Abstract

We demonstrate that cylindrical vector beams with radial and azimuthal polarization states can be generated by leaky emission from photoexcited molecules embedded in slab-optical-waveguides which are formed on thin metal films on glass. Mirrorless lasing action in the optical waveguide leads to an order-of-magnitude collapse of the emission energy bandwidth and an emission directionality enhancement exceeding three-fold. This leads to the creation of fine rings of quasi-coherent light with radial and azimuthal polarizations. We study the effect of the leakage loss on the amplified spontaneous emission process and on the photon yield. We find a critical value of metal film thickness for the observation of mirrorless lasing action and optimal values for enhancing photon extraction.

© 2012 OSA

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2011 (4)

T. Ellenbogen, P. Steinvurzel, and K. B. Crozier, “Strong coupling between excitons in j-aggregates and waveguide modes in thin polymer films,” Appl. Phys. Lett.98(26), 261103 (2011).
[CrossRef]

T. Ellenbogen and K. B. Crozier, “Exciton-polariton emission from organic semiconductor optical waveguides,” Phys. Rev. B84(16), 161304 (2011).
[CrossRef]

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics5(3), 166–169 (2011).
[CrossRef]

F. K. Fatemi, “Cylindrical vector beams for rapid polarization-dependent measurements in atomic systems,” Opt. Express19(25), 25143–25150 (2011).
[CrossRef] [PubMed]

2010 (1)

G. M. Akselrod, J. R. Tischler, E. R. Young, D. G. Nocera, and V. Bulovic, “Exciton-exciton annihilation in organic polariton microcavities,” Phys. Rev. B82(11), 113106 (2010).
[CrossRef]

2009 (1)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. in Opt. and Photon.1(1), 1–57 (2009).
[CrossRef]

2008 (1)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

2007 (1)

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A368(5), 402–407 (2007).
[CrossRef]

2006 (2)

2005 (1)

2004 (2)

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91(23), 233901 (2003).
[CrossRef] [PubMed]

2001 (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single Molecules,” Phys. Rev. Lett.86(23), 5251–5254 (2001).
[CrossRef] [PubMed]

2000 (3)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179(1-6), 1–7 (2000).
[CrossRef]

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett.77(21), 3322–3324 (2000).
[CrossRef]

P. Yang, G. Wirnsberger, H. C. Huang, S. R. Cordero, M. D. McGehee, B. Scott, T. Deng, G. M. Whitesides, B. F. Chmelka, S. K. Buratto, and G. D. Stucky, “Mirrorless lasing from mesostructured waveguides patterned by soft lithography,” Science287(5452), 465–467 (2000).
[CrossRef] [PubMed]

1993 (1)

1972 (1)

L. W. Casperson and A. Yariv, “Spectral narrowing in high-gain lasers,” IEEE J. Quantum Electron.8(2), 80–85 (1972).
[CrossRef]

Akselrod, G. M.

G. M. Akselrod, J. R. Tischler, E. R. Young, D. G. Nocera, and V. Bulovic, “Exciton-exciton annihilation in organic polariton microcavities,” Phys. Rev. B82(11), 113106 (2010).
[CrossRef]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single Molecules,” Phys. Rev. Lett.86(23), 5251–5254 (2001).
[CrossRef] [PubMed]

Blit, S.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett.77(21), 3322–3324 (2000).
[CrossRef]

Bomzon, Z.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett.77(21), 3322–3324 (2000).
[CrossRef]

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single Molecules,” Phys. Rev. Lett.86(23), 5251–5254 (2001).
[CrossRef] [PubMed]

Bulovic, V.

G. M. Akselrod, J. R. Tischler, E. R. Young, D. G. Nocera, and V. Bulovic, “Exciton-exciton annihilation in organic polariton microcavities,” Phys. Rev. B82(11), 113106 (2010).
[CrossRef]

Buratto, S. K.

P. Yang, G. Wirnsberger, H. C. Huang, S. R. Cordero, M. D. McGehee, B. Scott, T. Deng, G. M. Whitesides, B. F. Chmelka, S. K. Buratto, and G. D. Stucky, “Mirrorless lasing from mesostructured waveguides patterned by soft lithography,” Science287(5452), 465–467 (2000).
[CrossRef] [PubMed]

Casperson, L. W.

