Abstract

It is shown that thin films of dielectric, etched through with a suitably chosen lattice of holes, can support surface-emitting vertical resonances with very-high-cavity Q factors (105 in the case of AlxGaAs1-x on oxidized AlyGaAs1-y). A Bloch-wave expansion is used to develop a complete vector-field analysis of these resonances and to reveal their underlying physics. Since they do not require multilayer mirrors, such resonators are a practical and simple replacement for conventional vertical-cavity surface-emitting laser structures. Other applications include wavelength-division-multiplexing components and highly sensitive gas detectors.

© 2001 Optical Society of America

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References

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  1. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
    [CrossRef] [PubMed]
  2. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering,” Appl. Phys. Lett. 74, 1370–1372 (1999).
    [CrossRef]
  3. P. St. J. Russell, “Novel thick-grating beam-squeezing device in Ta2O5 corrugated planar waveguide,” Electron. Lett. 20, 72–73 (1984).
    [CrossRef]
  4. R. Zengerle, “Light propagation in single and doubly periodic planar waveguides,” J. Mod. Opt. 34, 1589–1617 (1987).
    [CrossRef]
  5. P. St, J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1995), pp. 585.
  6. P. St, J. Russell, D. M. Atkin, T. A. Birks, P. J. Roberts, “Bound modes of two-dimensional photonic crystal waveguides,” in Quantum Optics in Wavelength Scale Structures, J. G. Rarity, C. Weisbuch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996).
  7. D. M. Atkin, P. St. J. Russell, T. A. Birks, P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
    [CrossRef]
  8. V. N. Astratov, I. S. Culshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Resonant coupling of near-infrared radiation in photonic band structure waveguides,” J. Lightwave Technol. 17, 2050–2057 (1999).
    [CrossRef]
  9. P. St, J. Russell, T. A. Birks, “Bloch wave optics in photonic crystals: physics and applications,” in Photonic Band Gap Materials, C. M. Soukoulis ed., (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 71–91.
  10. P. St. J. Russell, “Photonic band-gaps,” Phys. World 5, 37–42 (1992).

1999 (3)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering,” Appl. Phys. Lett. 74, 1370–1372 (1999).
[CrossRef]

V. N. Astratov, I. S. Culshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, “Resonant coupling of near-infrared radiation in photonic band structure waveguides,” J. Lightwave Technol. 17, 2050–2057 (1999).
[CrossRef]

1996 (1)

D. M. Atkin, P. St. J. Russell, T. A. Birks, P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
[CrossRef]

1992 (1)

P. St. J. Russell, “Photonic band-gaps,” Phys. World 5, 37–42 (1992).

1987 (1)

R. Zengerle, “Light propagation in single and doubly periodic planar waveguides,” J. Mod. Opt. 34, 1589–1617 (1987).
[CrossRef]

1984 (1)

P. St. J. Russell, “Novel thick-grating beam-squeezing device in Ta2O5 corrugated planar waveguide,” Electron. Lett. 20, 72–73 (1984).
[CrossRef]

Astratov, V. N.

Atkin, D. M.

D. M. Atkin, P. St. J. Russell, T. A. Birks, P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
[CrossRef]

P. St, J. Russell, D. M. Atkin, T. A. Birks, P. J. Roberts, “Bound modes of two-dimensional photonic crystal waveguides,” in Quantum Optics in Wavelength Scale Structures, J. G. Rarity, C. Weisbuch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Birks, T. A.

D. M. Atkin, P. St. J. Russell, T. A. Birks, P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
[CrossRef]

P. St, J. Russell, D. M. Atkin, T. A. Birks, P. J. Roberts, “Bound modes of two-dimensional photonic crystal waveguides,” in Quantum Optics in Wavelength Scale Structures, J. G. Rarity, C. Weisbuch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996).

P. St, J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1995), pp. 585.

P. St, J. Russell, T. A. Birks, “Bloch wave optics in photonic crystals: physics and applications,” in Photonic Band Gap Materials, C. M. Soukoulis ed., (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 71–91.

Culshaw, I. S.

Dapkus, P. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

De La Rue, R. M.

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering,” Appl. Phys. Lett. 74, 1370–1372 (1999).
[CrossRef]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering,” Appl. Phys. Lett. 74, 1370–1372 (1999).
[CrossRef]

Kim, I.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering,” Appl. Phys. Lett. 74, 1370–1372 (1999).
[CrossRef]

Krauss, T. F.

Lee, R. K.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Lloyd-Lucas, F. D.

P. St, J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1995), pp. 585.

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering,” Appl. Phys. Lett. 74, 1370–1372 (1999).
[CrossRef]

O’Brien, J. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Painter, O.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Roberts, P. J.

