Abstract

The method of path integration is applied to the analysis of a model of a graded-index waveguide taper whose refractive index varies with position z along the guide and coordinates x and y transverse to the direction of propagation, as 1 − ½c(z)x2 − ½b2y2. Detailed calculations are presented for the case in which c(z) ∝ 1/z2, which describes the linear taper. Comments are made about tapers corresponding to other forms of c(z). We obtain an exact closed-form solution for the propagator and coupling efficiency of a linearly tapering graded-index waveguide.

© 1991 Optical Society of America

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References

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  1. E. Voges, A. Neyer, “Integrated-optic devices in LiNbO3for optical communication,” J. Lightwave Technol. LT-5, 1229–1238 (1987).
    [Crossref]
  2. R. Srivastava, C. K. Rao, R. V. Ramaswamy, “WKB analysis of planar surface waveguides with truncated index profiles,” J. Lightwave Technol. LT-5, 1605–1608 (1987).
    [Crossref]
  3. W. Mevenkamp, E. Voges, “Modeling and beam propagation analysis of integrated electro-optic devices,” Arch. Elektrotech. Übertragung 40, 289–296 (1986).
  4. D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982)
  5. R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
  6. G. Eichmann, “Quasi-geometrical optics of media with inhomogeneous index of refraction,”J. Opt. Soc. Am. 61, 161–168 (1971).
    [Crossref]
  7. M. Eve, “The use of path integrals in guided wave theory,” Proc. R. Soc. London Ser. A 347, 405–417 (1976).
    [Crossref]
  8. L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981).
  9. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978).
  10. C. C. Constantinou, “The study of paraxial scalar wave propagation in inhomogeneous media using path integrals,” masters thesis (University of Birmingham, Birmingham, UK, 1989).
  11. C. C. Constantinou, R. C. Jones, “Path-integral of an arbitrarily tapered graded-index waveguide: the inverse-square-law and parabolic tapers,” J. Opt. Soc. Am. A (to be published).
  12. D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell Syst. Tech. J. 49, 273–290 (1970); see in particular Fig. 8.
  13. T. J. Cullen, C. D. W. Wilkinson, “Radiation losses from single-mode optical Y junctions formed by silver–ion exchange in glass,” Opt. Lett. 10, 134–136 (1984).
    [Crossref]
  14. H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, New York, 1980).

1987 (2)

E. Voges, A. Neyer, “Integrated-optic devices in LiNbO3for optical communication,” J. Lightwave Technol. LT-5, 1229–1238 (1987).
[Crossref]

R. Srivastava, C. K. Rao, R. V. Ramaswamy, “WKB analysis of planar surface waveguides with truncated index profiles,” J. Lightwave Technol. LT-5, 1605–1608 (1987).
[Crossref]

1986 (1)

W. Mevenkamp, E. Voges, “Modeling and beam propagation analysis of integrated electro-optic devices,” Arch. Elektrotech. Übertragung 40, 289–296 (1986).

1984 (1)

1976 (1)

M. Eve, “The use of path integrals in guided wave theory,” Proc. R. Soc. London Ser. A 347, 405–417 (1976).
[Crossref]

1971 (1)

1970 (1)

D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell Syst. Tech. J. 49, 273–290 (1970); see in particular Fig. 8.

Constantinou, C. C.

C. C. Constantinou, “The study of paraxial scalar wave propagation in inhomogeneous media using path integrals,” masters thesis (University of Birmingham, Birmingham, UK, 1989).

C. C. Constantinou, R. C. Jones, “Path-integral of an arbitrarily tapered graded-index waveguide: the inverse-square-law and parabolic tapers,” J. Opt. Soc. Am. A (to be published).

Cullen, T. J.

Eichmann, G.

Eve, M.

M. Eve, “The use of path integrals in guided wave theory,” Proc. R. Soc. London Ser. A 347, 405–417 (1976).
[Crossref]

Feynman, R. P.

R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

Goldstein, H.

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, New York, 1980).

Hibbs, A. R.

R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

Jones, R. C.

C. C. Constantinou, R. C. Jones, “Path-integral of an arbitrarily tapered graded-index waveguide: the inverse-square-law and parabolic tapers,” J. Opt. Soc. Am. A (to be published).

Marchand, E. W.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978).

Marcuse, D.

D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell Syst. Tech. J. 49, 273–290 (1970); see in particular Fig. 8.

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982)

Mevenkamp, W.

