Abstract

The second-order coherence theory of partially spatially coherent light and the overlap integral method are applied to study the end-coupling of stationary multimode light beams into planar waveguides. A method is presented for the determination of the cross-spectral density function of the guided field. Examples are given on the effects of spatial coherence, lateral shift, angular tilt, and defocusing of the incident beam on the coupling efficiency, spatial coherence, and propagation characteristics of the guided field.

© 2015 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
    [Crossref]
  2. E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [Crossref]
  3. C. M. Warnky, B. L. Anderson, and C. A. Klein, “Determining spatial modes of lasers with spatial coherence measurements,” Appl. Opt. 39, 6109–6117 (2000).
    [Crossref]
  4. H. Partanen, J. Turunen, and J. Tervo, “Coherence measurement with digital micromirror device,” Opt. Lett. 39, 1034–1037 (2014).
    [Crossref] [PubMed]
  5. F. Gori, “Collet–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [Crossref]
  6. A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
    [Crossref]
  7. A. S. Ostrovsky, M. A. Olvera, C. Rickenstorff, G. Martínez-Niconoff, and V. Arrizón, “Generation of a secondary electromagnetic source with desired statistical properties,” Opt. Commun. 283, 4490–4493 (2010).
    [Crossref]
  8. P. Spano, “Connection between spatial coherence and modal structure in optical fibers and semiconductor lasers,” Opt. Commun. 33, 265–270 (1980).
    [Crossref]
  9. S. Piazzolla and S. De Marchis, “Spatial coherence in optical fibers,” Opt. Commun. 32, 380–382 (1980).
    [Crossref]
  10. M. Imai, K. Itoh, and Y. Ohtsuka, “Measurements of complex degree of spatial coherence at the end face of an optical fiber,” Opt. Commun. 42, 97–100 (1982).
    [Crossref]
  11. S. Piazzolla and P. Spano, “Spatial coherence in incoherently excited optical fibers,” Opt. Commun. 43, 175–179 (1982).
    [Crossref]
  12. S. Withington and G. Yassin, “Analyzing the power coupled between partially coherent waveguide fields in different states of coherence,” J. Opt. Soc. Am. A 19, 1376–1382 (2002).
    [Crossref]
  13. K. J. Tsanaktsidis, D. M. Paganin, and D. Pelliccia, “Analytical description of partially coherent propagation and absorption losses in x-ray planar waveguides,” Opt. Lett. 38, 1808–1810 (2013).
    [Crossref] [PubMed]
  14. M. Osterhoff and T. Salditt, “Coherence filtering of x-ray waveguides: analytical and numerical approach,” New J. Phys. 13, 103026 (2011).
    [Crossref]
  15. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), Chap. 7.
    [Crossref]
  16. T. Saastamoinen, M. Kuittinen, P. Vahimaa, J. Turunen, and J. Tervo, “Focusing of partially coherent light into planar waveguides,” Opt. Express 12, 4511–4522 (2004).
    [Crossref] [PubMed]
  17. J. Turunen, “Low coherence laser beams,” Chapter 10 in Laser Beam Propagation: Generation and Propagation of Customized Light, A. Forbes, ed. (CRC Press, 2014).
    [Crossref]
  18. P. Vahimaa and J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A: Pure Appl. Opt. 6, S41–S44 (2004).
    [Crossref]
  19. H. Kim, J. Park, and B. Lee, Fourier Modal Method and its Applications in Computational Nanophotonics (CRC Press, 2012).
  20. M. Bertolotti, L. Sereda, and A. Ferrari, “Application of the spectral representation of stochastic process to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
    [Crossref]
  21. H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22, 1536–1544 (2005).
    [Crossref]

2014 (1)

2013 (1)

2011 (1)

M. Osterhoff and T. Salditt, “Coherence filtering of x-ray waveguides: analytical and numerical approach,” New J. Phys. 13, 103026 (2011).
[Crossref]

2010 (1)

A. S. Ostrovsky, M. A. Olvera, C. Rickenstorff, G. Martínez-Niconoff, and V. Arrizón, “Generation of a secondary electromagnetic source with desired statistical properties,” Opt. Commun. 283, 4490–4493 (2010).
[Crossref]

2005 (1)

2004 (2)

T. Saastamoinen, M. Kuittinen, P. Vahimaa, J. Turunen, and J. Tervo, “Focusing of partially coherent light into planar waveguides,” Opt. Express 12, 4511–4522 (2004).
[Crossref] [PubMed]

P. Vahimaa and J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A: Pure Appl. Opt. 6, S41–S44 (2004).
[Crossref]

2002 (1)

2000 (1)

