Abstract

Two formalisms used in index profile reconstructions are compared. We develop a novel formulation of the WKB inverse method and show that this method is not sufficient to recover the entire index profile in ion-implanted waveguides. A reflectivity calculation involving a matrix formalism solves the problem. The index profile parameters are adjusted to fit experimental mode indices. With this formalism, the reconstruction of the entire index profile is achieved. An application is shown in a BaTiO3 sample.

© 1995 Optical Society of America

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  1. B. Fischer and M. Segev, “Photorefractive waveguides and non-linear mode coupling effects,” Appl. Phys. Lett. 54, 684–686 (1989).
    [Crossref]
  2. M. Zha, D. Fluck, P. Günter, M. Fleuster, and Ch. Buchal, “Two-wave mixing in photorefractive ion-implanted KNbO3planar waveguides at visible and near-infrared wavelengths,” Opt. Lett. 8, 577–579 (1993).
    [Crossref]
  3. K. E. Younden, S. W. James, R. W. Eason, P. J. Chandler, L. Zhang, and P. D. Townsend, “Photorefractive planar waveguides in BaTiO3fabricated by ion-beam implantation,” Opt. Lett. 17, 1509–1511 (1992).
    [Crossref]
  4. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1991), Chap. 35, p. 676.
  5. J. M. White and P. F. Heidrich, “Optical waveguide refractive index profiles determined from measurement of mode indices: a simple analysis,” Appl. Opt. 15, 151–155 (1976).
    [Crossref] [PubMed]
  6. G. B. Hocker and W. K. Burns, “Modes in diffused optical waveguides of arbitrary index profile,” IEEE J. Quantum Electron. QE-11, 270–276 (1975).
    [Crossref]
  7. D. Sarid, “Recovery of the refractive-index profile of an optical waveguide from the measured coupling angles,” Appl. Opt. 19, 1606–1608 (1980).
    [Crossref] [PubMed]
  8. A. N. Kaul and K. Thyagarajan, “Inverse WKB method for refractive index profile estimation of monomode graded index planar optical waveguides,” Opt. Commun. 48, 313–316 (1984).
    [Crossref]
  9. J. Noda, M. Minakata, S. Saito, and N. Uchida, “Precise determination of refractive index and thickness in the Ti-diffused LiNbO3waveguide,” J. Opt. Soc. Am. 68, 1690–1693 (1978).
    [Crossref]
  10. E. M. Conwell, “Modes in optical waveguides formed by diffusion,” Appl. Phys. Lett. 23, 328–329 (1973).
    [Crossref]
  11. P. K. Tien and R. Ulrich, “Theory of prism-film coupler and thin film light guides,” J. Opt. Soc. Am. 60, 1325–1337 (1970).
    [Crossref]
  12. R. Ulrich, “Theory of the prism-film coupler by plane wave analysis,” J. Opt. Soc. Am. 60, 1337–1350 (1970).
    [Crossref]
  13. D. Marcuse, “TE modes of graded index slab waveguide,” IEEE J. Quantum Electron. QE-9, 1000–1006 (1973).
    [Crossref]
  14. P. J. Chandler and F. L. Lama, “A new approach to the determination of planar waveguide profiles by means of a non-stationary mode index calculation,” Opt. Acta 33, 127–143 (1986).
    [Crossref]
  15. F. Abelès, “Recherches sur la propagation des ondes électromagnetiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. 5, 596–640, 706–782 (1950).
  16. P. J. B. Clarricoats and K. B. Chan, “Electromagnetic-wave propagation along radially inhomogeneous dielectric cylinders,” Electron. Lett. 6, 694–695 (1970).
    [Crossref]
  17. Y. Suematsu and K. Furuya, “Propagation mode and scattering loss of a two-dimensional dielectric waveguide with gradual distribution of refractive index,” IEEE Trans. Microwave Theory Tech. MTT20, 524–531 (1972).
    [Crossref]
  18. J. M. Arnold, “Stratification methods in the numerical analysis of optical-waveguide transmission parameters,” Electron. Lett. 13, 660–661 (1977).
    [Crossref]

