Abstract

A new analytical approximation for photonic array modes is presented. We consider the specific class of one-dimensional (1D) photonic crystals (encompassing large arrays of coupled identical planar waveguides, large arrays of identical phase-locked lasers, etc.), in which light propagates along the optical axis of the device. Approximate analytical expressions for the array modes (both spatial distribution and propagation constants) become available. This approach allows a fast, simple, and accurate analytical evaluation of the electromagnetic field in 1D photonic crystal devices.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 1995).
  2. E. Yablonovitch, T. J. Gmitter, and R. Bhat, “Inhibited and enhanced spontaneous emission from optically thin AlGaAs/GaAs double heterostructures,” Phys. Rev. Lett. 61, 2546–2549 (1988).
    [CrossRef]
  3. S. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162–165 (2001).
    [CrossRef]
  4. J. B. Pendry, “Calculating photonic band structure,” J. Phys. Condens. Matter 8, 1085–1108 (1996).
    [CrossRef]
  5. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  6. S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a plane-wave basis,” Opt. Express 8, 173–190 (2001).
    [CrossRef]
  7. E. Lidorikis, M. M. Sigalas, and C. M. Soukoulis, “Tight-binding parameterization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
    [CrossRef]
  8. H. Kogelnik, Theory of dielectric waveguides in integrated optics (Springer-Verlag, 1975).
  9. A. A. Hardy, W. Streifer, and M. Osinski, “Coupled mode equations for multiwaveguide systems in isotropic or anisotropic media,” Opt. Lett. 11, 742–744 (1986).
    [CrossRef]
  10. A. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
    [CrossRef]
  11. A. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972).
    [CrossRef]
  12. A. A. Hardy and E. Kapon, “Coupled-mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 966–971 (1996).
    [CrossRef]
  13. V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215–224 (2007).
    [CrossRef]
  14. E. Smith, V. Shteeman, E. Kapon, and A. A. Hardy, “Fast approximate derivation of photonic supermodes in one-dimensional photonic crystal devices,” in IEEE 27th Convention of Electrical & Electronics Engineers in Israel (IEEEI) (IEEE, 2012).
  15. C. Kittel, Quantum Theory of Solids (Wiley, 1987).
  16. L. Coldren and S. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

2007 (1)

V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215–224 (2007).
[CrossRef]

2001 (2)

1998 (1)

E. Lidorikis, M. M. Sigalas, and C. M. Soukoulis, “Tight-binding parameterization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[CrossRef]

1996 (2)

J. B. Pendry, “Calculating photonic band structure,” J. Phys. Condens. Matter 8, 1085–1108 (1996).
[CrossRef]

A. A. Hardy and E. Kapon, “Coupled-mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 966–971 (1996).
[CrossRef]

1988 (1)

E. Yablonovitch, T. J. Gmitter, and R. Bhat, “Inhibited and enhanced spontaneous emission from optically thin AlGaAs/GaAs double heterostructures,” Phys. Rev. Lett. 61, 2546–2549 (1988).
[CrossRef]

1986 (1)

1985 (1)

A. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
[CrossRef]

1972 (1)

Bhat, R.

E. Yablonovitch, T. J. Gmitter, and R. Bhat, “Inhibited and enhanced spontaneous emission from optically thin AlGaAs/GaAs double heterostructures,” Phys. Rev. Lett. 61, 2546–2549 (1988).
[CrossRef]

Boiko, D.

V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215–224 (2007).
[CrossRef]

Coldren, L.

L. Coldren and S. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

Corzine, S.

L. Coldren and S. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

Fan, S.

Gmitter, T. J.

E. Yablonovitch, T. J. Gmitter, and R. Bhat, “Inhibited and enhanced spontaneous emission from optically thin AlGaAs/GaAs double heterostructures,” Phys. Rev. Lett. 61, 2546–2549 (1988).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Hardy, A. A.

V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215–224 (2007).
[CrossRef]

A. A. Hardy and E. Kapon, “Coupled-mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 966–971 (1996).
[CrossRef]

A. A. Hardy, W. Streifer, and M. Osinski, “Coupled mode equations for multiwaveguide systems in isotropic or anisotropic media,” Opt. Lett. 11, 742–744 (1986).
[CrossRef]

A. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
[CrossRef]

E. Smith, V. Shteeman, E. Kapon, and A. A. Hardy, “Fast approximate derivation of photonic supermodes in one-dimensional photonic crystal devices,” in IEEE 27th Convention of Electrical & Electronics Engineers in Israel (IEEEI) (IEEE, 2012).

Haus, H. A.

Joannopoulos, J.

S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a plane-wave basis,” Opt. Express 8, 173–190 (2001).
[CrossRef]

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 1995).

