Abstract

We have investigated the formation of in-bandgap delocalized modes due to random lattice disorder as determined from the longitudinal mode spacing in a distributed Bragg laser. We were able to measure the penetration depth, and from transfer matrix simulations, determine how the localization length is altered for disordered lattices. Transfer matrix simulations and studies of the ensemble average were able to connect the gap delocalized modes to localized modes outside of the gap as expected from consideration of Anderson localization, as well as identify the controlling parameters.

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    [CrossRef]
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    [CrossRef] [PubMed]
  3. A. R. McGurn, K. T. Christensen, F. M. Mueller, and A. A. Maradudin, “Anderson Localization In One-Dimensional Randomly Disordered Optical-Systems That Are Periodic On Average,” Phys. Rev. B 47(20), 13120–13125 (1993).
    [CrossRef]
  4. J. B. Pendry, “Symmetry And Transport Of Waves In One-Dimensional Disordered-Systems,” Adv. Phys. 43(4), 461–542 (1994).
    [CrossRef]
  5. N. Garcia and A. Z. Genack, “Anomalous photon diffusion at the threshold of the Anderson Localization Transition,” Phys. Rev. Lett. 66(14), 1850–1853 (1991).
    [CrossRef] [PubMed]
  6. A. Z. Genack and N. Garcia, “Observation of photon localization in a three-dimensional disordered system,” Phys. Rev. Lett. 66(16), 2064–2067 (1991).
    [CrossRef] [PubMed]
  7. P.-E. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55(24), 2696–2699 (1985).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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2008

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

S. Zhang, J. Park, V. Milner, and A. Z. Genack, “Photon Delocalization Transition in Dimensional Crossover in Layered Media,” Phys. Rev. Lett. 101(18), 183901 (2008).
[CrossRef] [PubMed]

K. D. Singer, T. Kazmierczak, J. Lott, H. Song, Y. H. Wu, J. Andrews, E. Baer, A. Hiltner, and C. Weder, “Melt-processed all-polymer distributed Bragg reflector laser,” Opt. Express 16(14), 10358–10363 (2008).
[CrossRef] [PubMed]

2007

T. Kazmierczak, H. Song, A. Hiltner, and E. Baer,“Polymeric One-Dimensional Photonic Crystals by Continuous Coextrusion,” Macromol. Rapid Commun. 28(23), 2210–2216 (2007).
[CrossRef]

J. Topolancik, B. Ilic, and F. Vollmer, “Experimental observation of strong photon localization in disordered photonic crystal waveguides,” Phys. Rev. Lett. 99(25), 253901 (2007).
[CrossRef]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[CrossRef] [PubMed]

L. Sanchez-Palencia, D. Clément, P. Lugan, P. Bouyer, G. V. Shlyapnikov, and A. Aspect, “Anderson localization of expanding Bose-Einstein Condensates in Random Potentials,” Phys. Rev. Lett. 98(21), 210401 (2007).
[CrossRef] [PubMed]

2006

M. A. Kaliteevski, D. M. Beggs, S. Brand, R. A. Abram, and V. V. Nikolaev, “Statistics of the eigenmodes and optical properties of one-dimensional disordered photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(5), 056616 (2006).
[CrossRef] [PubMed]

2005

H. Cao, “Review on latest developments in random lasers with coherent feedback,” J. Phys. Math. Gen. 38(49), 10497–10535 (2005).
[CrossRef]

1997

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385(6611), 45–47 (1997).
[CrossRef]

1994

J. B. Pendry, “Symmetry And Transport Of Waves In One-Dimensional Disordered-Systems,” Adv. Phys. 43(4), 461–542 (1994).
[CrossRef]

1993

A. R. McGurn, K. T. Christensen, F. M. Mueller, and A. A. Maradudin, “Anderson Localization In One-Dimensional Randomly Disordered Optical-Systems That Are Periodic On Average,” Phys. Rev. B 47(20), 13120–13125 (1993).
[CrossRef]

D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, “Photonic Band-Structure And Defects In One And 2 Dimensions,” J. Opt. Soc. Am. B 10(2), 314–321 (1993).
[CrossRef]

1992

D. I. Babic and S. W. Corzine, “Analytic Expressions For The Reflection Delay, Penetration Depth, And Absorptance Of Quarter-Wave Dielectric Mirrors,” IEEE J. Quantum Electron. 28(2), 514–524 (1992).
[CrossRef]

