Abstract

Tilted fiber Bragg gratings are inscribed in non-photosensitized single mode fibers through the polyimide coating using a femtosecond infrared laser and a phase mask. The inscription technique used is based on simultaneously translating the fiber along its axis and the focusing cylindrical lens in the orthogonal direction by means of piezoelectric actuators. The grating plane tilt angles up to 10.3° are achieved with a 1.07 µm-pitch phase mask. The cladding modes reach ∼5 dB in strength in transmission despite the presence of the polyimide coating. The effectiveness of the fabricated tilted fiber Bragg gratings for refractive index sensing through the polyimide coating is also demonstrated. Additionally, we show that the classical approach for the inscription of tilted Bragg gratings, which is based on simply tilting the fiber with respect to the interference fringes, cannot be used in tight focusing geometries that are necessary for through-the-coating inscription.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
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    [Crossref]
  24. D. Grobnic, C. Hnatovsky, and S. J. Mihailov, “Thermally stable type II FBGs written through polyimide coatings of silica-based optical fiber,” IEEE Photonics Technol. Lett. 29(21), 1780–1783 (2017).
    [Crossref]
  25. C. Hnatovsky, D. Grobnic, and S. J. Mihailov, “Through-the-coating femtosecond laser inscription of very short fiber Bragg gratings for acoustic and high temperature sensing applications,” Opt. Express 25(21), 25435–25446 (2017).
    [Crossref]
  26. C. Hnatovsky, D. Grobnic, and S. J. Mihailov, “Nonlinear photoluminescence imaging applied to femtosecond laser manufacturing of fiber Bragg gratings,” Opt. Express 25(13), 14247–14259 (2017).
    [Crossref]
  27. N. Abdukerim, D. Grobnic, R. Lausten, C. Hnatovsky, and S. J. Mihailov, “Complex diffraction and dispersion effects in femtosecond laser writing of fiber Bragg gratings using the phase mask technique,” Opt. Express 27(22), 32536–32555 (2019).
    [Crossref]
  28. C. Hnatovsky, D. Grobnic, D. Coulas, M. Barnes, and S. J. Mihailov, “Self-organized nanostructure formation during femtosecond-laser inscription of fiber Bragg gratings,” Opt. Lett. 42(3), 399–402 (2017).
    [Crossref]

2019 (2)

2018 (4)

A. Wolf, M. Kotyushev, A. Dostovalov, and S. Babin, “Femtosecond core-scanning inscription of tilted fiber Bragg gratings,” Proc. SPIE 10681, 37 (2018).
[Crossref]

P. Lu, S. J. Mihailov, H. Ding, D. Grobnic, R. B. Walker, D. Coulas, C. Hnatovsky, and A. Y. Naumov, “Plane-be-plane inscription of grating structures in optical fibers,” J. Lightwave Technol. 36(4), 926–931 (2018).
[Crossref]

J. Tang, C. Fu, Z. Bai, C. Liao, and Y. Wang, “Sensing characteristics of tilted long period fiber gratings inscribed by infrared femtosecond laser,” Sensors 18(9), 3003 (2018).
[Crossref]

G. Bharathan, D. D. Hudson, R. I. Woodward, S. D. Jackson, and A. Fuerbach, “In-fiber polarizer based on a 45-degree tilted fluoride fiber Bragg grating for mid-infrared fiber laser technology,” OSA Continuum 1(1), 56–63 (2018).
[Crossref]

2017 (8)

M. Wang, Y. Zhang, Z. Wang, J. Sun, J. Cao, J. Leng, X. Gu, and X. Xu, “Fabrication of chirped and tilted fiber Bragg gratings and suppression of stimulated Raman scattering in fiber amplifiers,” Opt. Express 25(2), 1529–1534 (2017).
[Crossref]

C. Hnatovsky, D. Grobnic, D. Coulas, M. Barnes, and S. J. Mihailov, “Self-organized nanostructure formation during femtosecond-laser inscription of fiber Bragg gratings,” Opt. Lett. 42(3), 399–402 (2017).
[Crossref]

D. Grobnic, C. Hnatovsky, and S. J. Mihailov, “Thermally stable type II FBGs written through polyimide coatings of silica-based optical fiber,” IEEE Photonics Technol. Lett. 29(21), 1780–1783 (2017).
[Crossref]

