Abstract

A facile method is presented for the fabrication of microlenses at the facet of fused silica capillaries and microstructured fibers. After submersion in hydrogen fluoride solution water is pumped slowly through the center hole of the capillary microchannel to create an etchant gradient extending from the capillary axis. The desired axicon angle is generated by adjusting the etching time and/or concentration of the etchant. Similarly, flow- assisted HF etching of a custom microstructured fiber containing nine microchannels produces nine individual microlenses simultaneously at the fiber facet, where each microaxicon lens shows a similar focusing pattern. A theoretical model of the flow-assisted etching process is used to determine the axicon angle and post angle. Also, a simple ray-based model was applied to characterize the focusing properties of the microaxicons in good agreement with experimental observations.

© 2016 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  4. A. Marcinkevicius, S. Juodkazis, S. Matsuo, V. Mizeikis, and H. Misawa, “Application of Bessel beams for microfabrication of dielectrics by femtosecond laser,” Jpn. J. Appl. Phys. 2(40), L1197–L1199 (2001).
    [Crossref]
  5. Y. Song, D. Milam, and W. T. Hill, “Long, narrow all-light atom guide,” Opt. Lett. 24(24), 1805–1807 (1999).
    [Crossref] [PubMed]
  6. R. S. R. Ribeiro, O. Soppera, A. G. Oliva, A. Guerreiro, and P. A. S. Jorge, “New trends on optical fiber tweezers,” J. Lightwave Technol. 33(16), 3394–3405 (2015).
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  13. R. Stöckle, C. Fokas, V. Deckert, R. Zenobi, B. Sick, B. Hecht, and U. P. Wild, “High-quality near-field optical probes by tube etching,” Appl. Phys. Lett. 75(2), 160–162 (1999).
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    [Crossref]
  20. A. Kotsas, H. Ghafourishiraz, and T. S. M. Maclean, “Microlens fabrication on single-mode fibers for efficient coupling from laser-diodes,” Opt. Quantum Electron. 23(3), 367–378 (1991).
    [Crossref]

2015 (1)

2014 (1)

2013 (1)

2012 (1)

2008 (1)

2007 (1)

2006 (1)

S. Cabrini, C. Liberale, D. Cojoc, A. Carpentiero, M. Prasciolu, S. Mora, V. Degiorgio, F. De Angelis, and E. Di Fabrizio, “Axicon lens on optical fiber forming optical tweezers, made by focused ion beam milling,” Microelectron. Eng. 83(4-9), 804–807 (2006).
[Crossref]

2003 (1)

S. K. Eah, W. Jhe, and Y. Arakawa, “Nearly diffraction-limited focusing of a fiber axicon microlens,” Rev. Sci. Instrum. 74(11), 4969–4971 (2003).
[Crossref]

2001 (1)

A. Marcinkevicius, S. Juodkazis, S. Matsuo, V. Mizeikis, and H. Misawa, “Application of Bessel beams for microfabrication of dielectrics by femtosecond laser,” Jpn. J. Appl. Phys. 2(40), L1197–L1199 (2001).
[Crossref]

1999 (2)

R. Stöckle, C. Fokas, V. Deckert, R. Zenobi, B. Sick, B. Hecht, and U. P. Wild, “High-quality near-field optical probes by tube etching,” Appl. Phys. Lett. 75(2), 160–162 (1999).
[Crossref]

Y. Song, D. Milam, and W. T. Hill, “Long, narrow all-light atom guide,” Opt. Lett. 24(24), 1805–1807 (1999).
[Crossref] [PubMed]

1997 (1)

M. Erdelyi, Z. L. Horvath, G. Szabo, Z. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15(2), 287–292 (1997).
[Crossref]

1995 (1)

P. Hoffmann, B. Dutoit, and R. P. Salathe, “Comparison of mechanically drawn and protection layer chemically etched optical fiber tips,” Ultramicroscopy 61(1-4), 165–170 (1995).
[Crossref]

1993 (1)

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 71(1), 209 (1993).
[Crossref]

1991 (2)

G. He and F. W. Cuomo, “A light-intensity function suitable for multimode fiberoptic sensors,” J. Lightwave Technol. 9(4), 545–551 (1991).
[Crossref]

A. Kotsas, H. Ghafourishiraz, and T. S. M. Maclean, “Microlens fabrication on single-mode fibers for efficient coupling from laser-diodes,” Opt. Quantum Electron. 23(3), 367–378 (1991).
[Crossref]

1987 (1)

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

1982 (1)

1954 (1)

Arakawa, Y.

