Abstract

A novel demonstration of an all-optically controllable dye-doped liquid crystal infiltrated photonic crystal fiber (DDLCIPCF) is presented. Overall spectral transmittance of the DDLCIPCF can decrease and then increase with a concomitant red-shift of the spectrum curve with increasing irradiation time of one UV beam. Continuing irradiation of one green beam following UV illumination on the DDLCIPCF can cause the transmission spectrum to recover completely. The reversible all-optical controllability of the photonic band structure of the fiber is attributable to the isothermal planar nematic (PN) → scattering (S) → isotropic (I) and I → S → PN state transitions of the LCs via the UV-beam-induced transcis and green-beam-induced cistrans back isomerizations of the azo-dye, respectively, in the cladding of the DDLCIPCF. The photoinduced appearance of the S state and the variation of the index modulation between the core and the cladding of the fiber result in the variation of overall spectral transmittance and the shift of transmission spectrum, respectively.

© 2011 OSA

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2010 (1)

2009 (2)

2008 (4)

2007 (4)

D. Noordegraaf, L. Scolari, J. Laegsgaard, L. Rindorf, and T. T. Alkeskjold, “Electrically and mechanically induced long period gratings in liquid crystal photonic bandgap fibers,” Opt. Express 15(13), 7901–7912 (2007).
[CrossRef] [PubMed]

T. T. Alkeskjold and A. Bjarklev, “Electrically controlled broadband liquid crystal photonic bandgap fiber polarimeter,” Opt. Lett. 32(12), 1707–1709 (2007).
[CrossRef] [PubMed]

T. R. Wolínski, S. Ertman, A. Czapla, P. Lesiak, K. Nowecka, A. W. Domanski, E. Nowinowski-Kruszelnicki, R. Dabrowski, and J. Wojcik, “Polarization effects in photonic liquid crystal fibers,” Meas. Sci. Technol. 18(10), 3061–3069 (2007).
[CrossRef]

J. Tuominen, H. Hoffrén, and H. Ludvigsen, “All-optical switch based on liquid-crystal infiltrated photonic bandgap fiber in transverse configuration,” JEOS:RP 2, 07016 (2007).
[CrossRef]

2006 (1)

T. R. Wolínski, K. Szaniawska, S. Ertman, P. Lesiak, A. W. Domanski, R. Dabrowski, E. Nowinowski-Kruszelnicki, and J. Wojcik, “Influence of temperature and electrical fields on propagation properties of photonic liquid-crystal fibres,” Meas. Sci. Technol. 17(5), 985–991 (2006).
[CrossRef]

2005 (3)

L. Scolari, T. T. Alkeskjold, J. Riishede, A. Bjarklev, D. S. Hermann, A. Anawati, M. Nielsen, and P. Bassi, “Continuously tunable devices based on electrical control of dual-frequency liquid crystal filled photonic bandgap fibers,” Opt. Express 13(19), 7483–7496 (2005).
[CrossRef] [PubMed]

V. V. Presnyakov, Z. J. Liu, and V. G. Chigrinov, “Infiltration of photonic crystal fiber with liquid crystals,” Proc. SPIE 6017, 60170J, 60170J-7 (2005).
[CrossRef]

K. Szaniawska, T. R. Wolinski, S. Ertman, P. Lesiak, A. W. Domanski, R. Dabrowski, E. Nowinowski-Kruszelnicki, and J. Wojcik, “Temperature tuning in photonic liquid crystal fibers,” Proc. SPIE 5947, 594705, 594705-6 (2005).
[CrossRef]

2004 (2)

2003 (4)

2002 (1)

2001 (1)

1998 (2)

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998).
[CrossRef] [PubMed]

H.-K. Lee, A. Kanazawa, T. Shiono, T. Ikeda, T. Fujisawa, M. Aizawa, and B. Lee, “All-optically controllable polymer/liquid crystal composite films containing the azobenzene liquid crystal,” Chem. Mater. 10(5), 1402–1407 (1998).
[CrossRef]

Abeeluck, A. K.

Aizawa, M.

