Abstract

If a one-dimensional (1D) or two-dimensional (2D) photonic bandgap (PBG) structure is incorporated into a planar optical waveguide, the refractive-index nonuniformity in the direction perpendicular to the waveguide plane responsible for waveguiding may affect its behavior detrimentally. Such influence is demonstrated in the paper by numerical modeling of a deeply etched first-order waveguide Bragg grating. On the basis of physical considerations, a simple condition for the design of 1D and 2D waveguide PBG structures free of this degradation is formulated; it is, in fact the separability condition for the wave equation. Its positive effect is verified by numerical modeling of a modified waveguide Bragg grating that fulfills the separability condition.

© 2001 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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  7. The details can be found at URL http://www.ele.kth.se.COST268/ .
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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1999

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

W. C. Chew, J. M. Jin, E. Michelsen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microwave Opt. Technol. Lett. 16, 363–369 (1999).

1998

J. Čtyroký, S. Helfert, R. Pregla, “Analysis of a deep waveguide Bragg grating,” Opt. Quantum Electron. 30, 343–358 (1998).
[CrossRef]

1997

1996

1995

D. Cheng, R. Biswas, E. Ozbay, S. McCalmont, G. Tuttle, K.-M. Ho, “Optimized dipole antennas on photonic band gap crystals,” Appl. Phys. Lett. 67, 3399–3401 (1995).
[CrossRef]

1994

J-P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

A. S. Sudbø, “Improved formulation of the film mode matching method for mode field calculations in dielectric waveguides,” Pure Appl. Opt. 3, 381–388 (1994).
[CrossRef]

1993

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

1987

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

1984

Atkin, D. M.

Benisty, H.

H. Benisty, “Photonic bandgap structures in waveguides,” presented at the 9th European Conference on Integrated Optics, Turin, Italy, April 13–16, 1999.

Bérenger, J-P.

J-P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Birks, T. A.

Biswas, R.

D. Cheng, R. Biswas, E. Ozbay, S. McCalmont, G. Tuttle, K.-M. Ho, “Optimized dipole antennas on photonic band gap crystals,” Appl. Phys. Lett. 67, 3399–3401 (1995).
[CrossRef]

Cheng, D.

D. Cheng, R. Biswas, E. Ozbay, S. McCalmont, G. Tuttle, K.-M. Ho, “Optimized dipole antennas on photonic band gap crystals,” Appl. Phys. Lett. 67, 3399–3401 (1995).
[CrossRef]

Chew, W. C.

W. C. Chew, J. M. Jin, E. Michelsen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microwave Opt. Technol. Lett. 16, 363–369 (1999).

Chilwell, J.

Ctyroký, J.

J. Čtyroký, S. Helfert, R. Pregla, “Analysis of a deep waveguide Bragg grating,” Opt. Quantum Electron. 30, 343–358 (1998).
[CrossRef]

J. Čtyroký, J. Homola, M. Skalský, “Modelling of surface plasmon resonance waveguide sensor by complex mode expansion and propagation method,” Opt. Quantum Electron. 29, 301–311 (1997).
[CrossRef]

Fan, S.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

Helfert, S.

J. Čtyroký, S. Helfert, R. Pregla, “Analysis of a deep waveguide Bragg grating,” Opt. Quantum Electron. 30, 343–358 (1998).
[CrossRef]

Ho, K.-M.

D. Cheng, R. Biswas, E. Ozbay, S. McCalmont, G. Tuttle, K.-M. Ho, “Optimized dipole antennas on photonic band gap crystals,” Appl. Phys. Lett. 67, 3399–3401 (1995).
[CrossRef]

Hodgkinson, I.

Homola, J.

J. Čtyroký, J. Homola, M. Skalský, “Modelling of surface plasmon resonance waveguide sensor by complex mode expansion and propagation method,” Opt. Quantum Electron. 29, 301–311 (1997).
[CrossRef]

Jin, J. M.

W. C. Chew, J. M. Jin, E. Michelsen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microwave Opt. Technol. Lett. 16, 363–369 (1999).

Joannopoulos, J. D.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

Johnson, S. G.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

Knight, J. C.

