Abstract

Using the Kostenbauder matrix formalism, we analyze the complex spatio-temporal behaviour of focused femtosecond pulses propagating in three tunable phase mask interferometers for writing fiber Bragg gratings. In interferometers that use a second half-period mask to reverse the phase front, we show that at zero detuning, and with a proper amount of pre-dispersion, the pulses can be perfectly focused on the fiber, whereas with a Talbot interferometer, the pulses suffer from large, unavoidable spatial and temporal broadening. We quantify the enhancement of grating writing efficiency with focusing as a function of beam diameter and interferometer detuning, for a photoinduced index change that is due to a two-photon or higher order process. Based on previously published experimental results, we estimate the potential improvement in grating writing speed.

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

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References

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  1. A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, “Direct writing of fiber bragg gratings by femtosecond laser,” Electron. Lett. 40(19), 1170–1172 (2004).
    [Crossref]
  2. S. A. Slattery, D. N. Nikogosyan, and G. Brambilla, “Fiber bragg grating inscription by high-intensity femtosecond uv laser light: comparison with other existing methods of fabrication,” J. Opt. Soc. Am. B 22(2), 354–361 (2005).
    [Crossref]
  3. S. J. Mihailov, D. Grobnic, C. W. Smelser, P. Lu, R. B. Walker, and H. Ding, “Bragg grating inscription in various optical fibers with femtosecond infrared lasers and a phase mask,” Opt. Mater. Express 1(4), 754–765 (2011).
    [Crossref]
  4. A. Zagorulko, P. Kryukov, Y. Larionov, A. Rybaltovsky, and E. Dianov, “Fabrication of fiber bragg gratings with 267 nm femtosecond radiation,” Opt. Express 12(24), 5996–6001 (2004).
    [Crossref]
  5. M. Bernier, R. Vallée, B. Morasse, C. Desrosiers, A. Saliminia, and Y. Sheng, “Ytterbium fiber laser based on first-order fiber bragg gratings written with 400nm femtosecond pulses and a phase-mask,” Opt. Express 17(21), 18887–18893 (2009).
    [Crossref]
  6. M. Becker, L. Fernandes, M. Rothhardt, S. Brückner, K. Schuster, J. Kobelke, O. Frazão, H. Bartelt, and P. V. S. Marques, “Inscription of fiber bragg grating arrays in pure silica suspended core fibers,” IEEE Photon. Technol. Lett. 21(19), 1453–1455 (2009).
    [Crossref]
  7. A. Martinez, I. Y. Krushchev, and I. Bennion, “Direct inscription of bragg gratings in coated fibers by an infrared femtosecond laser,” Opt. Lett. 31(11), 1603–1605 (2006).
    [Crossref]
  8. A. Dragomir and D. N. Nikogosyan, “Inscription of fiber bragg gratings by ultraviolet femtosecond radiation,” Opt. Lett. 28(22), 2171–2173 (2003).
    [Crossref]
  9. M. Livitziis and S. Pissadakis, “Bragg grating recording in low-defect optical fibers using ultraviolet femtosecond radiation and a double-phase mask interferometer,” Opt. Lett. 33(13), 1449–1451 (2008).
    [Crossref]
  10. J. Thomas, E. Wikszak, T. Clausnitzer, U. Fuchs, U. Zeitner, S. Nolte, and A. Tünnermann, “Inscription of fiber bragg gratings with femtosecond pulses using a phase mask scanning technique,” Appl. Phys. A 86(2), 153–157 (2007).
    [Crossref]
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    [Crossref]
  12. R. Kashyap, J. Armitage, R. Campbell, D. L. Williams, G. Maxwell, and B. Ainslie, “Light sensitive optical fibres and planar waveguides,” BT Technol. J. 11, 150–158 (1993).
  13. M. Sozzi, A. Rahman, and S. Pissadakis, “Non-monotonous refractive index changes recorded in a phosphate glass optical fibre using 248nm, 500fs laser radiation,” Opt. Mater. Express 1(1), 121–127 (2011).
    [Crossref]
  14. R. Kashyap, Fiber Bragg Gratings, 2nd ed (Academic Press, 2010).
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    [Crossref]
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  17. A. Kostenbauder, “Ray-pulse matrices: A rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26(6), 1148–1157 (1990).
    [Crossref]
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    [Crossref]
  19. D. Li, X. Lv, S. Zeng, and Q. Luo, “A generalized analysis of femtosecond laser pulse broadening after angular dispersion,” Opt. Express 16(1), 237–247 (2008).
    [Crossref]
  20. C. G. Durfee, M. Greco, E. Block, D. Vitek, and J. A. Squier, “Intuitive analysis of space-time focusing with double-abcd calculation,” Opt. Express 20(13), 14244–14259 (2012).
    [Crossref]
  21. J. Martin and F. Ouellette, “Novel writing technique of long and highly reflective in-fibre gratings,” Electron. Lett. 30(10), 811–812 (1994).
    [Crossref]
  22. Z. Bor, “Distortion of femtosecond laser pulses in lenses,” Opt. Lett. 14(2), 119–121 (1989).
    [Crossref]

