Abstract

A detailed theoretical treatment is presented of bound-mode to bound-mode Bragg reflection and bound-mode to radiation-mode coupling loss in a tilted optical-fiber phase grating. Numerical predictions of the effects of grating tilt on the spectral characteristics of such a grating are calculated. These predictions are compared with experimentally measured spectra of strong gratings written by ultraviolet irradiation of deuterium-sensitized fiber with grating tilt angles ranging from 0° to 15°. Good agreement is obtained between the theoretical predictions and the experimental results.

© 1996 Optical Society of America

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References

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  1. G. Meltz, W. W. Morey, W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
    [CrossRef] [PubMed]
  2. V. Mizrahi, T. Erdogan, D. J. DiGiovanni, P. J. Lemaire, W. M. MacDonald, S. G. Kosinski, S. Cabot, J. E. Sipe, “Four channel fibre grating demultiplexer,” Electron. Lett. 30, 780–781 (1994).
    [CrossRef]
  3. G. Meltz, W. W. Morey, W. H. Glenn, “In-fiber Bragg grating tap,” presented at the Optical Fiber Communications Conference, San Francisco, Calif., January 22–26, 1990, paper TuG1.
  4. R. Kashyap, R. Wyatt, R. J. Campbell, “Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating,” Electron. Lett. 29, 154–156 (1993).
    [CrossRef]
  5. V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
    [CrossRef]
  6. K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
    [CrossRef]
  7. J. E. Sipe, G. I. Stegeman, “Comparison of normal mode and total field analysis techniques in planar integrated optics,” J. Opt. Soc. Am. 69, 1676–1683 (1979).
    [CrossRef]
  8. H. Kogelnik, “Theory of optical waveguides,” in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990).
    [CrossRef]
  9. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, Boston, 1991), Chap. 2.
  10. J. E. Sipe, L. Poladian, C. Martijn deSterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
    [CrossRef]
  11. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [CrossRef] [PubMed]
  12. H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
    [CrossRef]
  13. P. J. Lemaire, R. M. Atkins, V. Mizrahi, K. L. Walker, K. S. Kranz, W. A. Reed, “High pressure H2loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2doped optical fibres,” Electron. Lett. 29, 1191–1193 (1993).
    [CrossRef]
  14. A. S. Davydov, Quantum Mechanics (Pergamon, New York, 1965), Sect. 80.
  15. L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968), Sect. 35.

1994 (2)

V. Mizrahi, T. Erdogan, D. J. DiGiovanni, P. J. Lemaire, W. M. MacDonald, S. G. Kosinski, S. Cabot, J. E. Sipe, “Four channel fibre grating demultiplexer,” Electron. Lett. 30, 780–781 (1994).
[CrossRef]

J. E. Sipe, L. Poladian, C. Martijn deSterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
[CrossRef]

1993 (3)

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

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
[CrossRef]

P. J. Lemaire, R. M. Atkins, V. Mizrahi, K. L. Walker, K. S. Kranz, W. A. Reed, “High pressure H2loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2doped optical fibres,” Electron. Lett. 29, 1191–1193 (1993).
[CrossRef]

1990 (1)

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

1989 (1)

1979 (1)

1972 (1)

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

1971 (1)

Atkins, R. M.

P. J. Lemaire, R. M. Atkins, V. Mizrahi, K. L. Walker, K. S. Kranz, W. A. Reed, “High pressure H2loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2doped optical fibres,” Electron. Lett. 29, 1191–1193 (1993).
[CrossRef]

Bilodeau, F.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Cabot, S.

V. Mizrahi, T. Erdogan, D. J. DiGiovanni, P. J. Lemaire, W. M. MacDonald, S. G. Kosinski, S. Cabot, J. E. Sipe, “Four channel fibre grating demultiplexer,” Electron. Lett. 30, 780–781 (1994).
[CrossRef]

Campbell, R. J.

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

Davydov, A. S.

A. S. Davydov, Quantum Mechanics (Pergamon, New York, 1965), Sect. 80.

DiGiovanni, D. J.