Kolodziejski, L. A.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, San Diego, Calif., 1991).

McCalmont, S.

D. Cheng, R. Biswas, E. Ozbay, S. McCalmont, G. Tuttle, K.-M. Ho, “Optimized dipole antennas on photonic band gap crystals,” Appl. Phys. Lett. 67, 3399–3401 (1995).
[CrossRef]

Michelsen, E.

W. C. Chew, J. M. Jin, E. Michelsen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microwave Opt. Technol. Lett. 16, 363–369 (1999).

Nolting, H.-P.

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

Ozbay, E.

D. Cheng, R. Biswas, E. Ozbay, S. McCalmont, G. Tuttle, K.-M. Ho, “Optimized dipole antennas on photonic band gap crystals,” Appl. Phys. Lett. 67, 3399–3401 (1995).
[CrossRef]

Pregla, R.

J. Čtyroký, S. Helfert, R. Pregla, “Analysis of a deep waveguide Bragg grating,” Opt. Quantum Electron. 30, 343–358 (1998).
[CrossRef]

Russel, P. St. J.

Russell, J. St. J.

Skalský, M.

J. Čtyroký, J. Homola, M. Skalský, “Modelling of surface plasmon resonance waveguide sensor by complex mode expansion and propagation method,” Opt. Quantum Electron. 29, 301–311 (1997).
[CrossRef]

Sudbø, A. S.

A. S. Sudbø, “Improved formulation of the film mode matching method for mode field calculations in dielectric waveguides,” Pure Appl. Opt. 3, 381–388 (1994).
[CrossRef]

Sztefka, G.

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

Tuttle, G.

D. Cheng, R. Biswas, E. Ozbay, S. McCalmont, G. Tuttle, K.-M. Ho, “Optimized dipole antennas on photonic band gap crystals,” Appl. Phys. Lett. 67, 3399–3401 (1995).
[CrossRef]

Villeneuve, P. R.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Appl. Phys. Lett.

D. Cheng, R. Biswas, E. Ozbay, S. McCalmont, G. Tuttle, K.-M. Ho, “Optimized dipole antennas on photonic band gap crystals,” Appl. Phys. Lett. 67, 3399–3401 (1995).
[CrossRef]

IEEE Photonics Technol. Lett.

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

J. Comput. Phys.

J-P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. Opt. Soc. Am. A

Microwave Opt. Technol. Lett.

W. C. Chew, J. M. Jin, E. Michelsen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microwave Opt. Technol. Lett. 16, 363–369 (1999).

Opt. Lett.

Opt. Quantum Electron.

J. Čtyroký, S. Helfert, R. Pregla, “Analysis of a deep waveguide Bragg grating,” Opt. Quantum Electron. 30, 343–358 (1998).
[CrossRef]

J. Čtyroký, J. Homola, M. Skalský, “Modelling of surface plasmon resonance waveguide sensor by complex mode expansion and propagation method,” Opt. Quantum Electron. 29, 301–311 (1997).
[CrossRef]

Phys. Rev. B

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

Phys. Rev. Lett.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Pure Appl. Opt.

A. S. Sudbø, “Improved formulation of the film mode matching method for mode field calculations in dielectric waveguides,” Pure Appl. Opt. 3, 381–388 (1994).
[CrossRef]

Other

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

T. Itoh, ed., Numerical Techniques for Microwave and Millimeter Wave Passive Structures (Wiley, New York, 1989).

G. Guekos, ed., Photonic Devices for Telecommunications, How to Model and Measure (Springer-Verlag, Berlin, 1998), pp. 50–56.

H. Benisty, “Photonic bandgap structures in waveguides,” presented at the 9th European Conference on Integrated Optics, Turin, Italy, April 13–16, 1999.

The details can be found at URL http://www.ele.kth.se.COST268/ .

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, San Diego, Calif., 1991).

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Figures (10)

Fig. 1
Fig. 1

Two-dimensional photonic bandgap structure in a planar waveguide.

Fig. 2
Fig. 2

Deeply etched Bragg grating in a planar waveguide as a waveguide PBG structure.