2012 (1)

2011 (2)

2009 (2)

M. Bernier, R. Vallée, B. Morasse, C. Desrosiers, A. Saliminia, and Y. Sheng, “Ytterbium fiber laser based on first-order fiber bragg gratings written with 400nm femtosecond pulses and a phase-mask,” Opt. Express 17(21), 18887–18893 (2009).
[Crossref]

M. Becker, L. Fernandes, M. Rothhardt, S. Brückner, K. Schuster, J. Kobelke, O. Frazão, H. Bartelt, and P. V. S. Marques, “Inscription of fiber bragg grating arrays in pure silica suspended core fibers,” IEEE Photon. Technol. Lett. 21(19), 1453–1455 (2009).
[Crossref]

2008 (2)

2007 (1)

J. Thomas, E. Wikszak, T. Clausnitzer, U. Fuchs, U. Zeitner, S. Nolte, and A. Tünnermann, “Inscription of fiber bragg gratings with femtosecond pulses using a phase mask scanning technique,” Appl. Phys. A 86(2), 153–157 (2007).
[Crossref]

2006 (1)

2005 (2)

2004 (2)

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

A. Zagorulko, P. Kryukov, Y. Larionov, A. Rybaltovsky, and E. Dianov, “Fabrication of fiber bragg gratings with 267 nm femtosecond radiation,” Opt. Express 12(24), 5996–6001 (2004).
[Crossref]

2003 (1)

1998 (1)

P. Cortes, F. Ouellette, and S. LaRochelle, “Intrinsic apodization of bragg gratings written using uv-pulse interferometry,” Electron. Lett. 34(4), 396–397 (1998).
[Crossref]

1994 (1)

J. Martin and F. Ouellette, “Novel writing technique of long and highly reflective in-fibre gratings,” Electron. Lett. 30(10), 811–812 (1994).
[Crossref]

1993 (1)

R. Kashyap, J. Armitage, R. Campbell, D. L. Williams, G. Maxwell, and B. Ainslie, “Light sensitive optical fibres and planar waveguides,” BT Technol. J. 11, 150–158 (1993).

1990 (1)

A. Kostenbauder, “Ray-pulse matrices: A rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26(6), 1148–1157 (1990).
[Crossref]

1989 (1)

1959 (1)

F. J. Weinberg and N. Wood, “Interferometer based on four diffraction gratings,” J. Sci. Instrum. 36(5), 227–230 (1959).
[Crossref]

Ainslie, B.

R. Kashyap, J. Armitage, R. Campbell, D. L. Williams, G. Maxwell, and B. Ainslie, “Light sensitive optical fibres and planar waveguides,” BT Technol. J. 11, 150–158 (1993).

Akturk, S.

Armitage, J.

R. Kashyap, J. Armitage, R. Campbell, D. L. Williams, G. Maxwell, and B. Ainslie, “Light sensitive optical fibres and planar waveguides,” BT Technol. J. 11, 150–158 (1993).

Bartelt, H.

