Abstract

Leaky planar waveguides are critically important to the operation of present day and future integrated photonic circuits. However, to incorporate these waveguides successfully into practical photonic circuits requires an accurate knowledge of their attenuation and mode profile in operation. In contrast with previous numerical methods for obtaining leaky waveguide characteristics, which usually involve complicated algorithms to solve for the complex roots of boundary conditions, the transverse transmission/reflection (TTR) method presented here provides a straightforward and simple approach by simulating the corresponding coupled-waveguide structure. By adding a high-index layer adjacent to the cover to enable the coupling, the transmission/reflection coefficients are shown to be definitively expressed in the form of a Lorentzian that is directly related to the complex propagation constant of leaky/lossy mode. The TTR method simultaneously determines the mode profile of the leaky/lossy mode via the angle of incidence for resonant transmission/reflection. In the present work, the TTR method is applied to an antiresonant reflection optical waveguide (ARROW), a lossy waveguide structure, and a waveguide structure that is simultaneously leaky and lossy.

© 2009 Optical Society of America

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References

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  1. N. Marcuvitz, “On field representations in terms of leaky modes or eigenmodes,” IRE Trans. Antennas Propag. 4, 192-194 (1956).
    [CrossRef]
  2. T. Tamir and F. Y. Kou, “Variety of leaky waves and their excitation along multilayered structures,” J. Quantum Electron. 22, 544-551 (1986).
    [CrossRef]
  3. E. Anemogiannis and E. N. Glytsis, “Multilayer waveguide: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344-1351 (1992).
    [CrossRef]
  4. R. E. Smith and S. N. Houde-Walter, “Leaky guiding in nontransparent waveguides,” J. Opt. Soc. Am. A 12, 715-724 (1995).
    [CrossRef]
  5. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures,” J. Lightwave Technol. 12, 2080-2084 (1994).
    [CrossRef]
  6. M. Koshiba, H. Kumagami, and M. Suzuki, “Finite-element solution of planar arbitrarily anisotropic diffused optical waveguide,” J. Lightwave Technol. 3, 773-778 (1985).
    [CrossRef]
  7. A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. 5, 660-667 (1987).
    [CrossRef]
  8. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929-941 (1999).
    [CrossRef]
  9. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).
  10. P. Yeh, Optical Waves in Layered Media (Wiley, 1988).
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  13. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1977).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2008 (1)

1999 (1)

1995 (1)

1994 (1)

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures,” J. Lightwave Technol. 12, 2080-2084 (1994).
[CrossRef]

1992 (3)

E. Anemogiannis and E. N. Glytsis, “Multilayer waveguide: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344-1351 (1992).
[CrossRef]

T. Baba and Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides--numerical results and analytical expressions,” IEEE J. Quantum Electron. 28, 1689-1700 (1992).
[CrossRef]

J. Xia, A. K. Jordan, and J. A. Kong, “Inverse-scattering view of modal structures in inhomogeneous optical waveguides,” J. Opt. Soc. Am. A 9, 740-748 (1992).
[CrossRef]

1987 (1)

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. 5, 660-667 (1987).
[CrossRef]

1986 (2)

M. A. Duguay, Y. Kokubun, and T. L. Koch, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49, 13-15 (1986).
[CrossRef]

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

1985 (1)

M. Koshiba, H. Kumagami, and M. Suzuki, “Finite-element solution of planar arbitrarily anisotropic diffused optical waveguide,” J. Lightwave Technol. 3, 773-778 (1985).
[CrossRef]

1975 (1)

R. T. Kersten, “The prism-film coupler as a precision instrument Part I. Accuracy and capabilities of prism coupler as instruments,” Opt. Acta 22, 503-513 (1975).
[CrossRef]

1973 (1)

1971 (1)

1970 (2)

1956 (1)

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

Anemogiannis, E.

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929-941 (1999).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures,” J. Lightwave Technol. 12, 2080-2084 (1994).
[CrossRef]

E. Anemogiannis and E. N. Glytsis, “Multilayer waveguide: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344-1351 (1992).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1977).

Baba, T.

T. Baba and Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides--numerical results and analytical expressions,” IEEE J. Quantum Electron. 28, 1689-1700 (1992).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1977).

Cardin, J.

Duguay, M. A.

M. A. Duguay, Y. Kokubun, and T. L. Koch, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49, 13-15 (1986).
[CrossRef]

Gaylord, T. K.

