Abstract

We present a model of a graded-index taper for which the field solutions can be obtained directly by separation of variables. Those fields tightly concentrated about the axis of the taper are given very accurately by remarkably simple expressions which clearly illustrate the influence of the tapering. Frequent comparisons with a geometrical optics analysis demonstrate the link between the ray and field approaches and also assist in the physical interpretation of several results. Examples of the application of these field solutions are also described.

© 1987 Optical Society of America

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References

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  1. A. K. Agarwal, “Review of Optical Fiber Couplers,” Fiber Int. Opt. 6, 27 (1986).
    [CrossRef]
  2. T. Ozeki, T. Ito, T. Tamura, “Tapered Section of Multimode Cladded Fibers as Mode Filters and Mode Analysers,” Appl. Phy. Lett. 26, 386 (1975).
    [CrossRef]
  3. E. W. Marchand, Gradient Index Optics (Academic, New York1978).
  4. C. Gomez-Reino, J. Linares, E. Larrea, “Imaging and Transforming Transmission through Tapered Gradient-Index Rods: Analytical Solutions,” J. Opt. Soc. Am. A3, 1604 (1986).
    [CrossRef]
  5. J. C. Minano, “Design of Three-Dimensional Nonimaging Concentrators with Inhomogeneous Media,” J. Opt. Soc. Am. A3, 1345 (1986).
    [CrossRef]
  6. E. A. J. Marcatili, “Dielectric Tapers with Curved Axes and No Loss,” IEEE J. Quantum Electron. QE-21, 307 (1985).
    [CrossRef]
  7. D. Bertilone, C. Pask, “Exact Ray Paths in a Graded-Index Taper,” Appl. Opt. 26, 1159 (1987)
    [CrossRef]
  8. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  9. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1972).
  10. H. Jeffreys, B. S. Jeffreys, Methods of Mathematical Physics (Cambridge U. P., London, 1966).
  11. H. Kogelnik, “Theory of Dielectric Waveguides,” in Integrated Optics, T. Tamir, Ed., Topics in Applied Physics, Vol. 7 (Springer-Verlag, Berlin, 1975).
    [CrossRef]
  12. J. B. Keller, S. I. Rubinow, “Asymptotic Solution of Eigenvalue Problems,” Ann. Phys. 9, 24 (1960).
    [CrossRef]
  13. The Bessel function Jμ(x) has a point of inflection (i.e., changes from concave upward to concave downward) roughly where the argument equals the order (x = μ). In fact, if jμ″ denotes the exact location of the smallest positive zero of d2Jμ(x)/dx2, then μ(μ-1)<jμ″<μ2-1. See, for example, G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., London, 1966). Clearly, as μ gets larger jμ″/μ moves closer and closer to unity.

1987 (1)

D. Bertilone, C. Pask, “Exact Ray Paths in a Graded-Index Taper,” Appl. Opt. 26, 1159 (1987)
[CrossRef]

1986 (3)

A. K. Agarwal, “Review of Optical Fiber Couplers,” Fiber Int. Opt. 6, 27 (1986).
[CrossRef]

C. Gomez-Reino, J. Linares, E. Larrea, “Imaging and Transforming Transmission through Tapered Gradient-Index Rods: Analytical Solutions,” J. Opt. Soc. Am. A3, 1604 (1986).
[CrossRef]

J. C. Minano, “Design of Three-Dimensional Nonimaging Concentrators with Inhomogeneous Media,” J. Opt. Soc. Am. A3, 1345 (1986).
[CrossRef]

1985 (1)

E. A. J. Marcatili, “Dielectric Tapers with Curved Axes and No Loss,” IEEE J. Quantum Electron. QE-21, 307 (1985).
[CrossRef]

1975 (1)

T. Ozeki, T. Ito, T. Tamura, “Tapered Section of Multimode Cladded Fibers as Mode Filters and Mode Analysers,” Appl. Phy. Lett. 26, 386 (1975).
[CrossRef]

1960 (1)

J. B. Keller, S. I. Rubinow, “Asymptotic Solution of Eigenvalue Problems,” Ann. Phys. 9, 24 (1960).
[CrossRef]

Agarwal, A. K.

A. K. Agarwal, “Review of Optical Fiber Couplers,” Fiber Int. Opt. 6, 27 (1986).
[CrossRef]

Bertilone, D.

