Abstract

In the present work, a stable algorithm for the calculation of the electromagnetic field distributions of the eigenmodes of one-dimensional diffraction gratings is presented. The proposed approach is based on the method for the computation of the propagation constants of Bloch waves of such structures previously presented by Cao et al. [J. Opt. Soc. Am. A 19, 335 (2002)] and uses a modified S-matrix algorithm to ensure numerical stability.

© 2012 Optical Society of America

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References

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  1. M. G. Moharam, E. B. Grann, and D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  2. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  3. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  4. P. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. 25, 1092–1094 (2000).
    [CrossRef]
  5. 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]
  6. M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Aperiodic Fourier modal method in contrast-field from finite structures,” J. Opt. Soc. Am. A 27, 2423–2431 (2010).
    [CrossRef]
  7. M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Modified S-matrix algorithm for the aperiodic Fourier modal method in contrast-field formulation,” J. Opt. Soc. Am. A 28, 1364–1371 (2011).
    [CrossRef]
  8. Q. Cao, P. Lalanne, and J.-P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A 19, 335–338 (2002).
    [CrossRef]
  9. G. Lecamp, J. P. Hugonin, and P. Lalanne, “Theoretical and computational concepts for periodic optical waveguides,” Opt. Express 15, 11042–11060 (2007).
    [CrossRef]
  10. T. Vallius, J. Tervo, P. Vahimaa, and J. Turunen, “Electromagnetic field computation in semiconductor laser resonators,” J. Opt. Soc. Am. A 23, 906–911 (2006).
    [CrossRef]
  11. E. L. Tan, “Hybrid-matrix algorithm for rigorous coupled-wave analysis of multilayered diffraction gratings,” J. Mod. Opt. 53, 417–428 (2006).
    [CrossRef]
  12. J. Ning and E. L. Tan, “Generalized eigenproblem of hybrid matrix for Bloch–Floquet waves in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 26, 676–683 (2009).
    [CrossRef]
  13. Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-Fa Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
    [CrossRef]
  14. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
    [CrossRef]
  15. E. L. Tan, “Note on formulation of the enhanced scattering- (transmittance-) matrix approach,” J. Opt. Soc. Am. A 19, 1157–1161 (2002).
    [CrossRef]
  16. K. C. Chang, V. Shah, and T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. 70, 804–813 (1980).
    [CrossRef]
  17. E. A. Bezus, D. A. Bykov, L. L. Doskolovich, and I. I. Kadomin, “Diffraction gratings for generating varying-period interference patterns of surface plasmons,” J. Opt. A Pure Appl. Opt. 10, 095204 (2008).
    [CrossRef]
  18. P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. 45, 1357–1374 (1998).
    [CrossRef]
  19. E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Evanescent-wave interferometric nanoscale photolithography using guided-mode resonant gratings,” Microelectron. Eng. 88, 170–174 (2011).
    [CrossRef]

2011 (2)

E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Evanescent-wave interferometric nanoscale photolithography using guided-mode resonant gratings,” Microelectron. Eng. 88, 170–174 (2011).
[CrossRef]

M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Modified S-matrix algorithm for the aperiodic Fourier modal method in contrast-field formulation,” J. Opt. Soc. Am. A 28, 1364–1371 (2011).
[CrossRef]

2010 (1)

2009 (1)

2008 (1)

E. A. Bezus, D. A. Bykov, L. L. Doskolovich, and I. I. Kadomin, “Diffraction gratings for generating varying-period interference patterns of surface plasmons,” J. Opt. A Pure Appl. Opt. 10, 095204 (2008).
[CrossRef]

2007 (1)

2006 (2)

T. Vallius, J. Tervo, P. Vahimaa, and J. Turunen, “Electromagnetic field computation in semiconductor laser resonators,” J. Opt. Soc. Am. A 23, 906–911 (2006).
[CrossRef]

E. L. Tan, “Hybrid-matrix algorithm for rigorous coupled-wave analysis of multilayered diffraction gratings,” J. Mod. Opt. 53, 417–428 (2006).
[CrossRef]

2002 (2)

2001 (1)

2000 (1)

1998 (1)

P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. 45, 1357–1374 (1998).
[CrossRef]

1996 (2)

1995 (3)

1980 (1)

Bezus, E. A.

E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Evanescent-wave interferometric nanoscale photolithography using guided-mode resonant gratings,” Microelectron. Eng. 88, 170–174 (2011).
[CrossRef]

E. A. Bezus, D. A. Bykov, L. L. Doskolovich, and I. I. Kadomin, “Diffraction gratings for generating varying-period interference patterns of surface plasmons,” J. Opt. A Pure Appl. Opt. 10, 095204 (2008).
[CrossRef]

Bykov, D. A.

E. A. Bezus, D. A. Bykov, L. L. Doskolovich, and I. I. Kadomin, “Diffraction gratings for generating varying-period interference patterns of surface plasmons,” J. Opt. A Pure Appl. Opt. 10, 095204 (2008).
[CrossRef]

Cao, Q.