Ghatak, A. K.

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. 5, 660-667 (1987).
[CrossRef]

Glytsis, E. N.

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929-941 (1999).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures,” J. Lightwave Technol. 12, 2080-2084 (1994).
[CrossRef]

E. Anemogiannis and E. N. Glytsis, “Multilayer waveguide: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344-1351 (1992).
[CrossRef]

Houde-Walter, S. N.

Jordan, A. K.

Kersten, R. T.

R. T. Kersten, “The prism-film coupler as a precision instrument Part I. Accuracy and capabilities of prism coupler as instruments,” Opt. Acta 22, 503-513 (1975).
[CrossRef]

Koch, T. L.

M. A. Duguay, Y. Kokubun, and T. L. Koch, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49, 13-15 (1986).
[CrossRef]

Kokubun, Y.

T. Baba and Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides--numerical results and analytical expressions,” IEEE J. Quantum Electron. 28, 1689-1700 (1992).
[CrossRef]

M. A. Duguay, Y. Kokubun, and T. L. Koch, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49, 13-15 (1986).
[CrossRef]

Kong, J. A.

Koshiba, M.

M. Koshiba, H. Kumagami, and M. Suzuki, “Finite-element solution of planar arbitrarily anisotropic diffused optical waveguide,” J. Lightwave Technol. 3, 773-778 (1985).
[CrossRef]

Kou, F. Y.

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

Kumagami, H.

M. Koshiba, H. Kumagami, and M. Suzuki, “Finite-element solution of planar arbitrarily anisotropic diffused optical waveguide,” J. Lightwave Technol. 3, 773-778 (1985).
[CrossRef]

Leduc, D.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Marcuvitz, N.

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

Shenoy, M. R.

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. 5, 660-667 (1987).
[CrossRef]

Smith, R. E.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Suzuki, M.

M. Koshiba, H. Kumagami, and M. Suzuki, “Finite-element solution of planar arbitrarily anisotropic diffused optical waveguide,” J. Lightwave Technol. 3, 773-778 (1985).
[CrossRef]

Tamir, T.

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

Thyagarajan, K.

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. 5, 660-667 (1987).
[CrossRef]

Tien, P. K.

Torge, R.

Ulrich, R.

Xia, J.

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

Appl. Opt. (3)

Appl. Phys. Lett. (1)

M. A. Duguay, Y. Kokubun, and T. L. Koch, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49, 13-15 (1986).
[CrossRef]

IEEE J. Quantum Electron. (1)

T. Baba and Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides--numerical results and analytical expressions,” IEEE J. Quantum Electron. 28, 1689-1700 (1992).
[CrossRef]

IRE Trans. Antennas Propag. (1)

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

J. Lightwave Technol. (5)

E. Anemogiannis and E. N. Glytsis, “Multilayer waveguide: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344-1351 (1992).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures,” J. Lightwave Technol. 12, 2080-2084 (1994).
[CrossRef]

M. Koshiba, H. Kumagami, and M. Suzuki, “Finite-element solution of planar arbitrarily anisotropic diffused optical waveguide,” J. Lightwave Technol. 3, 773-778 (1985).
[CrossRef]

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” J. Lightwave Technol. 5, 660-667 (1987).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929-941 (1999).
[CrossRef]

J. Opt. Soc. Am. (2)

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

J. Quantum Electron. (1)

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

Opt. Acta (1)

R. T. Kersten, “The prism-film coupler as a precision instrument Part I. Accuracy and capabilities of prism coupler as instruments,” Opt. Acta 22, 503-513 (1975).
[CrossRef]

Other (3)

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1977).

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Figures (12)

Fig. 1
Fig. 1

Geometry of a multilayer structure.

Fig. 2
Fig. 2

Multilayer structure with a high-index layer ( n 0 = n c ) added above the original cover.

Fig. 3
Fig. 3

(a) ARROW structure of Ref. [20]. (b) ARROW structure with an added high-index layer.

Fig. 4
Fig. 4

TTR simulation results of the ARROW structure in Fig. 3b.

Fig. 5
Fig. 5

(a) TTR simulation results of the ARROW structure in Fig. 3b with various d a . (b) Absolute error in the real part of the normalized propagation constant, | Δ β | / k 0 , as a function of d a / λ for some selected TE modes for the ARROW structure. (c) Error in normalized attenuation coefficient, Δ α / k 0 , as a function of d a / λ for some selected TE modes for the ARROW structure.

