Abstract

Computational methods for determining the complex propagation constants of leaky waveguide modes have become so powerful and so readily available that it is possible to use these methods with little understanding of what they are calculating. We compare different computational methods for calculating the propagation constants of the leaky modes, focusing on the relatively simple context of a W-type slab waveguide. In a lossless medium with infinite transverse extent, a direct determination of the leaky mode by using mode matching is compared with complete mode decomposition. The mode matching method is analogous to the multipole method in two dimensions. We then compare these results with a simple finite-difference scheme in a transverse region with absorbing boundaries that is analogous to finite-difference or finite-element methods in two dimensions. While the physical meaning of the leaky modes in these different solution methods is different, they all predict a nearly identical evolution for an initial, nearly confined mode profile over a limited spatial region and a limited distance. Finally, we demonstrate that a waveguide that uses bandgap confinement with a central defect produces analogous results.

© 2009 Optical Society of America

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2007 (2)

2004 (1)

2002 (3)

2001 (1)

2000 (1)

1995 (1)

S.-L. Lee, Y. Chung, L. A. Coldren, N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[CrossRef]

1994 (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 110–117 (1994).
[CrossRef]

1992 (1)

D. Marcuse, “Solution of the vector wave equation for general dielectric waveguides by the Galerkin method,” IEEE J. Quantum Electron. 28, 459–465 (1992).
[CrossRef]

1990 (1)

1986 (2)

H. Haus, D. Miller, “Attenuation of cutoff modes and leaky modes of dielectric slab structures,” IEEE J. Quantum Electron. 22, 310–318 (1986).
[CrossRef]

T. Tamir, F. Y. Kou, “Varieties of leaky waves and their excitation along multilayered structures,” IEEE J. Quantum Electron. 22, 544–551 (1986).
[CrossRef]

1984 (2)

1981 (2)

S. T. Peng, A. A. Oliner, “Guidance and leakage properties of a class of open dielectric waveguides. I. Mathematical formulations,” IEEE Trans. Microwave Theory Tech. MTT-29, 843–855 (1981).
[CrossRef]

A. A. Oliner, S. T. Peng, T. I. Hsu, A. Sanchez, “Guidance and leakage properties of a class of open dielectric waveguides. II. New physical effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855–869 (1981).
[CrossRef]

1977 (3)

1975 (2)

Y. Suematsu, K. Furuya, “Quasi-guided modes and related radiation losses in optical dielectric waveguides with external higher index surroundings,” IEEE Trans. Microwave Theory Tech. 23, 170–175 (1975).
[CrossRef]

S. Kawakami, S. Nishida, “Perturbation theory of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. 11, 130–138 (1975).
[CrossRef]

1974 (6)

S. Kawakami, S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. 10, 879–887 (1974).
[CrossRef]

A. W. Snyder, “Leaky-ray theory of optical waveguides of circular cross section,” Appl. Phys. (N.Y.) 4, 273–298 (1974).
[CrossRef]

A. W. Snyder, D. J. Mitchell, “Leaky mode analysis of circular optical waveguides,” Opto-electronics (London) 6, 287–296 (1974).
[CrossRef]

A. W. Snyder, D. J. Mitchell, “Leaky rays on circular optical fibers,” J. Opt. Soc. Am. 64, 599–607 (1974).
[CrossRef]

A. W. Snyder, D. J. Mitchell, C. Pask, “Failure of geometric optics for analysis of circular optical fibers,” J. Opt. Soc. Am. 64, 608–614 (1974).
[CrossRef]

A. W. Snyder, D. J. Mitchell, “Ray attenuation in lossless dielectric structures,” J. Opt. Soc. Am. 64, 956–963 (1974).
[CrossRef]

1973 (1)

D. B. Hall, C. Yeh, “Leaky waves in a heteroepitaxial film,” J. Appl. Phys. 44, 2271–2274 (1973).
[CrossRef]

1972 (1)

V. V. Shevchenko, “On the behavior of wave numbers beyond the critical value for waves in dielectric waveguides (media with losses),” Radiophys. Quantum Electron. 15, 194–200 (1972).
[CrossRef]

1971 (1)