1997 (1)

M. Bertolotti, L. Sereda, and A. Ferrari, “Application of the spectral representation of stochastic process to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
[Crossref]

1982 (4)

A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
[Crossref]

E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[Crossref]

M. Imai, K. Itoh, and Y. Ohtsuka, “Measurements of complex degree of spatial coherence at the end face of an optical fiber,” Opt. Commun. 42, 97–100 (1982).
[Crossref]

S. Piazzolla and P. Spano, “Spatial coherence in incoherently excited optical fibers,” Opt. Commun. 43, 175–179 (1982).
[Crossref]

1980 (3)

P. Spano, “Connection between spatial coherence and modal structure in optical fibers and semiconductor lasers,” Opt. Commun. 33, 265–270 (1980).
[Crossref]

S. Piazzolla and S. De Marchis, “Spatial coherence in optical fibers,” Opt. Commun. 32, 380–382 (1980).
[Crossref]

F. Gori, “Collet–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

Anderson, B. L.

Arrizón, V.

A. S. Ostrovsky, M. A. Olvera, C. Rickenstorff, G. Martínez-Niconoff, and V. Arrizón, “Generation of a secondary electromagnetic source with desired statistical properties,” Opt. Commun. 283, 4490–4493 (2010).
[Crossref]

Bertolotti, M.

M. Bertolotti, L. Sereda, and A. Ferrari, “Application of the spectral representation of stochastic process to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
[Crossref]

De Marchis, S.

S. Piazzolla and S. De Marchis, “Spatial coherence in optical fibers,” Opt. Commun. 32, 380–382 (1980).
[Crossref]

Ferrari, A.

M. Bertolotti, L. Sereda, and A. Ferrari, “Application of the spectral representation of stochastic process to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
[Crossref]

Gori, F.

F. Gori, “Collet–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

Imai, M.

M. Imai, K. Itoh, and Y. Ohtsuka, “Measurements of complex degree of spatial coherence at the end face of an optical fiber,” Opt. Commun. 42, 97–100 (1982).
[Crossref]

Itoh, K.

M. Imai, K. Itoh, and Y. Ohtsuka, “Measurements of complex degree of spatial coherence at the end face of an optical fiber,” Opt. Commun. 42, 97–100 (1982).
[Crossref]

Kim, H.

H. Kim, J. Park, and B. Lee, Fourier Modal Method and its Applications in Computational Nanophotonics (CRC Press, 2012).

Klein, C. A.

Kuittinen, M.

Lajunen, H.

Lee, B.

H. Kim, J. Park, and B. Lee, Fourier Modal Method and its Applications in Computational Nanophotonics (CRC Press, 2012).

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
[Crossref]

Martínez-Niconoff, G.

A. S. Ostrovsky, M. A. Olvera, C. Rickenstorff, G. Martínez-Niconoff, and V. Arrizón, “Generation of a secondary electromagnetic source with desired statistical properties,” Opt. Commun. 283, 4490–4493 (2010).
[Crossref]

Ohtsuka, Y.

M. Imai, K. Itoh, and Y. Ohtsuka, “Measurements of complex degree of spatial coherence at the end face of an optical fiber,” Opt. Commun. 42, 97–100 (1982).
[Crossref]

Olvera, M. A.

A. S. Ostrovsky, M. A. Olvera, C. Rickenstorff, G. Martínez-Niconoff, and V. Arrizón, “Generation of a secondary electromagnetic source with desired statistical properties,” Opt. Commun. 283, 4490–4493 (2010).
[Crossref]

Osterhoff, M.

M. Osterhoff and T. Salditt, “Coherence filtering of x-ray waveguides: analytical and numerical approach,” New J. Phys. 13, 103026 (2011).
[Crossref]

Ostrovsky, A. S.

A. S. Ostrovsky, M. A. Olvera, C. Rickenstorff, G. Martínez-Niconoff, and V. Arrizón, “Generation of a secondary electromagnetic source with desired statistical properties,” Opt. Commun. 283, 4490–4493 (2010).
[Crossref]

Paganin, D. M.

Park, J.

H. Kim, J. Park, and B. Lee, Fourier Modal Method and its Applications in Computational Nanophotonics (CRC Press, 2012).

Partanen, H.

Pelliccia, D.

Piazzolla, S.

S. Piazzolla and P. Spano, “Spatial coherence in incoherently excited optical fibers,” Opt. Commun. 43, 175–179 (1982).
[Crossref]

S. Piazzolla and S. De Marchis, “Spatial coherence in optical fibers,” Opt. Commun. 32, 380–382 (1980).
[Crossref]

Rickenstorff, C.