W. Mevenkamp, E. Voges, “Modeling and beam propagation analysis of integrated electro-optic devices,” Arch. Elektrotech. Übertragung 40, 289–296 (1986).

Neyer, A.

E. Voges, A. Neyer, “Integrated-optic devices in LiNbO3for optical communication,” J. Lightwave Technol. LT-5, 1229–1238 (1987).
[Crossref]

Ramaswamy, R. V.

R. Srivastava, C. K. Rao, R. V. Ramaswamy, “WKB analysis of planar surface waveguides with truncated index profiles,” J. Lightwave Technol. LT-5, 1605–1608 (1987).
[Crossref]

Rao, C. K.

R. Srivastava, C. K. Rao, R. V. Ramaswamy, “WKB analysis of planar surface waveguides with truncated index profiles,” J. Lightwave Technol. LT-5, 1605–1608 (1987).
[Crossref]

Schulman, L. S.

L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981).

Srivastava, R.

R. Srivastava, C. K. Rao, R. V. Ramaswamy, “WKB analysis of planar surface waveguides with truncated index profiles,” J. Lightwave Technol. LT-5, 1605–1608 (1987).
[Crossref]

Voges, E.

E. Voges, A. Neyer, “Integrated-optic devices in LiNbO3for optical communication,” J. Lightwave Technol. LT-5, 1229–1238 (1987).
[Crossref]

W. Mevenkamp, E. Voges, “Modeling and beam propagation analysis of integrated electro-optic devices,” Arch. Elektrotech. Übertragung 40, 289–296 (1986).

Wilkinson, C. D. W.

Arch. Elektrotech. Übertragung (1)

W. Mevenkamp, E. Voges, “Modeling and beam propagation analysis of integrated electro-optic devices,” Arch. Elektrotech. Übertragung 40, 289–296 (1986).

Bell Syst. Tech. J. (1)

D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell Syst. Tech. J. 49, 273–290 (1970); see in particular Fig. 8.

J. Lightwave Technol. (2)

E. Voges, A. Neyer, “Integrated-optic devices in LiNbO3for optical communication,” J. Lightwave Technol. LT-5, 1229–1238 (1987).
[Crossref]

R. Srivastava, C. K. Rao, R. V. Ramaswamy, “WKB analysis of planar surface waveguides with truncated index profiles,” J. Lightwave Technol. LT-5, 1605–1608 (1987).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Proc. R. Soc. London Ser. A (1)

M. Eve, “The use of path integrals in guided wave theory,” Proc. R. Soc. London Ser. A 347, 405–417 (1976).
[Crossref]

Other (7)

L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981).

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978).

C. C. Constantinou, “The study of paraxial scalar wave propagation in inhomogeneous media using path integrals,” masters thesis (University of Birmingham, Birmingham, UK, 1989).

C. C. Constantinou, R. C. Jones, “Path-integral of an arbitrarily tapered graded-index waveguide: the inverse-square-law and parabolic tapers,” J. Opt. Soc. Am. A (to be published).

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982)

R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, New York, 1980).

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Figures (3)

Fig. 1
Fig. 1

Refractive-index distribution in the plane y = 0 for the linear taper.

Fig. 2
Fig. 2

Lowest-order mode coupling efficiency for the linear taper, plotted against taper angle ϑ and the ratio d, of the wide to the narrow width of the taper.

Fig. 3
Fig. 3

Lowest-order mode coupling efficiency prediction by Eq. (13), compared with the local normal-mode analysis for a linear taper with d = 2, as a function of the taper half-angle ϑ corresponding to the refractive-index contour n(x, y = 0, z) = 1/1.432.

Equations (24)

