Abstract

A coupled-mode analysis of 2-D generalized transverse Bragg waveguides (GTBW) with tilted distributed Bragg reflectors is presented. As a result of the absence of inversion symmetry about a plane perpendicular to the guiding stripe, the modes supported by these guides are not separable into the familiar form of transverse standing wave and longitudinal traveling-wave components. This fundamental change in the modal description yields new and potentially useful guided-mode behavior. Expressions for the spatial distribution of the optical field, phase and group velocity, and the dispersion relation as well as applications of GTBW are presented.

© 2005 Optical Society of America

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References

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  1. P. Yeh and A. Yariv, �??Bragg Reflection Waveguides,�?? Opt. Commun. 19, 427-430 (1976).
    [CrossRef]
  2. A. Y. Cho. A. Yariv and P. Yeh, �??Observation of confined propagation in Bragg waveguides,�?? Appl. Phys. Lett. 30, 471-473 (1977).
    [CrossRef]
  3. P. Russell, �??Photonic crystal fibers,�?? Science 299, 358-362 (2003).
    [CrossRef] [PubMed]
  4. S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, �??Linear waveguides in photonic-crystal slabs,�?? Phys. Rev. B62, 8212-8222, (2000).
  5. L. C. Andreani and M. Agio, �??Intrinsic diffraction losses in photonic crystal waveguides with line defects,�?? Appl. Phys. Lett. 82, 2011-2013 (2003).
    [CrossRef]
  6. A. Yariv, �??Coupled-wave formalism for optical waveguiding by transverse Bragg reflection,�?? Opt. Lett. 27, 936-938 (2002).
    [CrossRef]
  7. A. Yariv, Y. Xu, and S. Mookherjea, �??Transverse Bragg resonance laser amplifier,�?? Opt. Lett. 28, 176-178 (2003).
    [CrossRef] [PubMed]
  8. J. M. Choi, W. Lang, Y. Xu and A. Yariv, �??Loss optimization of transverse Bragg resonance waveguides,�?? J. Opt. Soc. Am. A 21, 426-429 (2004).
    [CrossRef]
  9. R. J. Lang, K. Dzurko, A. A. Hardy, S. Demars, A. Schoenfelder, and D. F. Welch, "Theory of grating confined broad-area lasers,�?? IEEE Journal of Quantum Electronics 34, 2196-2210 (1998)
    [CrossRef]
  10. I. Vurgaftman and J. R. Meyer, �??Photonic crystal distributed-feedback quantum cascade laser,�?? IEEE Journal of Quantum Electronics 38, 592-602, (2002).
    [CrossRef]
  11. A. Yariv and P. Yeh, Optical Waves in Crystals (John Wiley & Sons, New York, 1984).
  12. H. Kogelnik, �??Coupled wave theory for thick hologram gratings,�?? Bell Syst. Tech. J. 48, 2909-2947 (1969).
  13. L. Solymar and D. J. Cooke, Volume Holography and Volume Gratings (Academic Press, London, 1981).
  14. R. S. J. Brueck, V. A. Smagley and P. G. Eliseev, "Radiation from a dipole embedded in a multilayer dielectric slab," Phys. Rev. E68, 036608 (2003).
    [CrossRef] [PubMed]
  15. J. S. Lee and S. Y. Shin, �??Strong discrimination of transverse-modes in high-power laser diodes using Bragg channel waveguiding,�?? Opt. Lett. 14, 143-145 (1989).

Appl. Phys. Lett. (2)

A. Y. Cho. A. Yariv and P. Yeh, �??Observation of confined propagation in Bragg waveguides,�?? Appl. Phys. Lett. 30, 471-473 (1977).
[CrossRef]

L. C. Andreani and M. Agio, �??Intrinsic diffraction losses in photonic crystal waveguides with line defects,�?? Appl. Phys. Lett. 82, 2011-2013 (2003).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, �??Coupled wave theory for thick hologram gratings,�?? Bell Syst. Tech. J. 48, 2909-2947 (1969).

IEEE Journal of Quantum Electronics (2)

R. J. Lang, K. Dzurko, A. A. Hardy, S. Demars, A. Schoenfelder, and D. F. Welch, "Theory of grating confined broad-area lasers,�?? IEEE Journal of Quantum Electronics 34, 2196-2210 (1998)
[CrossRef]

I. Vurgaftman and J. R. Meyer, �??Photonic crystal distributed-feedback quantum cascade laser,�?? IEEE Journal of Quantum Electronics 38, 592-602, (2002).
[CrossRef]

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

Opt. Commun. (1)

P. Yeh and A. Yariv, �??Bragg Reflection Waveguides,�?? Opt. Commun. 19, 427-430 (1976).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. (2)

R. S. J. Brueck, V. A. Smagley and P. G. Eliseev, "Radiation from a dipole embedded in a multilayer dielectric slab," Phys. Rev. E68, 036608 (2003).
[CrossRef] [PubMed]

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, �??Linear waveguides in photonic-crystal slabs,�?? Phys. Rev. B62, 8212-8222, (2000).