A. S. Ostrovsky, M. A. Olvera, C. Rickenstorff, G. Martínez-Niconoff, and V. Arrizón, “Generation of a secondary electromagnetic source with desired statistical properties,” Opt. Commun. 283, 4490–4493 (2010).
[Crossref]

Saastamoinen, T.

Salditt, T.

M. Osterhoff and T. Salditt, “Coherence filtering of x-ray waveguides: analytical and numerical approach,” New J. Phys. 13, 103026 (2011).
[Crossref]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), Chap. 7.
[Crossref]

Sereda, L.

M. Bertolotti, L. Sereda, and A. Ferrari, “Application of the spectral representation of stochastic process to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
[Crossref]

Spano, P.

S. Piazzolla and P. Spano, “Spatial coherence in incoherently excited optical fibers,” Opt. Commun. 43, 175–179 (1982).
[Crossref]

P. Spano, “Connection between spatial coherence and modal structure in optical fibers and semiconductor lasers,” Opt. Commun. 33, 265–270 (1980).
[Crossref]

Starikov, A.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), Chap. 7.
[Crossref]

Tervo, J.

Tsanaktsidis, K. J.

Turunen, J.

Vahimaa, P.

Warnky, C. M.

Withington, S.

Wolf, E.

Yassin, G.

Appl. Opt. (1)

J. Opt. A: Pure Appl. Opt. (1)

P. Vahimaa and J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A: Pure Appl. Opt. 6, S41–S44 (2004).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

New J. Phys. (1)

M. Osterhoff and T. Salditt, “Coherence filtering of x-ray waveguides: analytical and numerical approach,” New J. Phys. 13, 103026 (2011).
[Crossref]

Opt. Commun. (6)

A. S. Ostrovsky, M. A. Olvera, C. Rickenstorff, G. Martínez-Niconoff, and V. Arrizón, “Generation of a secondary electromagnetic source with desired statistical properties,” Opt. Commun. 283, 4490–4493 (2010).
[Crossref]

P. Spano, “Connection between spatial coherence and modal structure in optical fibers and semiconductor lasers,” Opt. Commun. 33, 265–270 (1980).
[Crossref]

S. Piazzolla and S. De Marchis, “Spatial coherence in optical fibers,” Opt. Commun. 32, 380–382 (1980).
[Crossref]

M. Imai, K. Itoh, and Y. Ohtsuka, “Measurements of complex degree of spatial coherence at the end face of an optical fiber,” Opt. Commun. 42, 97–100 (1982).
[Crossref]

S. Piazzolla and P. Spano, “Spatial coherence in incoherently excited optical fibers,” Opt. Commun. 43, 175–179 (1982).
[Crossref]

F. Gori, “Collet–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Pure Appl. Opt. (1)

M. Bertolotti, L. Sereda, and A. Ferrari, “Application of the spectral representation of stochastic process to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
[Crossref]

Other (4)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
[Crossref]

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), Chap. 7.
[Crossref]

J. Turunen, “Low coherence laser beams,” Chapter 10 in Laser Beam Propagation: Generation and Propagation of Customized Light, A. Forbes, ed. (CRC Press, 2014).
[Crossref]

H. Kim, J. Park, and B. Lee, Fourier Modal Method and its Applications in Computational Nanophotonics (CRC Press, 2012).

Supplementary Material (5)

» Media 1: MP4 (744 KB)     
» Media 2: MP4 (661 KB)     
» Media 3: MP4 (515 KB)     
» Media 4: MP4 (471 KB)     
» Media 5: MP4 (555 KB)     

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Figures (12)