1993 (1)

M. Zha, D. Fluck, P. Günter, M. Fleuster, and Ch. Buchal, “Two-wave mixing in photorefractive ion-implanted KNbO3planar waveguides at visible and near-infrared wavelengths,” Opt. Lett. 8, 577–579 (1993).
[Crossref]

1992 (1)

1989 (1)

B. Fischer and M. Segev, “Photorefractive waveguides and non-linear mode coupling effects,” Appl. Phys. Lett. 54, 684–686 (1989).
[Crossref]

1986 (1)

P. J. Chandler and F. L. Lama, “A new approach to the determination of planar waveguide profiles by means of a non-stationary mode index calculation,” Opt. Acta 33, 127–143 (1986).
[Crossref]

1984 (1)

A. N. Kaul and K. Thyagarajan, “Inverse WKB method for refractive index profile estimation of monomode graded index planar optical waveguides,” Opt. Commun. 48, 313–316 (1984).
[Crossref]

1980 (1)

1978 (1)

1977 (1)

J. M. Arnold, “Stratification methods in the numerical analysis of optical-waveguide transmission parameters,” Electron. Lett. 13, 660–661 (1977).
[Crossref]

1976 (1)

1975 (1)

G. B. Hocker and W. K. Burns, “Modes in diffused optical waveguides of arbitrary index profile,” IEEE J. Quantum Electron. QE-11, 270–276 (1975).
[Crossref]

1973 (2)

D. Marcuse, “TE modes of graded index slab waveguide,” IEEE J. Quantum Electron. QE-9, 1000–1006 (1973).
[Crossref]

E. M. Conwell, “Modes in optical waveguides formed by diffusion,” Appl. Phys. Lett. 23, 328–329 (1973).
[Crossref]

1972 (1)

Y. Suematsu and K. Furuya, “Propagation mode and scattering loss of a two-dimensional dielectric waveguide with gradual distribution of refractive index,” IEEE Trans. Microwave Theory Tech. MTT20, 524–531 (1972).
[Crossref]

1970 (3)

1950 (1)

F. Abelès, “Recherches sur la propagation des ondes électromagnetiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. 5, 596–640, 706–782 (1950).

Abelès, F.

F. Abelès, “Recherches sur la propagation des ondes électromagnetiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. 5, 596–640, 706–782 (1950).

Arnold, J. M.

J. M. Arnold, “Stratification methods in the numerical analysis of optical-waveguide transmission parameters,” Electron. Lett. 13, 660–661 (1977).
[Crossref]

Buchal, Ch.

M. Zha, D. Fluck, P. Günter, M. Fleuster, and Ch. Buchal, “Two-wave mixing in photorefractive ion-implanted KNbO3planar waveguides at visible and near-infrared wavelengths,” Opt. Lett. 8, 577–579 (1993).
[Crossref]

Burns, W. K.

G. B. Hocker and W. K. Burns, “Modes in diffused optical waveguides of arbitrary index profile,” IEEE J. Quantum Electron. QE-11, 270–276 (1975).
[Crossref]

Chan, K. B.

P. J. B. Clarricoats and K. B. Chan, “Electromagnetic-wave propagation along radially inhomogeneous dielectric cylinders,” Electron. Lett. 6, 694–695 (1970).
[Crossref]

Chandler, P. J.

K. E. Younden, S. W. James, R. W. Eason, P. J. Chandler, L. Zhang, and P. D. Townsend, “Photorefractive planar waveguides in BaTiO3fabricated by ion-beam implantation,” Opt. Lett. 17, 1509–1511 (1992).
[Crossref]

P. J. Chandler and F. L. Lama, “A new approach to the determination of planar waveguide profiles by means of a non-stationary mode index calculation,” Opt. Acta 33, 127–143 (1986).
[Crossref]

Clarricoats, P. J. B.

P. J. B. Clarricoats and K. B. Chan, “Electromagnetic-wave propagation along radially inhomogeneous dielectric cylinders,” Electron. Lett. 6, 694–695 (1970).
[Crossref]

Conwell, E. M.

