).
[CrossRef]

P. Dansaṡ and N. Paraire, "Fast modeling of photonic bandgap structures by use of a diffraction-grating approach," J. Opt. Soc. Am. A 15, 1586-1598 (1998).
[CrossRef]

1996 (2)

1994 (2)

1981 (1)

R. J. Nelson, R. B. Wilson, P. D. Wright, P. A. Barnes, and N. K. Dutta, "CW electrooptical properties of InGaAsP buried-heterostructure lasers," IEEE J. Quantum Electron. QE-17, 202-207 (1981).
[CrossRef]

1978 (1)

Aalto, T.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, "Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer's star product," Opt. Commun. 198, 265-272 (2001).
[CrossRef]

Barnes, P. A.

R. J. Nelson, R. B. Wilson, P. D. Wright, P. A. Barnes, and N. K. Dutta, "CW electrooptical properties of InGaAsP buried-heterostructure lasers," IEEE J. Quantum Electron. QE-17, 202-207 (1981).
[CrossRef]

Benistry, H.

P. Lalanne and H. Benistry, "Out-of-plane losses of two-dimensional photonic crystal waveguides: electromagnetic analysis," J. Appl. Phys. 89, 1512-1524 (2001).
[CrossRef]

Dansa?, P.

Dutta, N. K.

R. J. Nelson, R. B. Wilson, P. D. Wright, P. A. Barnes, and N. K. Dutta, "CW electrooptical properties of InGaAsP buried-heterostructure lasers," IEEE J. Quantum Electron. QE-17, 202-207 (1981).
[CrossRef]

Greenwell, A.

M. G. Moharam and A. Greenwell, "Integrated output grating coupler in semiconductor lasers," in 2004 ICO International Conference on Optics and Photonics in Technology Frontier (International Commission for Optics, 2004), pp. 543-544.

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech, 2000).

Heimala, P.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, "Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer's star product," Opt. Commun. 198, 265-272 (2001).
[CrossRef]

Knop, K.

Kuittinen, M.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, "Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer's star product," Opt. Commun. 198, 265-272 (2001).
[CrossRef]

P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, "Guided-mode propagation through an ion-exchanged graded-index boundary," Opt. Commun. 14, 247-253 (1998).
[CrossRef]

Lalanne, P.

P. Lalanne and H. Benistry, "Out-of-plane losses of two-dimensional photonic crystal waveguides: electromagnetic analysis," J. Appl. Phys. 89, 1512-1524 (2001).
[CrossRef]

P. Lalanne and E. Silberstein, "Fourier-modal method applied to waveguide computational problems," Opt. Lett. 25, 1092-1094 (2000).
[CrossRef]

Leppihalme, M.

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, "Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer's star product," Opt. Commun. 198, 265-272 (2001).
[CrossRef]

Li, L.

Liñares, J.

P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, "Guided-mode propagation through an ion-exchanged graded-index boundary," Opt. Commun. 14, 247-253 (1998).
[CrossRef]

Lopez Lago, E.

P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, "Guided-mode propagation through an ion-exchanged graded-index boundary," Opt. Commun. 14, 247-253 (1998).
[CrossRef]

Moharam, M. G.

M. G. Moharam and A. Greenwell, "Integrated output grating coupler in semiconductor lasers," in 2004 ICO International Conference on Optics and Photonics in Technology Frontier (International Commission for Optics, 2004), pp. 543-544.

Nelson, R. J.

R. J. Nelson, R. B. Wilson, P. D. Wright, P. A. Barnes, and N. K. Dutta, "CW electrooptical properties of InGaAsP buried-heterostructure lasers," IEEE J. Quantum Electron. QE-17, 202-207 (1981).
[CrossRef]

Noponen, E.

Paraire, N.

Redheffer, R.

R. Redheffer, "Difference equations and functional equations in transmission-line theory," in Modern Mathematics for the Engineer, E.F.Beckenbach, ed. (McGraw-Hill, 1961), Chap. 12, pp. 282-337.

Saarinen, J.

P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, "Guided-mode propagation through an ion-exchanged graded-index boundary," Opt. Commun. 14, 247-253 (1998).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), Chap. 3.
[CrossRef]

Salmio, R.-P.

P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, "Guided-mode propagation through an ion-exchanged graded-index boundary," Opt. Commun. 14, 247-253 (1998).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Silberstein, E.

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech, 2000).

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), Chap. 3.
[CrossRef]

Tervo, J.

T. Vallius, J. Tervo, P. Vahimaa, and J. Turunen, "Electromagnetic approach to laser resonator analysis," Opt. Express 13, 5994-5999 (2005).
[CrossRef] [PubMed]

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, "Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer's star product," Opt. Commun. 198, 265-272 (2001).
[CrossRef]

Turunen, J.

T. Vallius, J. Tervo, P. Vahimaa, and J. Turunen, "Electromagnetic approach to laser resonator analysis," Opt. Express 13, 5994-5999 (2005).
[CrossRef] [PubMed]

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, "Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer's star product," Opt. Commun. 198, 265-272 (2001).
[CrossRef]

P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, "Guided-mode propagation through an ion-exchanged graded-index boundary," Opt. Commun. 14, 247-253 (1998).
[CrossRef]

E. Noponen and J. Turunen, "Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles," J. Opt. Soc. Am. A 11, 2494-2502 (1994).
[CrossRef]

P. Vahimaa and J. Turunen, "Electromagnetic analysis of waveguide Bragg reflectors," in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, 1998), pp. 69-71.

Vahimaa, P.