D. Bertilone, C. Pask, “Exact Ray Paths in a Graded-Index Taper,” Appl. Opt. 26, 1159 (1987)
[CrossRef]

Gomez-Reino, C.

C. Gomez-Reino, J. Linares, E. Larrea, “Imaging and Transforming Transmission through Tapered Gradient-Index Rods: Analytical Solutions,” J. Opt. Soc. Am. A3, 1604 (1986).
[CrossRef]

Ito, T.

T. Ozeki, T. Ito, T. Tamura, “Tapered Section of Multimode Cladded Fibers as Mode Filters and Mode Analysers,” Appl. Phy. Lett. 26, 386 (1975).
[CrossRef]

Jeffreys, B. S.

H. Jeffreys, B. S. Jeffreys, Methods of Mathematical Physics (Cambridge U. P., London, 1966).

Jeffreys, H.

H. Jeffreys, B. S. Jeffreys, Methods of Mathematical Physics (Cambridge U. P., London, 1966).

Keller, J. B.

J. B. Keller, S. I. Rubinow, “Asymptotic Solution of Eigenvalue Problems,” Ann. Phys. 9, 24 (1960).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Theory of Dielectric Waveguides,” in Integrated Optics, T. Tamir, Ed., Topics in Applied Physics, Vol. 7 (Springer-Verlag, Berlin, 1975).
[CrossRef]

Larrea, E.

C. Gomez-Reino, J. Linares, E. Larrea, “Imaging and Transforming Transmission through Tapered Gradient-Index Rods: Analytical Solutions,” J. Opt. Soc. Am. A3, 1604 (1986).
[CrossRef]

Linares, J.

C. Gomez-Reino, J. Linares, E. Larrea, “Imaging and Transforming Transmission through Tapered Gradient-Index Rods: Analytical Solutions,” J. Opt. Soc. Am. A3, 1604 (1986).
[CrossRef]

Love, J. D.

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

Marcatili, E. A. J.

E. A. J. Marcatili, “Dielectric Tapers with Curved Axes and No Loss,” IEEE J. Quantum Electron. QE-21, 307 (1985).
[CrossRef]

Marchand, E. W.

E. W. Marchand, Gradient Index Optics (Academic, New York1978).

Minano, J. C.

J. C. Minano, “Design of Three-Dimensional Nonimaging Concentrators with Inhomogeneous Media,” J. Opt. Soc. Am. A3, 1345 (1986).
[CrossRef]

Ozeki, T.

T. Ozeki, T. Ito, T. Tamura, “Tapered Section of Multimode Cladded Fibers as Mode Filters and Mode Analysers,” Appl. Phy. Lett. 26, 386 (1975).
[CrossRef]

Pask, C.

D. Bertilone, C. Pask, “Exact Ray Paths in a Graded-Index Taper,” Appl. Opt. 26, 1159 (1987)
[CrossRef]

Rubinow, S. I.

J. B. Keller, S. I. Rubinow, “Asymptotic Solution of Eigenvalue Problems,” Ann. Phys. 9, 24 (1960).
[CrossRef]

Snyder, A. W.

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

Tamura, T.

T. Ozeki, T. Ito, T. Tamura, “Tapered Section of Multimode Cladded Fibers as Mode Filters and Mode Analysers,” Appl. Phy. Lett. 26, 386 (1975).
[CrossRef]

Watson, G. N.

The Bessel function Jμ(x) has a point of inflection (i.e., changes from concave upward to concave downward) roughly where the argument equals the order (x = μ). In fact, if jμ″ denotes the exact location of the smallest positive zero of d2Jμ(x)/dx2, then μ(μ-1)<jμ″<μ2-1. See, for example, G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., London, 1966). Clearly, as μ gets larger jμ″/μ moves closer and closer to unity.