S. K. Eah, W. Jhe, and Y. Arakawa, “Nearly diffraction-limited focusing of a fiber axicon microlens,” Rev. Sci. Instrum. 74(11), 4969–4971 (2003).
[Crossref]

Berns, M. W.

Bescherer, K.

Bor, Z.

M. Erdelyi, Z. L. Horvath, G. Szabo, Z. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15(2), 287–292 (1997).
[Crossref]

Cabrini, S.

S. Cabrini, C. Liberale, D. Cojoc, A. Carpentiero, M. Prasciolu, S. Mora, V. Degiorgio, F. De Angelis, and E. Di Fabrizio, “Axicon lens on optical fiber forming optical tweezers, made by focused ion beam milling,” Microelectron. Eng. 83(4-9), 804–807 (2006).
[Crossref]

Carpentiero, A.

S. Cabrini, C. Liberale, D. Cojoc, A. Carpentiero, M. Prasciolu, S. Mora, V. Degiorgio, F. De Angelis, and E. Di Fabrizio, “Axicon lens on optical fiber forming optical tweezers, made by focused ion beam milling,” Microelectron. Eng. 83(4-9), 804–807 (2006).
[Crossref]

Cavallaro, J. R.

M. Erdelyi, Z. L. Horvath, G. Szabo, Z. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15(2), 287–292 (1997).
[Crossref]

Charraut, D.

Cojoc, D.

S. Cabrini, C. Liberale, D. Cojoc, A. Carpentiero, M. Prasciolu, S. Mora, V. Degiorgio, F. De Angelis, and E. Di Fabrizio, “Axicon lens on optical fiber forming optical tweezers, made by focused ion beam milling,” Microelectron. Eng. 83(4-9), 804–807 (2006).
[Crossref]

Cuomo, F. W.

G. He and F. W. Cuomo, “A light-intensity function suitable for multimode fiberoptic sensors,” J. Lightwave Technol. 9(4), 545–551 (1991).
[Crossref]

De Angelis, F.

S. Cabrini, C. Liberale, D. Cojoc, A. Carpentiero, M. Prasciolu, S. Mora, V. Degiorgio, F. De Angelis, and E. Di Fabrizio, “Axicon lens on optical fiber forming optical tweezers, made by focused ion beam milling,” Microelectron. Eng. 83(4-9), 804–807 (2006).
[Crossref]

Deckert, V.

R. Stöckle, C. Fokas, V. Deckert, R. Zenobi, B. Sick, B. Hecht, and U. P. Wild, “High-quality near-field optical probes by tube etching,” Appl. Phys. Lett. 75(2), 160–162 (1999).
[Crossref]

Degiorgio, V.

S. Cabrini, C. Liberale, D. Cojoc, A. Carpentiero, M. Prasciolu, S. Mora, V. Degiorgio, F. De Angelis, and E. Di Fabrizio, “Axicon lens on optical fiber forming optical tweezers, made by focused ion beam milling,” Microelectron. Eng. 83(4-9), 804–807 (2006).
[Crossref]

Di Fabrizio, E.

S. Cabrini, C. Liberale, D. Cojoc, A. Carpentiero, M. Prasciolu, S. Mora, V. Degiorgio, F. De Angelis, and E. Di Fabrizio, “Axicon lens on optical fiber forming optical tweezers, made by focused ion beam milling,” Microelectron. Eng. 83(4-9), 804–807 (2006).
[Crossref]

Durnin, J.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Dutoit, B.

P. Hoffmann, B. Dutoit, and R. P. Salathe, “Comparison of mechanically drawn and protection layer chemically etched optical fiber tips,” Ultramicroscopy 61(1-4), 165–170 (1995).
[Crossref]

Eah, S. K.

S. K. Eah, W. Jhe, and Y. Arakawa, “Nearly diffraction-limited focusing of a fiber axicon microlens,” Rev. Sci. Instrum. 74(11), 4969–4971 (2003).
[Crossref]

Eberly, J. H.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Eisenstein, G.

Erdelyi, M.

M. Erdelyi, Z. L. Horvath, G. Szabo, Z. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15(2), 287–292 (1997).
[Crossref]

Fokas, C.

R. Stöckle, C. Fokas, V. Deckert, R. Zenobi, B. Sick, B. Hecht, and U. P. Wild, “High-quality near-field optical probes by tube etching,” Appl. Phys. Lett. 75(2), 160–162 (1999).
[Crossref]

Ghafourishiraz, H.

A. Kotsas, H. Ghafourishiraz, and T. S. M. Maclean, “Microlens fabrication on single-mode fibers for efficient coupling from laser-diodes,” Opt. Quantum Electron. 23(3), 367–378 (1991).
[Crossref]

Grosjean, T.

