Abstract

We demonstrate mirrorless lasers based on all organic nanostructure fabricated by seven- and nine-beam interference using low contrast material, holographic polymer dispersed liquid crystals (H-PDLC). A finite-difference time-domain (FDTD) simulation is used to study the transmission of quasicrystal. The wavelengths of lasing peak are determined by both of local structure of quasicrystal that the pumping light experienced as well as the photoluminescence of laser dye doped. Features of mirrorless laser from quasicrystal based on H-PDLC include directional light source, low threshold, simple fabrication process, low cost and tunability. These properties make H-PDLC photonic quasicrystal promising for a new type of all organic miniature lasers.

© 2016 Optical Society of America

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References

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  1. D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Phys. Rev. Lett. 53(20), 1951–1953 (1984).
    [Crossref]
  2. E. Rotenberg, W. Theis, K. Horn, and P. Gille, “Quasicrystalline valence bands in decagonal AlNiCo,” Nature 406(6796), 602–605 (2000).
    [Crossref] [PubMed]
  3. S. P. Gorkhali, J. Qi, and G. P. Crawford, “Switchable quasi-crystal structures with five-, seven-, and ninefold symmetries,” J. Opt. Soc. Am. B 23(1), 149–158 (2006).
    [Crossref]
  4. Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. 80(5), 956–959 (1998).
    [Crossref]
  5. G. J. Parker, M. E. Zoorob, M. D. B. Charlton, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
    [Crossref] [PubMed]
  6. W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature 436(7053), 993–996 (2005).
    [Crossref] [PubMed]
  7. B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440(7088), 1166–1169 (2006).
    [Crossref] [PubMed]
  8. T. J. Bunning, L. V. Natarajan, V. P. Tondiglia, and R. L. Sutherland, “Holographic polymer-dispersed liquid crystals (H-PDLCs),” Annu. Rev. Mater. Sci. 30(1), 83–115 (2000).
    [Crossref]
  9. R. Sutherland, V. Tondiglia, L. Natarajan, S. Chandra, D. Tomlin, and T. Bunning, “Switchable orthorhombic F photonic crystals formed by holographic polymerization-induced phase separation of liquid crystal,” Opt. Express 10(20), 1074–1082 (2002).
    [Crossref] [PubMed]
  10. S. P. Gorkhali, J. Qi, and G. P. Crawford, “Switchable quasi-crystal structures with five-, seven-, and ninefold symmetries,” J. Opt. Soc. Am. B 23(1), 149–158 (2006).
    [Crossref]
  11. M. S. Li, A. Y. Fuh, and S. T. Wu, “Multimode lasing from the microcavity of an octagonal quasi-crystal based on holographic polymer-dispersed liquid crystals,” Opt. Lett. 37(15), 3249–3251 (2012).
    [Crossref] [PubMed]
  12. D. Luo, Q. G. Du, H. T. Dai, H. V. Demir, H. Z. Yang, W. Ji, and X. W. Sun, “Strongly linearly polarized low threshold lasing of all organic photonic quasicrystals,” Sci. Rep. 2, 627 (2012).
    [Crossref] [PubMed]
  13. D. Luo, Q. G. Du, H. T. Dai, X. H. Zhang, and X. W. Sun, “Temperature effect on lasing from Penrose photonic quasicrystal,” Opt. Mater. Express 4(6), 1172–1177 (2014).
    [Crossref]
  14. D. Luo, X. W. Sun, H. T. Dai, H. V. Demir, H. Z. Yang, and W. Ji, “Two-directional lasing from a dye-doped two-dimensional hexagonal photonic crystal made of holographic polymer-dispersed liquid crystals,” Appl. Phys. Lett. 95(15), 151115 (2009).
    [Crossref]

2014 (1)

2012 (2)

M. S. Li, A. Y. Fuh, and S. T. Wu, “Multimode lasing from the microcavity of an octagonal quasi-crystal based on holographic polymer-dispersed liquid crystals,” Opt. Lett. 37(15), 3249–3251 (2012).
[Crossref] [PubMed]