D. M. Atkin, P. St. J. Russell, T. A. Birks, P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
[CrossRef]

P. St, J. Russell, D. M. Atkin, T. A. Birks, P. J. Roberts, “Bound modes of two-dimensional photonic crystal waveguides,” in Quantum Optics in Wavelength Scale Structures, J. G. Rarity, C. Weisbuch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Russell, J.

P. St, J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1995), pp. 585.

P. St, J. Russell, D. M. Atkin, T. A. Birks, P. J. Roberts, “Bound modes of two-dimensional photonic crystal waveguides,” in Quantum Optics in Wavelength Scale Structures, J. G. Rarity, C. Weisbuch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996).

P. St, J. Russell, T. A. Birks, “Bloch wave optics in photonic crystals: physics and applications,” in Photonic Band Gap Materials, C. M. Soukoulis ed., (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 71–91.

Russell, P. St. J.

D. M. Atkin, P. St. J. Russell, T. A. Birks, P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
[CrossRef]

P. St. J. Russell, “Photonic band-gaps,” Phys. World 5, 37–42 (1992).

P. St. J. Russell, “Novel thick-grating beam-squeezing device in Ta2O5 corrugated planar waveguide,” Electron. Lett. 20, 72–73 (1984).
[CrossRef]

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering,” Appl. Phys. Lett. 74, 1370–1372 (1999).
[CrossRef]

Scherer, A.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Skolnick, M. S.

St, P.

P. St, J. Russell, T. A. Birks, “Bloch wave optics in photonic crystals: physics and applications,” in Photonic Band Gap Materials, C. M. Soukoulis ed., (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 71–91.

P. St, J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1995), pp. 585.

P. St, J. Russell, D. M. Atkin, T. A. Birks, P. J. Roberts, “Bound modes of two-dimensional photonic crystal waveguides,” in Quantum Optics in Wavelength Scale Structures, J. G. Rarity, C. Weisbuch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Stevenson, R. M.

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering,” Appl. Phys. Lett. 74, 1370–1372 (1999).
[CrossRef]

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering,” Appl. Phys. Lett. 74, 1370–1372 (1999).
[CrossRef]

Whittaker, D. M.

Yariv, A.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Zengerle, R.

R. Zengerle, “Light propagation in single and doubly periodic planar waveguides,” J. Mod. Opt. 34, 1589–1617 (1987).
[CrossRef]

Appl. Phys. Lett. (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering,” Appl. Phys. Lett. 74, 1370–1372 (1999).
[CrossRef]

Electron. Lett. (1)

P. St. J. Russell, “Novel thick-grating beam-squeezing device in Ta2O5 corrugated planar waveguide,” Electron. Lett. 20, 72–73 (1984).
[CrossRef]

J. Lightwave Technol. (1)

J. Mod. Opt. (2)

D. M. Atkin, P. St. J. Russell, T. A. Birks, P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
[CrossRef]

R. Zengerle, “Light propagation in single and doubly periodic planar waveguides,” J. Mod. Opt. 34, 1589–1617 (1987).
[CrossRef]

Phys. World (1)

P. St. J. Russell, “Photonic band-gaps,” Phys. World 5, 37–42 (1992).

Science (1)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Other (3)

P. St, J. Russell, T. A. Birks, “Bloch wave optics in photonic crystals: physics and applications,” in Photonic Band Gap Materials, C. M. Soukoulis ed., (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 71–91.

P. St, J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1995), pp. 585.

P. St, J. Russell, D. M. Atkin, T. A. Birks, P. J. Roberts, “Bound modes of two-dimensional photonic crystal waveguides,” in Quantum Optics in Wavelength Scale Structures, J. G. Rarity, C. Weisbuch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996).

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Figures (12)

Fig. 1
Fig. 1

Schematic of the structure analyzed. The holes penetrate the high-refractive-index film completely, stopping at the interface with the low-index substrate. The holes are filled with air and the cover is air.

Fig. 2
Fig. 2

Transmittance of the modeled photonic crystal film (see inset for geometry) against kp, featuring three main VCSER’s. For comparison the transmission of a layer of the same thickness, with the same refractive index as the dielectric but containing no holes, is also plotted. The Q values of the resonances are noted on the plot. The in-plane direction of incidence is (ΓJ), and the incident polarization is TM. These resonances correspond to the open circles in Fig. 3.

Fig. 3
Fig. 3

Calculated map of VCSER loci in kp space. Also shown are the isolated points at which the VCSER’s cannot be excited with pure TM light (solid circles) or with pure TM light (open circles). The hexagon is the boundary of the first Brillouin zone.