1970 (1)

1969 (4)

J. Arnbak, “Leaky waves on a dielectric rod,” Electron. Lett. 5, 41–42 (1969).
[CrossRef]

J. R. James, “Leaky waves on a dielectric rod,” Electron. Lett. 5, 252–254 (1969).
[CrossRef]

S. E. Miller, “Integrated optics: an introduction,” Bell Syst. Tech. J. 48, 2059–2069 (1969).
[CrossRef]

E. A. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[CrossRef]

1963 (2)

T. Tamir, A. A. Oliner, “Guided complex waves. Part 1: fields at an interface,” Proc. IEEE 110, 310–324 (1963).

T. Tamir, A. A. Oliner, “Guided complex waves. Part 2: relation to radiation pattern,” Proc. IEEE 110, 325–334 (1963).

1961 (2)

E. S. Cassedy, M. Cohn, “On the existence of leaky waves due to a line source above a grounded dielectric slab,” IRE Trans. Microwave Theory Tech. 9, 243–247 (1961).
[CrossRef]

E. Snitzer, “Cylindrical dielectric waveguide modes,” J. Opt. Soc. Am. 51, 491–198 (1961).
[CrossRef]

1956 (1)

N. Marcuvitz, “On field representations in terms of leaky modes or eigenmodes,” IRE Trans. Antennas Propag. 4, 192–194 (1956).
[CrossRef]

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, 1981), Chap. 2.6.

Albin, S.

Arfken, G. B.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

Arnbak, J.

J. Arnbak, “Leaky waves on a dielectric rod,” Electron. Lett. 5, 41–42 (1969).
[CrossRef]

Barone, S.

S. Barone, “Leaky wave contributions to the field of a line source above a dielectric slab,” Report R-532-546, PIB-462 (Microwave Research Institute, Polytechnic Institute of Brooklyn,Nov. 26, 1956).

S. Barone, A. Hessel, “Leaky wave contributions to the field of a line source above a dielectric slab—part II,” Report R-698-58, PIB-626 (Microwave Research Institute, Polytechnic Institute of Brooklyn, Dec. 1958).

Benisty, H.

J. M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, A. Tchelnokov, Photonic Crystals: towards Nanoscale Photonic Devices (Springer, 2005).

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 110–117 (1994).
[CrossRef]

Berger, V.

J. M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, A. Tchelnokov, Photonic Crystals: towards Nanoscale Photonic Devices (Springer, 2005).

Botten, L. C.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, 1999).

Brown, J. W.

R. V. Churchill, J. W. Brown, Complex Variables and Applications, 5th ed. (McGraw-Hill, 1990), Chap. 2.

Burke, J.

Canal, F.

Cassedy, E. S.

E. S. Cassedy, M. Cohn, “On the existence of leaky waves due to a line source above a grounded dielectric slab,” IRE Trans. Microwave Theory Tech. 9, 243–247 (1961).
[CrossRef]

Chilwell, J. T.

Chung, Y.

S.-L. Lee, Y. Chung, L. A. Coldren, N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[CrossRef]

Churchill, R. V.

R. V. Churchill, J. W. Brown, Complex Variables and Applications, 5th ed. (McGraw-Hill, 1990), Chap. 2.

Coddington, E. A.

E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, 1984).

Cohn, M.

E. S. Cassedy, M. Cohn, “On the existence of leaky waves due to a line source above a grounded dielectric slab,” IRE Trans. Microwave Theory Tech. 9, 243–247 (1961).
[CrossRef]

Coldren, L. A.

S.-L. Lee, Y. Chung, L. A. Coldren, N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[CrossRef]

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves, 2nd ed. (IEEE, 1991).

Dagli, N.

S.-L. Lee, Y. Chung, L. A. Coldren, N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[CrossRef]

de Sterke, C. M.

Felbacq, D.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, Foundations Of Photonic Crystal Fibres (Imperial College Press, 2005).

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Chap. 4.6.

Furuya, K.

Y. Suematsu, K. Furuya, “Quasi-guided modes and related radiation losses in optical dielectric waveguides with external higher index surroundings,” IEEE Trans. Microwave Theory Tech. 23, 170–175 (1975).
[CrossRef]

Gérard, J.-M.