D. Luo, Q. G. Du, H. T. Dai, H. V. Demir, H. Z. Yang, W. Ji, and X. W. Sun, “Strongly linearly polarized low threshold lasing of all organic photonic quasicrystals,” Sci. Rep. 2, 627 (2012).
[Crossref] [PubMed]

2009 (1)

D. Luo, X. W. Sun, H. T. Dai, H. V. Demir, H. Z. Yang, and W. Ji, “Two-directional lasing from a dye-doped two-dimensional hexagonal photonic crystal made of holographic polymer-dispersed liquid crystals,” Appl. Phys. Lett. 95(15), 151115 (2009).
[Crossref]

2006 (3)

2005 (1)

W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature 436(7053), 993–996 (2005).
[Crossref] [PubMed]

2002 (1)

2000 (3)

E. Rotenberg, W. Theis, K. Horn, and P. Gille, “Quasicrystalline valence bands in decagonal AlNiCo,” Nature 406(6796), 602–605 (2000).
[Crossref] [PubMed]

T. J. Bunning, L. V. Natarajan, V. P. Tondiglia, and R. L. Sutherland, “Holographic polymer-dispersed liquid crystals (H-PDLCs),” Annu. Rev. Mater. Sci. 30(1), 83–115 (2000).
[Crossref]

G. J. Parker, M. E. Zoorob, M. D. B. Charlton, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
[Crossref] [PubMed]

1998 (1)

Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. 80(5), 956–959 (1998).
[Crossref]

1984 (1)

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Phys. Rev. Lett. 53(20), 1951–1953 (1984).
[Crossref]

Bartal, G.

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440(7088), 1166–1169 (2006).
[Crossref] [PubMed]

Baumberg, J. J.

G. J. Parker, M. E. Zoorob, M. D. B. Charlton, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
[Crossref] [PubMed]

Blech, I.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Phys. Rev. Lett. 53(20), 1951–1953 (1984).
[Crossref]

Bunning, T.

Bunning, T. J.

T. J. Bunning, L. V. Natarajan, V. P. Tondiglia, and R. L. Sutherland, “Holographic polymer-dispersed liquid crystals (H-PDLCs),” Annu. Rev. Mater. Sci. 30(1), 83–115 (2000).
[Crossref]

Cahn, J. W.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Phys. Rev. Lett. 53(20), 1951–1953 (1984).
[Crossref]

Chaikin, P. M.

W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature 436(7053), 993–996 (2005).
[Crossref] [PubMed]

Chan, C. T.

Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. 80(5), 956–959 (1998).
[Crossref]

Chan, Y. S.

Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. 80(5), 956–959 (1998).
[Crossref]

Chandra, S.

Charlton, M. D. B.

G. J. Parker, M. E. Zoorob, M. D. B. Charlton, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
[Crossref] [PubMed]

Christodoulides, D. N.

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440(7088), 1166–1169 (2006).
[Crossref] [PubMed]

Crawford, G. P.

Dai, H. T.

D. Luo, Q. G. Du, H. T. Dai, X. H. Zhang, and X. W. Sun, “Temperature effect on lasing from Penrose photonic quasicrystal,” Opt. Mater. Express 4(6), 1172–1177 (2014).
[Crossref]

D. Luo, Q. G. Du, H. T. Dai, H. V. Demir, H. Z. Yang, W. Ji, and X. W. Sun, “Strongly linearly polarized low threshold lasing of all organic photonic quasicrystals,” Sci. Rep. 2, 627 (2012).
[Crossref] [PubMed]

D. Luo, X. W. Sun, H. T. Dai, H. V. Demir, H. Z. Yang, and W. Ji, “Two-directional lasing from a dye-doped two-dimensional hexagonal photonic crystal made of holographic polymer-dispersed liquid crystals,” Appl. Phys. Lett. 95(15), 151115 (2009).
[Crossref]

Demir, H. V.