Guerreiro, A.

Gurbatov, S.

He, G.

G. He and F. W. Cuomo, “A light-intensity function suitable for multimode fiberoptic sensors,” J. Lightwave Technol. 9(4), 545–551 (1991).
[Crossref]

Hecht, B.

R. Stöckle, C. Fokas, V. Deckert, R. Zenobi, B. Sick, B. Hecht, and U. P. Wild, “High-quality near-field optical probes by tube etching,” Appl. Phys. Lett. 75(2), 160–162 (1999).
[Crossref]

Herminghaus, S.

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 71(1), 209 (1993).
[Crossref]

Hill, W. T.

Hoffmann, P.

P. Hoffmann, B. Dutoit, and R. P. Salathe, “Comparison of mechanically drawn and protection layer chemically etched optical fiber tips,” Ultramicroscopy 61(1-4), 165–170 (1995).
[Crossref]

Horvath, Z. L.

M. Erdelyi, Z. L. Horvath, G. Szabo, Z. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15(2), 287–292 (1997).
[Crossref]

Ibrahim, I. A.

Jhe, W.

S. K. Eah, W. Jhe, and Y. Arakawa, “Nearly diffraction-limited focusing of a fiber axicon microlens,” Rev. Sci. Instrum. 74(11), 4969–4971 (2003).
[Crossref]

Jorge, P. A. S.

Juodkazis, S.

A. Marcinkevicius, S. Juodkazis, S. Matsuo, V. Mizeikis, and H. Misawa, “Application of Bessel beams for microfabrication of dielectrics by femtosecond laser,” Jpn. J. Appl. Phys. 2(40), L1197–L1199 (2001).
[Crossref]

Kotsas, A.

A. Kotsas, H. Ghafourishiraz, and T. S. M. Maclean, “Microlens fabrication on single-mode fibers for efficient coupling from laser-diodes,” Opt. Quantum Electron. 23(3), 367–378 (1991).
[Crossref]

Kuchmizhak, A.

Kulchin, Y.

Liberale, C.

S. Cabrini, C. Liberale, D. Cojoc, A. Carpentiero, M. Prasciolu, S. Mora, V. Degiorgio, F. De Angelis, and E. Di Fabrizio, “Axicon lens on optical fiber forming optical tweezers, made by focused ion beam milling,” Microelectron. Eng. 83(4-9), 804–807 (2006).
[Crossref]

Loock, H.-P.

Maclean, T. S. M.

A. Kotsas, H. Ghafourishiraz, and T. S. M. Maclean, “Microlens fabrication on single-mode fibers for efficient coupling from laser-diodes,” Opt. Quantum Electron. 23(3), 367–378 (1991).
[Crossref]

Marcinkevicius, A.

A. Marcinkevicius, S. Juodkazis, S. Matsuo, V. Mizeikis, and H. Misawa, “Application of Bessel beams for microfabrication of dielectrics by femtosecond laser,” Jpn. J. Appl. Phys. 2(40), L1197–L1199 (2001).
[Crossref]

Matsuo, S.

A. Marcinkevicius, S. Juodkazis, S. Matsuo, V. Mizeikis, and H. Misawa, “Application of Bessel beams for microfabrication of dielectrics by femtosecond laser,” Jpn. J. Appl. Phys. 2(40), L1197–L1199 (2001).
[Crossref]

McLeod, J. H.

Miceli, J.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Milam, D.

Misawa, H.

A. Marcinkevicius, S. Juodkazis, S. Matsuo, V. Mizeikis, and H. Misawa, “Application of Bessel beams for microfabrication of dielectrics by femtosecond laser,” Jpn. J. Appl. Phys. 2(40), L1197–L1199 (2001).
[Crossref]

Mizeikis, V.

A. Marcinkevicius, S. Juodkazis, S. Matsuo, V. Mizeikis, and H. Misawa, “Application of Bessel beams for microfabrication of dielectrics by femtosecond laser,” Jpn. J. Appl. Phys. 2(40), L1197–L1199 (2001).
[Crossref]

Mohanty, K. S.

Mohanty, S. K.

Mora, S.

S. Cabrini, C. Liberale, D. Cojoc, A. Carpentiero, M. Prasciolu, S. Mora, V. Degiorgio, F. De Angelis, and E. Di Fabrizio, “Axicon lens on optical fiber forming optical tweezers, made by focused ion beam milling,” Microelectron. Eng. 83(4-9), 804–807 (2006).
[Crossref]

Munzke, D.