1991

N. Garcia and A. Z. Genack, “Anomalous photon diffusion at the threshold of the Anderson Localization Transition,” Phys. Rev. Lett. 66(14), 1850–1853 (1991).
[CrossRef] [PubMed]

A. Z. Genack and N. Garcia, “Observation of photon localization in a three-dimensional disordered system,” Phys. Rev. Lett. 66(16), 2064–2067 (1991).
[CrossRef] [PubMed]

1988

J. L. Jewell, Y. H. Lee, S. L. McCall, J. P. Harbison, and L. T. Florez, “High-Finesse (Al,Ga)As Interference Filters Grown By Molecular-Beam Epitaxy,” Appl. Phys. Lett. 53(8), 640–642 (1988).
[CrossRef]

1987

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987).
[CrossRef] [PubMed]

1985

V. Baluni and J. Willemsen, “Transmission of acoustic waves in a random layered medium,” Phys. Rev. A 31(5), 3358–3363 (1985).
[CrossRef] [PubMed]

P.-E. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55(24), 2696–2699 (1985).
[CrossRef] [PubMed]

M. P. V. Albada and A. Lagendijk,“Observation of Weak Localization of Light in a Random Medium,” Phys. Rev. Lett. 55(24), 2692–2695 (1985).
[CrossRef] [PubMed]

1983

F. Koyama, Y. Suematsu, S. Arai, and T. E. Tawee, “1.5-1.6-Mu-M Galnasp/Inp Dynamic-Single-Mode (Dsm) Lasers With Distributed Bragg Reflector,” IEEE J. Quantum Electron. 19(6), 1042–1051 (1983).
[CrossRef]

1958

P. W. Anderson, “Absence of Diffusion in Certain Random Lattices,” Phys. Rev. 109(5), 1492–1505 (1958).
[CrossRef]

Abram, R. A.

M. A. Kaliteevski, D. M. Beggs, S. Brand, R. A. Abram, and V. V. Nikolaev, “Statistics of the eigenmodes and optical properties of one-dimensional disordered photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(5), 056616 (2006).
[CrossRef] [PubMed]

Albada, M. P. V.

M. P. V. Albada and A. Lagendijk,“Observation of Weak Localization of Light in a Random Medium,” Phys. Rev. Lett. 55(24), 2692–2695 (1985).
[CrossRef] [PubMed]

Anderson, P. W.

P. W. Anderson, “Absence of Diffusion in Certain Random Lattices,” Phys. Rev. 109(5), 1492–1505 (1958).
[CrossRef]

Andrews, J.

Arai, S.

F. Koyama, Y. Suematsu, S. Arai, and T. E. Tawee, “1.5-1.6-Mu-M Galnasp/Inp Dynamic-Single-Mode (Dsm) Lasers With Distributed Bragg Reflector,” IEEE J. Quantum Electron. 19(6), 1042–1051 (1983).
[CrossRef]

Aspect, A.

L. Sanchez-Palencia, D. Clément, P. Lugan, P. Bouyer, G. V. Shlyapnikov, and A. Aspect, “Anderson localization of expanding Bose-Einstein Condensates in Random Potentials,” Phys. Rev. Lett. 98(21), 210401 (2007).
[CrossRef] [PubMed]

Avidan, A.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

Babic, D. I.

D. I. Babic and S. W. Corzine, “Analytic Expressions For The Reflection Delay, Penetration Depth, And Absorptance Of Quarter-Wave Dielectric Mirrors,” IEEE J. Quantum Electron. 28(2), 514–524 (1992).
[CrossRef]

Baer, E.

K. D. Singer, T. Kazmierczak, J. Lott, H. Song, Y. H. Wu, J. Andrews, E. Baer, A. Hiltner, and C. Weder, “Melt-processed all-polymer distributed Bragg reflector laser,” Opt. Express 16(14), 10358–10363 (2008).
[CrossRef] [PubMed]

T. Kazmierczak, H. Song, A. Hiltner, and E. Baer,“Polymeric One-Dimensional Photonic Crystals by Continuous Coextrusion,” Macromol. Rapid Commun. 28(23), 2210–2216 (2007).
[CrossRef]

Baluni, V.