Chang, K. C.

Doskolovich, L. L.

E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Evanescent-wave interferometric nanoscale photolithography using guided-mode resonant gratings,” Microelectron. Eng. 88, 170–174 (2011).
[CrossRef]

E. A. Bezus, D. A. Bykov, L. L. Doskolovich, and I. I. Kadomin, “Diffraction gratings for generating varying-period interference patterns of surface plasmons,” J. Opt. A Pure Appl. Opt. 10, 095204 (2008).
[CrossRef]

Gaylord, T. K.

Grann, E. B.

Hugonin, J. P.

Hugonin, J.-P.

Jurek, M. P.

P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. 45, 1357–1374 (1998).
[CrossRef]

Kadomin, I. I.

E. A. Bezus, D. A. Bykov, L. L. Doskolovich, and I. I. Kadomin, “Diffraction gratings for generating varying-period interference patterns of surface plasmons,” J. Opt. A Pure Appl. Opt. 10, 095204 (2008).
[CrossRef]

Kazanskiy, N. L.

E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Evanescent-wave interferometric nanoscale photolithography using guided-mode resonant gratings,” Microelectron. Eng. 88, 170–174 (2011).
[CrossRef]

Kingsland, D. M.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-Fa Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

Lalanne, P.

Lecamp, G.

Lee, J.-Fa

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-Fa Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

Lee, R.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-Fa Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

Li, L.

Mattheij, R.

Maubach, J.

Moharam, M. G.

Ning, J.

Pisarenco, M.

Pommet, D. A.

Sacks, Z. S.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-Fa Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

Setija, I.

Shah, V.

Silberstein, E.

Tamir, T.

Tan, E. L.

Tervo, J.

Turunen, J.

Vahimaa, P.

Vallius, T.

IEEE Trans. Antennas Propag. (1)

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-Fa Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
[CrossRef]

J. Mod. Opt. (2)

P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. 45, 1357–1374 (1998).
[CrossRef]

E. L. Tan, “Hybrid-matrix algorithm for rigorous coupled-wave analysis of multilayered diffraction gratings,” J. Mod. Opt. 53, 417–428 (2006).
[CrossRef]

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

E. A. Bezus, D. A. Bykov, L. L. Doskolovich, and I. I. Kadomin, “Diffraction gratings for generating varying-period interference patterns of surface plasmons,” J. Opt. A Pure Appl. Opt. 10, 095204 (2008).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (10)

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]

Q. Cao, P. Lalanne, and J.-P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A 19, 335–338 (2002).
[CrossRef]

E. L. Tan, “Note on formulation of the enhanced scattering- (transmittance-) matrix approach,” J. Opt. Soc. Am. A 19, 1157–1161 (2002).
[CrossRef]

T. Vallius, J. Tervo, P. Vahimaa, and J. Turunen, “Electromagnetic field computation in semiconductor laser resonators,” J. Opt. Soc. Am. A 23, 906–911 (2006).
[CrossRef]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
[CrossRef]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[CrossRef]

M. G. Moharam, E. B. Grann, and D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
[CrossRef]

M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
[CrossRef]

M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Aperiodic Fourier modal method in contrast-field from finite structures,” J. Opt. Soc. Am. A 27, 2423–2431 (2010).
[CrossRef]

M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Modified S-matrix algorithm for the aperiodic Fourier modal method in contrast-field formulation,” J. Opt. Soc. Am. A 28, 1364–1371 (2011).
[CrossRef]

J. Opt. Soc. Am. B (1)

Microelectron. Eng. (1)

E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Evanescent-wave interferometric nanoscale photolithography using guided-mode resonant gratings,” Microelectron. Eng. 88, 170–174 (2011).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

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Figures (5)

Fig. 1.
Fig. 1.

Geometry of the structure.

Fig. 2.
Fig. 2.

Studied diffraction grating. Parameters of the structure: εsuper=1, εgr=εsub=2.56, εm=12.9222+0.4473i (Ag), d=1539nm, w=d/2, hgr=435nm, hm=65nm.

Fig. 3.
Fig. 3.

Electric field intensity distribution in the considered structure at normal incidence of TM-polarized plane wave with λ=550nm (the grating is shown with white dashed lines).

Fig. 4.
Fig. 4.

Distributions of the electric field intensity |E|2 corresponding to the interference of two counterpropagating eigenmodes of the considered structure calculated using the standard algorithm at (a) M=101 and (b) M=201. Dotted rectangle in (b) denotes the area where the numerical overflow occurs.

Fig. 5.
Fig. 5.

Distributions of the electric field intensity |E|2 corresponding to the interference of two counter-propagating eigenmodes of the considered structure calculated using the proposed stable modification of the algorithm at (a) M=401 and (b) M=801.