Fig. 6
Fig. 6

Comparison of exact mode profiles with the field profiles excited at the corresponding resonant angles for (a)  TE 1 , (b)  TE 2 , (c)  TE 3 , and (d)  TE 4 modes of the ARROW structure in Fig. 3b [20].

Fig. 7
Fig. 7

(a) Lossy thin-film waveguide structure. (b) Lossy waveguide structure with an added high-index layer.

Fig. 8
Fig. 8

TTR simulation results of the lossy waveguide structure in Fig. 7b with n I = 1 × 10 5 and 1 × 10 3 . The curve for n I = 1 × 10 3 has been multiplied by 50.

Fig. 9
Fig. 9

(a) Simultaneously leaky and lossy thin-film waveguide structure with a 1 μm thick cladding and a high-index substrate. (b) Leaky and lossy waveguide structure with an added high-index layer.

Fig. 10
Fig. 10

TTR simulation results of the leaky and lossy waveguide structure in Fig. 9b with n I = 1 × 10 5 and 1 × 10 3 . The curve for n I = 1 × 10 3 has been multiplied by 3.33.

Fig. 11
Fig. 11

Comparison of exact mode profiles with the field profiles excited at the corresponding resonant angles for (a)  TE 0 , (b)  TE 1 , and (c)  TE 2 modes of the leaky and lossy waveguide structure in Fig. 9b with n I = 1 × 10 5 .

Fig. 12
Fig. 12

(a) Thin-film waveguide structure. The refractive index of the film and substrate can be complex. (b) Thin-film waveguide structure with an added high-index layer.

Tables (3)

Tables Icon

Table 1 Calculated TE n Leaky Mode Normalized Propagation Constant γ n / k 0 = ( β n j α n ) / k 0 = N n j HWHM n for the ARROW Structure Shown in Fig. 3b [20]

Tables Icon

Table 2 Calculated TE n Leaky Mode Normalized Propagation Constant γ n / k 0 = ( β n j α n ) / k 0 = N n j HWHM n for the Lossy Waveguide Structure Shown in Fig. 7b

Tables Icon

Table 3 Calculated TE n Leaky Mode Normalized Propagation Constant γ n / k 0 = ( β n j α n ) / k 0 = N n j HWHM n for the Leaky and Lossy Waveguide Structure Shown in Fig. 9b

Equations (16)