H.-K. Lee, A. Kanazawa, T. Shiono, T. Ikeda, T. Fujisawa, M. Aizawa, and B. Lee, “All-optically controllable polymer/liquid crystal composite films containing the azobenzene liquid crystal,” Chem. Mater. 10(5), 1402–1407 (1998).
[CrossRef]

Alkeskjold, T. T.

L. Scolari, S. Gauza, H. Xianyu, L. Zhai, L. Eskildsen, T. T. Alkeskjold, S.-T. Wu, and A. Bjarklev, “Frequency tunability of solid-core photonic crystal fibers filled with nanoparticle-doped liquid crystals,” Opt. Express 17(5), 3754–3764 (2009).
[CrossRef] [PubMed]

L. Wei, L. Eskildsen, J. Weirich, L. Scolari, T. T. Alkeskjold, and A. Bjarklev, “Continuously tunable all-in-fiber devices based on thermal and electrical control of negative dielectric anisotropy liquid crystal photonic bandgap fibers,” Appl. Opt. 48(3), 497–503 (2009).
[CrossRef] [PubMed]

D. Noordegraaf, L. Scolari, J. Laegsgaard, L. Rindorf, and T. T. Alkeskjold, “Electrically and mechanically induced long period gratings in liquid crystal photonic bandgap fibers,” Opt. Express 15(13), 7901–7912 (2007).
[CrossRef] [PubMed]

T. T. Alkeskjold and A. Bjarklev, “Electrically controlled broadband liquid crystal photonic bandgap fiber polarimeter,” Opt. Lett. 32(12), 1707–1709 (2007).
[CrossRef] [PubMed]

L. Scolari, T. T. Alkeskjold, J. Riishede, A. Bjarklev, D. S. Hermann, A. Anawati, M. Nielsen, and P. Bassi, “Continuously tunable devices based on electrical control of dual-frequency liquid crystal filled photonic bandgap fibers,” Opt. Express 13(19), 7483–7496 (2005).
[CrossRef] [PubMed]

T. T. Alkeskjold, J. Laegsgaard, A. Bjarklev, D. S. Hermann, A. Anawati, J. Broeng, J. Li, and S. T. Wu, “All-optical modulation in dye-doped nematic liquid crystal photonic bandgap fibers,” Opt. Express 12(24), 5857–5871 (2004).
[CrossRef] [PubMed]

Anawati, A.

Bartelt, H.

Bassi, P.

Birks, T. A.

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998).
[CrossRef] [PubMed]

Bjarklev, A.

L. Wei, L. Eskildsen, J. Weirich, L. Scolari, T. T. Alkeskjold, and A. Bjarklev, “Continuously tunable all-in-fiber devices based on thermal and electrical control of negative dielectric anisotropy liquid crystal photonic bandgap fibers,” Appl. Opt. 48(3), 497–503 (2009).
[CrossRef] [PubMed]

L. Scolari, S. Gauza, H. Xianyu, L. Zhai, L. Eskildsen, T. T. Alkeskjold, S.-T. Wu, and A. Bjarklev, “Frequency tunability of solid-core photonic crystal fibers filled with nanoparticle-doped liquid crystals,” Opt. Express 17(5), 3754–3764 (2009).
[CrossRef] [PubMed]

T. T. Alkeskjold and A. Bjarklev, “Electrically controlled broadband liquid crystal photonic bandgap fiber polarimeter,” Opt. Lett. 32(12), 1707–1709 (2007).
[CrossRef] [PubMed]

L. Scolari, T. T. Alkeskjold, J. Riishede, A. Bjarklev, D. S. Hermann, A. Anawati, M. Nielsen, and P. Bassi, “Continuously tunable devices based on electrical control of dual-frequency liquid crystal filled photonic bandgap fibers,” Opt. Express 13(19), 7483–7496 (2005).
[CrossRef] [PubMed]

T. T. Alkeskjold, J. Laegsgaard, A. Bjarklev, D. S. Hermann, A. Anawati, J. Broeng, J. Li, and S. T. Wu, “All-optical modulation in dye-doped nematic liquid crystal photonic bandgap fibers,” Opt. Express 12(24), 5857–5871 (2004).
[CrossRef] [PubMed]

T. T. Larsen, A. Bjarklev, D. S. Hermann, and J. Broeng, “Optical devices based on liquid crystal photonic bandgap fibres,” Opt. Express 11(20), 2589–2596 (2003).
[CrossRef] [PubMed]

Borelli, E.