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n p ( x , y , z ) = n 0 n ( x , y , z ) = n 0 [ 1 - ½ c ( z ) x 2 - ½ b 2 y 2 ] ,
i k ψ z + 1 2 k 2 ( 2 x 2 + 2 y 2 ) ψ + n ( x , y , z ) ψ = 0 ,
K ( x , y , z ; x 0 , y 0 , z 0 ) = x ( z 0 ) = x 0 x ( z ) = x δ x ( z ) y ( z 0 ) = y 0 y ( z ) = y δ y ( z ) × exp ( i k z 0 z d ζ { 1 2 x ˙ 2 ( ζ ) + 1 2 y ˙ 2 ( ζ ) + n [ x ( ζ ) , y ( ζ ) , ζ ] } ) ,
K ( x , z ; x 0 , z 0 ) = [ k 2 π i f ( z , z 0 ) ] 1 / 2 exp ( i k S GO ) .
2 z 2 f ( z , z 0 ) + c ( z ) f ( z , z 0 ) = 0 ,
f ( z = z 0 , z 0 ) = 0 ,             [ z f ( z , z 0 ) ] z = z 0 = 1.
S GO = 1 2 x 2 z [ ln f ( z , z 0 ) ] - 1 2 x 0 2 z 0 [ ln f ( z , z 0 ) ] - x x 0 f ( z , z 0 ) .
f ( z , z 0 ) = ( z z 0 ) 1 / 2 sin [ q ln ( z / z 0 ) ] q ,
q = ( 8 - tan 2 ϑ ) 1 / 2 2 tan ϑ .
K ( x , z ; x 0 , z 0 ) = { k q 2 π i ( z z 0 ) 1 / 2 sin [ q ln ( z / z 0 ) ] } 1 / 2 × exp [ i k ( x 2 2 z { 1 2 + cot [ q ln ( z / z 0 ) ] } + x 0 2 2 z 0 × { 1 2 - cot [ q ln ( z / z 0 ) ] } - q x x 0 ( z z 0 ) 1 / 2 sin [ q ln ( z / z 0 ) ] ) ] .
ψ ( x , ζ ; z 0 ) = - + d x 0 K ( x , ζ ; x 0 , z 0 ) ψ n ( x 0 , z 0 ) .
C n m ( z , z 0 ) = - + d x ψ m * ( x ) ψ ( x , z ; z 0 ) .
C 00 2 = 1 / ( 1 + { sin [ q ln ( z / z 0 ) ] 2 q } 2 ) 1 / 2 ,
C 02 2 = C 20 2 = { sin [ q ln ( z / z 0 ) ] 2 q } 2 / 2 [ ( 1 + { sin [ q ln ( z / z 0 ) ] 2 q } 2 ) 1 / 2 ] 3 .
C 00 2 = [ 1 / ( 1 + { sin [ q ln ( z / z 0 ) ] 2 q } 2 ) 1 / 2 ] × [ 1 / ( 1 + { sin [ q ln ( z / z 0 ) ] 2 q } 2 ) 1 / 2 ] ,
ϑ max = arcsin [ 8 9 + ( 2 π / ln d ) 2 ] 1 / 2 .
w ( z ) = w ( z 0 ) ( z z 0 ) 1 / 2 { 1 + 1 4 q 2 - 1 2 q sin [ 2 q ln ( z z 0 ) ] - 1 4 q 2 cos [ 2 q ln ( z z 0 ) ] } 1 / 2 ,
1 R ( z ) = 1 z sin 2 [ q ln ( z / z 0 ) ] { 1 - [ 1 / ( 4 q 2 + 1 ) ] cos [ 2 q ln ( z / z 0 ) ] - [ 2 q / ( 4 q 2 + 1 ) ] sin [ 2 q ln ( z / z 0 ) ] } .
K ( x , z ; x 0 z 0 ) = x ( z 0 ) = x 0 x ( z ) = x δ x ( z ) × exp { i k z 0 z d ζ [ 1 2 x ˙ 2 ( ζ ) - 1 2 c ( ζ ) x 2 ( ζ ) ] } .
L ( x , z ) = 1 2 x ˙ 2 ( z ) - 1 2 c ( z ) x 2 ( z )
d 2 x d z 2 ( z ) + c ( z ) x ( z ) = 0
K ( x , z ; x 0 , z 0 ) = exp { i k z 0 z d ζ [ 1 2 X ˙ 2 ( ζ ) - 1 2 c ( ζ ) X 2 ( ζ ) ] } × δ ξ ( z ) exp { i k z 0 z d ζ [ 1 2 ξ ˙ 2 ( ζ ) - 1 2 c ( ζ ) ξ 2 ( ζ ) ] } ,
δ ξ ( z ) exp { i k z 0 z d ζ [ 1 2 ξ ˙ 2 ( ζ ) - 1 2 c ( ζ ) ξ 2 ( ζ ) ] } = lim N - + - + d ξ 1 d ξ N - 1 ( k 2 π i ) N / 2 × exp { i k j = 0 N - 1 [ ( ξ i + 1 - ξ i ) 2 2 - 1 2 c j ξ j 2 ] } ,
K ( x , z ; x 0 , z 0 ) = [ k 2 π i f ( z , z 0 ) ] 1 / 2 × exp { i k z 0 z d ζ [ 1 2 X ˙ 2 ( ζ ) - 1 2 c ( ζ ) X 2 ( ζ ) ] } ,

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