Joannopoulos, J. D.

Johnson, S.

Johnson, S. G.

Kapon, E.

V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215–224 (2007).
[CrossRef]

A. A. Hardy and E. Kapon, “Coupled-mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 966–971 (1996).
[CrossRef]

E. Smith, V. Shteeman, E. Kapon, and A. A. Hardy, “Fast approximate derivation of photonic supermodes in one-dimensional photonic crystal devices,” in IEEE 27th Convention of Electrical & Electronics Engineers in Israel (IEEEI) (IEEE, 2012).

Kittel, C.

C. Kittel, Quantum Theory of Solids (Wiley, 1987).

Kogelnik, H.

H. Kogelnik, Theory of dielectric waveguides in integrated optics (Springer-Verlag, 1975).

Lidorikis, E.

E. Lidorikis, M. M. Sigalas, and C. M. Soukoulis, “Tight-binding parameterization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[CrossRef]

Manolatou, C.

Meade, R.

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 1995).

Osinski, M.

Pendry, J. B.

J. B. Pendry, “Calculating photonic band structure,” J. Phys. Condens. Matter 8, 1085–1108 (1996).
[CrossRef]

Shteeman, V.

V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215–224 (2007).
[CrossRef]

E. Smith, V. Shteeman, E. Kapon, and A. A. Hardy, “Fast approximate derivation of photonic supermodes in one-dimensional photonic crystal devices,” in IEEE 27th Convention of Electrical & Electronics Engineers in Israel (IEEEI) (IEEE, 2012).

Sigalas, M. M.

E. Lidorikis, M. M. Sigalas, and C. M. Soukoulis, “Tight-binding parameterization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[CrossRef]

Smith, E.

E. Smith, V. Shteeman, E. Kapon, and A. A. Hardy, “Fast approximate derivation of photonic supermodes in one-dimensional photonic crystal devices,” in IEEE 27th Convention of Electrical & Electronics Engineers in Israel (IEEEI) (IEEE, 2012).

Snyder, A.

Soukoulis, C. M.

E. Lidorikis, M. M. Sigalas, and C. M. Soukoulis, “Tight-binding parameterization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[CrossRef]

Streifer, W.

A. A. Hardy, W. Streifer, and M. Osinski, “Coupled mode equations for multiwaveguide systems in isotropic or anisotropic media,” Opt. Lett. 11, 742–744 (1986).
[CrossRef]

A. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Winn, J.

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 1995).

Yablonovitch, E.

E. Yablonovitch, T. J. Gmitter, and R. Bhat, “Inhibited and enhanced spontaneous emission from optically thin AlGaAs/GaAs double heterostructures,” Phys. Rev. Lett. 61, 2546–2549 (1988).
[CrossRef]

IEEE J. Quantum Electron. (2)

A. A. Hardy and E. Kapon, “Coupled-mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 966–971 (1996).
[CrossRef]

V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215–224 (2007).
[CrossRef]

J. Lightwave Technol. (1)

A. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3, 1135–1146 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. Condens. Matter (1)

J. B. Pendry, “Calculating photonic band structure,” J. Phys. Condens. Matter 8, 1085–1108 (1996).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. Lett. (2)

E. Lidorikis, M. M. Sigalas, and C. M. Soukoulis, “Tight-binding parameterization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[CrossRef]

E. Yablonovitch, T. J. Gmitter, and R. Bhat, “Inhibited and enhanced spontaneous emission from optically thin AlGaAs/GaAs double heterostructures,” Phys. Rev. Lett. 61, 2546–2549 (1988).
[CrossRef]

Other (6)

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 1995).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

H. Kogelnik, Theory of dielectric waveguides in integrated optics (Springer-Verlag, 1975).

E. Smith, V. Shteeman, E. Kapon, and A. A. Hardy, “Fast approximate derivation of photonic supermodes in one-dimensional photonic crystal devices,” in IEEE 27th Convention of Electrical & Electronics Engineers in Israel (IEEEI) (IEEE, 2012).

C. Kittel, Quantum Theory of Solids (Wiley, 1987).

L. Coldren and S. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

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 (7)

Fig. 1.
Fig. 1.

Schematics of a 1D periodic array of parallel coupled lossless waveguides (1D photonic crystal). Injected light is confined in the transverse direction (x axis) and propagates along the optical axis of the device (z axis).

Fig. 2.
Fig. 2.

(a) Schematics of 1D photonic crystal of parabolic profile waveguides (curve 1) and “envelope planar waveguide,” being the envelope of the overall photonic crystal profile (curve 2). (b) First array mode of the photonic crystal (curve 3) and fundamental guided mode of the “envelope planar waveguide” (curve 4), depicted in subplot (a); circles—partial amplitudes—components of the envelope function of the array mode.

