## Mode delocalization in 1D photonic crystal lasers

Optics Express, Vol. 17, Issue 20, pp. 18038-18043 (2009)

http://dx.doi.org/10.1364/OE.17.018038

Acrobat PDF (181 KB)

### 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.

© 2009 OSA

## 1. Introduction

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

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

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]

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]

10. 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]

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]

11. 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]

9. 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]

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

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

14. 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]

8. 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]

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

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]

16. 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]

17. 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]

18. 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]

19. 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]

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

## 2. Experiment

19. 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]

*n*= 1.49) and polystyrene (

*n*= 1.585). The core layer consists of Rhodamine 6G dye doped into a 30/70 blend of PMMA and poly(vinylidenefluoride-co-hexafluoropropylene) with a refractive index of 1.40. The core films were fabricated in various thicknesses with a dye concentration of 5.4×10

^{−3}M.

20. 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]

## 3. Simulation and discussion

^{(2s)}>, which is the average of the symmetric direct product of the transfer matrix with itself. One of the components of the product of

*N*such average transfer matrices is the average inverse transmittance for a stack of

*N*bilayers. For sufficiently large

*N*all components of this product are dominated by the eigenvalue of this matrix having the largest absolute value, as well as its eigenvector. In consequence a definition of the localization length can be taken to be,

*l*, where

_{loc}=P/t*P*is the (average) period of the system or average bilayer thickness and

^{(2s)}> for which

*t*has the largest real part.

*t*/2) of a perfect and real film as calculated from Eq. (4) is given in Fig. 3(d). This figure clearly depicts both the increased localization outside of the gap and the delocalization inside the gap. Note that significant enhanced localization is observed only over a limited region near each band edge.

## 4. Conclusion

## Acknowledgement

## References and links

1. | P. W. Anderson, “Absence of Diffusion in Certain Random Lattices,” Phys. Rev. |

2. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

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 |

4. | J. B. Pendry, “Symmetry And Transport Of Waves In One-Dimensional Disordered-Systems,” Adv. Phys. |

5. | N. Garcia and A. Z. Genack, “Anomalous photon diffusion at the threshold of the Anderson Localization Transition,” Phys. Rev. Lett. |

6. | A. Z. Genack and N. Garcia, “Observation of photon localization in a three-dimensional disordered system,” Phys. Rev. Lett. |

7. | P.-E. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. |

8. | 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. |

9. | J. Topolancik, B. Ilic, and F. Vollmer, “Experimental observation of strong photon localization in disordered photonic crystal waveguides,” Phys. Rev. Lett. |

10. | T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature |

11. | M. P. V. Albada and A. Lagendijk,“Observation of Weak Localization of Light in a Random Medium,” Phys. Rev. Lett. |

12. | J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature |

13. | H. Cao, “Review on latest developments in random lasers with coherent feedback,” J. Phys. Math. Gen. |

14. | 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. |

15. | V. Baluni and J. Willemsen, “Transmission of acoustic waves in a random layered medium,” Phys. Rev. A |

16. | 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 |

17. | 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. |

18. | S. Zhang, J. Park, V. Milner, and A. Z. Genack, “Photon Delocalization Transition in Dimensional Crossover in Layered Media,” Phys. Rev. Lett. |

19. | 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 |

20. | T. Kazmierczak, H. Song, A. Hiltner, and E. Baer,“Polymeric One-Dimensional Photonic Crystals by Continuous Coextrusion,” Macromol. Rapid Commun. |

21. | E. Baer, J. Kerns, and A. Hiltner, |

22. | 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. |

23. | 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. |

24. | 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. |

25. | P. Yeh, |

26. | R. G. Petschek, Y. Wu, and K. D. Singer, (unpublished). |

**OCIS Codes**

(160.5470) Materials : Polymers

(230.1480) Optical devices : Bragg reflectors

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: July 21, 2009

Revised Manuscript: September 15, 2009

Manuscript Accepted: September 15, 2009

Published: September 23, 2009

**Citation**

Yeheng Wu, Kenneth D. Singer, Rolfe G. Petschek, Hyunmin Song, Eric Baer, and Anne Hiltner, "Mode delocalization in 1D photonic crystal lasers," Opt. Express **17**, 18038-18043 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-20-18038

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### References

- P. W. Anderson, “Absence of Diffusion in Certain Random Lattices,” Phys. Rev. 109(5), 1492–1505 (1958). [CrossRef]
- S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987). [CrossRef] [PubMed]
- 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]
- J. B. Pendry, “Symmetry And Transport Of Waves In One-Dimensional Disordered-Systems,” Adv. Phys. 43(4), 461–542 (1994). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385(6611), 45–47 (1997). [CrossRef]
- H. Cao, “Review on latest developments in random lasers with coherent feedback,” J. Phys. Math. Gen. 38(49), 10497–10535 (2005). [CrossRef]
- 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]
- V. Baluni and J. Willemsen, “Transmission of acoustic waves in a random layered medium,” Phys. Rev. A 31(5), 3358–3363 (1985). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- E. Baer, J. Kerns, and A. Hiltner, Processing and Properties of Polymer Microlayered Systems, Structure Development During Polymer Processing (Kluwer Academic Publishers, The Netherlands, 2000), pp. 327–344.
- 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]
- 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]
- 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]
- P. Yeh, Optical Waves in Layered Media, Wiley Series in Pure and Applied Optics (Wiley, 1998).
- R. G. Petschek, Y. Wu, and K. D. Singer, (unpublished).

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