Fig. 3
Fig. 3

Stack of layers with alternating refractive indices as a 1D PBG structure.

Fig. 4
Fig. 4

Spectral dependencies of (a) modal reflectance and (b) transmittance of the waveguide Bragg grating in Fig. 2 with 50 periods: solid curves, DBR grating; dashed curves, stack of layers with alternating refractive indices as in Fig. 3 (shown for comparison).

Fig. 5
Fig. 5

Spectral dependencies of relative modal power loss PL in the waveguide Bragg grating in Fig. 2.

Fig. 6
Fig. 6

Bragg grating in a symmetric planar waveguide as a waveguide PBG structure.

Fig. 7
Fig. 7

Spectral dependencies of (a) modal reflectance and (b) transmittance of the symmetric waveguide grating in Fig. 6 with 50 periods; solid curves, DBR grating; dashed curves, stack of layers with alternating refractive indices as in Fig. 3 (shown for comparison).

Fig. 8
Fig. 8

Spectral dependencies of the relative modal power loss PL of the waveguide Bragg grating in the symmetric waveguide shown in Fig. 6.

Fig. 9
Fig. 9

Perfect waveguide 1D PBG structure without coupling of the waveguide modes.

Fig. 10
Fig. 10

Spectral dependencies of (a) modal reflectance and (b) transmittance of the separable waveguide grating structure of Fig. 9 (solid curves). For comparison, the 1D PBG structure of Fig. 3 is shown (dashed curves).

Equations (39)

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

sβm=k0sNm=k01Nm,
sfm(x)=1fm(x)forodds,
sβm=k0s.Nm=k02Nm,
sfm(x)=2fm(x)forevens,
xminxmaxsfm(x)sfn(x)dx=δmn,
sEy(x, z)=2μ0/0mspm(z)sfm(x),
sHx(x, z)=-20/μ0msNmsqm(z)sfm(x),
sp(z+Δz)sq(z+Δz)=cos(k0sNΔz)i sin(k0sNΔz)i sin(k0sNΔz)cos(k0sNΔz)sp(z)sq(z),
s±1p=s±1,sOsp,
s±1q=s±1N-1s±1,sOsNsq,
s±1Omn=xminxmaxs±1fm(x)sfn(x)dx.
sp(z+Δz)=sa+exp(ik0sNΔz)+sa-exp(-ik0sNΔz),
sq(z+Δz)=sa+exp(ik0sNΔz)-sa-exp(-ik0sNΔz).
1a-=R1a+,2G+1a+=T1a+.
R=aa0-1a0+2=|R00|2,T=2G+1a0+1a0+=|T00|2.
PL=1-(R+T).
uL=λB/4uN=215.6nm,eL=λB/4eN=253.7nm,
d2sf(x)dx2+k02[sn2(x)-sN2]sf(x)=0,
s±1n2(x)-sn2(x)=c,
1k02d2sf(x)dx2+n2(x)sf(x)=sN2sf(x).
s±1Nm2-sNm2=c,
n2(x, z)=nx2(x)+nz2(z);
nz2(z)=0forodds(sectionse)cforevens(sectionsu),
2x2+2z2E(x, z)+k02n2(x, z)E(x, z)=0,
E(x, z)=eyf(x)exp(ik0sNz),
2f(x)x2+k02[nx2(x)-Nx2]f(x)=0,
sN2=Nx2forodds(sectionse)Nx2+cforevens(sectionsu).
n2(x, y, z)=nx2+n2(y, z),
××E-k02n2(x, y, z)E(x, y, z)=0.
E(x, y, z)=e(y, z)f(x),
d2f(x)dx2+k02[nx2(x)-Nx2]f(x)=0,
e(y, z)=0
=x+,x=x0/x,
=y0/y+z0/z.
××e(y, z)-k02N2(y, z)e(y, z)=0.
n2(x, y, z)=n2(x, y)+nz2(z).
E(x, y, z)=e(x, y)g(z),
××e(x, y)-k02[n2(x, y)-N2]e(x, y)=0,
2z2g(z)+k02[N2+nz2(z)]g(z)=0.

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