V. Mizrahi, T. Erdogan, D. J. DiGiovanni, P. J. Lemaire, W. M. MacDonald, S. G. Kosinski, S. Cabot, J. E. Sipe, “Four channel fibre grating demultiplexer,” Electron. Lett. 30, 780–781 (1994).
[CrossRef]

Erdogan, T.

V. Mizrahi, T. Erdogan, D. J. DiGiovanni, P. J. Lemaire, W. M. MacDonald, S. G. Kosinski, S. Cabot, J. E. Sipe, “Four channel fibre grating demultiplexer,” Electron. Lett. 30, 780–781 (1994).
[CrossRef]

Glenn, W. H.

G. Meltz, W. W. Morey, W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
[CrossRef] [PubMed]

G. Meltz, W. W. Morey, W. H. Glenn, “In-fiber Bragg grating tap,” presented at the Optical Fiber Communications Conference, San Francisco, Calif., January 22–26, 1990, paper TuG1.

Gloge, D.

Hill, K. O.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Johnson, D. C.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Kashyap, R.

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

Kogelnik, H.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

H. Kogelnik, “Theory of optical waveguides,” in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990).
[CrossRef]

Kosinski, S. G.

V. Mizrahi, T. Erdogan, D. J. DiGiovanni, P. J. Lemaire, W. M. MacDonald, S. G. Kosinski, S. Cabot, J. E. Sipe, “Four channel fibre grating demultiplexer,” Electron. Lett. 30, 780–781 (1994).
[CrossRef]

Kranz, K. S.

P. J. Lemaire, R. M. Atkins, V. Mizrahi, K. L. Walker, K. S. Kranz, W. A. Reed, “High pressure H2loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2doped optical fibres,” Electron. Lett. 29, 1191–1193 (1993).
[CrossRef]

Lemaire, P. J.

V. Mizrahi, T. Erdogan, D. J. DiGiovanni, P. J. Lemaire, W. M. MacDonald, S. G. Kosinski, S. Cabot, J. E. Sipe, “Four channel fibre grating demultiplexer,” Electron. Lett. 30, 780–781 (1994).
[CrossRef]

P. J. Lemaire, R. M. Atkins, V. Mizrahi, K. L. Walker, K. S. Kranz, W. A. Reed, “High pressure H2loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2doped optical fibres,” Electron. Lett. 29, 1191–1193 (1993).
[CrossRef]

MacDonald, W. M.

V. Mizrahi, T. Erdogan, D. J. DiGiovanni, P. J. Lemaire, W. M. MacDonald, S. G. Kosinski, S. Cabot, J. E. Sipe, “Four channel fibre grating demultiplexer,” Electron. Lett. 30, 780–781 (1994).
[CrossRef]

Malo, B.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, Boston, 1991), Chap. 2.

Martijn deSterke, C.

Meltz, G.

G. Meltz, W. W. Morey, W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
[CrossRef] [PubMed]

G. Meltz, W. W. Morey, W. H. Glenn, “In-fiber Bragg grating tap,” presented at the Optical Fiber Communications Conference, San Francisco, Calif., January 22–26, 1990, paper TuG1.

Mizrahi, V.

V. Mizrahi, T. Erdogan, D. J. DiGiovanni, P. J. Lemaire, W. M. MacDonald, S. G. Kosinski, S. Cabot, J. E. Sipe, “Four channel fibre grating demultiplexer,” Electron. Lett. 30, 780–781 (1994).
[CrossRef]

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
[CrossRef]

P. J. Lemaire, R. M. Atkins, V. Mizrahi, K. L. Walker, K. S. Kranz, W. A. Reed, “High pressure H2loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2doped optical fibres,” Electron. Lett. 29, 1191–1193 (1993).
[CrossRef]

Morey, W. W.

G. Meltz, W. W. Morey, W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
[CrossRef] [PubMed]

G. Meltz, W. W. Morey, W. H. Glenn, “In-fiber Bragg grating tap,” presented at the Optical Fiber Communications Conference, San Francisco, Calif., January 22–26, 1990, paper TuG1.

Poladian, L.

Reed, W. A.