C. Hnatovsky, D. Grobnic, and S. J. Mihailov, “Through-the-coating femtosecond laser inscription of very short fiber Bragg gratings for acoustic and high temperature sensing applications,” Opt. Express 25(21), 25435–25446 (2017).
[Crossref]

C. Hnatovsky, D. Grobnic, and S. J. Mihailov, “Nonlinear photoluminescence imaging applied to femtosecond laser manufacturing of fiber Bragg gratings,” Opt. Express 25(13), 14247–14259 (2017).
[Crossref]

J. Habel, T. Boilard, J. S. Frenière, F. Trépanier, and M. Bernier, “Femtosecond FBG written through the coating for sensing applications,” Sensors 17(11), 2519 (2017).
[Crossref]

A. Ioannou, A. Theodosiou, C. Caucheteur, and K. Kalli, “Direct writing of plane-by-plane tilted fiber Bragg gratings using a femtosecond laser,” Opt. Lett. 42(24), 5198–5201 (2017).
[Crossref]

R. Wang, J. Si, T. Chen, L. Yan, H. Cao, X. Pham, and X. Hou, “Fabrication of high-temperature tilted fiber Bragg gratings using a femtosecond laser,” Opt. Express 25(20), 23684–23689 (2017).
[Crossref]

2013 (2)

2012 (1)

S. J. Mihailov, “Fiber Bragg grating sensors for harsh environments,” Sensors 12(2), 1898–1918 (2012).
[Crossref]

2011 (1)

X. Dong, H. Zhang, B. Liu, and Y. Miao, “Tilted fiber Bragg gratings: principle and sensing applications,” Photonic Sens. 1(1), 6–30 (2011).
[Crossref]

2010 (1)

2007 (1)

S. J. Mihailov, D. Grobnic, and C. W. Smelser, “Efficient grating writing through fibre coating with femtosecond IR radiation and phase mask,” Electron. Lett. 43(8), 442–443 (2007).
[Crossref]

2006 (1)

2005 (1)

2004 (1)

A. Martinez, M. Dubov, I. Y. Khrushchev, and I. Bennion, “Direct writing of fibre Bragg gratings by femtosecond laser,” Electron. Lett. 40(19), 1170–1172 (2004).
[Crossref]

2003 (1)

2001 (1)

G. Laffont and P. Ferdinand, “Tilted short-period fibre-Bragg-grating-induced coupling to cladding modes for accurate refractometry,” Meas. Sci. Technol. 12(7), 765–770 (2001).
[Crossref]

1996 (1)

1993 (2)

R. Kashyap, R. Wyatt, and R. Campbell, “Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating,” Electron. Lett. 29(2), 154–156 (1993).
[Crossref]

K. O. Hill, B. Malo, F. Bilodeau, D. C. Johnson, and J. Albert, “Bragg gratings fabricated in monomode photosensitive optical fiber by UV exposure through a phase mask,” Appl. Phys. Lett. 62(10), 1035–1037 (1993).
[Crossref]

Abdukerim, N.

Albert, J.

J. Albert, L. Y. Shao, and C. Caucheteur, “Tilted fiber Bragg grating sensors,” Laser Photonics Rev. 7(1), 83–108 (2013).
[Crossref]

K. O. Hill, B. Malo, F. Bilodeau, D. C. Johnson, and J. Albert, “Bragg gratings fabricated in monomode photosensitive optical fiber by UV exposure through a phase mask,” Appl. Phys. Lett. 62(10), 1035–1037 (1993).
[Crossref]

Babin, S.

A. Wolf, M. Kotyushev, A. Dostovalov, and S. Babin, “Femtosecond core-scanning inscription of tilted fiber Bragg gratings,” Proc. SPIE 10681, 37 (2018).
[Crossref]

Bai, Z.

J. Tang, C. Fu, Z. Bai, C. Liao, and Y. Wang, “Sensing characteristics of tilted long period fiber gratings inscribed by infrared femtosecond laser,” Sensors 18(9), 3003 (2018).
[Crossref]

Bale, B. G.

Barnes, M.

Bennion, I.

Bernier, M.

J. Habel, T. Boilard, J. S. Frenière, F. Trépanier, and M. Bernier, “Femtosecond FBG written through the coating for sensing applications,” Sensors 17(11), 2519 (2017).
[Crossref]

Bharathan, G.

Bilodeau, F.