D. Luo, Q. G. Du, H. T. Dai, H. V. Demir, H. Z. Yang, W. Ji, and X. W. Sun, “Strongly linearly polarized low threshold lasing of all organic photonic quasicrystals,” Sci. Rep. 2, 627 (2012).
[Crossref] [PubMed]

D. Luo, X. W. Sun, H. T. Dai, H. V. Demir, H. Z. Yang, and W. Ji, “Two-directional lasing from a dye-doped two-dimensional hexagonal photonic crystal made of holographic polymer-dispersed liquid crystals,” Appl. Phys. Lett. 95(15), 151115 (2009).
[Crossref]

Du, Q. G.

D. Luo, Q. G. Du, H. T. Dai, X. H. Zhang, and X. W. Sun, “Temperature effect on lasing from Penrose photonic quasicrystal,” Opt. Mater. Express 4(6), 1172–1177 (2014).
[Crossref]

D. Luo, Q. G. Du, H. T. Dai, H. V. Demir, H. Z. Yang, W. Ji, and X. W. Sun, “Strongly linearly polarized low threshold lasing of all organic photonic quasicrystals,” Sci. Rep. 2, 627 (2012).
[Crossref] [PubMed]

Fleischer, J. W.

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440(7088), 1166–1169 (2006).
[Crossref] [PubMed]

Freedman, B.

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440(7088), 1166–1169 (2006).
[Crossref] [PubMed]

Fuh, A. Y.

Gille, P.

E. Rotenberg, W. Theis, K. Horn, and P. Gille, “Quasicrystalline valence bands in decagonal AlNiCo,” Nature 406(6796), 602–605 (2000).
[Crossref] [PubMed]

Gorkhali, S. P.

Gratias, D.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Phys. Rev. Lett. 53(20), 1951–1953 (1984).
[Crossref]

Horn, K.

E. Rotenberg, W. Theis, K. Horn, and P. Gille, “Quasicrystalline valence bands in decagonal AlNiCo,” Nature 406(6796), 602–605 (2000).
[Crossref] [PubMed]

Ji, W.

D. Luo, Q. G. Du, H. T. Dai, H. V. Demir, H. Z. Yang, W. Ji, and X. W. Sun, “Strongly linearly polarized low threshold lasing of all organic photonic quasicrystals,” Sci. Rep. 2, 627 (2012).
[Crossref] [PubMed]

D. Luo, X. W. Sun, H. T. Dai, H. V. Demir, H. Z. Yang, and W. Ji, “Two-directional lasing from a dye-doped two-dimensional hexagonal photonic crystal made of holographic polymer-dispersed liquid crystals,” Appl. Phys. Lett. 95(15), 151115 (2009).
[Crossref]

Li, M. S.

Lifshitz, R.

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440(7088), 1166–1169 (2006).
[Crossref] [PubMed]

Liu, Z. Y.

Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. 80(5), 956–959 (1998).
[Crossref]

Luo, D.

D. Luo, Q. G. Du, H. T. Dai, X. H. Zhang, and X. W. Sun, “Temperature effect on lasing from Penrose photonic quasicrystal,” Opt. Mater. Express 4(6), 1172–1177 (2014).
[Crossref]

D. Luo, Q. G. Du, H. T. Dai, H. V. Demir, H. Z. Yang, W. Ji, and X. W. Sun, “Strongly linearly polarized low threshold lasing of all organic photonic quasicrystals,” Sci. Rep. 2, 627 (2012).
[Crossref] [PubMed]

D. Luo, X. W. Sun, H. T. Dai, H. V. Demir, H. Z. Yang, and W. Ji, “Two-directional lasing from a dye-doped two-dimensional hexagonal photonic crystal made of holographic polymer-dispersed liquid crystals,” Appl. Phys. Lett. 95(15), 151115 (2009).
[Crossref]

Man, W.

W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature 436(7053), 993–996 (2005).
[Crossref] [PubMed]

Megens, M.

W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature 436(7053), 993–996 (2005).
[Crossref] [PubMed]

Natarajan, L.

Natarajan, L. V.