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E y i ( x ) = A i exp [ κ x , i ( x x i 1 ) ] + B i exp [ + κ x , i ( x x i 1 ) ] ,
( A s B s ) = M j 1 M j 2 M 1 M 0 ( A c B c ) = ( m 11 m 12 m 21 m 22 ) ( A c B c ) ,
M i = 1 2 ( [ 1 + f i κ x , i κ x , i + 1 ] exp ( κ x , i d i ) [ 1 f i κ x , i κ x , i + 1 ] exp ( κ x , i d i ) [ 1 f i κ x , i κ x , i + 1 ] exp ( κ x , i d i ) [ 1 + f i κ x , i κ x , i + 1 ] exp ( κ x , i d i ) ) for     i = 0 , , j 1 ,
( t s 0 ) = ( m 11 m 12 m 21 m 22 ) ( 1 r c ) ,
( r s 1 ) = ( m 11 m 12 m 21 m 22 ) ( 0 t c ) ,
r c = m 21 m 22 , r s = m 12 m 22 , t c = 1 m 22 , t s = m 11 m 22 m 12 m 21 m 22 .
R c = | r c | 2 , R s = | r s | 2 .
T c = T s = | t c | 2 n c cos ( θ c ) n s cos ( θ s ) = | t s | 2 n s cos ( θ s ) n c cos ( θ c ) = | t c t s | ,
T = | t c | 2 n c cos ( θ c ) n s cos ( θ s ) = | 1 m 22 | 2 n c cos ( θ c ) n s cos ( θ s ) S ( β ) | 1 l = 1 M p ( β γ l ) | 2 n c cos ( θ c ) n s cos ( θ s ) ,
M = ( m 11 m 12 m 21 m 22 ) ,
M = 1 / 2 ( m 11 m 12 m 21 m 22 ) ( ( 1 + f i = 0 κ c κ c ) exp ( κ c d c ) ( 1 f i = 0 κ c κ c ) exp ( κ c d c ) ( 1 f i = 0 κ c κ c ) exp ( κ c d c ) ( 1 + f i = 0 κ c κ c ) exp ( κ c d c ) ) = 1 / 2 ( m 12 ( 1 f i = 0 κ c κ c ) exp ( κ c d c ) + m 11 ( 1 + f i = 0 κ c κ c ) exp ( κ c d c ) m 12 ( 1 + f i = 0 κ c κ c ) exp ( κ c d c ) + m 11 ( 1 f i = 0 κ c κ c ) exp ( κ c d c ) m 22 ( 1 f i = 0 κ c κ c ) exp ( κ c d c ) + m 21 ( 1 + f i = 0 κ c κ c ) exp ( κ c d c ) m 22 ( 1 + f i = 0 κ c κ c ) exp ( κ c d c ) + m 21 ( 1 f i = 0 κ c κ c ) exp ( κ c d c ) ) .
r c = m 22 ( 1 f i = 0 κ c κ c ) exp ( κ c d c ) + m 21 ( 1 + f i = 0 κ c κ c ) exp ( κ c d c ) m 22 ( 1 + f i = 0 κ c κ c ) exp ( κ c d c ) + m 21 ( 1 f i = 0 κ c κ c ) exp ( κ c d c ) ( 1 f i = 0 κ c κ c ) ( 1 + f i = 0 κ c κ c ) ( 1 + 4 m 21 f i = 0 κ c κ c exp ( 2 κ c d c ) m 22 ( 1 f i = 0 2 κ c 2 κ c 2 ) ) exp ( j ϕ ) ( 1 s ( β ) m 21 exp ( 2 κ c d c ) ( β γ l ) ) ,
M i ( β ) = 1 2 ( [ 1 + f i κ x , i κ x , i + 1 ] exp ( κ x , i d i ) [ 1 f i κ x , i κ x , i + 1 ] exp ( κ x , i d i ) [ 1 f i κ x , i κ x , i + 1 ] exp ( κ x , i d i ) [ 1 + f i κ x , i κ x , i + 1 ] exp ( κ x , i d i ) ) for     i = 0 , , j 1 .
M ( β ) = 1 / 4 ( C exp ( κ f d f ) + C ++ exp ( κ f d f ) C + exp ( κ f d f ) + C + exp ( κ f d f ) C + exp ( κ f d f ) + C + exp ( κ f d f ) C ++ exp ( κ f d f ) + C exp ( κ f d f ) ) ,
r c = m 21 m 22 = ( 1 f p κ p / κ c ) ( 1 + f p κ p / κ c ) ( 1 f p 2 κ p 2 / κ c 2 ) C + κ c d c exp ( κ c d c ) + ( 1 + f p κ p / κ c ) 2 C κ c d c exp ( κ c d c ) ( 1 f p 2 κ p 2 / κ c 2 ) C + κ c d c exp ( κ c d c ) + ( 1 f p κ p / κ c ) 2 C κ c d c exp ( κ c d c ) exp ( j ϕ ) β γ l 2 i Δ β γ l ,
M i ( β m j α m ) 1 2 ( [ ( 1 + f i κ x , i κ x , i + 1 ) + f i α m β m κ x , i + 1 2 ( κ x , i κ x , i + 1 κ x , i + 1 κ x , i ) + j ( 1 + f i κ x , i κ x , i + 1 ) α m β m κ x , i d i ] exp ( κ x , i d i ) [ ( 1 f i κ x , i κ x , i + 1 ) f i α m β m κ x , i + 1 2 ( κ x , i κ x , i + 1 κ x , i + 1 κ x , i ) + j ( 1 f i κ x , i κ x , i + 1 ) α m β m κ x , i d i ] exp ( κ x , i d i ) [ ( 1 f i κ x , i κ x , i + 1 ) f i α m β m κ x , i + 1 2 ( κ x , i κ x , i + 1 κ x , i + 1 κ x , i ) j ( 1 f i κ x , i κ x , i + 1 ) α m β m κ x , i d i ] exp ( κ x , i d i ) [ ( 1 + f i κ x , i κ x , i + 1 ) + f i α m β m κ x , i + 1 2 ( κ x , i κ x , i + 1 κ x , i + 1 κ x , i ) j ( 1 + f i κ x , i κ x , i + 1 ) α m β m κ x , i d i ] exp ( κ x , i d i ) ) .

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