Fig. 4
Fig. 4

Corresponding band structure diagram β versus kp along the ΓJ direction for the case in Fig. 2. Several propagating Bloch waves exist at every angle of incidence. The positions of each of the five sharp VCSER’s are indicated by the vertical dotted lines. Only a small number (typically two to four) of these Bloch waves are significantly excited at each VCSER. Their relative intensities (expressed as a percentage of the sum of the intensities of all the Bloch waves involved in each resonance) are as follows: VCSER 1, a=65%, b=33%; VCSER 2, a=14%, b=36%, c=29%; VCSER 3, a=10%, b=83%; VCSER 4, a=47%, b=49%; VCSER 5, a=70%, b=26%.

Fig. 5
Fig. 5

Electric-field intensity profiles of (a) VCSER 1 and (b) VCSER 2 in the plane y=0. The white dashed lines are the hole boundaries, and the white solid lines are the film interfaces.

Fig. 6
Fig. 6

Transmittance (T) and reflectance (R=1-T) of the photonic crystal film against kp in the locality of the first VCSER in Fig. 2, plotted together with the intensity of the strongest coresonant Bloch mode.

Fig. 7
Fig. 7

Photonic crystal film in which three Bloch waves are excited by the incidence of a single plane wave. Note that the pitch is small enough to ensure that there are no diffracted rays—only transmitted and reflected ones. The Bloch waves are in general both negatively and positively refracted. By applying reciprocity to a toy model with two significant internal Bloch waves, it is possible to reproduce accurately the VCSER shapes in Fig. 2. In a typical case only one Bloch wave is resonant (experiencing high reflection at the interfaces), the others providing low-finesse background resonances (experiencing only weak reflection at the interfaces). The combination of these two yields the behavior in Fig. 2.

Fig. 8
Fig. 8

Solution generated by the toy model for two internal waves. The parameter values used (chosen to satisfy reciprocity) are τ10=0.1, τ20=0.9, ρ00=-0.42, ρ11=-0.98, ρ22=0.41, and β1/β2=1.5. The similarity with the behavior of a photonic crystal layer is striking. More complex behavior results if more than two internal waves are included in the analysis.

Fig. 9
Fig. 9

Wave-vector diagram in the plane of the film. The shaded circles in the center indicate the zones where the substrate and cover regions are able to support traveling waves; beyond the largest of these circles the light is evanescent both in the air and in the substrate. The hexagonal regions are the tiled first Brillouin zones of the two-dimensional photonic crystal. The in-plane wave-vector components of one of the coresonant Bloch modes for VCSER 1 are indicated by small black dots; each is displaced from the center of its Brillouin zone by δkp. The intensities of each associated plane wave (expressed as a percentage of the sum of the intensities of all the plane waves) are indicated by the numbers alongside each dot. Note that almost 95% of the power is carried by two plane waves that are unable to radiate into the substrate or the cover. The only plane wave that can radiate into both substrate and cover carries only 0.01% of the power. This explains why such high Q factors are obtainable: The incident plane wave is only very weakly coupled to the Bloch wave resonances.

Fig. 10
Fig. 10

Transmittance of a film 15 nm thick (and otherwise the same parameters as in Fig. 2), plotted versus kp/k in the ΓJ direction. It supports only one VCSER for TM-polarized light at 980 nm. This VCSER has Q=200,000.

Fig. 11
Fig. 11

TM transmission characteristics of a film with a hole radius of 120 nm, but otherwise the same parameters as in Fig. 2, plotted versus kp/k in the ΓJ direction. A single low-finesse VCSER appears with Q=120.

Fig. 12
Fig. 12

Wave amplitudes in the single-interface transfer matrix for a structure in which two internal waves are present on one side of the interface. All three incident waves are coupled to each other at the interface as well as being reflected and refracted.

Equations (18)

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

p2+k2+1 p×p×Hp=β2Hp,
iUνiexp(jGirp)exp[j(kprp±βz-kct)].
i,μMνμjiUμi=β2Uνj,
(r)=b+(a-b)exp[-(r/w)2n],
Hpinc=ihinciexp[j(Gi+kp)rp-jαcoviz-jkct],
Hpref=ihrefiexp[j(Gi+kp)rp+jαcoviz-jkct],
αi=+[k2-(Gi+kp)2]1/2.
Hptra=ihtraiexp[j(Gi+kp)rp-jαsubiz-jkct].
i(Hpinc,i+Hpref,i)|z=+L/2=iHppc,i|z=+L/2,
i(Epinc,i+Epref,i)|z=+L/2=iEppc,i|z=+L/2,
iHptra,i|z=-L/2=iHppc,i|z=-L/2,
iEptra,i|z=-L/2=iEppc,i|z=-L/2.
b0b1b2=MTa0a1a2=ρ00τ01τ02τ01ρ11ρ12τ02ρ12ρ22a0a1a2,
MTMT=I.
T=QL[I-(NL)2]-1B0,
R=ρ00+QLNL[I-(NL)2]-1B0,
Q=(τ01, τ01),N=ρ11ρ12ρ12ρ22,
L=exp(-jβ1L)00exp(-jβ2L),B0=τ10τ20.

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