Science (1)

P. Russell, �??Photonic crystal fibers,�?? Science 299, 358-362 (2003).
[CrossRef] [PubMed]

Other (2)

A. Yariv and P. Yeh, Optical Waves in Crystals (John Wiley & Sons, New York, 1984).

L. Solymar and D. J. Cooke, Volume Holography and Volume Gratings (Academic Press, London, 1981).

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Figures (6)

Fig. 1.
Fig. 1.

(a) Geometry of the tilted Bragg grating waveguide defining θ +, θ and θ B . The core of width W is between two Bragg grating regions of period Λ and width L (not shown) tilted by an angle α from the guiding direction. (b) The wave vector diagram and (c) the sinusoidal variation of the index along the direction ζ perpendicular to the variation of the Bragg grating.

Fig. 2.
Fig. 2.

(a) Geometry of the transverse Bragg waveguide. (b) Geometry of the α-DFB Laser. (c) Geometry of the PC-DFB Laser. (d) Geometry of the GTBW. (e) Family of Bragg planes used to analyze the 2-D PC waveguide in [6–8]. (f) Family of Bragg planes in the same 2-D PC waveguide which cannot be analyzed using the method outlined in [6–8].

Fig. 3.
Fig. 3.

(a) Angular variation of the reflectivity around the Bragg peak for Λ = 450 nm, λ = 1.3μm, a = 45°, δ = 0.005 and L = 250 μm. The peak reflectivity is a function of angle of incidence and the maximum is ~50%; (b) and (c) Absolute value and phase of the round trip reflectivity. The maximum amplitude is unity and the phase shift varies from 0 to 2π across the bandpass; (d) Absolute value of the self-consistency expression showing the presence of a low loss mode.

Fig. 4.
Fig. 4.

(a) Mode intensity, |E|2, as a function of position; (b) Variation of |E|2 along x; (c) Variation of |E|2 along z; (d) x-component of the Poynting vector; (e) z-component of the Poynting vector; (f) wave vector diagram.

Fig. 5.
Fig. 5.

Modal dispersion relation (bottom) and expanded view (top). For α = 0, the mode is present at all frequencies above the cutoff (in the absence of material dispersion). For α

Fig. 6.
Fig. 6.

(a) Wave vector diagram showing momentum relationship for guiding in the +x direction. Wave vectors k + and k coupled by ± K . (b) Wave vector diagram showing the momentum relationship for guiding in the - x direction after 180° rotation. Wave vectors - k + and - k are coupled by ± K . (c) Wave vector diagram showing the result of a reflection from a surface normal to the waveguide axis. The reflected waves Mx ( k +) and Mx ( k ) are not coupled by the grating.

Equations (17)

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E ( x , z ) = e ̂ y { A t ( z W 2 θ + ) exp [ i k n ( x sin θ + + ( z W 2 ) cos θ + ) ] + B t ( z W 2 θ + ) exp [ i k n ( x sin θ ( z W 2 ) cos θ ) ] } z > W 2
E ( x , z ) = e ̂ y { exp [ i k n ( x sin θ + + ( z W 2 ) cos θ + ) ] + R t ( θ + ) exp [ i k n ( x sin θ ( z W 2 ) cos θ ) ] } −W 2 < z < W 2
E ( x , z ) = e ̂ y R t ( θ + ) exp [ i k n cos θ ] ×
{ B b ( z + W 2 θ ) exp [ i k n ( x sin θ + + ( z + W 2 ) cos θ + ) ] + A b ( z + W 2 θ ) exp [ i k n ( x sin θ ( z + W 2 ) cos θ ) ] } z < −W 2
A t ( z W 2 ) = { i Δ β sinh [ H ( z W 2 L ) ] + 2 H cosh [ H ( z W 2 L ) ] } [ i Δ β sinh ( HL ) + 2 H cosh ( HL ) ] exp [ i Δ β 2 ( z W 2 ) ]
A t ( 0 ) = 1
B t ( z W 2 ) = 2 κ * cos ( θ ) sinh [ H ( z W 2 L ) ] [ i Δ β sinh ( HL ) + 2 H cosh ( HL ) ] exp [ i Δ β 2 ( z W 2 ) ]
R t = B t ( 0 ) 2 κ * cos ( θ ) sinh ( HL ) [ i Δ β sinh ( HL ) + 2 H cosh ( HL ) ]
Δ β [ k n ( cos θ + + cos θ ) K cos α ] , κ k n δ
H κ 2 cos θ + cos θ ( Δ β 2 ) 2
1 R t ( θ + ) R b ( θ ) exp [ i k n ( cos θ + + cos θ ) W ] = 0
θ + = θ B + α + δθ α + θ B , θ = α θ B + δθ α θ B
θ B = cos 1 ( λ 2 n Λ ) = cos 1 ( π c ω n Λ )
1 R t ( θ B + α ) R b ( θ B α ) exp [ i 2 πnW λ [ cos ( θ B + α ) + cos ( θ B α ) ] ]
= 1 R t ( θ B + α ) R b ( θ B α ) exp [ i 2 πW Λ cos ( α ) ] = 0
v phase = ω k = c n sin ( θ B ) cos ( α ) ;
v group = ω k = c n sin ( θ B ) n cos ( α ) .

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