T. Vallius, J. Tervo, P. Vahimaa, and J. Turunen, "Electromagnetic approach to laser resonator analysis," Opt. Express 13, 5994-5999 (2005).
[CrossRef] [PubMed]

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, "Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer's star product," Opt. Commun. 198, 265-272 (2001).
[CrossRef]

P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, "Guided-mode propagation through an ion-exchanged graded-index boundary," Opt. Commun. 14, 247-253 (1998).
[CrossRef]

P. Vahimaa and J. Turunen, "Electromagnetic analysis of waveguide Bragg reflectors," in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, 1998), pp. 69-71.

Vallius, T.

Wilson, R. B.

R. J. Nelson, R. B. Wilson, P. D. Wright, P. A. Barnes, and N. K. Dutta, "CW electrooptical properties of InGaAsP buried-heterostructure lasers," IEEE J. Quantum Electron. QE-17, 202-207 (1981).
[CrossRef]

Wright, P. D.

R. J. Nelson, R. B. Wilson, P. D. Wright, P. A. Barnes, and N. K. Dutta, "CW electrooptical properties of InGaAsP buried-heterostructure lasers," IEEE J. Quantum Electron. QE-17, 202-207 (1981).
[CrossRef]

Yariv, A.

A. Yariv, Optical Electronics, 3rd ed. (Holt, Rinehart and Winston, 1985).

IEEE J. Quantum Electron. (1)

R. J. Nelson, R. B. Wilson, P. D. Wright, P. A. Barnes, and N. K. Dutta, "CW electrooptical properties of InGaAsP buried-heterostructure lasers," IEEE J. Quantum Electron. QE-17, 202-207 (1981).
[CrossRef]

J. Appl. Phys. (1)

P. Lalanne and H. Benistry, "Out-of-plane losses of two-dimensional photonic crystal waveguides: electromagnetic analysis," J. Appl. Phys. 89, 1512-1524 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Opt. Commun. (2)

P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, "Guided-mode propagation through an ion-exchanged graded-index boundary," Opt. Commun. 14, 247-253 (1998).
[CrossRef]

J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, "Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer's star product," Opt. Commun. 198, 265-272 (2001).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Other (9)

M. G. Moharam and A. Greenwell, "Integrated output grating coupler in semiconductor lasers," in 2004 ICO International Conference on Optics and Photonics in Technology Frontier (International Commission for Optics, 2004), pp. 543-544.

P. Vahimaa and J. Turunen, "Electromagnetic analysis of waveguide Bragg reflectors," in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, 1998), pp. 69-71.

A. Yariv, Optical Electronics, 3rd ed. (Holt, Rinehart and Winston, 1985).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech, 2000).

A. E. Siegman, Lasers (University Science, 1986).

R.Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
[CrossRef]

R. Redheffer, "Difference equations and functional equations in transmission-line theory," in Modern Mathematics for the Engineer, E.F.Beckenbach, ed. (McGraw-Hill, 1961), Chap. 12, pp. 282-337.

L. Li, "Mathematical reflections on the Fourier modal method in grating theory," in Mathematical Modeling in Optical Science, G.Bao, L.Cowsar, and W.Masters, eds. (Society for Industrial and Applied Mathematics, 2001), pp. 111-139.
[CrossRef]

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), Chap. 3.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

General y-invariant periodic resonator configuration with absorbing boundaries (shaded regions around x = 0 and x = d ).

Fig. 2
Fig. 2

Amplification of the InGaAsP layer as a function of the wavelength.

Fig. 3
Fig. 3

Intracavity and near-field distributions of the InGaAsP resonator at λ = 1.3 μ m .

Fig. 4
Fig. 4

Distribution of the resonator modes of the InGaAsP resonator. Resonance takes place where the curves equal zero.

Fig. 5
Fig. 5

Intracavity and near-field distributions of the periodic resonator.

Equations (21)

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K U = Γ U ,
E y , 0 ( x , z ) = m B m , 0 exp [ i k m , x x i k m , z , 0 ( z z 1 ) ]
E y , J ( x , z ) = m A m , J exp [ i k m , x x + i k m , z , J ( z z J ) ]
k m , z , j = { [ ( k n j ) 2 k m , x 2 ] 1 2 , k n j > k m , x i [ k m , x 2 ( k n j ) 2 ] 1 2 , otherwise } , j = ( 0 , J ) .
M j E j = E j γ j 2 ,
E y , j ( x , z ) = l { A l , j exp [ i γ l , j ( z z j ) ] + B l , j exp [ i γ l , j ( z z j + 1 ) ] } m E m l , j exp ( i 2 π m x d ) .
ξ j 1 N j H j = H j γ j 2 ,
[ A j B j ] = S j j [ A j B j ] ,
S j j = [ T j j ( u u ) R j j ( u d ) R j j ( d u ) T j j ( d d ) ] .
[ A p B 0 ] = S 0 p [ 0 B p ] ,
[ A J B p ] = S p J [ A p 0 ] ,
A p = R 0 p ( u d ) B p ,
B 0 = T 0 p ( d d ) B p ,
B p = R p J ( d u ) A p ,
A J = T p J ( u u ) A p .
K 0 p J A p = A p .
K 0 p J A p = A p Γ p ,
[ S ( 1 ) S ( 2 ) ] S ( 3 ) = S ( 1 ) [ S ( 2 ) S ( 3 ) ]
[ S ( 1 ) S ( j 1 ) ] S ( j ) [ S ( j + 1 ) S ( J ) ] .
B j = [ I R j j + 1 ( d u ) R 0 j ( u d ) ] 1 T j j + 1 ( d d ) B j + 1
A j = [ I R j 1 j ( u d ) R j J ( d u ) ] 1 T j 1 j ( u u ) A j 1 .

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