Abstract

A new numerical analysis is proposed to investigate accurately the propagation characteristics of a dielectric thin-film waveguide, and its algorithm is presented. In this method, the Rayleigh principle previously used in conventional mode-matching techniques is extended to the Fourier transform of the wave field in the boundary-value problem for an unbounded object. Successful numerical results for dispersion relations and field distributions in a rib waveguide are obtained in detail, accompanied by their error estimations. The propagation characteristics, including field distributions in an optical rib waveguide, are investigated precisely for several geometries, and it is confirmed that the mode-matching method is accurate and effective for numerical analysis, not only with bounded objects, but also with unbounded objects such as thin-film waveguides in integrated optics.

© 1980 Optical Society of America

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References

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  1. E. A. J. Marcatili, “Slab-coupled waveguides,” Bell Syst. Tech. J. 53, 645–674 (1974).
    [Crossref]
  2. K. Petermann, “Theory of single-mode single-material fibers,” AEÜ, Arch, fur Elektron. und Uebertragungstech. Electron. und Commun. 30, 147–153 (1976).
  3. H. Furuta, H. Noda, and A. Ihaya, “Novel optical waveguide for integrated optics,” Appl. Opt. 13, 322–326 (1974).
    [Crossref] [PubMed]
  4. Y. Miyazaki, “Optical Modes in Dielectric Thin Film Fiber with Convex Surface,” Trans. IECE Jpn,  10, 1–6 (1976).
  5. P. M. Pelosi, P. Vandenbulcke, C. D. W. Wilkinson, and R. M. De La Rue, “Propagation characteristics of trapezoidal cross-section ridge optical waveguides: An experimental and theoretical investigation,” Appl. Opt. 17, 1187–1193 (1978).
    [Crossref] [PubMed]
  6. K. Yasuura, “Numerical analysis of thin-film fiber—Rayleigh principle in Fourier transform—,” IOOC’77 A2-3, 25–28 (1977).
  7. H. Ikuno and K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
    [Crossref]
  8. R. F. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problem for periodic surfaces and other scatters,” Radio Sci. 8, 785–796 (1973).
    [Crossref]
  9. K. Yasuura and T. Miyamoto, “Numerical analysis on isotropic elastic waveguides by mode-matching method—I,II,” IEEE Trans. Sonics Ultrason. SU-24, 359–375 (1977).
    [Crossref]
  10. K. Yasuura, “A view of numerical method in diffraction problems,” Progress in Radio Science 1966–1969, edited by W. V. Tilsten and M. Sauzade (URSI, Brussels, 1971) pp. 257–270.
  11. A. P. Calderón, “The multipole expansion of radiation fields,” J. Ration. Mech. Anal. 3, 523–537 (1954).

1978 (1)

1977 (2)

K. Yasuura, “Numerical analysis of thin-film fiber—Rayleigh principle in Fourier transform—,” IOOC’77 A2-3, 25–28 (1977).

K. Yasuura and T. Miyamoto, “Numerical analysis on isotropic elastic waveguides by mode-matching method—I,II,” IEEE Trans. Sonics Ultrason. SU-24, 359–375 (1977).
[Crossref]

1976 (2)

K. Petermann, “Theory of single-mode single-material fibers,” AEÜ, Arch, fur Elektron. und Uebertragungstech. Electron. und Commun. 30, 147–153 (1976).

Y. Miyazaki, “Optical Modes in Dielectric Thin Film Fiber with Convex Surface,” Trans. IECE Jpn,  10, 1–6 (1976).

1974 (2)

1973 (2)

H. Ikuno and K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
[Crossref]

R. F. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problem for periodic surfaces and other scatters,” Radio Sci. 8, 785–796 (1973).
[Crossref]

1954 (1)

A. P. Calderón, “The multipole expansion of radiation fields,” J. Ration. Mech. Anal. 3, 523–537 (1954).

Calderón, A. P.

A. P. Calderón, “The multipole expansion of radiation fields,” J. Ration. Mech. Anal. 3, 523–537 (1954).

De La Rue, R. M.

Furuta, H.

Ihaya, A.

Ikuno, H.

H. Ikuno and K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
[Crossref]

Marcatili, E. A. J.

E. A. J. Marcatili, “Slab-coupled waveguides,” Bell Syst. Tech. J. 53, 645–674 (1974).
[Crossref]

Millar, R. F.

R. F. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problem for periodic surfaces and other scatters,” Radio Sci. 8, 785–796 (1973).
[Crossref]

Miyamoto, T.

K. Yasuura and T. Miyamoto, “Numerical analysis on isotropic elastic waveguides by mode-matching method—I,II,” IEEE Trans. Sonics Ultrason. SU-24, 359–375 (1977).
[Crossref]

Miyazaki, Y.