Fig. 3.
Fig. 3.

(a) Fundamental array mode of 1D photonic crystal of 20 identical single-mode planar waveguides (see also Table 2). Black curve, CMT formalism; gray curve, numerical solution of Helmholtz equation. (b) Gray curve, fundamental mode of the envelope planar waveguide ψ1(x) [Eq. (4)]; black circles ψ1(x), evaluated at the centers of the photonic crystal sites (i.e., ψ1(xp)). (c) Envelope of the first array mode, as computed with the standard CMT (gray circles) and with the analytical approximation (black daggers).

Fig. 4.
Fig. 4.

Propagation constants of 1D photonic crystal of 20 identical single-mode waveguides (see also Table 2). Gray circles, standard CMT formalism; black wildcards, analytical approximation.

Fig. 5.
Fig. 5.

Dominant subsets of the envelopes of the three specific array modes of the array of 20 identical multimode planar waveguides (see also Table 2), as computed with the standard CMT (gray circles) and with the analytical approximation (black daggers) (a) [u0{1,1}{1,1},,u0{1,1}{N,1}] and [u0{1,1}(approx){1,1},,u0{1,1}(approx){N,1}] (contribution of the first order solitary modes to the first array mode of the first group); (b) [u0{2,2}{1,1},,u0{2,2}{N,1}] and [u0{2,2}(approx){1,1},,u0{2,2}(approx){N,1}] (contribution of the second-order solitary modes to the first array mode of the second group); (c) [u0{3,3}{1,1},,u0{3,3}{N,1}] and [u0{3,3}(approx){1,1},,u0{3,3}(approx){N,1}] (contribution of the third-order solitary modes to the first array mode of the third band). The total modal fields are shown in Fig. 6.

Fig. 6.
Fig. 6.

Modal fields of the first array modes of the first [subplot (a)], second [subplot (b)] and third [subplot (c)] groups of 1D photonic crystal of 20 identical multimode planar waveguides (see also Table 2). Black curves, CMT formalism; gray curves, analytical approximation. The dominant subsets of the corresponding envelopes are shown in Fig. 5.

Fig. 7.
Fig. 7.

Propagation constants of 1D photonic crystal of 20 identical multimode waveguides (see also Table 2). Gray circles, standard CMT formalism; black wildcards, analytical approximation.

Tables (4)

Tables Icon

Table 1. Estimated Computation Time Required for the Evaluation of the Envelopes, Propagation Constants, and Array Modes of Two Specific 1D Photonic Crystalsa

Tables Icon

Table 2. Parameters of Solitary Single- and Multimode Waveguides Comprising 1D Photonic Crystals of 20 Identical Waveguides

Tables Icon

Table 3. Cumulative Error [see Eq. (8)] in the Computation of Electric Field of the Specific Array Modes, Depicted in Figs. 3(a) (Array of Single-Mode Waveguides) and 6(a)6(c) (Array of Multimode Waveguides): Standard CMT and Analytical Approximation vs Helmholtz Equation

Tables Icon

Table 4. Cumulative Error [see Eq. (9)] in the Computation of Propagation Constants of the Array Modes: Standard CMT and Analytical Approximation Versus Helmholtz Equation

Equations (9)

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

(Et(x,z)Ht(x,z))=eiσzp=1Nm=1Gu0m{p}(Etm{p}(x)Htm{p}(x)),
σI^U0=M^U0,
σ(kx)I^U^̲̲0=M^̲̲(kx)U^̲̲0.
ψq(x)=aq(x)sin(πqNΛx)aq(x)sin(kx(q)x),(kx(q)=πqNΛ,1qN),
aq(x)={1,1q<Nsin(πNΛx),q=N.
[u0{m,g}(approx){1,q},,u0{m,g}(approx){N,q}]={b(m,g)·[ψq(x1),,ψq(xN)],m=g,g±2,g±4,,goddb(m,g)·[ψN+1q(x1),,ψN+1q(xN)],m=g,g±2,g±4,,geven[0,,0,,0],m=g±1,g±3,,g
b(m,g)={1,m=g2Pm,g(l,l+1),m=g±2,g±4,.
Errorg(q)[a.m.]=p=1N|Et(xp,0)g(method)(q)Et(xp,0)g(Helmholtz)(q)|2p=1N|Et(xp,0)g(Helmholtz)(q)|2(1qN1gGp=1,,N),
Errorg[σ]=q=1N|σg(method)(q)σg(Helmholtz)(q)|2q=1N|σg(Helmholtz)(q)|2(1gG).

Metrics