Nepomniaschii, A.

Oliva, A. G.

Omrani, H.

Piquerey, V.

Prasciolu, M.

S. Cabrini, C. Liberale, D. Cojoc, A. Carpentiero, M. Prasciolu, S. Mora, V. Degiorgio, F. De Angelis, and E. Di Fabrizio, “Axicon lens on optical fiber forming optical tweezers, made by focused ion beam milling,” Microelectron. Eng. 83(4-9), 804–807 (2006).
[Crossref]

Reich, O.

Ribeiro, R. S. R.

Salathe, R. P.

P. Hoffmann, B. Dutoit, and R. P. Salathe, “Comparison of mechanically drawn and protection layer chemically etched optical fiber tips,” Ultramicroscopy 61(1-4), 165–170 (1995).
[Crossref]

Saleh, S. S.

Sandoz, P.

Saunders, J.

Sick, B.

R. Stöckle, C. Fokas, V. Deckert, R. Zenobi, B. Sick, B. Hecht, and U. P. Wild, “High-quality near-field optical probes by tube etching,” Appl. Phys. Lett. 75(2), 160–162 (1999).
[Crossref]

Smayling, M. C.

M. Erdelyi, Z. L. Horvath, G. Szabo, Z. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15(2), 287–292 (1997).
[Crossref]

Song, Y.

Soppera, O.

Stöckle, R.

R. Stöckle, C. Fokas, V. Deckert, R. Zenobi, B. Sick, B. Hecht, and U. P. Wild, “High-quality near-field optical probes by tube etching,” Appl. Phys. Lett. 75(2), 160–162 (1999).
[Crossref]

Suarez, M. A.

Szabo, G.

M. Erdelyi, Z. L. Horvath, G. Szabo, Z. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15(2), 287–292 (1997).
[Crossref]

Tittel, F. K.

M. Erdelyi, Z. L. Horvath, G. Szabo, Z. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15(2), 287–292 (1997).
[Crossref]

Vitello, D.

Vitrik, O.

Wild, U. P.

R. Stöckle, C. Fokas, V. Deckert, R. Zenobi, B. Sick, B. Hecht, and U. P. Wild, “High-quality near-field optical probes by tube etching,” Appl. Phys. Lett. 75(2), 160–162 (1999).
[Crossref]

Wulle, T.

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 71(1), 209 (1993).
[Crossref]

Zenobi, R.

R. Stöckle, C. Fokas, V. Deckert, R. Zenobi, B. Sick, B. Hecht, and U. P. Wild, “High-quality near-field optical probes by tube etching,” Appl. Phys. Lett. 75(2), 160–162 (1999).
[Crossref]

Appl. Opt. (5)

Appl. Phys. Lett. (1)

R. Stöckle, C. Fokas, V. Deckert, R. Zenobi, B. Sick, B. Hecht, and U. P. Wild, “High-quality near-field optical probes by tube etching,” Appl. Phys. Lett. 75(2), 160–162 (1999).
[Crossref]

J. Lightwave Technol. (2)

G. He and F. W. Cuomo, “A light-intensity function suitable for multimode fiberoptic sensors,” J. Lightwave Technol. 9(4), 545–551 (1991).
[Crossref]

R. S. R. Ribeiro, O. Soppera, A. G. Oliva, A. Guerreiro, and P. A. S. Jorge, “New trends on optical fiber tweezers,” J. Lightwave Technol. 33(16), 3394–3405 (2015).
[Crossref]

J. Opt. Soc. Am. (1)

J. Vac. Sci. Technol. B (1)

M. Erdelyi, Z. L. Horvath, G. Szabo, Z. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15(2), 287–292 (1997).
[Crossref]

Jpn. J. Appl. Phys. (1)

A. Marcinkevicius, S. Juodkazis, S. Matsuo, V. Mizeikis, and H. Misawa, “Application of Bessel beams for microfabrication of dielectrics by femtosecond laser,” Jpn. J. Appl. Phys. 2(40), L1197–L1199 (2001).
[Crossref]

Microelectron. Eng. (1)

S. Cabrini, C. Liberale, D. Cojoc, A. Carpentiero, M. Prasciolu, S. Mora, V. Degiorgio, F. De Angelis, and E. Di Fabrizio, “Axicon lens on optical fiber forming optical tweezers, made by focused ion beam milling,” Microelectron. Eng. 83(4-9), 804–807 (2006).
[Crossref]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

A. Kotsas, H. Ghafourishiraz, and T. S. M. Maclean, “Microlens fabrication on single-mode fibers for efficient coupling from laser-diodes,” Opt. Quantum Electron. 23(3), 367–378 (1991).
[Crossref]