V. Baluni and J. Willemsen, “Transmission of acoustic waves in a random layered medium,” Phys. Rev. A 31(5), 3358–3363 (1985).
[CrossRef] [PubMed]

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Beggs, D. M.

M. A. Kaliteevski, D. M. Beggs, S. Brand, R. A. Abram, and V. V. Nikolaev, “Statistics of the eigenmodes and optical properties of one-dimensional disordered photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(5), 056616 (2006).
[CrossRef] [PubMed]

Bouyer, P.

L. Sanchez-Palencia, D. Clément, P. Lugan, P. Bouyer, G. V. Shlyapnikov, and A. Aspect, “Anderson localization of expanding Bose-Einstein Condensates in Random Potentials,” Phys. Rev. Lett. 98(21), 210401 (2007).
[CrossRef] [PubMed]

Brand, S.

M. A. Kaliteevski, D. M. Beggs, S. Brand, R. A. Abram, and V. V. Nikolaev, “Statistics of the eigenmodes and optical properties of one-dimensional disordered photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(5), 056616 (2006).
[CrossRef] [PubMed]

Cao, H.

H. Cao, “Review on latest developments in random lasers with coherent feedback,” J. Phys. Math. Gen. 38(49), 10497–10535 (2005).
[CrossRef]

Christensen, K. T.

A. R. McGurn, K. T. Christensen, F. M. Mueller, and A. A. Maradudin, “Anderson Localization In One-Dimensional Randomly Disordered Optical-Systems That Are Periodic On Average,” Phys. Rev. B 47(20), 13120–13125 (1993).
[CrossRef]

Christodoulides, D. N.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

Clément, D.

L. Sanchez-Palencia, D. Clément, P. Lugan, P. Bouyer, G. V. Shlyapnikov, and A. Aspect, “Anderson localization of expanding Bose-Einstein Condensates in Random Potentials,” Phys. Rev. Lett. 98(21), 210401 (2007).
[CrossRef] [PubMed]

Corzine, S. W.

D. I. Babic and S. W. Corzine, “Analytic Expressions For The Reflection Delay, Penetration Depth, And Absorptance Of Quarter-Wave Dielectric Mirrors,” IEEE J. Quantum Electron. 28(2), 514–524 (1992).
[CrossRef]

Dalichaouch, R.

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Florez, L. T.

J. L. Jewell, Y. H. Lee, S. L. McCall, J. P. Harbison, and L. T. Florez, “High-Finesse (Al,Ga)As Interference Filters Grown By Molecular-Beam Epitaxy,” Appl. Phys. Lett. 53(8), 640–642 (1988).
[CrossRef]

Garcia, N.

A. Z. Genack and N. Garcia, “Observation of photon localization in a three-dimensional disordered system,” Phys. Rev. Lett. 66(16), 2064–2067 (1991).
[CrossRef] [PubMed]

N. Garcia and A. Z. Genack, “Anomalous photon diffusion at the threshold of the Anderson Localization Transition,” Phys. Rev. Lett. 66(14), 1850–1853 (1991).
[CrossRef] [PubMed]

Genack, A. Z.

S. Zhang, J. Park, V. Milner, and A. Z. Genack, “Photon Delocalization Transition in Dimensional Crossover in Layered Media,” Phys. Rev. Lett. 101(18), 183901 (2008).
[CrossRef] [PubMed]

N. Garcia and A. Z. Genack, “Anomalous photon diffusion at the threshold of the Anderson Localization Transition,” Phys. Rev. Lett. 66(14), 1850–1853 (1991).
[CrossRef] [PubMed]

A. Z. Genack and N. Garcia, “Observation of photon localization in a three-dimensional disordered system,” Phys. Rev. Lett. 66(16), 2064–2067 (1991).
[CrossRef] [PubMed]

Harbison, J. P.

J. L. Jewell, Y. H. Lee, S. L. McCall, J. P. Harbison, and L. T. Florez, “High-Finesse (Al,Ga)As Interference Filters Grown By Molecular-Beam Epitaxy,” Appl. Phys. Lett. 53(8), 640–642 (1988).
[CrossRef]

Hiltner, A.