T. J. Bunning, L. V. Natarajan, V. P. Tondiglia, and R. L. Sutherland, “Holographic polymer-dispersed liquid crystals (H-PDLCs),” Annu. Rev. Mater. Sci. 30(1), 83–115 (2000).
[Crossref]

Netti, M. C.

G. J. Parker, M. E. Zoorob, M. D. B. Charlton, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
[Crossref] [PubMed]

Parker, G. J.

G. J. Parker, M. E. Zoorob, M. D. B. Charlton, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
[Crossref] [PubMed]

Qi, J.

Rotenberg, E.

E. Rotenberg, W. Theis, K. Horn, and P. Gille, “Quasicrystalline valence bands in decagonal AlNiCo,” Nature 406(6796), 602–605 (2000).
[Crossref] [PubMed]

Segev, M.

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440(7088), 1166–1169 (2006).
[Crossref] [PubMed]

Shechtman, D.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Phys. Rev. Lett. 53(20), 1951–1953 (1984).
[Crossref]

Steinhardt, P. J.

W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature 436(7053), 993–996 (2005).
[Crossref] [PubMed]

Sun, X. W.

D. Luo, Q. G. Du, H. T. Dai, X. H. Zhang, and X. W. Sun, “Temperature effect on lasing from Penrose photonic quasicrystal,” Opt. Mater. Express 4(6), 1172–1177 (2014).
[Crossref]

D. Luo, Q. G. Du, H. T. Dai, H. V. Demir, H. Z. Yang, W. Ji, and X. W. Sun, “Strongly linearly polarized low threshold lasing of all organic photonic quasicrystals,” Sci. Rep. 2, 627 (2012).
[Crossref] [PubMed]

D. Luo, X. W. Sun, H. T. Dai, H. V. Demir, H. Z. Yang, and W. Ji, “Two-directional lasing from a dye-doped two-dimensional hexagonal photonic crystal made of holographic polymer-dispersed liquid crystals,” Appl. Phys. Lett. 95(15), 151115 (2009).
[Crossref]

Sutherland, R.

Sutherland, R. L.

T. J. Bunning, L. V. Natarajan, V. P. Tondiglia, and R. L. Sutherland, “Holographic polymer-dispersed liquid crystals (H-PDLCs),” Annu. Rev. Mater. Sci. 30(1), 83–115 (2000).
[Crossref]

Theis, W.

E. Rotenberg, W. Theis, K. Horn, and P. Gille, “Quasicrystalline valence bands in decagonal AlNiCo,” Nature 406(6796), 602–605 (2000).
[Crossref] [PubMed]

Tomlin, D.

Tondiglia, V.

Tondiglia, V. P.

T. J. Bunning, L. V. Natarajan, V. P. Tondiglia, and R. L. Sutherland, “Holographic polymer-dispersed liquid crystals (H-PDLCs),” Annu. Rev. Mater. Sci. 30(1), 83–115 (2000).
[Crossref]

Wu, S. T.

Yang, H. Z.

D. Luo, Q. G. Du, H. T. Dai, H. V. Demir, H. Z. Yang, W. Ji, and X. W. Sun, “Strongly linearly polarized low threshold lasing of all organic photonic quasicrystals,” Sci. Rep. 2, 627 (2012).
[Crossref] [PubMed]

D. Luo, X. W. Sun, H. T. Dai, H. V. Demir, H. Z. Yang, and W. Ji, “Two-directional lasing from a dye-doped two-dimensional hexagonal photonic crystal made of holographic polymer-dispersed liquid crystals,” Appl. Phys. Lett. 95(15), 151115 (2009).
[Crossref]

Zhang, X. H.

Zoorob, M. E.