Broeng, J.

Chen, C.-H.

Cheng, W.-H.

Chigrinov, V. G.

V. V. Presnyakov, Z. J. Liu, and V. G. Chigrinov, “Infiltration of photonic crystal fiber with liquid crystals,” Proc. SPIE 6017, 60170J, 60170J-7 (2005).
[CrossRef]

Czapla, A.

T. R. Wolínski, S. Ertman, A. Czapla, P. Lesiak, K. Nowecka, A. W. Domanski, E. Nowinowski-Kruszelnicki, R. Dabrowski, and J. Wojcik, “Polarization effects in photonic liquid crystal fibers,” Meas. Sci. Technol. 18(10), 3061–3069 (2007).
[CrossRef]

Dabrowski, R.

T. R. Wolínski, S. Ertman, A. Czapla, P. Lesiak, K. Nowecka, A. W. Domanski, E. Nowinowski-Kruszelnicki, R. Dabrowski, and J. Wojcik, “Polarization effects in photonic liquid crystal fibers,” Meas. Sci. Technol. 18(10), 3061–3069 (2007).
[CrossRef]

T. R. Wolínski, K. Szaniawska, S. Ertman, P. Lesiak, A. W. Domanski, R. Dabrowski, E. Nowinowski-Kruszelnicki, and J. Wojcik, “Influence of temperature and electrical fields on propagation properties of photonic liquid-crystal fibres,” Meas. Sci. Technol. 17(5), 985–991 (2006).
[CrossRef]

K. Szaniawska, T. R. Wolinski, S. Ertman, P. Lesiak, A. W. Domanski, R. Dabrowski, E. Nowinowski-Kruszelnicki, and J. Wojcik, “Temperature tuning in photonic liquid crystal fibers,” Proc. SPIE 5947, 594705, 594705-6 (2005).
[CrossRef]

de Sterke, C. M.

Domanski, A. W.

T. R. Wolínski, S. Ertman, A. Czapla, P. Lesiak, K. Nowecka, A. W. Domanski, E. Nowinowski-Kruszelnicki, R. Dabrowski, and J. Wojcik, “Polarization effects in photonic liquid crystal fibers,” Meas. Sci. Technol. 18(10), 3061–3069 (2007).
[CrossRef]

T. R. Wolínski, K. Szaniawska, S. Ertman, P. Lesiak, A. W. Domanski, R. Dabrowski, E. Nowinowski-Kruszelnicki, and J. Wojcik, “Influence of temperature and electrical fields on propagation properties of photonic liquid-crystal fibres,” Meas. Sci. Technol. 17(5), 985–991 (2006).
[CrossRef]

K. Szaniawska, T. R. Wolinski, S. Ertman, P. Lesiak, A. W. Domanski, R. Dabrowski, E. Nowinowski-Kruszelnicki, and J. Wojcik, “Temperature tuning in photonic liquid crystal fibers,” Proc. SPIE 5947, 594705, 594705-6 (2005).
[CrossRef]

Dong, X.

Du, J.

Dunn, S. C.

Eggleton, B. J.

Ertman, S.

T. R. Wolínski, S. Ertman, A. Czapla, P. Lesiak, K. Nowecka, A. W. Domanski, E. Nowinowski-Kruszelnicki, R. Dabrowski, and J. Wojcik, “Polarization effects in photonic liquid crystal fibers,” Meas. Sci. Technol. 18(10), 3061–3069 (2007).
[CrossRef]

T. R. Wolínski, K. Szaniawska, S. Ertman, P. Lesiak, A. W. Domanski, R. Dabrowski, E. Nowinowski-Kruszelnicki, and J. Wojcik, “Influence of temperature and electrical fields on propagation properties of photonic liquid-crystal fibres,” Meas. Sci. Technol. 17(5), 985–991 (2006).
[CrossRef]

K. Szaniawska, T. R. Wolinski, S. Ertman, P. Lesiak, A. W. Domanski, R. Dabrowski, E. Nowinowski-Kruszelnicki, and J. Wojcik, “Temperature tuning in photonic liquid crystal fibers,” Proc. SPIE 5947, 594705, 594705-6 (2005).
[CrossRef]

Eskildsen, L.