Fig. 1
Fig. 1 Geometry and notations. A Gaussian Schell-model beam, propagating at an angle θ0 with respect to the z axis, is incident to a planar waveguide with diameter d at z = 0. The waist of the beam is located at (x, z) = (x0, −z0).
Fig. 2
Fig. 2 Coupling into single-mode waveguides. (a) Total coupling efficiency η and (b) most significant coefficients η n m if either the total beam size w0 (black lines) or the size of the lowest-order coherent mode w 0 β (red lines) is matched to the waveguide mode. In (b) we plot η n m instead of ηnm to show the m = 2 contributions more clearly.
Fig. 3
Fig. 3 Two-mode waveguide with variable coherence width of the input light. The two leftmost columns illustrate the absolute value and the phase, respectively, of the CSD of the incident field at plane z = 0. Corresponding distributions for the guided field at z = 0 are shown in the two rightmost columns. The middle column shows the evolution of Sg(x, z). The top three rows respectively show the results for the fully coherent case σ0 = ∞, for a partially coherent case σ0 = 8λ, and the case σ0 = 2.2λ, which is close to the transition into non-paraxial domain. Finally, (p) shows the intensity profiles of the incident field and the guided field at the plane z = 0. The movie ( Media 1) shows the evolution of the illustrated quantities when the coherence is varied.
Fig. 4
Fig. 4 Effect of decreasing degree of spatial coherence in (a) the total coupling efficiency, (b) the overall degree of coherence of the coupled field, and (c) the modal coupling coefficients η n m. In (b) the black solid line shows that overall degree of coherence μ̄2 = β of the incident field and the red dashed line illustrates μ ¯ g 2 for the guided field.
Fig. 5
Fig. 5 Two mode waveguide with partially coherent input field and variable lateral shift with values x0 = 0, x0 = 0.5λ, and x0 = 5λ. We keep σ0 constant at 4λ, considering the cases x0 = 0, x0 = 0.5λ, and x0 = 5λ in the top three rows, respectively. The structure of the figure is otherwise similar to Fig. 3. The movie ( Media 2) shows the evolution of the illustrated quantities when the shift is varied.
Fig. 6
Fig. 6 Effect of variable lateral shift in (a) the total coupling efficiency, (b) the overall degree of coherence of the coupled field, and (c) the modal coupling coefficients η n m.
Fig. 7
Fig. 7 Two-mode waveguide with partially coherent input field and variable angular tilt. The cases θ0 = 0, θ0 = 1°, and θ0 = 5° are considered in the three top rows, respectively. The structure of the figure is otherwise similar to Figs. 3 and 5. The movie ( Media 3) shows the evolution of the illustrated quantities when the tilt is varied.
Fig. 8
Fig. 8 Effect of variable angular tilt in (a) the total coupling efficiency, (b) the overall degree of coherence of the coupled field, and (c) the modal coupling coefficients η m n.
Fig. 9
Fig. 9 Two-mode waveguide with partially coherent defocused incident field, with values z0 = 0 (top row), z0 = 50λ (second row), and z0 = 200λ (third row). The structure of the figure is otherwise similar to Fig. 3. The movie ( Media 4) shows the evolution of the illustrated quantities when the defocus is varied.
Fig. 10
Fig. 10 Effect of defocusing in (a) the total coupling efficiency, (b) the overall degree of coherence of the coupled field, and (c) the modal coupling coefficients η m n.
Fig. 11
Fig. 11 Three-mode waveguide with partially coherent input field and variable defocus with values z0 = 0 (first row), z0 = 50λ (second row), and z0 = 200λ (third row), and a constant lateral shift x0 = 2λ. The structure of the figure is otherwise similar to Fig. 3. The movie ( Media 5) shows the evolution of the illustrated quantities when the defocus is varied.
Fig. 12
Fig. 12 Effect of defocusing with constant lateral shift in (a) the total coupling efficiency, (b) the overall degree of coherence of the coupled field, and (c) the modal coupling coefficients η m n.

Equations (31)