Y. Miyazaki, “Optical Modes in Dielectric Thin Film Fiber with Convex Surface,” Trans. IECE Jpn,  10, 1–6 (1976).

Noda, H.

Pelosi, P. M.

Petermann, K.

K. Petermann, “Theory of single-mode single-material fibers,” AEÜ, Arch, fur Elektron. und Uebertragungstech. Electron. und Commun. 30, 147–153 (1976).

Vandenbulcke, P.

Wilkinson, C. D. W.

Yasuura, K.

K. Yasuura, “Numerical analysis of thin-film fiber—Rayleigh principle in Fourier transform—,” IOOC’77 A2-3, 25–28 (1977).

K. Yasuura and T. Miyamoto, “Numerical analysis on isotropic elastic waveguides by mode-matching method—I,II,” IEEE Trans. Sonics Ultrason. SU-24, 359–375 (1977).
[Crossref]

H. Ikuno and K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
[Crossref]

K. Yasuura, “A view of numerical method in diffraction problems,” Progress in Radio Science 1966–1969, edited by W. V. Tilsten and M. Sauzade (URSI, Brussels, 1971) pp. 257–270.

AEÜ, Arch, fur Elektron. und Uebertragungstech. Electron. und Commun. (1)

K. Petermann, “Theory of single-mode single-material fibers,” AEÜ, Arch, fur Elektron. und Uebertragungstech. Electron. und Commun. 30, 147–153 (1976).

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, “Slab-coupled waveguides,” Bell Syst. Tech. J. 53, 645–674 (1974).
[Crossref]

IEEE Trans. Antennas Propag. (1)

H. Ikuno and K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
[Crossref]

IEEE Trans. Sonics Ultrason. (1)

K. Yasuura and T. Miyamoto, “Numerical analysis on isotropic elastic waveguides by mode-matching method—I,II,” IEEE Trans. Sonics Ultrason. SU-24, 359–375 (1977).
[Crossref]

IOOC’77 (1)

K. Yasuura, “Numerical analysis of thin-film fiber—Rayleigh principle in Fourier transform—,” IOOC’77 A2-3, 25–28 (1977).

J. Ration. Mech. Anal. (1)

A. P. Calderón, “The multipole expansion of radiation fields,” J. Ration. Mech. Anal. 3, 523–537 (1954).

Radio Sci. (1)

R. F. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problem for periodic surfaces and other scatters,” Radio Sci. 8, 785–796 (1973).
[Crossref]

Trans. IECE Jpn (1)

Y. Miyazaki, “Optical Modes in Dielectric Thin Film Fiber with Convex Surface,” Trans. IECE Jpn,  10, 1–6 (1976).

Other (1)

K. Yasuura, “A view of numerical method in diffraction problems,” Progress in Radio Science 1966–1969, edited by W. V. Tilsten and M. Sauzade (URSI, Brussels, 1971) pp. 257–270.

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Figures (11)

FIG. 1
FIG. 1

Domain Sand contour C.

FIG. 2
FIG. 2

Cross section of a rib waveguide.

FIG. 3
FIG. 3

Dependence on the number of divisions N in calculations of Ωw and βw for the lowest-order mode TEy0, where δ = 1, a/d = 1, and wd = 3.

FIG. 4
FIG. 4

Dependence on the number of divisions M in calculation of Ωw and βw for the lowest-order mode TEy0, where kd = 7, δ = 1, a/d = 1, and wd = 3.

FIG. 5
FIG. 5

Convergence of the mean-square error Ωw and phase constant βw for the TEyn modes when wd is increased, where kd = 7, δ = 1, and solid lines indicate the TEy0 mode, chain lines the TEy1 mode, and dotted lines the TEy2 mode, respectively. (a) mean square error Ωw, (b) phase constant βw.

FIG. 6
FIG. 6

Convergence of a field E x w 1 ( x , y ) at a fixed, point on the cross section when wd is increased, where kd = 7, δ = 1. (a) TEy0 mode (a/d = 1), (b) TEy1 mode (a/d = 3).

FIG. 7
FIG. 7

Dispersion relations of several lower-order modes, where δ = 1, a/d = 3, and the chain line indicates the lowest-order TE mode of the uniform slab waveguide. (a) TEyn, TMyn modes, (b) EHyn, HEyn modes.

FIG. 8
FIG. 8

Dispersion relations of each lowest-order mode when δ and a/d are altered, where the chain line indicates the lowest-order TE mode of the uniform slab waveguide. (a) TEy0 mode (α/d = 1), (b) EHy0, HEy0 modes (δ = 1).

FIG. 9
FIG. 9

Comparision of dispersion relations calculated by three methods in the lowest-order mode TEy0, where δ = 1.

FIG. 10
FIG. 10

Panoramic field distribution of E x w 1 ( x , y ) for the lowest-order TEy0, where kd = 7, a/d = 1, and δ = 1.