P. J. Lemaire, R. M. Atkins, V. Mizrahi, K. L. Walker, K. S. Kranz, W. A. Reed, “High pressure H2loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2doped optical fibres,” Electron. Lett. 29, 1191–1193 (1993).
[CrossRef]

Schiff, L. I.

L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968), Sect. 35.

Shank, C. V.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Sipe, J. E.

J. E. Sipe, L. Poladian, C. Martijn deSterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
[CrossRef]

V. Mizrahi, T. Erdogan, D. J. DiGiovanni, P. J. Lemaire, W. M. MacDonald, S. G. Kosinski, S. Cabot, J. E. Sipe, “Four channel fibre grating demultiplexer,” Electron. Lett. 30, 780–781 (1994).
[CrossRef]

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
[CrossRef]

J. E. Sipe, G. I. Stegeman, “Comparison of normal mode and total field analysis techniques in planar integrated optics,” J. Opt. Soc. Am. 69, 1676–1683 (1979).
[CrossRef]

Skinner, I.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Stegeman, G. I.

Vineberg, K. A.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Walker, K. L.

P. J. Lemaire, R. M. Atkins, V. Mizrahi, K. L. Walker, K. S. Kranz, W. A. Reed, “High pressure H2loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2doped optical fibres,” Electron. Lett. 29, 1191–1193 (1993).
[CrossRef]

Wyatt, R.

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

Appl. Opt. (1)

Electron. Lett. (4)

P. J. Lemaire, R. M. Atkins, V. Mizrahi, K. L. Walker, K. S. Kranz, W. A. Reed, “High pressure H2loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2doped optical fibres,” Electron. Lett. 29, 1191–1193 (1993).
[CrossRef]

V. Mizrahi, T. Erdogan, D. J. DiGiovanni, P. J. Lemaire, W. M. MacDonald, S. G. Kosinski, S. Cabot, J. E. Sipe, “Four channel fibre grating demultiplexer,” Electron. Lett. 30, 780–781 (1994).
[CrossRef]

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

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

J. Appl. Phys. (1)

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

J. Lightwave Technol. (1)

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Lett. (1)

Other (5)

A. S. Davydov, Quantum Mechanics (Pergamon, New York, 1965), Sect. 80.

L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968), Sect. 35.

H. Kogelnik, “Theory of optical waveguides,” in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990).
[CrossRef]

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, Boston, 1991), Chap. 2.

G. Meltz, W. W. Morey, W. H. Glenn, “In-fiber Bragg grating tap,” presented at the Optical Fiber Communications Conference, San Francisco, Calif., January 22–26, 1990, paper TuG1.

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Figures (17)

Fig. 1
Fig. 1

Diagram of the core of a step-index optical fiber showing a tilted fiber phase grating and definitions used in the analysis.

Fig. 2
Fig. 2

Plots of the two terms g and g ˜ that contribute to the forward and backward coupling coefficients gf and gb, respectively, defined in Eqs. (3.1). Note that g ˜g.

Fig. 3
Fig. 3

Plot of backward coupling coefficient versus grating tilt angle for the fiber parameters described in the text.

Fig. 4
Fig. 4

Calculated maximum grating reflectivity versus grating tilt angle for gratings with three different index modulations and assuming s-polarized incident light and the fiber parameters listed.

Fig. 5
Fig. 5

Calculated grating reflectivity spectra versus tilt angle for gratings with three different index modulations and assuming s-polarized incident light and the fiber parameters listed.

Fig. 6
Fig. 6

Diagram showing rays that represent the two uv beams incident on the fiber during grating writing and their associated angles inside and outside the fiber.

Fig. 7
Fig. 7

Experimentally measured maximum grating reflectivity (circles) for gratings written with tilt angles of 0°–15°. The solid line shows the calculated result.

Fig. 8
Fig. 8

(a) Experimentally measured grating reflectivity spectra versus tilt angle, (b) corresponding calculated reflectivity spectra assuming a 0.2-nm resolution-limited smoothing.

Fig. 9
Fig. 9

Comparison of (a) the calculated reflectivity spectrum and (b) the measured reflectivity spectrum for a grating with a 5° tilt angle. No smoothing has been used on the calculated spectrum.