E. M. Conwell, “Modes in optical waveguides formed by diffusion,” Appl. Phys. Lett. 23, 328–329 (1973).
[Crossref]

Eason, R. W.

Fischer, B.

B. Fischer and M. Segev, “Photorefractive waveguides and non-linear mode coupling effects,” Appl. Phys. Lett. 54, 684–686 (1989).
[Crossref]

Fleuster, M.

M. Zha, D. Fluck, P. Günter, M. Fleuster, and Ch. Buchal, “Two-wave mixing in photorefractive ion-implanted KNbO3planar waveguides at visible and near-infrared wavelengths,” Opt. Lett. 8, 577–579 (1993).
[Crossref]

Fluck, D.

M. Zha, D. Fluck, P. Günter, M. Fleuster, and Ch. Buchal, “Two-wave mixing in photorefractive ion-implanted KNbO3planar waveguides at visible and near-infrared wavelengths,” Opt. Lett. 8, 577–579 (1993).
[Crossref]

Furuya, K.

Y. Suematsu and K. Furuya, “Propagation mode and scattering loss of a two-dimensional dielectric waveguide with gradual distribution of refractive index,” IEEE Trans. Microwave Theory Tech. MTT20, 524–531 (1972).
[Crossref]

Günter, P.

M. Zha, D. Fluck, P. Günter, M. Fleuster, and Ch. Buchal, “Two-wave mixing in photorefractive ion-implanted KNbO3planar waveguides at visible and near-infrared wavelengths,” Opt. Lett. 8, 577–579 (1993).
[Crossref]

Heidrich, P. F.

Hocker, G. B.

G. B. Hocker and W. K. Burns, “Modes in diffused optical waveguides of arbitrary index profile,” IEEE J. Quantum Electron. QE-11, 270–276 (1975).
[Crossref]

James, S. W.

Kaul, A. N.

A. N. Kaul and K. Thyagarajan, “Inverse WKB method for refractive index profile estimation of monomode graded index planar optical waveguides,” Opt. Commun. 48, 313–316 (1984).
[Crossref]

Lama, F. L.

P. J. Chandler and F. L. Lama, “A new approach to the determination of planar waveguide profiles by means of a non-stationary mode index calculation,” Opt. Acta 33, 127–143 (1986).
[Crossref]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1991), Chap. 35, p. 676.

Marcuse, D.

D. Marcuse, “TE modes of graded index slab waveguide,” IEEE J. Quantum Electron. QE-9, 1000–1006 (1973).
[Crossref]

Minakata, M.

Noda, J.

Saito, S.

Sarid, D.

Segev, M.

B. Fischer and M. Segev, “Photorefractive waveguides and non-linear mode coupling effects,” Appl. Phys. Lett. 54, 684–686 (1989).
[Crossref]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1991), Chap. 35, p. 676.

Suematsu, Y.

Y. Suematsu and K. Furuya, “Propagation mode and scattering loss of a two-dimensional dielectric waveguide with gradual distribution of refractive index,” IEEE Trans. Microwave Theory Tech. MTT20, 524–531 (1972).
[Crossref]

Thyagarajan, K.

A. N. Kaul and K. Thyagarajan, “Inverse WKB method for refractive index profile estimation of monomode graded index planar optical waveguides,” Opt. Commun. 48, 313–316 (1984).
[Crossref]

Tien, P. K.

Townsend, P. D.

Uchida, N.

Ulrich, R.

White, J. M.

Younden, K. E.

Zha, M.

M. Zha, D. Fluck, P. Günter, M. Fleuster, and Ch. Buchal, “Two-wave mixing in photorefractive ion-implanted KNbO3planar waveguides at visible and near-infrared wavelengths,” Opt. Lett. 8, 577–579 (1993).
[Crossref]

Zhang, L.

Ann. Phys. (1)

F. Abelès, “Recherches sur la propagation des ondes électromagnetiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. 5, 596–640, 706–782 (1950).