G. J. Parker, M. E. Zoorob, M. D. B. Charlton, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
[Crossref] [PubMed]

Annu. Rev. Mater. Sci. (1)

T. J. Bunning, L. V. Natarajan, V. P. Tondiglia, and R. L. Sutherland, “Holographic polymer-dispersed liquid crystals (H-PDLCs),” Annu. Rev. Mater. Sci. 30(1), 83–115 (2000).
[Crossref]

Appl. Phys. Lett. (1)

D. Luo, X. W. Sun, H. T. Dai, H. V. Demir, H. Z. Yang, and W. Ji, “Two-directional lasing from a dye-doped two-dimensional hexagonal photonic crystal made of holographic polymer-dispersed liquid crystals,” Appl. Phys. Lett. 95(15), 151115 (2009).
[Crossref]

J. Opt. Soc. Am. B (2)

Nature (4)

E. Rotenberg, W. Theis, K. Horn, and P. Gille, “Quasicrystalline valence bands in decagonal AlNiCo,” Nature 406(6796), 602–605 (2000).
[Crossref] [PubMed]

G. J. Parker, M. E. Zoorob, M. D. B. Charlton, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
[Crossref] [PubMed]

W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature 436(7053), 993–996 (2005).
[Crossref] [PubMed]

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440(7088), 1166–1169 (2006).
[Crossref] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

Opt. Mater. Express (1)

Phys. Rev. Lett. (2)

Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. 80(5), 956–959 (1998).
[Crossref]

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Phys. Rev. Lett. 53(20), 1951–1953 (1984).
[Crossref]

Sci. Rep. (1)

D. Luo, Q. G. Du, H. T. Dai, H. V. Demir, H. Z. Yang, W. Ji, and X. W. Sun, “Strongly linearly polarized low threshold lasing of all organic photonic quasicrystals,” Sci. Rep. 2, 627 (2012).
[Crossref] [PubMed]

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Figures (5)

Fig. 1
Fig. 1 (a) Prism used for seven-beam generation and seven-beam interference configuration. θ represents the angle between the beam and vertical z axis. Φ represents the side-bottom plane angle of the prism. (b) A laser beam impinging on the prism will split into seven beams and interference with each other, the interfered pattern will be recorded on the liquid crystal cell filled with LC/polymer mixture, or sample, which is adhered on the bottom of prism through index-match liquid. (c) Optical setup of lasing. A Q-switch frequency-doubled Nd: YAG pulsed laser operating at 532 nm is used to pump the sample with two-dimensional quasicrystal nanostructure. A cylinder lens is used to shape laser beam to a narrow line.
Fig. 2
Fig. 2 (a) Simulated intensity distribution of the seven-beam interference pattern. Scale bar: 4 μm. (b) AFM image of the surface morphology of seven-beam interference pattern. Scale bar: 2 μm. (c) Diffraction image of seven-beam interference pattern. (d) Simulated intensity distribution of the nine-beam interference pattern. Scale bar: 4 μm. (b) AFM image of the surface morphology of nine-beam interference pattern. Scale bar: 2 μm. (c) Diffraction image of nine-beam interference pattern.
Fig. 3
Fig. 3 (a) Schematic of the lasing generated along x axis and cross the center of quasicrystal formed by seven-beam interference. (b) Spectrum of lasing from quasicrystal formed by seven-beam interference measured from 0.028 to 0.121 mJ/pulse. (c) Schematic of the lasing generated along x axis and cross the center of quasicrystal formed by nine-beam interference. (d) Spectrum of lasing from quasicrystal formed by nine-beam interference measured from 0.028 to 0.121 mJ/pulse.
Fig. 4
Fig. 4 (a) The intensity and line width verse pumping energy (a) 617 nm for seven-beam formed quasicrystal, the threshold is 0.027 mJ/pulse. (b) 564 nm for nine-beam formed quasicrystal, the threshold is 0.025 mJ/pulse.
Fig. 5
Fig. 5 Calculated transmittance spectrum along x axis of quasicrystal formed by (a) seven-beam and (b) nine-beam interference. The band gap centers at 610 nm and 540 nm, respectively. The lasing position is illustrated by blue region that is located at the band edge.

Equations (2)

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K n =k( cos 2(n1)π p sinθ,sin 2(n1)π p sinθ,cosθ ),
I(r)=( l,m=1 p E l E m exp[i( k l k m )r] ),

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