J. M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, A. Tchelnokov, Photonic Crystals: towards Nanoscale Photonic Devices (Springer, 2005).

Gloge, D.

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

Guenneau, S.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, Foundations Of Photonic Crystal Fibres (Imperial College Press, 2005).

Guo, S.

Hagness, S. C.

A. Taflove, S. C. Hagness, Computational Electrodynamics, 2nd ed (Artech House, 2000).

Hall, D. B.

D. B. Hall, C. Yeh, “Leaky waves in a heteroepitaxial film,” J. Appl. Phys. 44, 2271–2274 (1973).
[CrossRef]

Haus, H.

H. Haus, D. Miller, “Attenuation of cutoff modes and leaky modes of dielectric slab structures,” IEEE J. Quantum Electron. 22, 310–318 (1986).
[CrossRef]

Hernandez-Marco, J.

Hessel, A.

S. Barone, A. Hessel, “Leaky wave contributions to the field of a line source above a dielectric slab—part II,” Report R-698-58, PIB-626 (Microwave Research Institute, Polytechnic Institute of Brooklyn, Dec. 1958).

Hodgkinson, I. J.

Hoffman, K.

K. Hoffman, R. Kunze, Linear Algebra, 2nd ed. (Prentice Hall, 1971), Chap. 8.5.

Hong, C.-S.

Hsu, T. I.

A. A. Oliner, S. T. Peng, T. I. Hsu, A. Sanchez, “Guidance and leakage properties of a class of open dielectric waveguides. II. New physical effects,” IEEE Trans. Microwave Theory Tech. MTT-29, 855–869 (1981).
[CrossRef]

Hsue, C. W.

Hu, J.

James, J. R.

J. R. James, “Leaky waves on a dielectric rod,” Electron. Lett. 5, 252–254 (1969).
[CrossRef]

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).

Joannopoulos, J.

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).

Johnson, S.

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).

Kawakami, S.

S. Kawakami, S. Nishida, “Perturbation theory of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. 11, 130–138 (1975).
[CrossRef]

S. Kawakami, S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. 10, 879–887 (1974).
[CrossRef]

Kawano, K.

K. Kawano, T. Kitoh, Introduction to Optical Waveguide Analysis Solving Maxwell’s Equations and the Schrödinger Equation (Wiley, 2001).
[CrossRef]

Kitoh, T.

K. Kawano, T. Kitoh, Introduction to Optical Waveguide Analysis Solving Maxwell’s Equations and the Schrödinger Equation (Wiley, 2001).
[CrossRef]

Klaus, W.

Koshiba, M.

Kou, F. Y.

T. Tamir, F. Y. Kou, “Varieties of leaky waves and their excitation along multilayered structures,” IEEE J. Quantum Electron. 22, 544–551 (1986).
[CrossRef]

Kuhlmey, B.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, Foundations Of Photonic Crystal Fibres (Imperial College Press, 2005).

Kuhlmey, B. T.

Kunze, R.

K. Hoffman, R. Kunze, Linear Algebra, 2nd ed. (Prentice Hall, 1971), Chap. 8.5.

Lee, S.-L.

S.-L. Lee, Y. Chung, L. A. Coldren, N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[CrossRef]

Leeb, W. R.

Levinson, N.

E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, 1984).

Lourtioz, J. M.

J. M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, A. Tchelnokov, Photonic Crystals: towards Nanoscale Photonic Devices (Springer, 2005).

Love, J. D.

Maeda, M.

Marcatili, E. A.

E. A. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[CrossRef]

Marcuse, D.

D. Marcuse, “Solution of the vector wave equation for general dielectric waveguides by the Galerkin method,” IEEE J. Quantum Electron. 28, 459–465 (1992).
[CrossRef]

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

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1972), Chap. 8.4.

Marcuvitz, N.

N. Marcuvitz, “On field representations in terms of leaky modes or eigenmodes,” IRE Trans. Antennas Propag. 4, 192–194 (1956).
[CrossRef]

Maystre, D.

McPhedran, R. C.

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).