K. O. Hill, B. Malo, F. Bilodeau, D. C. Johnson, and J. Albert, “Bragg gratings fabricated in monomode photosensitive optical fiber by UV exposure through a phase mask,” Appl. Phys. Lett. 62(10), 1035–1037 (1993).
[Crossref]

Boilard, T.

J. Habel, T. Boilard, J. S. Frenière, F. Trépanier, and M. Bernier, “Femtosecond FBG written through the coating for sensing applications,” Sensors 17(11), 2519 (2017).
[Crossref]

Campbell, R.

R. Kashyap, R. Wyatt, and R. Campbell, “Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating,” Electron. Lett. 29(2), 154–156 (1993).
[Crossref]

Cao, H.

Cao, J.

Caucheteur, C.

Chen, C.

Chen, Q. D.

Chen, T.

Chen, X.

Coulas, D.

Ding, H.

Dong, X.

X. Dong, H. Zhang, B. Liu, and Y. Miao, “Tilted fiber Bragg gratings: principle and sensing applications,” Photonic Sens. 1(1), 6–30 (2011).
[Crossref]

Dostovalov, A.

A. Wolf, M. Kotyushev, A. Dostovalov, and S. Babin, “Femtosecond core-scanning inscription of tilted fiber Bragg gratings,” Proc. SPIE 10681, 37 (2018).
[Crossref]

Dubov, M.

A. Martinez, M. Dubov, I. Y. Khrushchev, and I. Bennion, “Direct writing of fibre Bragg gratings by femtosecond laser,” Electron. Lett. 40(19), 1170–1172 (2004).
[Crossref]

Erdogan, T.

Ferdinand, P.

G. Laffont and P. Ferdinand, “Tilted short-period fibre-Bragg-grating-induced coupling to cladding modes for accurate refractometry,” Meas. Sci. Technol. 12(7), 765–770 (2001).
[Crossref]

Frenière, J. S.

J. Habel, T. Boilard, J. S. Frenière, F. Trépanier, and M. Bernier, “Femtosecond FBG written through the coating for sensing applications,” Sensors 17(11), 2519 (2017).
[Crossref]

Fu, C.

J. Tang, C. Fu, Z. Bai, C. Liao, and Y. Wang, “Sensing characteristics of tilted long period fiber gratings inscribed by infrared femtosecond laser,” Sensors 18(9), 3003 (2018).
[Crossref]

Fuerbach, A.

Grobnic, D.

N. Abdukerim, D. Grobnic, R. Lausten, C. Hnatovsky, and S. J. Mihailov, “Complex diffraction and dispersion effects in femtosecond laser writing of fiber Bragg gratings using the phase mask technique,” Opt. Express 27(22), 32536–32555 (2019).
[Crossref]

P. Lu, S. J. Mihailov, H. Ding, D. Grobnic, R. B. Walker, D. Coulas, C. Hnatovsky, and A. Y. Naumov, “Plane-be-plane inscription of grating structures in optical fibers,” J. Lightwave Technol. 36(4), 926–931 (2018).
[Crossref]

C. Hnatovsky, D. Grobnic, D. Coulas, M. Barnes, and S. J. Mihailov, “Self-organized nanostructure formation during femtosecond-laser inscription of fiber Bragg gratings,” Opt. Lett. 42(3), 399–402 (2017).
[Crossref]

C. Hnatovsky, D. Grobnic, and S. J. Mihailov, “Through-the-coating femtosecond laser inscription of very short fiber Bragg gratings for acoustic and high temperature sensing applications,” Opt. Express 25(21), 25435–25446 (2017).
[Crossref]

D. Grobnic, C. Hnatovsky, and S. J. Mihailov, “Thermally stable type II FBGs written through polyimide coatings of silica-based optical fiber,” IEEE Photonics Technol. Lett. 29(21), 1780–1783 (2017).
[Crossref]

C. Hnatovsky, D. Grobnic, and S. J. Mihailov, “Nonlinear photoluminescence imaging applied to femtosecond laser manufacturing of fiber Bragg gratings,” Opt. Express 25(13), 14247–14259 (2017).
[Crossref]

S. J. Mihailov, D. Grobnic, and C. W. Smelser, “Efficient grating writing through fibre coating with femtosecond IR radiation and phase mask,” Electron. Lett. 43(8), 442–443 (2007).
[Crossref]

S. J. Mihailov, C. W. Smelser, P. Lu, R. B. Walker, D. Grobnic, H. Ding, G. Henderson, and J. Unruh, “Fiber Bragg gratings made with a phase mask and 800-nm femtosecond radiation,” Opt. Lett. 28(12), 995–997 (2003).
[Crossref]

Gu, X.