M. Becker, L. Fernandes, M. Rothhardt, S. Brückner, K. Schuster, J. Kobelke, O. Frazão, H. Bartelt, and P. V. S. Marques, “Inscription of fiber bragg grating arrays in pure silica suspended core fibers,” IEEE Photon. Technol. Lett. 21(19), 1453–1455 (2009).
[Crossref]

Becker, M.

M. Becker, L. Fernandes, M. Rothhardt, S. Brückner, K. Schuster, J. Kobelke, O. Frazão, H. Bartelt, and P. V. S. Marques, “Inscription of fiber bragg grating arrays in pure silica suspended core fibers,” IEEE Photon. Technol. Lett. 21(19), 1453–1455 (2009).
[Crossref]

Bennion, I.

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

Bernier, M.

Block, E.

Bor, Z.

Brambilla, G.

Brückner, S.

M. Becker, L. Fernandes, M. Rothhardt, S. Brückner, K. Schuster, J. Kobelke, O. Frazão, H. Bartelt, and P. V. S. Marques, “Inscription of fiber bragg grating arrays in pure silica suspended core fibers,” IEEE Photon. Technol. Lett. 21(19), 1453–1455 (2009).
[Crossref]

Campbell, R.

R. Kashyap, J. Armitage, R. Campbell, D. L. Williams, G. Maxwell, and B. Ainslie, “Light sensitive optical fibres and planar waveguides,” BT Technol. J. 11, 150–158 (1993).

Clausnitzer, T.

J. Thomas, E. Wikszak, T. Clausnitzer, U. Fuchs, U. Zeitner, S. Nolte, and A. Tünnermann, “Inscription of fiber bragg gratings with femtosecond pulses using a phase mask scanning technique,” Appl. Phys. A 86(2), 153–157 (2007).
[Crossref]

Cortes, P.

P. Cortes, F. Ouellette, and S. LaRochelle, “Intrinsic apodization of bragg gratings written using uv-pulse interferometry,” Electron. Lett. 34(4), 396–397 (1998).
[Crossref]

Desrosiers, C.

Dianov, E.

Ding, H.

Dragomir, A.

Dubov, M.

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

Durfee, C. G.

Fernandes, L.

M. Becker, L. Fernandes, M. Rothhardt, S. Brückner, K. Schuster, J. Kobelke, O. Frazão, H. Bartelt, and P. V. S. Marques, “Inscription of fiber bragg grating arrays in pure silica suspended core fibers,” IEEE Photon. Technol. Lett. 21(19), 1453–1455 (2009).
[Crossref]

Frazão, O.

M. Becker, L. Fernandes, M. Rothhardt, S. Brückner, K. Schuster, J. Kobelke, O. Frazão, H. Bartelt, and P. V. S. Marques, “Inscription of fiber bragg grating arrays in pure silica suspended core fibers,” IEEE Photon. Technol. Lett. 21(19), 1453–1455 (2009).
[Crossref]

Fuchs, U.

J. Thomas, E. Wikszak, T. Clausnitzer, U. Fuchs, U. Zeitner, S. Nolte, and A. Tünnermann, “Inscription of fiber bragg gratings with femtosecond pulses using a phase mask scanning technique,” Appl. Phys. A 86(2), 153–157 (2007).
[Crossref]

Gabolde, P.

Greco, M.

Grobnic, D.

Grunnet-Jepsen, A.

P. St-Hilaire and A. Grunnet-Jepsen, US Patent 6,904,201 (2002).

Gu, X.

Kashyap, R.

R. Kashyap, J. Armitage, R. Campbell, D. L. Williams, G. Maxwell, and B. Ainslie, “Light sensitive optical fibres and planar waveguides,” BT Technol. J. 11, 150–158 (1993).

R. Kashyap, Fiber Bragg Gratings, 2nd ed (Academic Press, 2010).

Khrushchev, I.

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

Kobelke, J.