Menyuk, C. R.

Miller, D.

H. Haus, D. Miller, “Attenuation of cutoff modes and leaky modes of dielectric slab structures,” IEEE J. Quantum Electron. 22, 310–318 (1986).
[CrossRef]

Miller, S. E.

S. E. Miller, “Integrated optics: an introduction,” Bell Syst. Tech. J. 48, 2059–2069 (1969).
[CrossRef]

Mitchell, D. J.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Chap. 4.6.

Nicolet, A.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, Foundations Of Photonic Crystal Fibres (Imperial College Press, 2005).

Nishida, S.

S. Kawakami, S. Nishida, “Perturbation theory of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. 11, 130–138 (1975).
[CrossRef]

S. Kawakami, S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. 10, 879–887 (1974).
[CrossRef]

Okamoto, K.

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G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).

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R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, 1999).

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

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[CrossRef]

M. N. O. Sadiku, Numerical Techniques in Electromagnetics, 2nd ed. (CRC Press, 2001).

A. Taflove, S. C. Hagness, Computational Electrodynamics, 2nd ed (Artech House, 2000).

G. H. Golub, C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

Supplementary Material (9)

» Media 1: MOV (571 KB)     
» Media 2: MOV (572 KB)     
» Media 3: MOV (572 KB)     
» Media 4: MOV (5732 KB)     
» Media 5: MOV (577 KB)     
» Media 6: MOV (5780 KB)     
» Media 7: MOV (5137 KB)     
» Media 8: MOV (1066 KB)     
» Media 9: ZIP (55 KB)     

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Figures (30)

Fig. 1
Fig. 1

The refractive index profile for (a) a three-layer waveguide and (b) a W-type waveguide.

Fig. 2
Fig. 2

Comparison of (a) guided modes, (b) radiation modes, and (c) leaky modes in a one-dimensional waveguide. The solid curves show the mode power outside a center region, which depends on the details of the waveguide index variation.

Fig. 3
Fig. 3

The normalized spectral power density P ( k x ) = | A ̃ ( k x ) | 2 max [ | A ̃ ( k x ) | 2 ] and the real part of n eff as a function of k x .

Fig. 4
Fig. 4

Wave propagation in a uniform medium. Light is injected into a uniform medium at z = 0 . The movie (Media 1) shows the real part of the electric field.

Fig. 5
Fig. 5

I norm = | A ( z , x = 0 ) | 2 | A ( z = 0 , x = 0 ) | 2 as a function of z λ for a Gaussian beam. Blue circles represent the complete solution, while the red solid curve represents the lowest-order asymptotic approximation.

Fig. 6
Fig. 6

The normalized spectral power density P ( k x ) = | A ̃ ( k x ) | 2 max [ | A ̃ ( k x ) | 2 ] and the Re ( n eff ) as a function of k x .

Fig. 7
Fig. 7

Wave propagation in a three-layer waveguide. The light is injected into the waveguide at z = 0 . The movie (Media 2) shows the real part of the electric field. The black dashed lines indicate x = ± a .

Fig. 8
Fig. 8

I norm = | A ( z , x = 0 ) | 2 | A ( z = 0 , x = 0 ) | 2 as a function of z λ for a Gaussian beam. The blue circles represent the power calculated by solving the propagation equation, Eq. (4), by using the complete decomposition, while the red dashed curve represents the sum of the asymptotic approximation from Eq. (39) and the guided mode contribution.

Fig. 9
Fig. 9

Logarithm of the magnitude of the difference between the left- and the right-hand side of Eq. (42).

Fig. 10
Fig. 10

Schematic illustration of the flux flow.

Fig. 11
Fig. 11

The normalized spectral power density P ( k x ) = | A ̃ ( k x ) | 2 max [ | A ̃ ( k x ) | 2 ] and the real part of effective index Re ( n eff ) as a function of k x k 0 .

Fig. 12
Fig. 12

Wave propagation in a W-type waveguide. Light is injected into the W-type waveguide at z = 0 . The black dashed–dotted lines and black dashed lines indicate x = ± a and x = ± b , respectively. The movie (Media 3) shows the real part of the electric field.