Guo, J. C.

Habel, J.

J. Habel, T. Boilard, J. S. Frenière, F. Trépanier, and M. Bernier, “Femtosecond FBG written through the coating for sensing applications,” Sensors 17(11), 2519 (2017).
[Crossref]

Henderson, G.

Hill, K. O.

K. O. Hill, B. Malo, F. Bilodeau, D. C. Johnson, and J. Albert, “Bragg gratings fabricated in monomode photosensitive optical fiber by UV exposure through a phase mask,” Appl. Phys. Lett. 62(10), 1035–1037 (1993).
[Crossref]

Hnatovsky, C.

Hou, X.

Hudson, D. D.

Ioannou, A.

Jackson, S. D.

Johnson, D. C.

K. O. Hill, B. Malo, F. Bilodeau, D. C. Johnson, and J. Albert, “Bragg gratings fabricated in monomode photosensitive optical fiber by UV exposure through a phase mask,” Appl. Phys. Lett. 62(10), 1035–1037 (1993).
[Crossref]

Kalli, K.

Kashyap, R.

R. Kashyap, R. Wyatt, and R. Campbell, “Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating,” Electron. Lett. 29(2), 154–156 (1993).
[Crossref]

Khrushchev, I. Y.

A. Martinez, I. Y. Khrushchev, and I. Bennion, “Direct inscription of Bragg gratings in coated fibers by an infrared femtosecond laser,” Opt. Lett. 31(11), 1603–1605 (2006).
[Crossref]

A. Martinez, M. Dubov, I. Y. Khrushchev, and I. Bennion, “Direct writing of fibre Bragg gratings by femtosecond laser,” Electron. Lett. 40(19), 1170–1172 (2004).
[Crossref]

Kotyushev, M.

A. Wolf, M. Kotyushev, A. Dostovalov, and S. Babin, “Femtosecond core-scanning inscription of tilted fiber Bragg gratings,” Proc. SPIE 10681, 37 (2018).
[Crossref]

Laffont, G.

G. Laffont and P. Ferdinand, “Tilted short-period fibre-Bragg-grating-induced coupling to cladding modes for accurate refractometry,” Meas. Sci. Technol. 12(7), 765–770 (2001).
[Crossref]

Lausten, R.

Leng, J.

Liao, C.

J. Tang, C. Fu, Z. Bai, C. Liao, and Y. Wang, “Sensing characteristics of tilted long period fiber gratings inscribed by infrared femtosecond laser,” Sensors 18(9), 3003 (2018).
[Crossref]

Liu, B.

X. Dong, H. Zhang, B. Liu, and Y. Miao, “Tilted fiber Bragg gratings: principle and sensing applications,” Photonic Sens. 1(1), 6–30 (2011).
[Crossref]

Lu, P.

Malo, B.

K. O. Hill, B. Malo, F. Bilodeau, D. C. Johnson, and J. Albert, “Bragg gratings fabricated in monomode photosensitive optical fiber by UV exposure through a phase mask,” Appl. Phys. Lett. 62(10), 1035–1037 (1993).
[Crossref]

Martinez, A.

A. Martinez, I. Y. Khrushchev, and I. Bennion, “Direct inscription of Bragg gratings in coated fibers by an infrared femtosecond laser,” Opt. Lett. 31(11), 1603–1605 (2006).
[Crossref]

A. Martinez, M. Dubov, I. Y. Khrushchev, and I. Bennion, “Direct writing of fibre Bragg gratings by femtosecond laser,” Electron. Lett. 40(19), 1170–1172 (2004).
[Crossref]

Miao, Y.

X. Dong, H. Zhang, B. Liu, and Y. Miao, “Tilted fiber Bragg gratings: principle and sensing applications,” Photonic Sens. 1(1), 6–30 (2011).
[Crossref]

Mihailov, S. J.