M. Becker, L. Fernandes, M. Rothhardt, S. Brückner, K. Schuster, J. Kobelke, O. Frazão, H. Bartelt, and P. V. S. Marques, “Inscription of fiber bragg grating arrays in pure silica suspended core fibers,” IEEE Photon. Technol. Lett. 21(19), 1453–1455 (2009).
[Crossref]

Kostenbauder, A.

A. Kostenbauder, “Ray-pulse matrices: A rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26(6), 1148–1157 (1990).
[Crossref]

Krushchev, I. Y.

Kryukov, P.

Larionov, Y.

LaRochelle, S.

P. Cortes, F. Ouellette, and S. LaRochelle, “Intrinsic apodization of bragg gratings written using uv-pulse interferometry,” Electron. Lett. 34(4), 396–397 (1998).
[Crossref]

Li, D.

Livitziis, M.

Lu, P.

Luo, Q.

Lv, X.

Marques, P. V. S.

M. Becker, L. Fernandes, M. Rothhardt, S. Brückner, K. Schuster, J. Kobelke, O. Frazão, H. Bartelt, and P. V. S. Marques, “Inscription of fiber bragg grating arrays in pure silica suspended core fibers,” IEEE Photon. Technol. Lett. 21(19), 1453–1455 (2009).
[Crossref]

Martin, J.

J. Martin and F. Ouellette, “Novel writing technique of long and highly reflective in-fibre gratings,” Electron. Lett. 30(10), 811–812 (1994).
[Crossref]

Martinez, A.

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

Maxwell, G.

R. Kashyap, J. Armitage, R. Campbell, D. L. Williams, G. Maxwell, and B. Ainslie, “Light sensitive optical fibres and planar waveguides,” BT Technol. J. 11, 150–158 (1993).

Mihailov, S. J.

Morasse, B.

Nikogosyan, D. N.

Nolte, S.

J. Thomas, E. Wikszak, T. Clausnitzer, U. Fuchs, U. Zeitner, S. Nolte, and A. Tünnermann, “Inscription of fiber bragg gratings with femtosecond pulses using a phase mask scanning technique,” Appl. Phys. A 86(2), 153–157 (2007).
[Crossref]

Ouellette, F.

P. Cortes, F. Ouellette, and S. LaRochelle, “Intrinsic apodization of bragg gratings written using uv-pulse interferometry,” Electron. Lett. 34(4), 396–397 (1998).
[Crossref]

J. Martin and F. Ouellette, “Novel writing technique of long and highly reflective in-fibre gratings,” Electron. Lett. 30(10), 811–812 (1994).
[Crossref]

Pissadakis, S.

Rahman, A.

Rothhardt, M.

M. Becker, L. Fernandes, M. Rothhardt, S. Brückner, K. Schuster, J. Kobelke, O. Frazão, H. Bartelt, and P. V. S. Marques, “Inscription of fiber bragg grating arrays in pure silica suspended core fibers,” IEEE Photon. Technol. Lett. 21(19), 1453–1455 (2009).
[Crossref]

Rybaltovsky, A.

Saliminia, A.

Schuster, K.

M. Becker, L. Fernandes, M. Rothhardt, S. Brückner, K. Schuster, J. Kobelke, O. Frazão, H. Bartelt, and P. V. S. Marques, “Inscription of fiber bragg grating arrays in pure silica suspended core fibers,” IEEE Photon. Technol. Lett. 21(19), 1453–1455 (2009).
[Crossref]

Sheng, Y.

Slattery, S. A.

Smelser, C. W.

Sozzi, M.

Squier, J. A.

St-Hilaire, P.

P. St-Hilaire and A. Grunnet-Jepsen, US Patent 6,904,201 (2002).

Thomas, J.

J. Thomas, E. Wikszak, T. Clausnitzer, U. Fuchs, U. Zeitner, S. Nolte, and A. Tünnermann, “Inscription of fiber bragg gratings with femtosecond pulses using a phase mask scanning technique,” Appl. Phys. A 86(2), 153–157 (2007).
[Crossref]

Trebino, R.

Tünnermann, A.