Ann. Phys. (1)

J. B. Keller, S. I. Rubinow, “Asymptotic Solution of Eigenvalue Problems,” Ann. Phys. 9, 24 (1960).
[CrossRef]

Appl. Opt. (1)

D. Bertilone, C. Pask, “Exact Ray Paths in a Graded-Index Taper,” Appl. Opt. 26, 1159 (1987)
[CrossRef]

Appl. Phy. Lett. (1)

T. Ozeki, T. Ito, T. Tamura, “Tapered Section of Multimode Cladded Fibers as Mode Filters and Mode Analysers,” Appl. Phy. Lett. 26, 386 (1975).
[CrossRef]

Fiber Int. Opt. (1)

A. K. Agarwal, “Review of Optical Fiber Couplers,” Fiber Int. Opt. 6, 27 (1986).
[CrossRef]

IEEE J. Quantum Electron. (1)

E. A. J. Marcatili, “Dielectric Tapers with Curved Axes and No Loss,” IEEE J. Quantum Electron. QE-21, 307 (1985).
[CrossRef]

J. Opt. Soc. Am. (2)

C. Gomez-Reino, J. Linares, E. Larrea, “Imaging and Transforming Transmission through Tapered Gradient-Index Rods: Analytical Solutions,” J. Opt. Soc. Am. A3, 1604 (1986).
[CrossRef]

J. C. Minano, “Design of Three-Dimensional Nonimaging Concentrators with Inhomogeneous Media,” J. Opt. Soc. Am. A3, 1345 (1986).
[CrossRef]

Other (6)

E. W. Marchand, Gradient Index Optics (Academic, New York1978).

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

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1972).

H. Jeffreys, B. S. Jeffreys, Methods of Mathematical Physics (Cambridge U. P., London, 1966).

H. Kogelnik, “Theory of Dielectric Waveguides,” in Integrated Optics, T. Tamir, Ed., Topics in Applied Physics, Vol. 7 (Springer-Verlag, Berlin, 1975).
[CrossRef]

The Bessel function Jμ(x) has a point of inflection (i.e., changes from concave upward to concave downward) roughly where the argument equals the order (x = μ). In fact, if jμ″ denotes the exact location of the smallest positive zero of d2Jμ(x)/dx2, then μ(μ-1)<jμ″<μ2-1. See, for example, G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., London, 1966). Clearly, as μ gets larger jμ″/μ moves closer and closer to unity.

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Figures (7)

Fig. 1
Fig. 1

Coordinate systems. The origin of the Rθ plane-polar coordinate system is located at the point x = 0, z = −D of the Cartesian coordinate system.

Fig. 2
Fig. 2

Constant refractive-index contours for the graded-index taper defined by Eq. (4). Index values are ordered n0 > n1 > n2 > n3.

Fig. 3
Fig. 3

Plots of Bn(θ) for n = 0 (solid line) and n = 5 (broken line) and with α = 100. (Note that the curves have been nomalized to give a maximum value of unity.) Also shown are θ ˜ 0 and θ ˜ 5 where the second derivative disappears.

Fig. 4
Fig. 4

Schematic drawing of a typical ray trajectory inside the graded-index taper. Ray caustics are shown at R = Rmin and θ = ±θmax, confining the ray to the region inside a truncated wedge. Far from the taper apex the ray tends asymptotically to a straight line, here denoted θ−∞ (as the ray comes in) and θ+∞ (as the ray leaves).

Fig. 5
Fig. 5

Single period of a typical ray path is shown from S0 to S2. Also illustrated is the local plane-wave vector kloc(R,θ) at a point (R,θ) on the trajectory and its radial component kR and transverse-radial component kθ.

Fig. 6
Fig. 6

Schematic illustrations of the examples discussed in Sec. V.A (a), V.B (b), and V.C (c). Also shown are some typical ray trajectories encountered in these cases.

Fig. 7
Fig. 7

Coordinate systems. The origin of the Rθϕ spherical–polar coordinate system is located at the point x = 0, y = 0, z = −D of the Cartesian system. We follow the usual convention: 0 ≤ R < ∞; 0 ≤ θπ;0 ≤ ϕ < 2π.