FIG. 11
FIG. 11

Field distributions of E x w 1 ( x , 0.5 ) and E x w 1 ( 0 , y ) for the TEy0 mode when δ and a/d are altered in the case of kd = 7.

Equations (34)

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y = ξ ( x ) 0 for a < | x | < , y = ξ ( x ) 0 for | x | a .
ϕ h ( x , y ) = exp [ j h x + j κ ( h ) y ] ,
( x 2 + x 2 ) Ψ ( x , y ) + k 2 Ψ ( x , y ) = 0
Ψ ( x , y ) = 1 2 π ψ ( h ) ϕ h ( x , y ) d h , < x < and y > max ξ ( x ) .
Ψ w ( x , y ) = 1 2 π w w ψ w ( h ) ϕ h ( x , y ) d h , ( x , y ) S ,
Ψ N ( x , y ) = m = N N a m ( N ) ϕ m ( x , y ) ,
lim w ψ w ( h ) = ψ ( h ) .
y = ξ ( x ) = { d δ 1 ( x / a ) 10 + d for | x | a , d for | x | > a ,
E z ( x , y ) = j β E x ( x , y ) x , H x ( x , y ) = 1 ω μ β 2 E x ( x , y ) x y , H y ( x , y ) = 1 ω μ β ( 2 x β 2 ) E x ( x , y ) , H z ( x , y ) = j ω μ E x ( x , y ) y ,
( 2 x 2 + 2 y 2 ) E x i ( x , y ) + ( n i 2 k 2 β 2 ) E x i ( x , y ) = 0 , i = 0 , 1 , 2.
E x w 0 ( x , y ) = 1 2 π w w e j κ 0 ( h ) y e j h x ψ 0 w ( h ) d h ,
E x w 1 ( x , y ) = 1 2 π w w [ e j κ 1 ( h ) y + R TE ( h ) e j κ 1 ( h ) y ] × e j h x ψ 1 w ( h ) d h
R TE ( h ) = κ 1 ( h ) κ 2 ( h ) κ 1 ( h ) + κ 2 ( h ) , κ i ( h ) = n i 2 k 2 β 2 h 2 , i = 0 , 1 , 2
[ E x w i , E y w i , E z w i ] T = 1 2 π w w [ e i x , e i y , e i z ] T e j h x ψ i w ( h ) d h , [ H x w i , H y w i , H z w i ] T = 1 2 π w w [ m i x , m i y , m i z ] T × e j h x ψ i w ( h ) d h , i = 0 , 1
e 0 x = e j κ 0 ( h ) y , e 0 y = 0 , e 0 z = ( h / β ) e 0 x , e 1 x = e j κ 1 ( h ) y + R TE ( h ) e j κ 1 ( h ) y , e 1 y = 0 , e 1 z = ( h / β ) e 1 x , m 0 x = κ 0 ( h ) h ω μ β e 0 x , m 0 y = h 2 + β 2 ω μ β e 0 x , m 0 z = β h m 0 x , m 1 x = κ 1 ( h ) h ω μ β [ e j κ 1 ( h ) y + R TE ( h ) e j κ 1 ( h ) y ] , m 1 x = κ 1 ( h ) h ω μ β [ e j κ 1 ( h ) y + R TE ( h ) e j κ 1 ( h ) y ] , m 1 y = h 2 + β 2 ω μ β e 1 x , m 1 z = β h m 1 x .
H z ( x , y ) = j β H x ( x , y ) x , E x ( x , y ) = 1 ω β 2 H x ( x , y ) x y , E y ( x , y ) = 1 ω β ( 2 x 2 β 2 ) H x ( x , y ) , E z ( x , y ) = j ω H x ( x , y ) y
R TM ( h ) = n 2 2 κ 1 ( h ) n 1 2 κ 2 ( h ) n 2 2 κ 1 ( h ) + n 1 2 κ 2 ( h ) .
( E x w i , E y w i , E z w i ) T = 1 2 π w w [ e i x , e i y , e i z ] T e j h x ϕ i w ( h ) d h , ( H x w i , H y w i , H z w i ) T = 1 2 π w w [ m i x , m i y , m i z ] T e j h x ϕ i w ( h ) d h i = 0 , 1