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W ( x 1 , x 2 , z ) = S 0 w 0 w ( z ) exp [ ( x 1 x 0 θ 0 Δ z ) 2 + ( x 2 x 0 θ 0 Δ z ) 2 w 2 ( z ) ] exp [ ( x 1 x 2 ) 2 2 σ 2 ( z ) ] × exp { i k 0 2 R ( z ) [ ( x 1 x 0 θ 0 Δ z ) 2 ( x 2 x 0 θ 0 Δ z ) 2 ] } exp [ i k 0 θ 0 ( x 2 x 1 ) ] ,
w ( z ) = w 0 ( 1 + Δ z 2 z R 2 ) 1 / 2 , σ ( z ) = σ 0 w 0 w ( z ) , and z R = 1 2 k 0 w 0 2 β
R ( z ) = Δ z + z R 2 Δ z
β = [ 1 + ( w 0 / σ 0 ) 2 ] 1 / 2 .
W ( x 1 , x 2 , z ) = n = 0 a n ϕ n * ( x 1 , z ) ϕ n ( x 2 , z ) ,
ϕ n ( x , z ) = ( 2 π w 0 2 β ) 1 / 4 1 2 n n ! w 0 w ( z ) exp [ i ψ ( z ) / 2 ] H n [ 2 w ( z ) β ( x x 0 θ 0 Δ z ) ] × exp [ ( x x 0 θ 0 Δ z ) 2 w 2 ( z ) β ] exp [ i k 0 2 R ( z ) ( x x 0 θ 0 Δ z ) 2 ] exp [ i k 0 θ 0 ( x x 0 θ 0 Δ z ) ]
a n = S 0 2 π w 0 β 1 + β ( 1 β 1 + β ) n
μ ¯ 2 = | W ( x 1 , x 2 , z ) | 2 d x 1 d x 2 [ S ( x , z ) d x ] 2 = n = 0 a n 2 [ n = 0 a n ] 2 ,
φ m ( x , z ) = X m ( x ) exp ( i β m z )
X m ( x ) X n ( x ) d x = δ m n ,
a n ϕ n ( x , 0 ) = m 0 M 1 p n m X m ( x ) + radiation mode contribution
p n m = a n ϕ n ( x , 0 ) X m ( x ) d x
W g ( x 1 , x 2 , 0 ) = n = 0 m = 0 M 1 q = 0 M 1 p n m * p n q X m ( x 1 ) X q ( x 2 ) .
W g ( x 1 , x 2 , z ) = n = 0 m = 0 M 1 q = 0 M 1 p n m * p n q X m ( x 1 ) X q ( x 2 ) exp [ i ( β m β q ) z ]
S g ( x , z ) = n = 0 | m = 0 M 1 p n m X m ( x ) exp ( i β m z ) | 2
μ g ( x 1 , x 2 , z ) = W ( x 1 , x 2 , z ) S ( x 1 , z ) S ( x 2 , z ) .
η = S g ( x , 0 ) d x S ( x , 0 ) d x = n = 0 m = 0 M 1 η n m ,
η n m = | p n m | 2 m = 0 a n
μ ¯ g 2 = m = 0 M 1 q = 0 M 1 | n = 0 p n m * p n q | 2 [ m = 0 M 1 n = 0 | p n m | 2 ] 2 .
W g ( x 1 , x 2 , z ) = n = 0 | p n 0 | 2 X 0 ( x 1 ) X 0 ( x 2 ) .
η = n = 0 | p n 0 | 2 n = 0 a n = n = 0 a n | ϕ n ( x , 0 ) X 0 ( x ) d x | 2 n = 0 a n .
W g ( x 1 , x 2 , z ) = m = 0 M 1 q = 0 M 1 p n m * p n q X m ( x 1 ) X q ( x 2 ) exp [ i ( β m β q ) z ] .
μ g ( x 1 , x 2 , z ) = m = 0 M 1 p n m * X m ( x 1 ) exp ( i β m z ) q = 0 M 1 p n q X q ( x 2 ) exp ( i β q z ) | m = 0 M 1 p n m X m ( x 1 ) exp ( i β m z ) | | q = 0 M 1 p n q X q ( x 2 ) exp ( i β q z ) | ,
W g ( x 1 , x 2 , z ) = X 0 ( x 1 ) X 0 ( x 2 ) n = 0 | p n 0 | 2 + X 1 ( x 1 ) X 1 ( x 2 ) n = 0 | p n 1 | 2 + X 0 ( x 1 ) X 1 ( x 2 ) exp ( i Δ β z ) n = 0 p n 0 * p n 1 + X 1 ( x 1 ) X 0 ( x 2 ) exp ( i Δ β z ) n = 0 p n 0 p n 1 * ,
n = 0 p n 0 * p n 1 = 0
μ ¯ g 2 ( z ) = ( n = 0 | p n 0 | 2 ) 2 + ( n = 0 | p n 1 | 2 ) 2 + 2 ( n = 0 p n 0 * p n 1 ) 2 [ n = 0 ( | p n 0 | 2 + | p n 1 | 2 ) ] 2 .
n ( x ) = { n g when | x | < d / 2 n c when | x | > d / 2 .
X m ( x ) = { C m cos ( α m d / 2 ) exp [ γ m ( x d / 2 ) ] , when x > d / 2 C m cos ( α m x ) , when d / 2 x d / 2 C m cos ( α m d / 2 ) exp [ γ m ( x + d / 2 ) ] , when x < d / 2
X m ( x ) = { S m sin ( α m d / 2 ) exp [ γ m ( x d / 2 ) ] , when x > d / 2 S m sin ( α m x ) , when d / 2 x d / 2 S m sin ( α m d / 2 ) exp [ γ m ( x + d / 2 ) ] , when x < d / 2
α m = ( k 0 2 n g 2 β m 2 ) 1 / 2 , γ m = ( β m 2 k 0 2 n c 2 ) 1 / 2 , tan ( α m d / 2 m π / 2 ) = γ m α m ,
C m = [ d 2 + sin ( α m d ) 2 α m + cos 2 ( α m d / 2 ) γ m ] 1 / 2 , S m = [ d 2 sin ( α m d ) 2 α m + sin 2 ( α m d / 2 ) γ m ] 1 / 2 .

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