Fujisawa, T.

H.-K. Lee, A. Kanazawa, T. Shiono, T. Ikeda, T. Fujisawa, M. Aizawa, and B. Lee, “All-optically controllable polymer/liquid crystal composite films containing the azobenzene liquid crystal,” Chem. Mater. 10(5), 1402–1407 (1998).
[CrossRef]

Gauza, S.

Grozhik, V. A.

Hale, A.

Headley, C.

Hermann, D. S.

Hoffrén, H.

J. Tuominen, H. Hoffrén, and H. Ludvigsen, “All-optical switch based on liquid-crystal infiltrated photonic bandgap fiber in transverse configuration,” JEOS:RP 2, 07016 (2007).
[CrossRef]

Hsiao, V. K. S.

Ikeda, T.

H.-K. Lee, A. Kanazawa, T. Shiono, T. Ikeda, T. Fujisawa, M. Aizawa, and B. Lee, “All-optically controllable polymer/liquid crystal composite films containing the azobenzene liquid crystal,” Chem. Mater. 10(5), 1402–1407 (1998).
[CrossRef]

Kanazawa, A.

H.-K. Lee, A. Kanazawa, T. Shiono, T. Ikeda, T. Fujisawa, M. Aizawa, and B. Lee, “All-optically controllable polymer/liquid crystal composite films containing the azobenzene liquid crystal,” Chem. Mater. 10(5), 1402–1407 (1998).
[CrossRef]

Kao, C.-L.

Kerbage, C.

Kitzerow, H.-S.

Knight, J. C.

J. C. Knight, “Photonic crystal fibres,” Nature 424(6950), 847–851 (2003).
[CrossRef] [PubMed]

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998).
[CrossRef] [PubMed]

Ko, C.-Y.

Kobelke, J.

Laegsgaard, J.

Larsen, T. T.

Lee, B.

H.-K. Lee, A. Kanazawa, T. Shiono, T. Ikeda, T. Fujisawa, M. Aizawa, and B. Lee, “All-optically controllable polymer/liquid crystal composite films containing the azobenzene liquid crystal,” Chem. Mater. 10(5), 1402–1407 (1998).
[CrossRef]

Lee, C.-H.

Lee, H.-K.

H.-K. Lee, A. Kanazawa, T. Shiono, T. Ikeda, T. Fujisawa, M. Aizawa, and B. Lee, “All-optically controllable polymer/liquid crystal composite films containing the azobenzene liquid crystal,” Chem. Mater. 10(5), 1402–1407 (1998).
[CrossRef]

Lesiak, P.

T. R. Wolínski, S. Ertman, A. Czapla, P. Lesiak, K. Nowecka, A. W. Domanski, E. Nowinowski-Kruszelnicki, R. Dabrowski, and J. Wojcik, “Polarization effects in photonic liquid crystal fibers,” Meas. Sci. Technol. 18(10), 3061–3069 (2007).
[CrossRef]

T. R. Wolínski, K. Szaniawska, S. Ertman, P. Lesiak, A. W. Domanski, R. Dabrowski, E. Nowinowski-Kruszelnicki, and J. Wojcik, “Influence of temperature and electrical fields on propagation properties of photonic liquid-crystal fibres,” Meas. Sci. Technol. 17(5), 985–991 (2006).
[CrossRef]

K. Szaniawska, T. R. Wolinski, S. Ertman, P. Lesiak, A. W. Domanski, R. Dabrowski, E. Nowinowski-Kruszelnicki, and J. Wojcik, “Temperature tuning in photonic liquid crystal fibers,” Proc. SPIE 5947, 594705, 594705-6 (2005).
[CrossRef]

Li, J.

Lin, T.-H.

Litchinitser, N. M.

Liu, B.

Liu, Y.

Liu, Z. J.

V. V. Presnyakov, Z. J. Liu, and V. G. Chigrinov, “Infiltration of photonic crystal fiber with liquid crystals,” Proc. SPIE 6017, 60170J, 60170J-7 (2005).
[CrossRef]

Lorenz, A.