m 0 x = e 0 x , m 0 y = 0 , m 0 z = ( h / β ) m 0 x , m 1 x = e j κ 1 ( h ) y + R TM ( h ) e j κ 1 ( h ) y , m 1 y = 0 , m 1 z = ( h / β ) m 1 x , e 0 x = κ 0 ( h ) h ω 0 β e 0 x , e 0 y = h 2 + β 2 ω 0 β e 0 x , e 0 z = β h e 0 x , e 1 x = κ 1 ( h ) h ω 1 β [ e j κ 1 ( h ) y + R TM ( h ) e j κ 1 ( h ) y ] , e 1 y = h 2 + β 2 ω 1 β m 1 x , e 1 z = β h e 1 x .
n ˆ × ( E 0 E 1 ) = i ˆ x n y ( E z 0 E z 1 ) i ˆ y n x ( E z 0 E z 1 ) + i ˆ z [ n x ( E y 0 E y 1 ) n y ( E x 0 E x 1 ) ] = 0 , n ˆ × ( H 0 H 1 ) = i ˆ x n y ( H z 0 H z 1 ) i ˆ y n x ( H z 0 H z 1 ) + i ˆ z [ n x ( H y 0 H y 1 ) n y ( H x 0 H x 1 ) ] = 0
n x ( x , ξ ( x ) ) = [ d y / d x 1 + ( d y / d x ) 2 ] y = ξ ( x ) = n x ( ξ ( x ) ) , n y ( x , ξ ( x ) ) = [ 1 1 + ( d y / d x ) 2 ] y = ξ ( x ) = n y ( ξ ( x ) ) .
Ω w = { | n ˆ ( x , ξ ( x ) ) × [ E w 0 ( x , ξ ( x ) ) E w 1 ( x , ξ ( x ) ) ] | 2 + α 2 | n ˆ ( x , ξ ( x ) ) × [ H w 0 ( x , ξ ( x ) ) H w 1 ( x , ξ ( x ) ) ] | 2 } d x
Ω w = 1 ( 2 π ) 2 d x w w d h w w d h e j ( h h ) x × X T * ( h ) ϕ ( h , h , ξ ( x ) , ξ ( x ) ) X ( h ) ,
X ( h ) = [ ψ 1 w , ψ 0 w , ϕ 1 w , ϕ 0 w ] T , ϕ ( h , h , ξ , ξ ) = ϕ 1 h * ( ξ ) ϕ 1 h T ( ξ ) + ϕ 2 h * ( ξ , ξ ) ϕ 2 h T ( ξ , ξ ) + α 2 { ϕ 3 h * ( ξ ) ϕ 3 h T ( ξ ) + ϕ 4 h * ( ξ , ξ ) ϕ 4 h T ( ξ , ξ ) } ϕ 1 h ( ξ ) = [ e 1 z , e 0 z , e 1 z , e 0 z ] T , ϕ 2 h ( ξ , ξ ) = [ n y e 1 x , n y e 0 x , n y e 1 x n x e 1 y , n y e 0 x n x e 0 y ] T , ϕ 3 h ( ξ ) = [ m 1 z , m 0 z , m 1 z , m 0 z ] T , ϕ 4 h ( ξ , ξ ) = [ n y m 1 x n x m 1 y , n y m 0 x n x m 0 y , n y m 1 x , n y m 0 x ] T ,
{ E x w 1 ( 0 , d / 2 ) H x w 1 ( 0 , d / 2 ) } = 1 2 π w w d h ( e j κ 1 ( h ) d / 2 + { R TE R TM } e j κ 1 ( h ) d / 2 ) { ψ 1 w ( h ) ϕ 1 w ( h ) } = 1 for { even mode in TE y or HE y odd mode in TM y or EH y } , { E x w 1 ( 0 , d / 2 ) / x H x w 1 ( 0 , d / 2 ) / x } = π / a for { odd mode in TM y or EH y even mode in TE y or HE y }
Ω ˜ w = Ω w + λ * [ w w X T ( h ) F ( h ) d h C ] + λ [ w w X T * ( h ) F * ( h ) d h C * ] ,
F ( h ) = [ { e 1 x m 1 x } , 0 ] T for { T E y T M y } , F ( h ) = [ { e 1 x 0 } , 0 , { 0 m 1 x } , 0 ] T for { HE y EH y } ,
1 ( 2 π ) 2 d x w w d h e j ( h h ) x ϕ ( h , h , ξ , ξ ) X ( h ) + λ F * ( h ) = 0 , w w X T ( h ) F ( h ) d h = C .
d x e j ( h h ) x ϕ ( h , h , ξ , ξ ) = a a d x e j ( h h ) x × [ ϕ ( h , h , ξ , ξ ) ϕ ( h , h , d , 0 ) ] + d x e j ( h h ) x × ϕ ( h , h , d , 0 ) ,
δ ( h h ) = 1 2 π d x e j ( h h ) x ,
ϕ ( h , h , d , 0 ) X ( h ) + 1 2 π w w M ( h , h ) X ( h ) d h = 2 π λ F * ( h ) ,
M ( h , h ) = a a [ ϕ ( h , h , ξ , ξ ) ϕ ( h , h , d , 0 ) ] × e j ( h h ) x d x .
λ ( w , β , k ) = Ω w , min ( β , k ) / C *
λ ( w , β , k ) / β = 0