Fig. 13
Fig. 13

Im ( n eff ) as a function of b a . The blue solid curve, green dashed–dotted curve, and red dashed curve represent the leakage loss calculated from the direct determination of the leaky mode solution, the perturbation method, and the determination from the radiation mode solution, respectively.

Fig. 14
Fig. 14

Movie (Media 4) of the transverse mode evolution as the mode propagates along a W-type slab waveguide. The red dashed curve and blue solid curve represent the transverse mode power from the leaky mode and the actual profile that is found by integrating Eq. (4). The black dashed lines indicate x = ± b .

Fig. 15
Fig. 15

I norm = | A ( z , x = 0 ) | 2 | A ( z = 0 , x = 0 ) | 2 as a function of z λ for a Gaussian beam with b a = 2.5 . The red dashed curve shows the power of the field obtained by using numerical integration. The green dashed–dotted and green solid curves show, respectively, the steepest descent analysis for the evolution at x = 0 and the leaky mode evolution. The blue solid curve shows the I norm that is calculated by summing the fields from the steepest descent analysis that were used to produce I and II.

Fig. 16
Fig. 16

The refractive index profile for (a) an infinite periodic structure, (b) an infinite periodic structure with a center defect, and (c) a leaky bandgap waveguide.

Fig. 17
Fig. 17

Band structure as a function of normalized frequency and propagation constant. The dark areas are the allowed bands.

Fig. 18
Fig. 18

Refractive index and real part of E ( x ) as a function of x.

Fig. 19
Fig. 19

The real part of n eff and the normalized coefficient P ( K x ) = | A ̃ ( K x ) | 2 max [ | A ̃ ( K x ) | 2 ] with (a) a linear scale and (b) a logarithmic scale as a function of K x .

Fig. 20
Fig. 20

Wave propagation in a leaky bandgap waveguide. A beam is injected into the waveguide at z = 0 . The movie (Media 5) shows the real part of the electric field. The black dashed–dotted lines and black dashed lines indicate x = ± d and x = ± ( d + M Λ ) , respectively.

Fig. 21
Fig. 21

Movie (Media 6) of the transverse mode evolution as it propagates along a leaky bandgap slab waveguide. The red dashed curve and blue solid curve represent the power of the leaky mode and the computational solution of Eq. (4). The boundary lines are not shown in this figure, since they are very close to the center, as shown in Fig. 20.

Fig. 22
Fig. 22

Im ( n eff ) as a function of the number of periods. The blue solid curve, green dashed–dotted curve, and red dashed curve represent the leakage loss calculated from the direct determination of the leaky mode solution, the perturbation method, and the determination from the radiation mode solution, respectively.

Fig. 23
Fig. 23

Refractive index profile for a W-type waveguide with absorbing layers.

Fig. 24
Fig. 24

Im ( n eff ) = Im ( β ) k 0 as a function of L λ . The blue dashed–dotted curve, dashed curve, and dotted curve show the results with normalized absorbing layer widths d λ = 5 , 10 , 15 , respectively. The red solid line shows the results from Eq. (42).

Fig. 25
Fig. 25

Im ( n eff ) = Im ( β ) k 0 as a function of L λ . The blue dashed–dotted curve, dashed curve, and dotted curve show the results with s = 5 , 2 , 1 , respectively. The red solid line shows the results from Eq. (42).

Fig. 26
Fig. 26

Average value of Im ( n eff ) and the standard deviation for 100 evenly spaced values of L λ as we allow it to vary from 5 to 10.

Fig. 27
Fig. 27

Re ( n eff ) and Im ( n eff ) for all 500 eigenmodes.

Fig. 28
Fig. 28

Movie (Media 7) of the transverse normalized power of the same initial Gaussian beam that we considered in Subsection 4.4. We also show the power of the leaky mode as a red dashed curve. We have normalized the peak of the mode power profiles to 1. The black dashed–dotted lines, black dashed line, and black dotted lines indicate x = ± a , x = ± b , and x = ± L , respectively.