L. W. Casperson and A. Yariv, “Spectral narrowing in high-gain lasers,” IEEE J. Quantum Electron.8(2), 80–85 (1972).
[CrossRef]

Chen, X. W.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics5(3), 166–169 (2011).
[CrossRef]

Chmelka, B. F.

P. Yang, G. Wirnsberger, H. C. Huang, S. R. Cordero, M. D. McGehee, B. Scott, T. Deng, G. M. Whitesides, B. F. Chmelka, S. K. Buratto, and G. D. Stucky, “Mirrorless lasing from mesostructured waveguides patterned by soft lithography,” Science287(5452), 465–467 (2000).
[CrossRef] [PubMed]

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Choudhury, A.

Cordero, S. R.

P. Yang, G. Wirnsberger, H. C. Huang, S. R. Cordero, M. D. McGehee, B. Scott, T. Deng, G. M. Whitesides, B. F. Chmelka, S. K. Buratto, and G. D. Stucky, “Mirrorless lasing from mesostructured waveguides patterned by soft lithography,” Science287(5452), 465–467 (2000).
[CrossRef] [PubMed]

Crozier, K. B.

T. Ellenbogen and K. B. Crozier, “Exciton-polariton emission from organic semiconductor optical waveguides,” Phys. Rev. B84(16), 161304 (2011).
[CrossRef]

T. Ellenbogen, P. Steinvurzel, and K. B. Crozier, “Strong coupling between excitons in j-aggregates and waveguide modes in thin polymer films,” Appl. Phys. Lett.98(26), 261103 (2011).
[CrossRef]

Davidson, N.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett.77(21), 3322–3324 (2000).
[CrossRef]

Deng, T.

P. Yang, G. Wirnsberger, H. C. Huang, S. R. Cordero, M. D. McGehee, B. Scott, T. Deng, G. M. Whitesides, B. F. Chmelka, S. K. Buratto, and G. D. Stucky, “Mirrorless lasing from mesostructured waveguides patterned by soft lithography,” Science287(5452), 465–467 (2000).
[CrossRef] [PubMed]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91(23), 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179(1-6), 1–7 (2000).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179(1-6), 1–7 (2000).
[CrossRef]

Eghlidi, H.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics5(3), 166–169 (2011).
[CrossRef]

Ellenbogen, T.

T. Ellenbogen and K. B. Crozier, “Exciton-polariton emission from organic semiconductor optical waveguides,” Phys. Rev. B84(16), 161304 (2011).
[CrossRef]

T. Ellenbogen, P. Steinvurzel, and K. B. Crozier, “Strong coupling between excitons in j-aggregates and waveguide modes in thin polymer films,” Appl. Phys. Lett.98(26), 261103 (2011).
[CrossRef]

Fainman, Y.

Fatemi, F. K.

Friesem, A. A.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett.77(21), 3322–3324 (2000).
[CrossRef]

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179(1-6), 1–7 (2000).
[CrossRef]

Götzinger, S.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics5(3), 166–169 (2011).
[CrossRef]

Gupta, D. N.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A368(5), 402–407 (2007).
[CrossRef]

Hasman, E.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett.77(21), 3322–3324 (2000).
[CrossRef]

Huang, H. C.

P. Yang, G. Wirnsberger, H. C. Huang, S. R. Cordero, M. D. McGehee, B. Scott, T. Deng, G. M. Whitesides, B. F. Chmelka, S. K. Buratto, and G. D. Stucky, “Mirrorless lasing from mesostructured waveguides patterned by soft lithography,” Science287(5452), 465–467 (2000).
[CrossRef] [PubMed]

Kant, N.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A368(5), 402–407 (2007).
[CrossRef]

Kim, D. E.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A368(5), 402–407 (2007).
[CrossRef]

Kim, G. H.

Kimura, W. D.

Kozawa, Y.

Kukura, P.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics5(3), 166–169 (2011).
[CrossRef]

Lee, K. G.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics5(3), 166–169 (2011).
[CrossRef]

Lettow, R.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics5(3), 166–169 (2011).
[CrossRef]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91(23), 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179(1-6), 1–7 (2000).
[CrossRef]

Levy, U.