Ludvigsen, H.

J. Tuominen, H. Hoffrén, and H. Ludvigsen, “All-optical switch based on liquid-crystal infiltrated photonic bandgap fiber in transverse configuration,” JEOS:RP 2, 07016 (2007).
[CrossRef]

McPhedran, R. C.

Nielsen, M.

Noordegraaf, D.

Nowecka, K.

T. R. Wolínski, S. Ertman, A. Czapla, P. Lesiak, K. Nowecka, A. W. Domanski, E. Nowinowski-Kruszelnicki, R. Dabrowski, and J. Wojcik, “Polarization effects in photonic liquid crystal fibers,” Meas. Sci. Technol. 18(10), 3061–3069 (2007).
[CrossRef]

Nowinowski-Kruszelnicki, E.

T. R. Wolínski, S. Ertman, A. Czapla, P. Lesiak, K. Nowecka, A. W. Domanski, E. Nowinowski-Kruszelnicki, R. Dabrowski, and J. Wojcik, “Polarization effects in photonic liquid crystal fibers,” Meas. Sci. Technol. 18(10), 3061–3069 (2007).
[CrossRef]

T. R. Wolínski, K. Szaniawska, S. Ertman, P. Lesiak, A. W. Domanski, R. Dabrowski, E. Nowinowski-Kruszelnicki, and J. Wojcik, “Influence of temperature and electrical fields on propagation properties of photonic liquid-crystal fibres,” Meas. Sci. Technol. 17(5), 985–991 (2006).
[CrossRef]

K. Szaniawska, T. R. Wolinski, S. Ertman, P. Lesiak, A. W. Domanski, R. Dabrowski, E. Nowinowski-Kruszelnicki, and J. Wojcik, “Temperature tuning in photonic liquid crystal fibers,” Proc. SPIE 5947, 594705, 594705-6 (2005).
[CrossRef]

Presnyakov, V. V.

V. V. Presnyakov, Z. J. Liu, and V. G. Chigrinov, “Infiltration of photonic crystal fiber with liquid crystals,” Proc. SPIE 6017, 60170J, 60170J-7 (2005).
[CrossRef]

Riishede, J.

Rindorf, L.

Russell, P.

P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003).
[CrossRef] [PubMed]

Russell, P. S. J.

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998).
[CrossRef] [PubMed]

Schwuchow, A.

Scolari, L.

Serak, S. V.

Shiono, T.

H.-K. Lee, A. Kanazawa, T. Shiono, T. Ikeda, T. Fujisawa, M. Aizawa, and B. Lee, “All-optically controllable polymer/liquid crystal composite films containing the azobenzene liquid crystal,” Chem. Mater. 10(5), 1402–1407 (1998).
[CrossRef]

Steinvurzel, P. E.

Szaniawska, K.

T. R. Wolínski, K. Szaniawska, S. Ertman, P. Lesiak, A. W. Domanski, R. Dabrowski, E. Nowinowski-Kruszelnicki, and J. Wojcik, “Influence of temperature and electrical fields on propagation properties of photonic liquid-crystal fibres,” Meas. Sci. Technol. 17(5), 985–991 (2006).
[CrossRef]

K. Szaniawska, T. R. Wolinski, S. Ertman, P. Lesiak, A. W. Domanski, R. Dabrowski, E. Nowinowski-Kruszelnicki, and J. Wojcik, “Temperature tuning in photonic liquid crystal fibers,” Proc. SPIE 5947, 594705, 594705-6 (2005).
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Figures (11)

Fig. 1
Fig. 1

(a) The CCD image of the cross section of the DDLCIPCF. Lateral transmitted images of the DDLCIPCF are observed under the POM with crossed polarizer and analyzer (P⊥A), in which the fiber axis is oriented at (b) 45° and (c) 90° relative to the transmission axis of the polarizer.

