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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 20 — Sep. 28, 2009
  • pp: 18038–18043
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Mode delocalization in 1D photonic crystal lasers

Yeheng Wu, Kenneth D. Singer, Rolfe G. Petschek, Hyunmin Song, Eric Baer, and Anne Hiltner  »View Author Affiliations


Optics Express, Vol. 17, Issue 20, pp. 18038-18043 (2009)
http://dx.doi.org/10.1364/OE.17.018038


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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

Anderson localization is a concept in the physics of solids dealing, for example, with a transition from metal to insulator due to multiple scattering of electronic wave functions from random disorder in an otherwise periodic potential resulting from interference of locally scattered waves [1

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

]. The analogous, photon localization in disordered photonic crystals (PhC) has also received considerable theoretical attention during the last decade [2

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

4

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

]. Early microwave measurements demonstrated photon localization in a disordered system [5

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

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]

]. A number of experiments on photon localization have been carried out on 2 or 3 dimensional systems [5

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

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]

]. Observation of both weak [7

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

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]

] and strong photon localization [9

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]

] have been found in coherent back scattering experiments. Localization in random media have found application in the study of random lasers [12

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

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

].

In one dimension (1D), Anderson localization has been studied in electrons, matter waves [14

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]

], photons [8

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]

], and acoustic waves [15

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]

]. In a perfectly periodic dielectric material in 1D PhC, a reflection band appears as a photonic bandgap. Within the bandgap, light is localized as coherent reflection attenuates the light penetrating the structure [3

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

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]

]. Defects in a PhC structure lead to localized defect modes within the bandgap [16

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]

]. In contrast, theory and simulations have predicted that a loss of coherent reflection in disordered structures resulting in delocalization of the optical mode as is penetrates the PhC structure [17

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

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]

].

In this work, we study Anderson localization in the case of a 1D disordered lattice by studying the phase delay associated with the longitudinal mode separation in a polymer distributed Bragg laser [19

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]

]. We have confirmed that, indeed, the optical mode in the Bragg mirrors becomes delocalized in the presence of disorder, and we were able to quantify the corresponding delocalization. Through transfer matrix theory and simulations, we were able to study this delocalized mode and its connection to Anderson localization outside of the bandgap in keeping with Pendry’s analysis [4

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

]. We are able to observe agreement between transfer matrix theory and experiment for disorder induced delocalization within the bandgap by studying the phase delay of the reflected wave. We were also able to quantify the disorder induced localization outside of the bandgap by studying the transmission spectrum of the Bragg mirrors, and to theoretically identify three parameters that control the localization in such reflectors.

2. Experiment

Recently, we reported on all-polymer surface emitting distributed Bragg reflector lasers fabricated using a roll-to-roll compatible melt process [19

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]

]. In addition to their fundamental interest, such reflectors are of interest as they may find practical application in low cost photonic devices. The lasers are produced from two 128-layer (i.e. 64 bilayers each) co-extruded polymer films and a polymer core layers containing laser dyes sandwiched between the Bragg reflectors. The polymers used to make multilayer Bragg mirrors were poly(methyl methacrylate) (PMMA, 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 5.4×10−3M.

The layer thicknesses comprising the polymer Bragg mirrors were measured using atomic force microscopy (AFM) of the cross section, with a typical cross section shown in Fig. 1(a)
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%.
. Even though the multilayer film has considerable thickness fluctuations (Fig. 1b), it exhibits a clear reflection band as shown in Fig. 2a
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.
or Fig. 3a
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.
(the difference in these two figures is that there dye absorption centered at around 525nm contributes in Fig. 2a, but not 3a.) These fluctuations occur due to the co-extrusion process, where layer multiplication is used to make the large number of layers [20

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]

]. The viscosities of the two melted polymer fluids were well matched by controlling their temperature. However, remaining rheological differences, edge effects during melt spreading in the layer-multiplication dies, and path-length differences in the dies create fluctuations in the layer thickness [21

21. 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.

].

3. Simulation and discussion

Since our measurement was based on many different films, it is suitable to study the large ensemble average. Pendry indicated how to relate the average of the inverse transmittance to a 3×3 matrix <X(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, lloc=P/t, where P is the (average) period of the system or average bilayer thickness and t=ln(ymax(2)) where ymax(2)is the eigenvalue of <X(2s)> for which t has the largest real part.

In Fig. 1b, the statistics on the multilayer film cross section gives σ=0.22x¯. The inverse localization length (e.g. 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.

We can also examine the intensity distribution throughout the layers. Assuming the light is incident from the left. Figure 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).
depicts the results of our numerical transfer matrix calculations on a perfect film (Fig. 4a), and a specific realization of a disordered film (Fig. 4(b)). In Fig. 4(a), intense bands near the band edge can be easily seen reflecting the enhanced density of photonic states at the band-edge. In Fig. 4(b), the large intensity regions (red spots) are seen to be more localized, reflecting Anderson localization. The appearance of the localized fields outside of the gap is consistent with the results depicted in Fig. 3(d), except that in Fig. 3(d) average values are given. This is more clearly verified in Fig. 4(c) as photon localization is enhanced in the presence of disorder, which is shown in the figure by narrowed peaks. Inside the reflection band, the decay is non-exponential as would be generally expected in a disordered system as, shown in Fig. 4(d). The mode penetrates further and thus is less localized, which is also consistent with conclusion drawn from Fig. 3(d). The phase delay calculated from the laser modes includes the phase accumulated over the nonexponential section beyond layer 60 in Fig. 4(d).

Thus, we have found both experimentally and theoretically that the disorder introduced into the periodic structure causes the penetration depth to increase well inside the bandgap of the corresponding perfect mirror. We note that the penetration into the layers within the reflection band as deduced from the phase delay reflected in the longitudinal mode spacing has, qualitatively, the same trend as photon delocalization/localization as determined by the energy distribution derived from two types of transfer matrix calculations. The details of this calculation and the relationship between them will be described in a later publication [26

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

].

4. Conclusion

Acknowledgement

The authors are grateful to the National Science Foundation for financial support under the Science and Technology Center for Layered Polymer Systems under grant number 0423914 and the National Science Foundation Materials World Network under grant number DMR-0602767.

References and links

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

21.

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.

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. 19(6), 1042–1051 (1983). [CrossRef]

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. 53(8), 640–642 (1988). [CrossRef]

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. 28(2), 514–524 (1992). [CrossRef]

25.

P. Yeh, Optical Waves in Layered Media, Wiley Series in Pure and Applied Optics (Wiley, 1998).

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

  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]
  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]
  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]
  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]
  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]
  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]
  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]
  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]
  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]
  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]
  21. 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.
  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. 19(6), 1042–1051 (1983). [CrossRef]
  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. 53(8), 640–642 (1988). [CrossRef]
  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. 28(2), 514–524 (1992). [CrossRef]
  25. P. Yeh, Optical Waves in Layered Media, Wiley Series in Pure and Applied Optics (Wiley, 1998).
  26. R. G. Petschek, Y. Wu, and K. D. Singer, (unpublished).

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