Appl. Opt. (2)

Appl. Phys. Lett. (2)

B. Fischer and M. Segev, “Photorefractive waveguides and non-linear mode coupling effects,” Appl. Phys. Lett. 54, 684–686 (1989).
[Crossref]

E. M. Conwell, “Modes in optical waveguides formed by diffusion,” Appl. Phys. Lett. 23, 328–329 (1973).
[Crossref]

Electron. Lett. (2)

P. J. B. Clarricoats and K. B. Chan, “Electromagnetic-wave propagation along radially inhomogeneous dielectric cylinders,” Electron. Lett. 6, 694–695 (1970).
[Crossref]

J. M. Arnold, “Stratification methods in the numerical analysis of optical-waveguide transmission parameters,” Electron. Lett. 13, 660–661 (1977).
[Crossref]

IEEE J. Quantum Electron. (2)

D. Marcuse, “TE modes of graded index slab waveguide,” IEEE J. Quantum Electron. QE-9, 1000–1006 (1973).
[Crossref]

G. B. Hocker and W. K. Burns, “Modes in diffused optical waveguides of arbitrary index profile,” IEEE J. Quantum Electron. QE-11, 270–276 (1975).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

Y. Suematsu and K. Furuya, “Propagation mode and scattering loss of a two-dimensional dielectric waveguide with gradual distribution of refractive index,” IEEE Trans. Microwave Theory Tech. MTT20, 524–531 (1972).
[Crossref]

J. Opt. Soc. Am. (3)

Opt. Acta (1)

P. J. Chandler and F. L. Lama, “A new approach to the determination of planar waveguide profiles by means of a non-stationary mode index calculation,” Opt. Acta 33, 127–143 (1986).
[Crossref]

Opt. Commun. (1)

A. N. Kaul and K. Thyagarajan, “Inverse WKB method for refractive index profile estimation of monomode graded index planar optical waveguides,” Opt. Commun. 48, 313–316 (1984).
[Crossref]

Opt. Lett. (2)

M. Zha, D. Fluck, P. Günter, M. Fleuster, and Ch. Buchal, “Two-wave mixing in photorefractive ion-implanted KNbO3planar waveguides at visible and near-infrared wavelengths,” Opt. Lett. 8, 577–579 (1993).
[Crossref]

K. E. Younden, S. W. James, R. W. Eason, P. J. Chandler, L. Zhang, and P. D. Townsend, “Photorefractive planar waveguides in BaTiO3fabricated by ion-beam implantation,” Opt. Lett. 17, 1509–1511 (1992).
[Crossref]

Other (1)

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1991), Chap. 35, p. 676.

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Figures (8)

Fig. 1
Fig. 1

Geometry of a four-dielectric lossless medium resulting from the association of a coupling prism and a step-index guide.

Fig. 2
Fig. 2

Principle of index profile stratification into multilayers. Profile shape 1 is representative of infinite barrier waveguides. Curve 2 depicts a typical profile in ion-implanted waveguides.

Fig. 3
Fig. 3

Difference Δnm between the effective indices of a free guide and the system prism coupler + air gap + waveguide as a function of mode number m. Parameters are λ = 633 nm, d = 9 μm, n1 = 2.41, n2 = 1, n3 = 2.31, and np = 2.83 (prism index). (a) Results obtained through Eqs. (1) and (9) for the following air-gap widths: S = 0.05 μm (circles), S = 0.1 μm (squares), S = 0.2 μm (triangles), and S = 0.6 μm (crosses). (b) Results obtained through matrix formalism for the same air-gap values S = 0.05 μm (circles), S = 0.1 μm (squares), S = 0.2 μm (triangles), and S = 0.6 μm (crosses).

Fig. 4
Fig. 4

Reflectivity curves versus the effective indices for four values of the air gap: S = 0.6 μm (curve 1), S = 0.3 μm (curve 2), S = 0.1 μm (curve 3), and S = 0.05 μm (curve 4). The spikes stand for the dark lines observed by the m-lines technique. Parameters are λ = 633 nm, s = 300, nd = 2.32, Δ = 0.65, hf = 9 μm, a = 0.09 μm, and γ = 0.22 μm. The contrast and the intensity of the dark lines are optimum for one air-gap value equal to 0.1 μm (curve 3).