Equations (31)

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2Hy(n)z2+εxx(n)(x)x[1εzz(n)(x)Hy(n)x]+k02μ(n)(x)εxx(n)(x)Hy(n)=0,
Hy(n)(x,z)=l{cl,n,L+exp[k0ql,n(zzn1)]mwm,l,nexp(ikx,mx)+cl,n,Lexp[k0ql,n(zzn1)]mwm,l,nexp(ikx,mx)},
(cn,L+c1,L)=SL(n)(c1,L+cn,L),
Ex(n)(x,z)=il{cl,n,L+exp[k0ql,n(zzn1)]mvm,l,nexp(ikx,mx)cl,n,Lexp[k0ql,n(zzn1)]mvm,l,nexp(ikx,mx)},
(Wn1Wn1Vn1Vn1)(Xn100Xn11)(cn1,L+cn1,L)=(WnWnVnVn)(cn,L+cn,L),
sL(n)=(I00Xn1)(WnWn1VnVn1)1(Wn1WnVn1Vn)(Xn100I).
T++,L(n)=t++,L(n)(IR+,L(n1)r+,L(n))1T++,L(n1),R+,L(n)=r+,L(n)+t++,L(n)R+,L(n1)(Ir+,L(n)R+,L(n1))1t,L(n),R+,L(n)=R+,L(n1)+T,L(n1)r+,L(n)(IR+,L(n1)r+,L(n))1T++,L(n1),T,L(n)=T,L(n1)(Ir+,L(n)R+,L(n1))1t,L(n),
SL(i)=(T++,L(i)R+,L(i)R+,L(i)T,L(i)),sL(i)=(t++,L(i)r+,L(i)r+,L(i)t,L(i)).
(cN+1,L+c1,L)=SL(c1,L+cN+1,L).
(cN+1,L+cN+1,L)=β(c1,L+c1,L),
(T++,L0R+,LI)(c1,L+c1,L)=(IR+,L0T,L)(cN+1,L+cN+1,L).
(T++,L0R+,LI)(c1,L+c1,L)=β(IR+,L0T,L)(c1,L+c1,L).
β=exp[ik0(neff+neffi)d]=exp(ik0neffd)exp(k0neffd).
neff=ln|β|k0d.
neff=2πl+argβk0d,
cn,L=12Xn{(Wn1Wn+1Vn1Vn+1)cn+1,L++(Wn1Wn+1+Vn1Vn+1)cn+1,L}.
cn,L+=T++,L(n)c1,L++R+,L(n)cn,L.
Hy(n)(x,z)=l{cl,n+exp[k0ql,n(zzn1)]mwm,l,nexp(ikx,mx)+cl,nexp[k0ql,n(zzn)]mwm,l,nexp(ikx,mx)}.
cn+=cn,L+,cn=cn,R,
(cN+1,R+cn,R)=SR(n)(cn,R+cN+1,R).
T++,R(n)=T++,R(n+1)(Ir+,R(n)R+,R(n+1))1t++,R(n+1),R+,R(n)=R+,R(n+1)+T++,R(n+1)r+,R(n)(IR+,R(n+1)r+,R(n))1T,R(n+1),R+,R(n)=r+,R(n)+t,R(n)R+,R(n+1)(Ir+,R(n)R+,R(n+1))1t++,R(n),T,R(n)=t,R(n)(IR+,R(n+1)r+,R(n))1T,R(n+1),
(I00X1)(cn+c1)=(T++,L(n)R+,L(n)R+,L(n)T,L(n))(I00Xn)(c1+cn),
(XN+100I)(cN+1+cn)=(T++,R(n)R+,R(n)R+,R(n)T,R(n))(Xn00I)(cn+cN+1).
cn+=(IR+,L(n)XnR+,R(n)Xn)1(T++,L(n)c1++R+,L(n)XnT,R(n)cN+1).
cn=(IR+,R(n)XnR+,L(n)Xn)1(R+,R(n)XnT++,L(n)c1++T,R(n)cN+1).
cn+=(IR+,L(n)XnR+,R(n)Xn)1(T++,L(n)c1++R+,L(n)XnT,R(n)c1β),cn=(IR+,R(n)XnR+,L(n)Xn)1(R+,R(n)XnT++,L(n)c1++T,R(n)c1β).
cN+1+=(IR+,L(N+1)XN+1R+,R(N+1)XN+1)1(T++,L(N+1)R+,L(N+1)XN+1T,R(N+1))(c1+cN+1).
c1=(IR+,R(1)X1R+,L(1)X1)1(R+,R(1)X1T++,L(1)T,R(1))(c1+cN+1).
(cN+1+c1)=(T++,L(N+1)R+,L(N+1)XN+1R+,R(1)X1T,R(1))(c1+cN+1).
(cN+1+cN+1)=β(c1+c1).
(T++,L(N+1)0R+,R(1)X1I)(c1+c1)=β(IR+,L(N+1)X10T,R(1))(c1+c1).

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