Fig. 10
Fig. 10

Schematic illustration of three regimes of grating coupling in propagation-constant space: (a) forward bound-mode to backward bound-mode coupling, (b) forward bound-mode to backward radiation-mode coupling, (c) forward bound-mode to forward radiation-mode coupling.

Fig. 11
Fig. 11

Plot of the longest wavelength allowed for forward bound-mode to backward radiation-mode coupling (solid), the shortest wavelength allowed for forward bound-mode to forward radiation-mode coupling (dotted), and the boundary wavelength between forward and backward coupling (dashed), all as a function of grating period. The fiber parameters used to generate this plot are those used throughout the manuscript.

Fig. 12
Fig. 12

Calculated extinction coefficients for s-polarized bound-mode to s-polarized LPq radiation-mode coupling. Plots are made for grating tilt angles of (a) 0°, (b) 5°, and (c) 15°.

Fig. 13
Fig. 13

Calculated transmission loss spectra versus grating tilt angle for s-polarized bound-mode to s-polarized bound-mode and radiation-mode coupling, where all LPq radiation modes for q ≤ 5 have been included. The waterfall of spectra is plotted in both (a) ascending and (b) descending order of tilt angle.

Fig. 14
Fig. 14

Same as Fig. 13, but for p polarization.

Fig. 15
Fig. 15

Calculated absolute minimum transmission (maximum loss) that is due solely to radiation-mode coupling for s-polarized bound-mode to s-polarized radiation-mode coupling (solid) and for p-polarized coupling (dashed).

Fig. 16
Fig. 16

(a) Experimentally measured transmission loss spectra versus grating tilt angle for gratings written with tilt angles of 0°–15°, (b) corresponding calculated transmission loss spectra, where loss that is due to Bragg scattering has been excluded.

Fig. 17
Fig. 17

Measured wavelength of minimum transmission (circles) and calculated wavelength of minimum transmission (solid) versus tilt angle, where loss that is due to Bragg scattering has been excluded.

Equations (76)