Fig. 29
Fig. 29

I norm = | A ( z , x = 0 ) | 2 | A ( z = 0 , x = 0 ) | 2 as a function of z λ for a Gaussian beam. The red solid curve shows the result keeping all 500 modes, while the blue dashed curve shows the result keeping only the leaky mode.

Fig. 30
Fig. 30

Slide show (Media 8) for the input wave (blue solid curves) and its decomposition into the eigenmodes (red dashed curves). In (b), we show the central region from (a). The black dashed–dotted lines, black dashed line, and black dotted lines indicate x = ± a , x = ± b , and x = ± L , respectively.

Equations (77)

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2 A ( z , x ) z 2 + 2 A ( z , x ) x 2 + k 0 2 n 2 ( x ) A ( z , x ) = 0 ,
A ( z , x ) = E ( x ) exp ( i β z i ω t ) ,
d 2 E ( x ) d x 2 + [ k 0 2 n 2 ( x ) β 2 ] E ( x ) = 0 ,
A ( z = 0 , x ) = l = 1 N A ̃ l E l ( x ) + 1 π 0 A ̃ e ( k x ) E e ( k x , x ) d k x + 1 π 0 A ̃ o ( k x ) E o ( k x , x ) d k x ,
E l ( x ) E m ( x ) d x = δ l m ,
E e ( k x , x ) E e ( k x , x ) d x = π δ ( k x k x ) ,
E o ( k x , x ) E o ( k x , x ) d x = π δ ( k x k x ) ,
A ̃ l = A ( z = 0 , x ) E l ( x ) d x ,
A ̃ e ( k x ) = A ( z = 0 , x ) E e ( k x , x ) d x ,
A ̃ o ( k x ) = A ( z = 0 , x ) E o ( k x , x ) d x .
A ( z = 0 , x ) = l = 1 N A ̃ l E l ( x ) + 1 2 π A ̃ ( k x ) E ( k x , x ) d k x ,
A ̃ ( k x ) = A ( z = 0 , x ) E * ( k x , x ) d x .
A ( z , x ) = l = 1 N A ̃ l E l ( x ) exp ( i β l z ) + 1 2 π A ̃ ( k x ) E ( k x , x ) exp [ i β ( k x ) z ] d k x ,
A ( z = 0 , x ) = l = 1 A ̃ l ( x ) E l ( x ) ,
A ( z , x ) = l = 1 A ̃ l ( x ) E l ( x ) exp ( i β l z ) .
f ( x ) { d 2 d x 2 + [ k 0 2 n ( x ) β 2 ] } g ( x ) d x = g ( x ) { d 2 d x 2 + [ k 0 2 n ( x ) β 2 ] } f ( x ) d x
A ̃ l = A ( z = 0 , x ) E l ( x ) d x .
E l 2 ( x ) d x = 1 ,
A ( z , x ) = 1 2 π A ̃ ( k x ) exp [ i k x x + i β ( k x ) z ] d k x ,
A ̃ ( k x ) = A 0 exp ( x 2 2 w 2 ) exp ( i k x x ) d x = 2 π w A 0 exp ( k x 2 w 2 2 ) .
A ( z , x ) = w A 0 exp ( i k 0 n 0 z ) 2 π exp [ 1 2 ( w 2 + i z k 0 n 0 ) k x 2 + i k x x ] d k x = w A 0 exp ( i k 0 n 0 z ) ( w 2 + i z k 0 n 0 ) 1 2 exp [ x 2 2 ( w 2 + i z k 0 n 0 ) ] .
| A ( z , x ) | 2 = w 2 A 0 2 ( w 4 + z 2 k 0 2 n 0 2 ) 1 2 exp [ w 2 x 2 ( w 4 + z 2 k 0 2 n 0 2 ) ] .