K. D. Singer, T. Kazmierczak, J. Lott, H. Song, Y. H. Wu, J. Andrews, E. Baer, A. Hiltner, and C. Weder, “Melt-processed all-polymer distributed Bragg reflector laser,” Opt. Express 16(14), 10358–10363 (2008).
[CrossRef] [PubMed]

T. Kazmierczak, H. Song, A. Hiltner, and E. Baer,“Polymeric One-Dimensional Photonic Crystals by Continuous Coextrusion,” Macromol. Rapid Commun. 28(23), 2210–2216 (2007).
[CrossRef]

Ilic, B.

J. Topolancik, B. Ilic, and F. Vollmer, “Experimental observation of strong photon localization in disordered photonic crystal waveguides,” Phys. Rev. Lett. 99(25), 253901 (2007).
[CrossRef]

Jewell, J. L.

J. L. Jewell, Y. H. Lee, S. L. McCall, J. P. Harbison, and L. T. Florez, “High-Finesse (Al,Ga)As Interference Filters Grown By Molecular-Beam Epitaxy,” Appl. Phys. Lett. 53(8), 640–642 (1988).
[CrossRef]

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987).
[CrossRef] [PubMed]

Kaliteevski, M. A.

M. A. Kaliteevski, D. M. Beggs, S. Brand, R. A. Abram, and V. V. Nikolaev, “Statistics of the eigenmodes and optical properties of one-dimensional disordered photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(5), 056616 (2006).
[CrossRef] [PubMed]

Kazmierczak, T.

K. D. Singer, T. Kazmierczak, J. Lott, H. Song, Y. H. Wu, J. Andrews, E. Baer, A. Hiltner, and C. Weder, “Melt-processed all-polymer distributed Bragg reflector laser,” Opt. Express 16(14), 10358–10363 (2008).
[CrossRef] [PubMed]

T. Kazmierczak, H. Song, A. Hiltner, and E. Baer,“Polymeric One-Dimensional Photonic Crystals by Continuous Coextrusion,” Macromol. Rapid Commun. 28(23), 2210–2216 (2007).
[CrossRef]

Koyama, F.

F. Koyama, Y. Suematsu, S. Arai, and T. E. Tawee, “1.5-1.6-Mu-M Galnasp/Inp Dynamic-Single-Mode (Dsm) Lasers With Distributed Bragg Reflector,” IEEE J. Quantum Electron. 19(6), 1042–1051 (1983).
[CrossRef]

Kroll, N.

Lagendijk, A.

M. P. V. Albada and A. Lagendijk,“Observation of Weak Localization of Light in a Random Medium,” Phys. Rev. Lett. 55(24), 2692–2695 (1985).
[CrossRef] [PubMed]

Lahini, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

Lee, Y. H.

J. L. Jewell, Y. H. Lee, S. L. McCall, J. P. Harbison, and L. T. Florez, “High-Finesse (Al,Ga)As Interference Filters Grown By Molecular-Beam Epitaxy,” Appl. Phys. Lett. 53(8), 640–642 (1988).
[CrossRef]

Lott, J.

Lugan, P.

L. Sanchez-Palencia, D. Clément, P. Lugan, P. Bouyer, G. V. Shlyapnikov, and A. Aspect, “Anderson localization of expanding Bose-Einstein Condensates in Random Potentials,” Phys. Rev. Lett. 98(21), 210401 (2007).
[CrossRef] [PubMed]

Maradudin, A. A.

A. R. McGurn, K. T. Christensen, F. M. Mueller, and A. A. Maradudin, “Anderson Localization In One-Dimensional Randomly Disordered Optical-Systems That Are Periodic On Average,” Phys. Rev. B 47(20), 13120–13125 (1993).
[CrossRef]

Maret, G.

P.-E. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55(24), 2696–2699 (1985).
[CrossRef] [PubMed]

McCall, S. L.

D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, “Photonic Band-Structure And Defects In One And 2 Dimensions,” J. Opt. Soc. Am. B 10(2), 314–321 (1993).
[CrossRef]

J. L. Jewell, Y. H. Lee, S. L. McCall, J. P. Harbison, and L. T. Florez, “High-Finesse (Al,Ga)As Interference Filters Grown By Molecular-Beam Epitaxy,” Appl. Phys. Lett. 53(8), 640–642 (1988).
[CrossRef]

McGurn, A. R.