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ɛ ( R , t ) = E ( R ) exp ( - i ω t ) + c . c .
E t ( x , y , z ) = x ^ E x ( x , y , z ) + y ^ E y ( x , y , z ) .
F ( x , y , z ) = ( E z ( x , y , z ) H z ( x , y , z ) E t ( x , y , z ) H t ( x , y , z ) ) .
( E α p + z ( x , y ) H α p + z ( x , y ) E α p + t ( x , y ) H α p + t ( x , y ) ) exp ( i β σ z ) ( e α p z ( x , y ) h α p z ( x , y ) e α p t ( x , y ) h α p t ( x , y ) ) exp ( i β α z ) f α p + ( x , y ) exp ( i β α z ) ,
( E α p - z ( x , y ) H α p - z ( x , y ) E α p - t ( x , y ) H α p - t ( x , y ) ) exp ( - i β σ z ) ( - e α p z ( x , y ) h α p z ( x , y ) e a p t ( x , y ) - h α p t ( x , y ) ) exp ( - i β α z ) f α p - ( x , y ) exp ( - i β α z ) .
( e m p t × h m p t * ) · z ^ ds = 1 2 P m p δ m m δ p p ,
( e ρ p t × h ρ p t * ) · z ^ ds = 1 2 P p ( ρ ) δ ( ρ - ρ ) δ p p ,
( e m p t × h ρ p t * ) · z ^ ds = 0 ,             etc .
ρ = n cl 2 k 2 - β 2 ,
E d ( x , y , z ) = - z ^ P z ( x , y , z ) 0 n 0 2 ( x , y ) ,
F ( x , y , z ) = α p a ˜ α p + ( z ) f α p + ( x , y ) exp ( i β α z ) + α p a ˜ α p - ( z ) f α p - ( x , y ) exp ( - i β α z ) ,
d a ˜ α p + ( z ) d z = i ω P α p exp ( - i β α z ) P ( x , y , z ) · E α p + * ( x , y ) d s , d a ˜ α p - ( z ) d z = - i ω P α p exp ( i β α z ) P ( x , y , z ) · E α p - * ( x , y ) d s ,
P ( x , y , z ) = 0 Δ χ E ( x , y , z ) ,
P ( x , y , z ) = 2 0 n ¯ Δ n ( x , y , z ) E ( x , y , z ) = 2 0 n ¯ Δ n ( x , y , z ) [ E modes ( x , y , z ) + E d ( x , y , z ) ] ,
P t ( x , y , z ) = 2 0 n ¯ Δ n ( x , y , z ) E modes t ( x , y , z ) , P z ( x , y , z ) = 2 0 n ¯ Δ n ( x , y , z ) E modes z ( x , y , z ) 1 + 2 n ¯ Δ n ( x , y , z ) n 0 2 ( x , y ) .
P ( x , y , z ) 2 0 n ¯ Δ n ( x , y , z ) E modes ( x , y , z ) ,
Δ n ( x , y , z ) n ¯ = ζ ( x , y ) η ( z ) ,
η ( z ) = σ ¯ ( z ) + 2 κ ¯ ( z ) cos [ 2 K g z + ϕ ¯ ( z ) ] .
σ ( z ) σ ¯ ( z cos θ ) , ϕ ( z ) ϕ ¯ ( z cos θ ) , κ ( z ) κ ¯ ( z cos θ ) , κ c ( z ) κ ( z ) exp [ i ϕ ( z ) ] ,
P ( x , y , z ) = 2 0 n ¯ 2 ζ ( x , y ) { σ ( z ) + κ c ( z ) × exp [ 2 i K ( z + x tan θ ) ] + κ c * ( z ) exp [ - 2 i K ( z + x tan θ ) ] } × E modes ( x , y , z ) .
β ¯ - 1 d a α + d z = i σ ( z ) { α g α α + + a α + ( z ) exp [ i ( β α - β α ) z ] + α g α α + - a α - ( z ) exp [ i ( - β α - β α ) z ] } + i κ c ( z ) exp ( 2 i K z ) { α μ α α + + a α + ( z ) exp [ i ( β α - β α ) z ] + α μ α α + - a α - ( z ) exp [ i ( - β α - β α ) z ] } + i κ c * ( z ) exp ( - 2 i K z ) { α ν α α + + a α + ( z ) exp [ i ( β α - β α ) z ] + α ν α α + - a α - ( z ) exp [ i ( - β α - β α ) z ] } ,