N. Abdukerim, D. Grobnic, R. Lausten, C. Hnatovsky, and S. J. Mihailov, “Complex diffraction and dispersion effects in femtosecond laser writing of fiber Bragg gratings using the phase mask technique,” Opt. Express 27(22), 32536–32555 (2019).
[Crossref]

P. Lu, S. J. Mihailov, H. Ding, D. Grobnic, R. B. Walker, D. Coulas, C. Hnatovsky, and A. Y. Naumov, “Plane-be-plane inscription of grating structures in optical fibers,” J. Lightwave Technol. 36(4), 926–931 (2018).
[Crossref]

C. Hnatovsky, D. Grobnic, and S. J. Mihailov, “Through-the-coating femtosecond laser inscription of very short fiber Bragg gratings for acoustic and high temperature sensing applications,” Opt. Express 25(21), 25435–25446 (2017).
[Crossref]

D. Grobnic, C. Hnatovsky, and S. J. Mihailov, “Thermally stable type II FBGs written through polyimide coatings of silica-based optical fiber,” IEEE Photonics Technol. Lett. 29(21), 1780–1783 (2017).
[Crossref]

C. Hnatovsky, D. Grobnic, D. Coulas, M. Barnes, and S. J. Mihailov, “Self-organized nanostructure formation during femtosecond-laser inscription of fiber Bragg gratings,” Opt. Lett. 42(3), 399–402 (2017).
[Crossref]

C. Hnatovsky, D. Grobnic, and S. J. Mihailov, “Nonlinear photoluminescence imaging applied to femtosecond laser manufacturing of fiber Bragg gratings,” Opt. Express 25(13), 14247–14259 (2017).
[Crossref]

S. J. Mihailov, “Fiber Bragg grating sensors for harsh environments,” Sensors 12(2), 1898–1918 (2012).
[Crossref]

S. J. Mihailov, D. Grobnic, and C. W. Smelser, “Efficient grating writing through fibre coating with femtosecond IR radiation and phase mask,” Electron. Lett. 43(8), 442–443 (2007).
[Crossref]

S. J. Mihailov, C. W. Smelser, P. Lu, R. B. Walker, D. Grobnic, H. Ding, G. Henderson, and J. Unruh, “Fiber Bragg gratings made with a phase mask and 800-nm femtosecond radiation,” Opt. Lett. 28(12), 995–997 (2003).
[Crossref]

Mou, C.

Naumov, A. Y.

Pham, X.

Qin, F.

Shao, L. Y.

J. Albert, L. Y. Shao, and C. Caucheteur, “Tilted fiber Bragg grating sensors,” Laser Photonics Rev. 7(1), 83–108 (2013).
[Crossref]

Si, J.

Simpson, G.

Sipe, J. E.

Smelser, C. W.

S. J. Mihailov, D. Grobnic, and C. W. Smelser, “Efficient grating writing through fibre coating with femtosecond IR radiation and phase mask,” Electron. Lett. 43(8), 442–443 (2007).
[Crossref]

S. J. Mihailov, C. W. Smelser, P. Lu, R. B. Walker, D. Grobnic, H. Ding, G. Henderson, and J. Unruh, “Fiber Bragg gratings made with a phase mask and 800-nm femtosecond radiation,” Opt. Lett. 28(12), 995–997 (2003).
[Crossref]

Sun, H. B.

Sun, J.

Tang, J.

J. Tang, C. Fu, Z. Bai, C. Liao, and Y. Wang, “Sensing characteristics of tilted long period fiber gratings inscribed by infrared femtosecond laser,” Sensors 18(9), 3003 (2018).
[Crossref]

Theodosiou, A.

Trépanier, F.

J. Habel, T. Boilard, J. S. Frenière, F. Trépanier, and M. Bernier, “Femtosecond FBG written through the coating for sensing applications,” Sensors 17(11), 2519 (2017).
[Crossref]

Unruh, J.

Walker, R. B.

Wang, C.

Wang, H.

Wang, M.

Wang, R.

Wang, Y.

J. Tang, C. Fu, Z. Bai, C. Liao, and Y. Wang, “Sensing characteristics of tilted long period fiber gratings inscribed by infrared femtosecond laser,” Sensors 18(9), 3003 (2018).
[Crossref]

Wang, Z.

Wolf, A.

A. Wolf, M. Kotyushev, A. Dostovalov, and S. Babin, “Femtosecond core-scanning inscription of tilted fiber Bragg gratings,” Proc. SPIE 10681, 37 (2018).
[Crossref]

Woodward, R. I.

Wyatt, R.

R. Kashyap, R. Wyatt, and R. Campbell, “Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating,” Electron. Lett. 29(2), 154–156 (1993).
[Crossref]

Xu, X.

Xue, Y.

Yan, L.

Yang, R.

Yu, Y. S.

Zhang, H.