J. Thomas, E. Wikszak, T. Clausnitzer, U. Fuchs, U. Zeitner, S. Nolte, and A. Tünnermann, “Inscription of fiber bragg gratings with femtosecond pulses using a phase mask scanning technique,” Appl. Phys. A 86(2), 153–157 (2007).
[Crossref]

Vallée, R.

Vitek, D.

Walker, R. B.

Weinberg, F. J.

F. J. Weinberg and N. Wood, “Interferometer based on four diffraction gratings,” J. Sci. Instrum. 36(5), 227–230 (1959).
[Crossref]

Wikszak, E.

J. Thomas, E. Wikszak, T. Clausnitzer, U. Fuchs, U. Zeitner, S. Nolte, and A. Tünnermann, “Inscription of fiber bragg gratings with femtosecond pulses using a phase mask scanning technique,” Appl. Phys. A 86(2), 153–157 (2007).
[Crossref]

Williams, D. L.

R. Kashyap, J. Armitage, R. Campbell, D. L. Williams, G. Maxwell, and B. Ainslie, “Light sensitive optical fibres and planar waveguides,” BT Technol. J. 11, 150–158 (1993).

Wood, N.

F. J. Weinberg and N. Wood, “Interferometer based on four diffraction gratings,” J. Sci. Instrum. 36(5), 227–230 (1959).
[Crossref]

Zagorulko, A.

Zeitner, U.

J. Thomas, E. Wikszak, T. Clausnitzer, U. Fuchs, U. Zeitner, S. Nolte, and A. Tünnermann, “Inscription of fiber bragg gratings with femtosecond pulses using a phase mask scanning technique,” Appl. Phys. A 86(2), 153–157 (2007).
[Crossref]

Zeng, S.

Appl. Phys. A (1)

J. Thomas, E. Wikszak, T. Clausnitzer, U. Fuchs, U. Zeitner, S. Nolte, and A. Tünnermann, “Inscription of fiber bragg gratings with femtosecond pulses using a phase mask scanning technique,” Appl. Phys. A 86(2), 153–157 (2007).
[Crossref]

BT Technol. J. (1)

R. Kashyap, J. Armitage, R. Campbell, D. L. Williams, G. Maxwell, and B. Ainslie, “Light sensitive optical fibres and planar waveguides,” BT Technol. J. 11, 150–158 (1993).

Electron. Lett. (3)

P. Cortes, F. Ouellette, and S. LaRochelle, “Intrinsic apodization of bragg gratings written using uv-pulse interferometry,” Electron. Lett. 34(4), 396–397 (1998).
[Crossref]

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

J. Martin and F. Ouellette, “Novel writing technique of long and highly reflective in-fibre gratings,” Electron. Lett. 30(10), 811–812 (1994).
[Crossref]

IEEE J. Quantum Electron. (1)

A. Kostenbauder, “Ray-pulse matrices: A rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26(6), 1148–1157 (1990).
[Crossref]

IEEE Photon. Technol. Lett. (1)

M. Becker, L. Fernandes, M. Rothhardt, S. Brückner, K. Schuster, J. Kobelke, O. Frazão, H. Bartelt, and P. V. S. Marques, “Inscription of fiber bragg grating arrays in pure silica suspended core fibers,” IEEE Photon. Technol. Lett. 21(19), 1453–1455 (2009).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Sci. Instrum. (1)

F. J. Weinberg and N. Wood, “Interferometer based on four diffraction gratings,” J. Sci. Instrum. 36(5), 227–230 (1959).
[Crossref]

Opt. Express (5)

Opt. Lett. (4)

Opt. Mater. Express (2)

Other (2)

P. St-Hilaire and A. Grunnet-Jepsen, US Patent 6,904,201 (2002).

R. Kashyap, Fiber Bragg Gratings, 2nd ed (Academic Press, 2010).

Supplementary Material (3)

NameDescription
» Visualization 1       Pulse evolution in the Talbot interferometer
» Visualization 2       Pulse evolution in the achromatic and ring interferometers at zero detuning.
» Visualization 3       Pulse evolution in the achromatic interferometer for 200 nm detuning.