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

McGehee, M. D.

P. Yang, G. Wirnsberger, H. C. Huang, S. R. Cordero, M. D. McGehee, B. Scott, T. Deng, G. M. Whitesides, B. F. Chmelka, S. K. Buratto, and G. D. Stucky, “Mirrorless lasing from mesostructured waveguides patterned by soft lithography,” Science287(5452), 465–467 (2000).
[CrossRef] [PubMed]

Nocera, D. G.

G. M. Akselrod, J. R. Tischler, E. R. Young, D. G. Nocera, and V. Bulovic, “Exciton-exciton annihilation in organic polariton microcavities,” Phys. Rev. B82(11), 113106 (2010).
[CrossRef]

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single Molecules,” Phys. Rev. Lett.86(23), 5251–5254 (2001).
[CrossRef] [PubMed]

Oron, R.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett.77(21), 3322–3324 (2000).
[CrossRef]

Pang, L.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91(23), 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179(1-6), 1–7 (2000).
[CrossRef]

Renn, A.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics5(3), 166–169 (2011).
[CrossRef]

Sandoghdar, V.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics5(3), 166–169 (2011).
[CrossRef]

Sato, S.

Scott, B.

P. Yang, G. Wirnsberger, H. C. Huang, S. R. Cordero, M. D. McGehee, B. Scott, T. Deng, G. M. Whitesides, B. F. Chmelka, S. K. Buratto, and G. D. Stucky, “Mirrorless lasing from mesostructured waveguides patterned by soft lithography,” Science287(5452), 465–467 (2000).
[CrossRef] [PubMed]

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Sheppard, C. J. R.

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Steinvurzel, P.

T. Ellenbogen, P. Steinvurzel, and K. B. Crozier, “Strong coupling between excitons in j-aggregates and waveguide modes in thin polymer films,” Appl. Phys. Lett.98(26), 261103 (2011).
[CrossRef]

Stucky, G. D.

P. Yang, G. Wirnsberger, H. C. Huang, S. R. Cordero, M. D. McGehee, B. Scott, T. Deng, G. M. Whitesides, B. F. Chmelka, S. K. Buratto, and G. D. Stucky, “Mirrorless lasing from mesostructured waveguides patterned by soft lithography,” Science287(5452), 465–467 (2000).
[CrossRef] [PubMed]

Suk, H.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, “Electron acceleration to GeV energy by a radially polarized laser,” Phys. Lett. A368(5), 402–407 (2007).
[CrossRef]

Tidwell, S. C.

Tischler, J. R.

G. M. Akselrod, J. R. Tischler, E. R. Young, D. G. Nocera, and V. Bulovic, “Exciton-exciton annihilation in organic polariton microcavities,” Phys. Rev. B82(11), 113106 (2010).
[CrossRef]

Tsai, C. H.

Wang, H.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Whitesides, G. M.

P. Yang, G. Wirnsberger, H. C. Huang, S. R. Cordero, M. D. McGehee, B. Scott, T. Deng, G. M. Whitesides, B. F. Chmelka, S. K. Buratto, and G. D. Stucky, “Mirrorless lasing from mesostructured waveguides patterned by soft lithography,” Science287(5452), 465–467 (2000).
[CrossRef] [PubMed]

Wirnsberger, G.

P. Yang, G. Wirnsberger, H. C. Huang, S. R. Cordero, M. D. McGehee, B. Scott, T. Deng, G. M. Whitesides, B. F. Chmelka, S. K. Buratto, and G. D. Stucky, “Mirrorless lasing from mesostructured waveguides patterned by soft lithography,” Science287(5452), 465–467 (2000).
[CrossRef] [PubMed]

Yang, P.

P. Yang, G. Wirnsberger, H. C. Huang, S. R. Cordero, M. D. McGehee, B. Scott, T. Deng, G. M. Whitesides, B. F. Chmelka, S. K. Buratto, and G. D. Stucky, “Mirrorless lasing from mesostructured waveguides patterned by soft lithography,” Science287(5452), 465–467 (2000).
[CrossRef] [PubMed]

Yariv, A.