β ¯ - 1 d a α - d z = - i σ ( z ) { α g α α - + a α + ( z ) exp [ i ( β α + β α ) z ] + α g α α - - a α - ( z ) exp [ i ( - β α + β α ) z ] } - i κ c ( z ) exp ( 2 i K z ) { α μ α α - + a α + ( z ) exp [ i ( β α + β α ) z ] + α μ α α - - a α - ( z ) exp [ i ( - β α + β α ) z ] } - i κ c * ( z ) exp ( - 2 i K z ) { α ν α α - + a α + ( z ) exp [ i ( β α + β α ) z ] + α ν α α - - a α - ( z ) exp [ i ( - β α + β α ) z ] } .
a α ± 1 2 P α a ˜ α ± ,
β ¯ ω c n ¯ .
g α α i j 2 n ¯ c 0 P α P α E α i * ( x , y ) · ζ ( x , y ) E α j ( x , y ) d s ,
μ α α i j 2 n ¯ c 0 P α P α E α i * ( x , y ) exp [ 2 i K x ( tan θ ) ] · ζ ( x , y ) E α j ( x , y ) d s ,
ν α α i j 2 n ¯ c 0 P α P α E α i * ( x , y ) exp [ - 2 i K x ( tan θ ) ] · ζ ( x , y ) E α j ( x , y ) d s .
g 01 ; 01 - - = g 01 ; 01 + + = 2 n ¯ c 0 P 01 E 01 + * ( x , y ) · ζ ( x , y ) E 01 + ( x , y ) d s g f , μ 01 ; 01 + - = ( ν 01 ; 01 - + ) * = 2 n ¯ c 0 P 01 E 01 + * ( x , y ) · ζ ( x , y ) E 01 - ( x , y ) d s g b ,
β ¯ - 1 d a 01 + d z = i g f σ ( z ) a 01 + ( z ) + i g b κ ( z ) exp [ i ϕ ( z ) ] × exp [ 2 i ( K - β 01 ) z ] a 01 - ( z ) , β ¯ - 1 d a 01 - d z = - i g f σ ( z ) a 01 - ( z ) - i g b κ ( z ) exp [ - i ϕ ( z ) ] × exp [ - 2 i ( K - β 01 ) z ] a 01 + ( z ) .
u ( z ) a 01 + ( z ) exp [ 1 2 i ϕ ( z ) ] exp [ - i ( K - β 01 ) z ] , v ( z ) a 01 - ( z ) exp [ 1 2 i ϕ ( z ) ] exp [ i ( K - β 01 ) z ] ,
β ¯ - 1 d u d z = i [ g f σ ( z ) + δ - 1 2 β ¯ - 1 d ϕ d z ] u ( z ) + i g b κ ( z ) v ( z ) , β ¯ - 1 d v d z = - i [ g f σ ( z ) + δ - 1 2 β ¯ - 1 d ϕ d z ] v ( z ) - i g b κ ( z ) u ( z ) ,
δ β 01 - K β ¯
δ ω - ω Bragg ω Bragg ,
g = b 01 [ J 0 2 ( κ 01 a ) J 1 2 ( κ 01 a ) + 1 ] , g ˜ = Δ b 01 ( 1 - b 01 ) 1 + 2 Δ b 01 [ 1 - J 0 ( κ 01 a ) J 2 ( κ 01 a ) J 1 2 ( κ 01 a ) ] .
μ 01 ; 01 + - = ( ν 01 ; 01 - + ) * = 2 n ¯ c 0 P 01 E 01 + * ( x , y ) exp [ 2 i K x ( tan θ ) ] · ζ ( x , y ) E 01 - ( x , y ) d s g b .
( κ 01 a ) 2 J 1 2 ( κ 01 a ) g b ( s - polarized ) 2 b 01 = 0 κ 01 a J 0 2 ( u ) J 0 ( Ω u ) u d u - 2 Δ ( 1 - b 01 ) 1 + 2 Δ b 01 × 0 κ 01 a J 1 ( Ω u ) Ω u J 1 2 ( u ) u d u
( κ 01 a ) 2 J 1 2 ( κ 01 a ) g b ( p - polarized ) 2 b 01 = 0 κ 01 a J 0 2 ( u ) J 0 ( Ω u ) u d u - 2 Δ ( 1 - b 01 ) 1 + 2 Δ b 01 × 0 κ 01 a J 1 2 ( u ) J 0 ( Ω u ) u d u + 2 Δ ( 1 - b 01 ) 1 + 2 Δ b 01 × 0 κ 01 a J 1 2 ( u ) J 1 ( Ω u ) Ω u u d u .
d d z [ u ( z ) 2 - v ( z ) 2 ] = 0.