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Figures (11)

Fig. 1.
Fig. 1. (a) Talbot interferometer; (b) Achromatic interferometer; (c) Ring interferometer.
Fig. 2.
Fig. 2. Bragg wavelength as a function of the phase mask tilt angle $\theta _{M}$ for the achromatic interferometer in Fig. 1 (b).
Fig. 3.
Fig. 3. Path delay per mm of grating for the three interferometers
Fig. 4.
Fig. 4. Pulse front behaviour when focusing the beam at zero detuning in the direction parallel to the fiber axis.
Fig. 5.
Fig. 5. (a) spectral spread in the Talbot interferometer (b) spectral spread in the achromatic interferometer at zero detuning, and (c) for non-zero detuning
Fig. 6.
Fig. 6. Pulse evolution animation previews : (a) Talbot interferometer (see Visualization 1); (b) Achromatic and Ring at zero detuning (Visualization 2); (c) Achromatic for 200 nm detuning (Visualization 3). On these previews, only the final pulse dimensions are shown.
Fig. 7.
Fig. 7. (a) Evolution of the pulse width and duration and (b) enhancement parameters for a Talbot interferometer.
Fig. 8.
Fig. 8. Pulse evolution and enhancement parameters at zero detuning without (a) and (b) and with pre-dispersion (c) and (d) for the ring and achromatic interferometers
Fig. 9.
Fig. 9. Pulse front evolution for detuned achromatic and ring interferometer
Fig. 10.
Fig. 10. Pulse width, duration and enhancement parameters for 200 nm detuning with the Achromatic (a) and (b) and Ring interferometer (c) and (d).
Fig. 11.
Fig. 11. Enhancement factor $e_{g}$ as a function of detuning for 3 beam diameters: (a) Achromatic ; (b) Achromatic with pre-dispersion; (c) Ring; (d) Ring with pre-dispersion

Equations (35)

Equations on this page are rendered with MathJax. Learn more.

δntot=NpkIpknτg=kIpkn1Ftot
k=δntotIpkn1Ftot
Ipk=Kpwtwpτp
Tloc=NpR=(wpwtτp)nKpnτg(δntotk)
Tmm=Tlocwg=wtnKpn1Pav(wpτp)n1wgτgwpτpA
Tmm0Tmm=(1wpτp)n1wgτgwpτp=epn2eg
ep=τp0wpτp
eg=ep(wgτgwpτp)
km1=(sin(θd)00001sin(θd)0λ0cot(θd)ccot(θd)c0100001)
km2=(sin(θd2)sin(θi)0000sin(θi)sin(θd2)02ϵλ0csin(θd2)2ϵcsin(θi)0100001)
θd2=πarccos(2ϵ+cos(θi))
E=exp[iπλ0(Qi(1,1)η2+2Qi(1,2)ηζQi(2,2)ζ2)]
E(x,t)=E0exp(η2w02)exp(ζ2τ02)
Qin(1,1)=iπw02λ0
Qin(2,2)=iπτ02λ0
Qout=kl2km2kl1km1kdfklenskDQin
ηR=xsinθB
ηL=xsinθB
ζR=t+xcosθBc
ζL=t+xcosθBc
ER(x,t)=exp[i(αx2+2βxtγt2)]
EL(x,t)=exp[i(αx22βxtγt2)]
α=πλ0(Qouti(1,1)sin2θB2Qouti(1,2)sinθBcosθBcQouti(2,2)cos2θBc2)
β=πλ0(Qouti(1,2)sinθB+Qouti(2,2)cosθBc)
γ=πλ0Qouti(2,2)
|ER|2exp((xst)2wp2)exp(t2τp2)
|EL|2exp((x+st)2wp2)exp(t2τp2)
wp=12αi
τg=12γi
τp=τg1(2βiwpτg)2
EREL=exp(x2wp2)exp(t2τg2)exp(4iβrxt)
ERELdtexp(x2wg2)
wg=wp1+(2βrwpτg)2
Δt=zc(1sin(θdδ)1sin(θd+δ)),
Lf=(tan(θd+δ)tan(θdδ))sin(θd)[tan(θd+δ)tan(θdδ)]w

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