Equations (72)

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n 2 = n 0 2 [ 1 - 2 Δ G ( θ ) / R 2 ] .
G ( θ ) = sin 2 θ / cos 4 θ .
x = R sin θ ,             z = R cos θ - D .
n 2 = n 0 2 [ 1 - 2 Δ x 2 / ( D + z ) 4 ] .
n 2 = n 0 2 [ 1 - 2 δ x 2 / ρ 2 ( z ) ] ,
ρ ( z ) = ρ 0 ( 1 + z / D ) 2
2 δ = 2 Δ ρ 0 2 / D 2
( 2 + k 2 n 2 ) ψ = 0.
ψ ( R , θ ) = A ( R ) B ( θ ) ,
A ( R ) = { J μ ( k n 0 R ) , Y μ ( k n 0 R ) ,
d 2 B / d θ 2 + [ μ 2 - α 2 G ( θ ) ] B ( θ ) = 0.
α = k n 0 2 Δ ,
d 2 B / d θ 2 + ( μ 2 - α 2 sin 2 θ / cos 4 θ ) B ( θ ) = 0.
G ( θ ) θ 2
d 2 B / d θ 2 + ( μ 2 - α 2 θ 2 ) B ( θ ) = 0.
μ 2 = α ( 2 n + 1 ) ,             n = 0 , 1 , 2 , ,
B n ( θ ) = exp ( - α θ 2 / 2 ) H n ( α θ ) ,
ψ n ( R , θ ) = exp ( - α θ 2 / 2 ) H n ( α θ ) { J α ( 2 n + 1 ) ( k n 0 R ) , Y α ( 2 n + 1 ) ( k n 0 R ) .
n 2 = n 0 2 ( 1 - 2 δ x 2 / ρ 0 2 )
ψ n ( inf . par . ) = exp ( - q x 2 / 2 ρ 0 ) H n [ ( q / ρ 0 ) 1 / 2 x ] exp ( ± i β n z ) n = 0 , 1 , 2 , ,
β n = [ k 2 n 0 2 - ( q / ρ 0 ) ( 2 n + 1 ) ] 1 / 2
q = k n 0 2 δ .
R D + z , θ x / ( D + z ) ,
ψ n ( x , z ) = exp [ - q x 2 / 2 ρ ( z ) ] H n { [ q / ρ ( z ) ] 1 / 2 x } × { J D q ( 2 n + 1 ) / ρ 0 [ k n 0 ( D + z ) ] , Y D q ( 2 n + 1 ) / ρ 0 [ k n 0 ( D + z ) ] .
- θ ˜ n θ θ ˜ n ,
θ ˜ n = [ ( 2 n + 1 ) / α ] 1 / 2 .
[ ( 2 n + 1 ) / α ] 1 / 2 1.
κ = R 2 ( t ) [ n 0 2 - R ˙ 2 ( t ) ] = R 2 ( 0 ) [ n 0 2 - R ˙ 2 ( 0 ) ]
R 4 θ ˙ 2 = κ - 2 Δ n 0 2 G ( θ ) .
- θ max θ θ max ,
R min R ,
R min = κ / n 0 ,
θ max = κ / n 0 2 Δ .
κ = α ( 2 n + 1 ) / k 2 ;
k loc ( R , θ ) = k n ( R , θ ) [ R ˙ ( R ˙ 2 + R 2 θ ˙ 2 ) e ^ R + R θ ˙ ( R ˙ 2 + R 2 θ ˙ 2 ) e ^ θ ]
Δ Φ + 2 ( - π / 2 ) = 2 π n , ( n an integer )
Δ Φ = ray path S 0 S 2 [ k n ( R , θ ) R θ ˙ / ( R ˙ 2 + R 2 θ ˙ 2 ) 1 / 2 ] R d θ .
Δ Φ = 2 α - θ max + θ max [ G ( θ max ) - G ( θ ) ] 1 / 2 d θ .
Δ Φ = π k 2 κ / α .
ψ n ( R , θ ) = exp ( - α θ 2 / 2 ) H n ( α θ ) J α ( 2 n + 1 ) ( k n 0 R ) .
R R ˜ n ,
R ˜ n = [ α ( 2 n + 1 ) ] 1 / 2 / k n 0 .
J μ ( k n 0 R ) = ( 1 / 2 ) [ H μ ( 2 ) ( k n 0 R ) + H μ ( 1 ) ( k n 0 R ) ]
β n = { k 2 n 0 2 - [ q / ρ ( z ) ] ( 2 n + 1 ) } 1 / 2 .