A. R. McGurn, K. T. Christensen, F. M. Mueller, and A. A. Maradudin, “Anderson Localization In One-Dimensional Randomly Disordered Optical-Systems That Are Periodic On Average,” Phys. Rev. B 47(20), 13120–13125 (1993).
[CrossRef]

Milner, V.

S. Zhang, J. Park, V. Milner, and A. Z. Genack, “Photon Delocalization Transition in Dimensional Crossover in Layered Media,” Phys. Rev. Lett. 101(18), 183901 (2008).
[CrossRef] [PubMed]

Morandotti, R.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

Mueller, F. M.

A. R. McGurn, K. T. Christensen, F. M. Mueller, and A. A. Maradudin, “Anderson Localization In One-Dimensional Randomly Disordered Optical-Systems That Are Periodic On Average,” Phys. Rev. B 47(20), 13120–13125 (1993).
[CrossRef]

Nikolaev, V. V.

M. A. Kaliteevski, D. M. Beggs, S. Brand, R. A. Abram, and V. V. Nikolaev, “Statistics of the eigenmodes and optical properties of one-dimensional disordered photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(5), 056616 (2006).
[CrossRef] [PubMed]

Nöckel, J. U.

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385(6611), 45–47 (1997).
[CrossRef]

Park, J.

S. Zhang, J. Park, V. Milner, and A. Z. Genack, “Photon Delocalization Transition in Dimensional Crossover in Layered Media,” Phys. Rev. Lett. 101(18), 183901 (2008).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, “Symmetry And Transport Of Waves In One-Dimensional Disordered-Systems,” Adv. Phys. 43(4), 461–542 (1994).
[CrossRef]

Platzman, P. M.

Pozzi, F.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

Sanchez-Palencia, L.

L. Sanchez-Palencia, D. Clément, P. Lugan, P. Bouyer, G. V. Shlyapnikov, and A. Aspect, “Anderson localization of expanding Bose-Einstein Condensates in Random Potentials,” Phys. Rev. Lett. 98(21), 210401 (2007).
[CrossRef] [PubMed]

Schultz, S.

Schwartz, T.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Segev, M.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Shlyapnikov, G. V.

L. Sanchez-Palencia, D. Clément, P. Lugan, P. Bouyer, G. V. Shlyapnikov, and A. Aspect, “Anderson localization of expanding Bose-Einstein Condensates in Random Potentials,” Phys. Rev. Lett. 98(21), 210401 (2007).
[CrossRef] [PubMed]

Silberberg, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

Singer, K. D.

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Figures (4)

Fig. 1
Fig. 1

(a) Atomic force micrograph of the cross section of the multilayer polymer film. (b) Statistic on the layer thicknesses. The layer thickness variation is 22%.

Fig. 2
Fig. 2

(a) Typical emission spectrum of the micro-resonator laser (black) superimposed on the DBR stack transmission spectrum(red). (b) The relationship between core layer thickness and the reciprocal of the mode spacing. Solid line is the linear fitting.

Fig. 3
Fig. 3

(a) Transmission spectrum of a perfect film (black) and a real film (red). (b) The effective penetration length (black) calculated from Eq. (3) and reflection spectrum (red) of a “perfect” film. (c) The effective penetration length (black) calculated from Eq. (3) and reflection spectrum (red) of the “real” film.(d) The inverse of localization length for a “perfect” (black) and “real” (red) disordered multilayer polymer film by calculating the largest eigenvalue of the average transfer matrix. P is the average bilayer thickness. The refractive indices used in the calculation are 1.49 and 1.585. In our real system, 2P~370nm.

Fig. 4
Fig. 4

Transfer matrix calculations of the intensity distribution in a (a) “perfect” 128 layer film, and (b) the “real” disordered 128 layer film. Part (c) plots the intensity against the position outside the band gap. Solid curves: perfect structure; dashed curves: real film with disorder; black curves for wavelength λ2 denoted in (b); blue curves: at wavelength λ1. Part (d) plots the exponential decay (perfect film, solid line) non-exponential decay(real film, dashed curve) behavior of the intensity inside band gap (λ3).

Equations (4)

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λ02/Δλ=2neffLeff
Leff=Lcore+2Lp
Lp=Δφ2k=λ24πnΔφλ
(1+16(p2κ2))t3+s(212κ2)t2+t(κ2p2)κ2s=0.

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