Fig. 5
Fig. 5

Difference δnm versus mode number m for S = 0.1 μm (circles), S = 0.3 μm (triangles), and S = 0.6 μm (squares). The index values correspond to the reflectivity spikes shown in Fig. 4.

Fig. 6
Fig. 6

Index profile parameters of mode index curves obtained by varying (a) parameter hf, (b) minimum peak index nd, (c) maximum index (nd2 + Δ2)1/2, (d) profile index stiffness a, and (e) Gaussian trailing edge γ. The modes are discrete points (m is an integer), but for easier understanding continuous representation is used.

Fig. 7
Fig. 7

Comparison of index profiles obtained by the WKB inverse method (squares) and its fit to a Fermi function (dashed curve) whose parameters are nd = 2.319, Δ = 0.655, hf = 8.85 μm, and a = 0.378 μm. The profile obtained by the matrix formalism is shown as a thick solid curve with the parameters nd = 2.319, Δ = 0.6495, hf = 9.28 μm, a = 0.1 μm, and γ = 0.17 μm. Only this last profile describes the entire index variation. Its shape is more abrupt and is in better agreement with the implantation profile (thin solid curve).

Fig. 8
Fig. 8

Experimental effective indices (squares) for BaTiO3 implanted with 1-MeV H+ ions at a dose of 4.6 × 1016 cm−2. The curve shows calculated mode indices deduced by the matrix formalism.

Equations (34)