Phys. Rev. Lett. (2)

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 71(1), 209 (1993).
[Crossref]

Rev. Sci. Instrum. (1)

S. K. Eah, W. Jhe, and Y. Arakawa, “Nearly diffraction-limited focusing of a fiber axicon microlens,” Rev. Sci. Instrum. 74(11), 4969–4971 (2003).
[Crossref]

Ultramicroscopy (1)

P. Hoffmann, B. Dutoit, and R. P. Salathe, “Comparison of mechanically drawn and protection layer chemically etched optical fiber tips,” Ultramicroscopy 61(1-4), 165–170 (1995).
[Crossref]

Other (1)

Y. Fu, S. Morency, K. Bachus, D. Simon, T. Hutama, G. T. T. Gibson, Y. Messaddeq, and R. D. Oleschuk, “A microstructured fiber with defined borosilicate regions to produce a radial micronozzle array for nanoelectrospray ionization,” Sci. Rep.-UK 6(2016).

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Figures (7)

Fig. 1
Fig. 1 (A) Optical micrograph and (B) scanning electron micrograph of the custom-designed microstructured fiber utilized in this study. The dark shaded regions of the fiber in image (A) correspond to regions of borosilicate (9 mol%) glass while the light shaded regions correspond to fused-silica. Scale bar in (B) is 100 µm
Fig. 2
Fig. 2 Schematic diagram of experimental set-up for imaging the emission patterns of microlenses on the facet of capillaries and microstructured fibers.
Fig. 3
Fig. 3 Schematic drawing of rays incident on the surface of the axicon microlens. (A) shows two rays that are contributing to the intensity increase at the optical axis, (B) shows two rays that contribute to the ring pattern observable at large cone angles. (C) Light ray showing internal reflection at the inner axicon surface.
Fig. 4
Fig. 4 Two-dimensional representations of the calculated intensity distribution inside and outside axicon lenses with different cone angles. Angles in the range of about 10-40 degrees produce a bright focal region in front of the axicon lens. At larger cone angles the light is predominantly reflected inside the cone and only then leaves the cone. These rays are not shown in the figure.
Fig. 5
Fig. 5 (A) Modelled emission pattern for 15° axicon lens, (B) Optical micrograph of 15° lens in front of a Ø320 μm capillary, (C) experimental emission pattern of 15° lens. (D)-(F) are the analogous images for a 35° lens.
Fig. 6
Fig. 6 Scanning electron micrographs of the micronozzles (microlenses) produced by etching the custom microstructured fiber for (A) 12 minutes and (B) for 17 minutes with ~80 nL/min water flow through the centre holes. Inset images show the entire fiber profile. All scale bars are 50 µm. (C) Schematic diagram of microlens (micronozzle) formation as the etching procedure occurs. (1) and (2) correspond to borosilicate and fused-silica compositions, respectively, while R0 is the distance from the channel wall to the borosilicate boundary (i.e. the base width) and R is the width of the top portion of the microlens.
Fig. 7
Fig. 7 Optical micrographs of (A) lens tips in focus and (B) from ~40 µm away from lens tip showing the focal point of a shallow etched MSF. These images show 3 of the 9 microlenses on the facet of the MSF of Fig. 6(A). The scale bars are 50 µm in both images.

Equations (14)

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sin ϕ s1,2 = n core n 0 sin ϕ f1,2
θ c = sin 1 ( n clad n core )
ϕ xy u = tan 1 ( y y ACL x x ACL ) ϕ xy d = tan 1 ( y+ y ACL x x ACL )
θ ACL = sin 1 ( n 0 n core )
ϕ xy = cos 1 ( a+bc 4ab ) a= ( x ACL x ) 2 + ( y ACL y ) 2 b= ( x ACL x p ) 2 + ( y ACL y p ) 2 c= ( x p x ) 2 + ( y p y ) 2 ,
dx dt = k i [HF] X
γ= tan 1 ( Δ x C Δ x D R )= tan 1 ( k 2 t R ( [ HF ] D [ HF ] C ) )
Δ x B = k 2 [ HF ] B ( t t A )
t A = Δ x A k 1 [ HF ] A
Δ x B = k 2 [ HF ] B t k 2 k 1 [ HF ] B [ HF ] A Δ x A .
x A = k 1 [ HF ] B τ k 1 k 2 x B
α= cos 1 ( k 1 k 2 )
R= R 0 Δ x B cos( α )
R= R 0 k 1 [ HF ] B t+Δ x A

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