A ( z , x ) = 1 2 π A ̃ ( k x ) exp [ i k x x + i β ( k x ) z ] d k x = 1 2 π A ̃ ( k x ) exp ( i φ ) d k x .
A ( z , x ) = 1 2 π exp [ i ( 1 + x 2 z 2 ) 1 2 k 0 n 0 z ] A ̃ ( k x ) exp [ i 2 ( 1 + x 2 z 2 ) 3 2 z k 0 n 0 ( k x k s ) 2 ] d k x .
A ( z , x ) = 1 2 π exp [ i ( 1 + x 2 z 2 ) 1 2 k 0 n 0 z ] A ̃ [ x ( x 2 + z 2 ) 1 2 k 0 n 0 ] exp [ i 2 ( 1 + x 2 z 2 ) 3 2 z k 0 n 0 ( k x k s ) 2 ] d k x = 1 i 2 π ( 1 + x 2 z 2 ) 3 4 ( k 0 n 0 z ) 1 2 exp [ i ( 1 + x 2 z 2 ) 1 2 k 0 n 0 z ] × A ̃ [ x ( x 2 + z 2 ) 1 2 k 0 n 0 ] .
A ( z , x ) = 1 i 2 π ( k 0 n 0 z ) 1 2 exp ( i k 0 n 0 z ) A ̃ ( x z k 0 n 0 ) ,
| A ( z , x ) | 2 = 1 2 π k 0 n 0 z | A ̃ ( x z k 0 n 0 ) | 2 .
E ( x ) = { C 1 cos ( k c x ) | x | a C 0 exp ( α | x | ) a | x | } ,
C 1 cos ( k c a ) = C 0 exp ( α a ) ,
k c C 1 sin ( k c a ) = α C 0 exp ( α a ) ,
k c tan ( k c a ) = α = [ ( k 0 2 ( n 1 2 n 0 2 ) k c 2 ) ] 1 2 .
1 = E 2 ( x ) d x = C 1 2 [ a + sin ( 2 k c a ) 2 k c + cos 2 ( k c a ) α ] .
E ( x ) = { C 1 sin ( k c x ) 0 x a C 0 exp ( α x ) a x } ,
k c cot ( k c a ) = α = [ k 0 2 ( n 1 2 n 0 2 ) k c 2 ] 1 2 ,
E e ( k x , x ) = { C 1 e cos ( k c x ) 0 x a C 0 e cos ( k x x + φ e ) a x } ,
E o ( k x , x ) = { C 1 o sin ( k c x ) 0 x a C 0 o sin ( k x x + φ o ) a x } ,
E ( k x , x ) = { C 1 exp ( i k c x ) 0 x a C 0 exp ( i k x x ) + D 0 exp ( i k x x ) a x } ,
C 1 exp ( i k c a ) = C 0 exp ( i k x a ) + D 0 exp ( i k x a ) ,
i k c C 1 exp ( i k c a ) = i k x C 0 exp ( i k x a ) i k x D 0 exp ( i k x a ) ,
tan [ k x ( B a ) ] + k c k x tan ( k c a ) = 0 .
tan [ k x ( B a ) ] k c k x cot ( k c a ) = 0 ,
A rad ( z , x ) = 1 2 π A ̃ ( k x ) E ( k x , x ) exp [ i β ( k x ) z ] d k x = 1 2 π A ̃ ( k x ) E ( k x , x ) exp ( i φ ) d k x .
A rad ( z , x = 0 ) = 1 2 π A ̃ ( k x ) E ( k x , 0 ) exp ( i φ ) d k x .
A rad ( z , x = 0 ) = 1 2 π exp [ i k 0 n 0 z ] A ̃ ( 0 ) E ( 0 , 0 ) k x 2 exp ( i 2 z k 0 n 0 k x 2 ) d k x = 1 + i 2 π exp ( i k 0 n 0 z ) A ̃ ( 0 ) E ( 0 , 0 ) ( k 0 n 0 z ) 3 2 .
| A rad ( z , x = 0 ) | 2 = 1 2 π ( k 0 n 0 z ) 3 | A ̃ ( 0 ) E ( 0 , 0 ) | 2 .
E ( x ) = { C cos k x x 0 x a C cos ( k x a ) cosh ( α a + φ ) cosh ( α x + φ ) a x b C cos ( k x a ) cosh ( α b + φ ) cosh ( α a + φ ) exp [ i k x ( x b ) ] b x } ,