2 α ext = arcsin [ n cl sin ( α + θ ) ] + arcsin [ n cl sin ( α - θ ) ] , θ ext = 1 2 arcsin [ n cl sin ( α + θ ) ] - 1 2 arcsin [ n cl sin ( α - θ ) ] .
β ¯ - 1 d a α - d z = - i κ ( z ) exp [ - 1 2 i ϕ ( z ) ] ν α ; 01 - + u ( z ) exp [ i ( β - K ) z ] , β ¯ - 1 d a α + d z = i κ ( z ) exp [ 1 2 i ϕ ( z ) ] μ α ; 01 + - v ( z ) exp [ - i ( β - K ) z ]
β ¯ - 1 d u d z = i γ ( z ) u ( z ) + i g b κ ( z ) v ( z ) + i κ ( z ) exp [ 1 2 i ϕ ( z ) ] × α μ 01 ; α a α - ( z ) exp [ i ( K - β ) z ] , β ¯ - 1 d v d z = - i γ ( z ) v ( z ) - i g b κ ( z ) u ( z ) - i κ ( z ) exp [ - 1 2 i ϕ ( z ) ] × α ν 01 ; α a α + ( z ) exp [ - i ( K - β ) z ] ,
γ ( z ) g f σ ( z ) + δ - 1 2 β ¯ - 1 d ϕ d z .
u ( z ) u ( z 0 ) exp [ i β ¯ γ 0 ( z - z 0 ) ] , v ( z ) v ( z 0 ) exp [ - i β ¯ γ 0 ( z - z 0 ) ] , ϕ ( z ) ϕ ( z 0 ) + ( d ϕ d z ) 0 ( z - z 0 ) ,
a α - ( z ) = - β ¯ κ ( z ) exp [ - 1 2 i ϕ ( z ) ] exp [ i ( β - K ) z ] ν a ; 01 - + u ( z ) β - β res ( z ) ,
β res ( z ) K - β ¯ γ ( z ) + 1 2 d ϕ d z .
β ¯ - 1 d u d z = [ i γ ( z ) - κ 2 ( z ) A ( z ) ] u ( z ) + i g b κ ( z ) v ( z ) , β ¯ - 1 d v d z = - [ i γ ( z ) - κ 2 ( z ) A ( z ) ] v ( z ) - i g b κ ( z ) u ( z ) ,
A ( z ) = i β ¯ ρ p ν ρ p ; 01 - + 2 β - β res ( z ) ,
μ 01 ; ρ p + - ν ρ p ; 01 - + = ν ρ p ; 01 - + 2
A ( z ) = i β ¯ p β d β ρ ν ρ p ; 01 - + 2 β + i - β res ( z ) ,
1 β + i - β res ( z ) P [ 1 β - β res ( z ) ] - i π δ [ β - β res ( z ) ] ,
A ( z ) = p ( β π ρ β ¯ ν ρ p ; 01 - + 2 ) β = β res ( z ) p A p ( z ) ,
β res ( z ) = ( 2 K + d ϕ d z ) - [ β 01 + β ¯ g f σ ( z ) ] .
d d z [ u ( z ) 2 - v ( z ) 2 ] = - 2 β ¯ κ 2 ( z ) A ( z ) [ u ( z ) 2 - v ( z ) 2 ,
A q j i = 4 ( β a ) 2 ( ρ a ) 2 q b 01 π e q J 1 2 ( κ 01 a ) Γ q c q j i 2 ,
Γ q [ τ a ( ρ a ) q J q + 1 ( τ a ) J q ( ρ a ) - ( ρ a ) q + 1 J q ( τ a ) J q + 1 ( ρ a ) ] 2 + [ τ a ( ρ a ) q J q + 1 ( τ a ) N q ( ρ a ) - ( ρ a ) q + 1 J q ( τ a ) N q + 1 ( ρ a ) ] 2 ,
τ n core 2 k 2 - β 2
δ β 01 - K K ,
F ± ( x , y , z ) = ( E ± z ( x , y , z ) H ± z ( x , y , z ) E ± t ( x , y , z ) H ± t ( x , y , z ) ) ,
F ± ( x , y , z ) = α p a ˜ α p ± f α p ± ( x , y ) exp ( ± i β α z ) m p a ˜ m p ± f m p ± ( x , y ) exp ( ± i β m z ) + p d ρ a ˜ ρ p ± f ρ p ± ( x , y ) exp [ ± i β ( ρ ) z ] ,
× E ± ( R ) - i ω μ 0 H ± ( R ) = 0 , × H ± ( R ) + i ω 0 n 0 2 ( x , y ) E ± ( R ) = 0 ,
P ( R ) = p ( x , y ) δ ( z - z ) , p ( x , y ) = p t ( x , y ) + z ^ p z ( x , y ) ,
× E ( R ) - i ω μ 0 H ( R ) = 0 , × H ( R ) + i ω 0 n 0 2 ( x , y ) E ( R ) = - i ω P ( R ) .