H μ ( 1 ) , ( 2 ) ( k n 0 R ) ~ ( 2 / π k n 0 R ) 1 / 2 exp { ± i [ k n 0 R - μ ( π / 2 ) - π / 4 ] }
ψ n ( R , θ ) = exp ( - α θ 2 / 2 ) H n ( α θ ) H α ( 2 n + 1 ) ( 1 ) ( k n 0 R ) .
E = n a n exp ( - α θ 2 / 2 ) H n ( α θ ) × [ J α ( 2 n + 1 ) ( k n 0 R ) + b n Y α ( 2 n + 1 ) ( k n 0 R ) ] ,
n 2 = n 0 2 [ 1 - 2 Δ G ( θ ) / R 2 ] ,
G ( θ ) = sin 2 θ / cos 4 θ .
x = R sin θ cos ϕ ; y = R sin θ sin ϕ ; z = R cos θ - D .
n 2 = n 0 2 [ 1 - 2 Δ r 2 / ( D + z ) 4 ] ,
r = ( x 2 + y 2 ) 1 / 2 .
n 2 = n 0 2 [ 1 - 2 δ r 2 / ρ 2 ( z ) ] ,
ψ ( R , θ , ϕ ) = A ( R ) B ( θ ) C ( ϕ ) ,
A ( R ) = { ( 1 / R ) J ν + ( 1 / 4 ) ( k n 0 R ) , ( 1 / R ) Y ν + ( 1 / 4 ) ( k n 0 R ) ,
C ( ϕ ) = { cos m ϕ , sin m ϕ ,             m = 0 , 1 , 2 , ,
sin 2 θ d 2 B / d θ 2 + sin θ cos θ d B / d θ + [ ν sin 2 θ - α 2 sin 2 θ G ( θ ) - m 2 ] B ( θ ) = 0.
θ 2 d 2 B / d θ 2 + θ d B / d θ + ( ν θ 2 - α 2 θ 4 - m 2 ) B ( θ ) = 0.
B n , m ( θ ) = exp ( - α θ 2 / 2 ) θ m L n ( m ) ( α θ 2 )             m , n = 0 , 1 , 2 , ,
ν = 2 α ( m + 2 n + 1 ) .
ψ n , m o , e = R - 1 / 2 θ m exp ( - α θ 2 / 2 ) L n ( m ) ( α θ 2 ) sin cos ( m ϕ ) { J ζ ( k n 0 R ) Y ζ ( k n 0 R )             m , n = 0 , 1 , 2 , ,
ζ = ( ½ ) [ 1 + 8 α ( m + 2 n + 1 ) ] 1 / 2 .
ψ n = exp [ - q x 2 / 2 ρ ( z ) ] H n { [ q / ρ ( z ) ] 1 / 2 x } { J μ ( μ sec δ ) , Y μ ( μ sec δ ) ,
μ = D [ q ( 2 n + 1 ) / ρ 0 ] 1 / 2 ,
sec δ = k n 0 [ ρ 0 / q ( 2 n + 1 ) ] 1 / 2 ( 1 + z / D ) .
k n 0 [ ρ 0 / q ( 2 n + 1 ) ] 1 / 2 1
J μ ( μ sec δ ) ~ ( 2 / π μ tan δ ) 1 / 2 cos ( μ tan δ - μ δ - π / 4 )
μ tan δ C 1 + D [ k n 0 + 2 δ ( 2 n + 1 ) / 2 ρ 0 ] ( z / D ) , μ δ C 2 + D [ 2 δ ( 2 n + 1 ) / ρ 0 ] ( z / D ) ,
J μ ( μ sec δ ) ~ A 1 cos { [ k n 0 - 2 δ ( 2 n + 1 ) / 2 ρ 0 ] z + ϕ 1 } ,
Y μ ( μ sec δ ) ~ A 1 sin { [ k n 0 - 2 δ ( 2 n + 1 ) / 2 ρ 0 ] z + ϕ 1 } .
ψ n ~ exp ( - q x 2 / 2 ρ 0 ) H n [ ( q / ρ 0 ) 1 / 2 x ] × { cos { [ k n 0 - 2 δ ( 2 n + 1 ) / 2 ρ 0 ] z + ϕ 1 } sin { [ k n 0 - 2 δ ( 2 n + 1 ) / 2 ρ 0 ] z + ϕ 1 } .
β n k n 0 - 2 δ ( 2 n + 1 ) / 2 ρ 0 ,

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