k x tan ( k x a ) = α tanh ( α a + φ ) ,
α tanh ( α b + φ ) = i k x .
tan ( k x a ) = α k x tanh [ tanh 1 ( i k x α ) + α ( b a ) ] .
k x = [ ( k 0 n 1 ) 2 ( β 0 + Δ β ) 2 ] 1 2 k x 0 β 0 Δ β k x 0 ,
α = [ ( β 0 + Δ β ) 2 ( k 0 n 0 ) 2 ] 1 2 α 0 + β 0 Δ β α 0 ,
Δ β = 2 exp [ 2 α 0 ( b a ) ] [ i k x 0 2 tan ( k x 0 a ) + α 0 2 ] β 0 M W ,
M W = 2 + ( α 0 k x 0 k x 0 α 0 ) [ i + tan ( k x 0 a ) ] + a ( α 0 i k x 0 ) [ 1 + tan 2 ( k x 0 a ) ] 2 i tan ( k x 0 a ) + { 4 α 0 ( b a ) 4 + 2 i a k x 0 [ 1 + tan 2 ( k x 0 a ) ] + 4 i tan ( k x 0 a ) + 4 i tan ( k x 0 a ) ( b a ) k x 0 α 0 } exp [ 2 α 0 ( b a ) ] .
F ( z , x ) = ( 1 2 i ω ) [ A * ( z , x ) A ( z , x ) A ( z , x ) A * ( z , x ) ] ,
E ( x ) = { C 2 exp ( i k x x ) 0 x a C 1 exp ( α x ) + D 1 exp ( α x ) a x b C 0 exp ( i k x x ) + D 0 exp ( i k x x ) b x } ,
A ̃ ( k x ) = A ( z = 0 , x ) E * ( k x , x ) d x ,
A ̃ ( k x ) E ( k x , x = 0 ) = i k i A ̃ ( k r ) E ( k r , x = 0 ) k x k r i k i + i k i A ̃ ( k r ) E ( k r , x = 0 ) k x + k r + i k i .
A ( z , x = 0 ) = 1 2 π A ̃ ( k x ) E ( k x , x = 0 ) exp [ i β ( k x ) z ] d k x 1 2 k i A ̃ ( k r ) E ( k r , x = 0 ) exp [ i β r z β i z ] + c.c. ,
( E 1 ( x + Λ ) E 2 ( x + Λ ) ) = ( A B C D ) ( E 1 ( x ) E 2 ( x ) ) ,
| A λ B C D λ | = 0 = λ 2 λ ( A + D ) + 1 .
λ ± = A + D 2 ± [ ( A + D 2 ) 2 1 ] 1 2 .
E ( x ) = { C 1 cos ( k c x ) 0 x d C 0 u + ( x ) exp ( i K + x ) + D 0 u ( x ) exp ( i K x ) d x } ,
E ( x ) = { C 1 cos ( k c x ) 0 x d C 0 u + ( x ) exp ( α x ) d x } .
C 1 cos ( k c d ) = C 0 u + ( d ) exp ( α d ) ,
C 1 k c sin ( k c d ) = C 0 [ u + ( d ) α u + ( d ) ] exp ( α d ) .
k c tan ( k c d ) = u + ( d ) α u + ( d ) u + ( d ) ,
E ( x ) = { C 2 cos ( k c x ) 0 < x d C 1 u + ( x ) exp ( α x ) + D 1 u ( x ) exp ( α x ) d x d + M Λ C 0 exp ( i K x x ) d + M Λ x } ,
1 Δ 2 [ E ( x k 1 ) 2 E ( x k ) + E ( x k + 1 ) ] + [ k 0 2 n 2 ( x k ) β 2 ] E ( x k ) = 0 ,
ε = n 0 2 [ 1 + i ( | x | L d ) 2 s ] ,
( M β 2 I ) E = 0 ,
M j , k = δ j 1 , k 2 δ j , k + δ j + 1 , k Δ 2 + k 0 2 n k 2 δ j , k
A ( x k ) = l = 1 N A ̃ l E l ( x k ) ,
( M β 2 I ) E l = 0.
F l T ( M β 2 I ) = 0 ,
A ̃ l = l = 1 N A ( x k ) F j ( x k ) .
A ( z , x k ) = l = 1 N A ̃ l E l ( x k ) exp ( i β l z ) ,
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