L. W. Casperson and A. Yariv, “Spectral narrowing in high-gain lasers,” IEEE J. Quantum Electron.8(2), 80–85 (1972).
[CrossRef]

Yonezawa, K.

Young, E. R.

G. M. Akselrod, J. R. Tischler, E. R. Young, D. G. Nocera, and V. Bulovic, “Exciton-exciton annihilation in organic polariton microcavities,” Phys. Rev. B82(11), 113106 (2010).
[CrossRef]

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single Molecules,” Phys. Rev. Lett.86(23), 5251–5254 (2001).
[CrossRef] [PubMed]

Zhan, Q.

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. in Opt. and Photon.1(1), 1–57 (2009).
[CrossRef]

Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett.31(11), 1726–1728 (2006).
[CrossRef] [PubMed]

Adv. in Opt. and Photon. (1)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. in Opt. and Photon.1(1), 1–57 (2009).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

T. Ellenbogen, P. Steinvurzel, and K. B. Crozier, “Strong coupling between excitons in j-aggregates and waveguide modes in thin polymer films,” Appl. Phys. Lett.98(26), 261103 (2011).
[CrossRef]

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett.77(21), 3322–3324 (2000).
[CrossRef]

IEEE J. Quantum Electron. (1)

L. W. Casperson and A. Yariv, “Spectral narrowing in high-gain lasers,” IEEE J. Quantum Electron.8(2), 80–85 (1972).
[CrossRef]

Nat. Photonics (2)

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Figures (5)

Fig. 1
Fig. 1

(a) Illustration of the sample and the conical leakage to rings of light with radial and azimuthal polarizations. (b) Calculation of the angular dispersion of TE (red line) and TM (black dashed) waveguide modes in a 1.1μm thick PMMA waveguide on top of a silver film. Photograph of leaky emission when the sample is pumped with a CW source without (c) and with (d) polarizer in beam path. (e) and (f) correspond to (c) and (d) respectively when the sample is pumped at high pumping rates.

Fig. 2
Fig. 2

Measured angular dispersion of emission from a sample having a silver film 45 nm thick for (a) low pumping rate and (b) high pumping rate. (c) Emission spectra of the T E 4 mode at low (blue) and high (red) pumping rate. (d) Angular spread of emission of the T E 4 mode at low (blue) and high (red) pumping rate. (e) Normalized emission spectra of T E 4 for different pump pulse energies showing narrowing of the emission down to 5.9 nm FWHM. (f) Collected emission from T E 4 mode for different pump pulse energies.

Fig. 3
Fig. 3

(a) Calculated leaky emission spectra for different pumping rates. Inset shows the data as intensity plot: spectral narrowing is evident. (b) Dependence of leaky emission on the leakage loss for different gain values.

Fig. 4
Fig. 4

(a) Measured emission dispersion characteristics of samples with different silver film thicknesses pumped at 2.1 mJ/cm2. (b) Emission from TM4 for sample with 25 nm silver thickness showing sublinear dependence on pulse energy and with 30 nm silver thickness showing superlinear dependence on the pump energy. Emission from TM4 (c) and TE4 (d) vs. pulse energy for samples with different silver thicknesses. (e) Normalized emission at high pumping rate from TE and TM leaky modes from samples with different silver thicknesses showing an optimum value of silver thickness for maximum photon extraction.

Fig. 5
Fig. 5

Calculated transient decay rate in inverse picoseconds for TE3 (blue line) and TM3 (red dashed line) waveguide modes vs. silver film thickness. The inset shows the region where the silver film thickness ranges from 50 to 120 nm in enlarged detail.

Equations (5)

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k glass (λ)sin( θ m p )= β m p (λ)
I(λ,z)= ηg gα { exp[ ( gα )z ]1 }
d I leak (λ,z) dz = α leak I(λ,z)
I leak (λ,z)= α leak ηg (gα) 2 { exp[ ( gα )z ]1 } α leak ηgz gα
γ 50nm γ 100nm = γ leak_50nm + γ m_50nm γ m_100nm < γ leak_50nm + γ m_50nm γ m_50 = γ leak_50nm γ m_50 +1

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