X. Dong, H. Zhang, B. Liu, and Y. Miao, “Tilted fiber Bragg gratings: principle and sensing applications,” Photonic Sens. 1(1), 6–30 (2011).
[Crossref]

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[Crossref]

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Figures (7)

Fig. 1.
Fig. 1. Illustration of focal tilt and split caused by rotating (a) the cylindrical lens CL only (rotation angle ξ in the xz-plane), (b) the phase mask M only (rotation angle γ in the xz-plane). M produces only 1st diffraction order (i.e., m = ± 1). The respective diffraction angles θ+1 and θ−1 are counted from the z-axis, which coincides with the beam propagation direction. The focal splits associated with CL and M rotations are denoted by ΔzCL and ΔzM, respectively. A small angle between the split focal lines in (a) and (b) is not shown (see Appendix A).
Fig. 2.
Fig. 2. (a) Experimental setup for writing TFBGs. (b) Refraction of the converging beam at the fiber surface (thin fiber coating is neglected).
Fig. 3.
Fig. 3. Transmission spectra of TFBGs written through the coating with different tilt angles of 4.6°, 6.8°, 8.0°, 9.0°, and 10.3°: (a) with the coating, (b) after the coating has been removed. (the spectra are offset along the vertical axis for clarity.
Fig. 4.
Fig. 4. Transformation of the transmission spectra of a 10.3° TFBG written through the coating in response to different surrounding refractive indices.
Fig. 5.
Fig. 5. Illustration of focal split and tilt caused by rotating (a) the cylindrical lens CL only, (b) the phase mask M only.
Fig. 6.
Fig. 6. Merging the focal lines by counter-rotating the phase mask M and cylindrical lens CL.
Fig. 7.
Fig. 7. Illustration of refraction of the converging fs-IR beam at the fiber surface (thin fiber coating is not shown).

Equations (25)

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ψ + 1 = arctan ( cos θ tan ξ 1 + sin θ tan ξ ) ,
ψ 1 = arctan ( cos θ tan ξ 1 sin θ tan ξ ) .
Δ z CL = z tan θ ( tan ψ + 1 + tan ψ 1 ) ,
Δ z CL 2 z ξ sin θ .
φ + 1 = arctan ( 1 cos θ + 1 tan γ 1 + sin θ + 1 tan γ ) ,
φ 1 = arctan ( 1 cos θ 1 tan γ 1 sin θ 1 tan γ ) .
Δ z M z ( tan φ 1 tan φ + 1 ) / cos θ
Δ z M 2 z γ sin 2 θ tan θ .
tan χ = sin ( ξ + γ ) cos ( θ + 1 γ ) cos ξ + sin ( ξ + γ ) sin ( θ + 1 γ ) = sin ( ξ + γ ) cos ( θ 1 + γ ) cos ξ sin ( ξ + γ ) sin ( θ 1 + γ ) .
χ γ / γ cos θ cos θ .
α = χ γ = γ / γ cos θ γ cos θ γ .
β = 1 2 ( θ 1 θ + 1 ) γ / γ cos θ cos θ γ .
z 1 = Δ L tan η R 2 + R 2 tan 2 η Δ L 2 1 + tan 2 η , y 1 = z 1 tan η ,
z 2 = Δ L tan η R 2 + R 2 tan 2 η Δ L 2 1 + tan 2 η , y 2 = z 2 tan η .
y 1 Δ L z 1 = tan η R 2 + R 2 tan 2 η Δ L 2 + Δ L Δ L tan η + R 2 + R 2 tan 2 η Δ L 2 ,
y 2 Δ L z 2 = tan η R 2 + R 2 tan 2 η Δ L 2 Δ L Δ L tan η R 2 + R 2 tan 2 η Δ L 2 .
tan ε 1 = tan ε 2 = Δ L R 2 + R 2 tan 2 η Δ L 2 .
tan ε 1 = tan ε 2 = Δ L n 2 R 2 + n 2 R 2 tan 2 η Δ L 2 ,
y 1 y 0 z 1 z 0 ( 1 1 n ) Δ L R cos η tan η ,
y 2 y 0 z 2 z 0 ( 1 1 n ) Δ L R cos η + tan η .
z 0 0 ,
y 0 ( n 1 ) n Δ L .
z 0 ( n 1 ) Δ L 2 cos η n 2 R ,
y 0 ( n 1 ) n Δ L .
Δ Y = Δ L y 0 Δ L n ,

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