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2 k d ( n 1 2 - n m 2 ) 1 / 2 - 2 ϕ 1 / 2 - 2 ϕ 1 / 3 = 2 m π ,             1 m M ,
n m 2 = n 1 2 - λ 2 4 d 2 ( m + 1 ) 2 .
2 z 0 z m [ k 2 n 2 ( z ) - β 2 ] 1 / 2 d z - 2 ϕ 1 / 2 - 2 ϕ 1 / 3 = 2 m π ,             1 m M ,
k = 1 k = m z k - 1 z k { [ n ( z ) + n m ] 1 / 2 [ n ( z ) - n m ] 1 / 2 } d z = k = 1 k = m z k - 1 z k { ( n k - 1 + n k 2 + n m ) × [ n k - n m + ( n k - 1 - n k z k - z k - 1 ) ( z k - z ) ] } 1 / 2 d z ,
k = 1 k = m z k - 1 z k { [ n k + ( n k - 1 - n k z k - z k - 1 ) ( z k - z ) ] 2 - n m 2 } 1 / 2 d z ,
z m = z m - 1 + 2 ( n m - 1 - n m ) / n m 2 { ln 1 n m - 1 n m + [ ( n m - 1 n m ) 2 - 1 ] 1 / 2 } + n m - 1 n m [ ( n m - 1 n m ) 2 - 1 ] 1 / 2 [ λ 4 π [ 2 m π + 2 ( ϕ 1 / 2 + ϕ 1 / 3 ) - 2 π ] - n m 2 2 k = 1 k = m = 1 ( z k - z k - 1 n k - 1 - n k ) ( ln { n k n m + [ ( n k n m ) 2 - 1 ] 1 / 2 n k - 1 n m + [ ( n k - 1 n m ) 2 - 1 ] 1 / 2 } + n k - 1 n m [ ( n k - 1 n m ) 2 - 1 ] 1 / 2 - n k n m [ ( n k n m ) 2 - 1 ] 1 / 2 ) ] ,
z 1 = λ π ( ϕ 1 / 2 + ϕ 1 / 3 ) ( n 0 - n 1 ) n 1 2 × 1 ln { 1 n 0 n 1 + [ ( n 0 n 1 ) 2 - 1 ] 1 / 2 } + n 0 n 1 [ ( n 0 n 1 ) 2 - 1 ] 1 / 2 .
n 2 ( z ) = n d 2 + Δ 2 1 + exp ( z - h f a ) ,
Δ 2 - n m d 2 ln ( Δ 2 - n m d 2 + n Fm Δ 2 - n m d 2 - n Fm ) - 2 n m d arctan ( n Fm n m ) = λ 8 a ( 4 m + 3 ) ,             0 m M - 1 ,
2 k d n 1 2 - n m 2 - 2 ϕ 1 / 3 - 2 ϕ 1 / 2 = 2 m π ,             1 m M ,
ϕ 1 / 2 = ϕ 1 / 2 + exp ( - 2 k S n m 2 - n 2 2 ) sin 2 ϕ 1 / 2 cos 2 ϕ p / 2 .
( 2 z 2 + 2 x 2 ) U ( x , z , t ) = n 2 ( z ) c 2 2 U ( x , z , t ) t 2 ,
U ( x , z , t ) = E ( x , z , t ) = e ( z ) exp ( i β x ) exp ( - i ω t ) , U ( x , z , t ) = H ( x , z , t ) = h ( z ) exp ( i β x ) exp ( - i ω t ) ,
[ d 2 d z 2 + k 2 n 2 ( z ) - β 2 ] u ( z ) = 0 ,
u j ( z ) = A j exp ( i k j z ) + B j exp ( - i k j z ) ,
[ A j B j ] = [ M j ] [ A j + 1 B j + 1 ] ,
[ M j ] = [ ( 1 + k j + 1 k j b ) ( 2 2 ξ j ) ( 1 - k j + 1 k j b ) ( 2 2 ξ j ) ( 1 - k j + 1 k j b ) ( ξ j 2 ) ( 1 - k j + 1 k j b ) ( ξ j 2 ) ] ,
[ A 1 B 1 ] = [ j = 1 s M j ] [ A s B s ] = [ P 11 P 12 P 21 P 22 ] [ A s B s ] ,
R = | B 1 A 1 | 2 = | P 21 P 11 | 2 .
h j ( z ) = g j e j ( z ) ,
g j = 1 μ 0 c k j k for TE modes , g j = n j 2 μ 0 c k j k for TM modes ,
[ e ( z ) h ( z ) ] = [ N j ] [ e ( z + l j ) h ( z + l j ) ] ,
[ N j ] = [ cos ( β j ) i g j sin ( β j ) i g j sin ( β j ) cos ( β j ) ] = [ n 11 j n 12 j n 21 j n 22 j ] , β j = k j l j .
r = B 1 A 1 = g 0 n 11 + g 0 g n + 1 n 12 - n 21 - g n + 1 n 22 g 0 n 11 + g 0 g n + 1 n 12 + n 21 + g n + 1 n 22 = numerator denominator ,
[ N ] = [ j = 1 s N j ] = [ n 11 n 12 n 21 n 22 ] .
R = r 2 .
[ N j ] = [ cos ( k j l j ) i μ 0 c k k j sin ( k j l j ) i k j μ 0 c k sin ( k j l j ) cos ( k j l j ) ] .
[ N j ] = [ ch ( k j l j ) i μ 0 c k k j sh ( k j l j ) - i k j μ 0 ck sh ( k j l j ) ch ( k j l j ) ] .
[ N j ] [ N j + 1 ] = [ N 11 N 12 N 21 N 22 ] ,
n 11 r - k n + 1 c μ 0 ck n 12 c = 0 ,             n 21 c + k n + 1 c μ 0 ck n 22 r = 0.
k 0 r n 11 r - k n + 1 r n 22 r = 0 , k 0 r k n + 1 r μ 0 ck n 12 c - μ 0 ck n 21 c = 0.
k 0 r n 11 r + k n + 1 r n 22 r = 0 , k 0 r k n + 1 r μ 0 ck n 12 c + μ 0 ck n 21 c = 0.
n ( z ) = [ n d 2 + Δ 2 1 + exp ( z - h f a ) ] 1 / 2 , n ( z ) = ( n d - n d 2 + Δ 2 ) exp [ - ( z - h b ) 2 γ ] + n d 2 + Δ 2 ,
h b = h f + a ln ( Δ 2 10 3 2 n d 2 - 1 )

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