E ( R ) = θ ( z - z ) E + ( R ) + θ ( z - z ) E - ( R ) + δ ( z - z ) z ^ E d ( x , y ) , H ( R ) = θ ( z - z ) H + ( R ) + θ ( z - z ) H - ( R ) ,
θ ( z - z ) = z ^ δ ( z - z ) , δ ( z - z ) × z ^ = 0 ,
z ^ × [ E + ( x , y , z ) - E - ( x , y , z ) ] - z ^ × E d ( x , y ) = 0 , z ^ × [ H + ( x , y , z ) - H - ( x , y , z ) ] + i ω 0 n 0 2 ( x , y ) z ^ E d ( x , y ) = - i ω p ( x , y ) .
E d ( x , y ) = - p z ( x , y ) 0 n 0 2 ( x , y ) .
1 2 P α p [ a ˜ α p + exp ( i β α z ) + a ˜ α p - exp [ - i β α z ) ] = i ω p t ( x , y ) · e α p t * ( x , y ) d s , 1 2 P α p [ a ˜ α p + exp ( i β α z ) - a ˜ α p - exp ( - i β α z ) ] = i ω p z ( x , y ) e α p z * ( x , y ) d s .
a ˜ α p + = i ω P α p exp ( - i β α z ) p ( x , y ) · E ˜ α p + * ( x , y ) d s , a ˜ α p - = i ω P α p exp ( i β α z ) p ( x , y ) · E ˜ α p + * ( x , y ) d s ,
P ( R ) = P ( x , y , z ) = P ( x , y , z ) δ ( z - z ) d z .
a ˜ α p + ( z ) = i ω P α p - z exp ( - i β α z ) P ( x , y , z ) · E α p + * ( x , y ) d s d z , a ˜ α p - ( z ) = i ω P α p z exp ( i β α z ) P ( x , y , z ) · E α p - * ( x , y ) d s d z ,
c q j i = 1 2 ( κ 01 a ) 2 0 κ 01 a I q j i ( u ) u d u ,
I q s s ( u ) = 2 ( n core k β + 1 ) J q ( T u ) J 0 ( u ) J q ( Ω u ) + ( n core k β - 1 ) [ J q + 2 ( T u ) J 0 ( u ) J q + 2 ( Ω u ) + J q - 2 ( T u ) J 0 ( u ) J q - 2 ( Ω u ) ] - κ 01 ρ β 01 β { J q + 1 ( T u ) J 1 ( u ) [ J q ( Ω u ) + J q + 2 ( Ω u ) ] - J q - 1 ( T u ) J 1 ( u ) [ J q ( Ω u ) + J q - 2 ( Ω u ) ] }
I q p s ( u ) = ( n core k β - 1 ) [ J q + 2 ( T u ) J 0 ( u ) J q + 2 ( Ω u ) - J q - 2 ( T u ) J 0 ( u ) J q - 2 ( Ω u ) ] - κ 01 ρ β 01 β { J q + 1 ( T u ) J 1 ( u ) [ J q ( Ω u ) + J q + 2 ( Ω u ) ] + J q - 1 ( T u ) J 1 ( u ) [ J q ( Ω u ) + J q - 2 ( Ω u ) ] } ,
I q p p ( u ) = 2 ( n core k β + 1 ) J q ( T u ) J 0 ( u ) J q ( Ω u ) - ( n core k β - 1 ) [ J q + 2 ( T u ) J 0 ( u ) J q + 2 ( Ω u ) + J q - 2 ( T u ) J 0 ( u ) J q - 2 ( Ω u ) ] - κ 01 ρ β 01 β { J q + 1 ( T u ) J 1 ( u ) [ J q ( Ω u ) - J q + 2 ( Ω u ) ] - J q - 1 ( T u ) J 1 ( u ) [ J q ( Ω u ) - J q - 2 ( Ω u ) ] } ,
I q s p ( u ) = ( n core k β - 1 ) [ J q + 2 ( T u ) J 0 ( u ) J q + 2 ( Ω u ) - J q - 2 ( T u ) J 0 ( u ) J q - 2 ( Ω u ) ] + κ 01 ρ β 01 β { J q + 1 ( T u ) J 1 ( u ) [ J q ( Ω u ) - J q + 2 ( Ω u ) ] + J q - 1 ( T u ) J 1 ( u ) [ J q ( Ω u ) - J q - 2 ( Ω u ) ] } .
Ω = 2 K tan θ κ 01 ,             T = τ κ 01 .

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