Fig. 2
Fig. 2

Top view of the experimental setups for examining the all-optical controllability of the DDLCIPCF. One white beam is guided via a single-mode fiber A and coupled into the input end of the DDLCIPCF. One UV beam and one green beam are guided to irradiate the DDLCIPCF at sequence. (a) The transmitted image of the cross section of the DDLCIPCF at its output end is collected by an objective lens (×10) and then recorded by a CCD camera. (b) The spectrum of the transmitted output signal at the output end of the DDLCIPCF is coupled into a single-mode fiber B and guided into a spectrometer for recording the transmission spectrum of the white beam via the DDLCIPCF. In (a) and (b), the green beam is reflected by an NBS into the objective lens and guided by the single-mode fiber B, respectively, and then irradiates the DDLCIPCF.

Fig. 3
Fig. 3

Blue and red absorption spectrum curves for the 4MAB azo-dye in a homogenously aligned DDLC plane cell with a 7 μm thickness before and after the irradiation of the UV light with 237.5 mW/cm2 on the cell for two minutes, respectively. The inset represents the transcis isomerization under UV irradiation and the cistrans back isomerization via green-light-irradiation or thermal relaxation (Δ) for the azo-dye.

Fig. 4
Fig. 4

Variations of (a) the CCD image of the cross section and (b) the normalized transmitted intensity in the output end of the DDLCIPCF upon increasing the irradiation time of the UV beam tUV from 0 to 660 s. Three different states of LCs in the cladding of the DDLCIPCF present in those relative extremities of the transmitted intensity curve.

Fig. 5
Fig. 5

Variations of (a) the CCD image of the cross section and (b) the normalized transmitted intensity in the output end of the DDLCIPCF upon increasing the irradiation time of the green beam tG from 0 to 60 s, following UV irradiation with 237.5 mW/cm2 for 660 s.

Fig. 6
Fig. 6

Mechanism for the isothermal phase transitions of LCs from N to I and I to N phases induced by the transcis and cistrans back isomerizations of the azo-dyes, respectively, under successive irradiations of one UV and one green beams, with increasing individual irradiation time of tUV and tG, respectively.

Fig. 8
Fig. 8

Lateral transmitted images of the DDLCIPCF are recorded under the POM with crossed polarizers (P⊥A) at ϕ=45° (first row) and 90° (second row) while the LCs in the cladding region of the fiber lies in (a) PN, (b) S, and (c) I states. Magnified images in the insets in (b) display the presentation of multi-domain-like LC texture that coexists with the PN and I states in the cladding region of the fiber under irradiation of the UV beam (at tUV=20 s) or the green beam (tG=15 s).

Fig. 7
Fig. 7

Model describing the reversible transformation of the LC state in the cladding region of the DDLCIPCF: (a) PN → (b) S → (c) I states with increasing tUV, and (c) I → (b) S → (a) PN states with increasing tG. In the S state, scattering is caused by the multi-domain-like LC texture that coexists with I and PN states in the cladding region of the fiber.

Fig. 9
Fig. 9

Variations of the transmission spectrum of the incident white beam through the DDLCIPCF with increasing tUV (a) from 0 to 20 s (PN → S states) and (b) from 20 to 660 s (S → I states) at a fixed irradiated intensity 237.5 mW/cm2 of the UV beam. The red-dotted vertical lines shown in (a) and (b) indicate the calculated cut-off wavelengths of the optical mode guided in a single LC hole based on Eq. (1) when the LCs in the hole is at PN and I state, respectively. These cut-off wavelengths coincide with the minima in the transmission spectrum of the DDLCIPCF.

Fig. 10
Fig. 10

Following UV irradiation for 660 s, as described in Fig. 9, the UV beam is turned off, and simultaneously the green beam with a fixed irradiated intensity of 955.4 mW/cm2 is turned on to irradiate the fiber based on the setup in Fig. 2(b). Variations of the transmission spectrum of the incident white beam through the DDLCIPCF with increasing tG (a) from 0 to 10 s (I → S states) and (b) from 10 to 50 s (S → PN states) are recorded.

Fig. 11
Fig. 11

Variations of the normalized transmitted intensity and corresponding CCD image of the cross section of the DDLCIPCF with the time via the I→S→PN state transition of LCs in the cladding of the fiber via the thermal cistrans back isomerization, following the UV irradiation with 237.5 mW/cm2 on the fiber for 660 s.

Equations (1)

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λ m = 2 d m + 1 / 2 n 2 2 n 1 2 ,

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