Abstract

We investigate the effect of slow axial tapering on the 3-D structure of the light distribution at the foci of a parabolic index medium. Our approach is based on an adiabatic local mode representation of the field and leads to simple expressions for the field distribution and intensity widths, for any taper shape provided it satisfies a slowness criterion. This slowness criterion unifies the results given here with previously published work.

© 1988 Optical Society of America

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References

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  1. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978).
  2. A. L. Mikaelian, “Self-Focusing Media with Variable Index of Refraction,” Prog. Opt. 17, 279 (1980).
    [CrossRef]
  3. K. Iga, Y. Kokubun, M. Oikawa, Fundamentals of Micro-Optics (Academic, New York, 1984).
  4. A. Yariv, “Transmission and Recovery of Three-Dimensional Image Information in Optical Waveguides,” J. Opt. Soc. Am. 66, 301 (1976).
    [CrossRef]
  5. A. A. Friesem, U. Levy, Y. Silberberg, “Parallel Transmission of Images Through Single Optical Fibers,” Proc. IEEE 71, 208 (1983).
    [CrossRef]
  6. K. Iga, “Theory for Gradient-Index Imaging,” Appl. Opt. 19, 1039 (1980).
    [CrossRef] [PubMed]
  7. D. Bertilone, “Light Distribution in the Focal Regions of a Lens in a Parabolic Index Medium,” Appl. Opt. 26, 306 (1987).
    [CrossRef] [PubMed]
  8. C. Gomez-Reino, E. Larrea, “Imaging and Transforming Transmission Through a Medium with Nonrotation-Symmetric Gradient Index,” Appl. Opt. 22, 387 (1983).
    [CrossRef] [PubMed]
  9. C. Gomez-Reino, E. Larrea, M. V. Perez, J. M. Cuadrado, “Transmittance Function and Modal Propagation in a Conical Gradient-Index Rod,” Appl. Opt. 23, 1107 (1984).
    [CrossRef] [PubMed]
  10. C. Gomez-Reino, M. V. Perez, J. Sochacki, M. Sochacka, “Image and Transform Transmission Through Divergent Conical GRIN Rods,” Opt. Commun. 55, 5 (1985).
    [CrossRef]
  11. C. Gomez-Reino, M. V. Perez, J. Sochacki, M. Sochacka, “Image and Transform Transmission Through Divergent Parabolical GRIN Rods,” Opt. Commun. 55, 8 (1985).
    [CrossRef]
  12. C. Gomez-Reino, J. Sochacki, “Imaging and Transforming Capabilities of GRIN Rods with Noncylindrical Surfaces of Constant Index: a Family of Exact Solutions,” Appl. Opt. 24, 4375 (1985).
    [CrossRef] [PubMed]
  13. C. Gomez-Reino, E. Larrea, M. V. Perez, J. M. Cuadrado, “Imaging, Transforming, and Modal Propagation in a Parabolic Gradient-Index Rod,” Appl. Opt. 24, 4379 (1985).
    [CrossRef] [PubMed]
  14. C. Gomez-Reino, J. Linares, J. Sochacki, “Image and Transform Transmission Through a GRIN Rod with Exponential Variation of the Equi-Index Surfaces,” Appl. Opt. 25, 1076 (1986).
    [CrossRef] [PubMed]
  15. C. Gomez-Reino, J. Linares, E. Larrea, “Imaging and Transforming Transmission Through Tapered Gradient-Index Rods: Analytical Solutions,” J. Opt. Soc. Am. A 3, 1604 (1986).
    [CrossRef]
  16. A. W. Synder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  17. D. Bertilone, A. Ankiewicz, C. Pask, “Wave Propagation in a Graded-Index Taper,” Appl. Opt. 26, 2213 (1987).
    [CrossRef] [PubMed]
  18. A. Erdelyi, Ed., Higher Transcendental Functions, Vol. 2, Bateman Manuscript Project (McGraw-Hill, New York, 1953).
  19. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).
  20. A. Ankiewicz, C. Pask, A. W. Snyder, “Slowly Varying Optical Fibers,” J. Opt. Soc. Am. 72, 198 (1982).
    [CrossRef]
  21. D. M. Milder, “Ray and Wave Invariants for SOFAR Channel Propagation,” J. Acoust. Soc. Am. 46, 1259 (1969).
    [CrossRef]
  22. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1972).
  23. C. Gomez-Reino, M. V. Perez, E. Larrea, “Imaging and Transforming Transmission Through Inhomogeneous Media with Revolution Symmetry,” Opt. Commun. 44, 8 (1982).
    [CrossRef]
  24. C. Gomez-Reino, E. Larrea, “Pupil Effect in GRIN Material,” Appl. Opt. 22, 970 (1983).
    [CrossRef] [PubMed]

1987

1986

1985

C. Gomez-Reino, J. Sochacki, “Imaging and Transforming Capabilities of GRIN Rods with Noncylindrical Surfaces of Constant Index: a Family of Exact Solutions,” Appl. Opt. 24, 4375 (1985).
[CrossRef] [PubMed]

C. Gomez-Reino, E. Larrea, M. V. Perez, J. M. Cuadrado, “Imaging, Transforming, and Modal Propagation in a Parabolic Gradient-Index Rod,” Appl. Opt. 24, 4379 (1985).
[CrossRef] [PubMed]

C. Gomez-Reino, M. V. Perez, J. Sochacki, M. Sochacka, “Image and Transform Transmission Through Divergent Conical GRIN Rods,” Opt. Commun. 55, 5 (1985).
[CrossRef]

C. Gomez-Reino, M. V. Perez, J. Sochacki, M. Sochacka, “Image and Transform Transmission Through Divergent Parabolical GRIN Rods,” Opt. Commun. 55, 8 (1985).
[CrossRef]

1984

1983

1982

A. Ankiewicz, C. Pask, A. W. Snyder, “Slowly Varying Optical Fibers,” J. Opt. Soc. Am. 72, 198 (1982).
[CrossRef]

C. Gomez-Reino, M. V. Perez, E. Larrea, “Imaging and Transforming Transmission Through Inhomogeneous Media with Revolution Symmetry,” Opt. Commun. 44, 8 (1982).
[CrossRef]

1980

K. Iga, “Theory for Gradient-Index Imaging,” Appl. Opt. 19, 1039 (1980).
[CrossRef] [PubMed]

A. L. Mikaelian, “Self-Focusing Media with Variable Index of Refraction,” Prog. Opt. 17, 279 (1980).
[CrossRef]

1976

1969

D. M. Milder, “Ray and Wave Invariants for SOFAR Channel Propagation,” J. Acoust. Soc. Am. 46, 1259 (1969).
[CrossRef]

Ankiewicz, A.

Bertilone, D.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

Cuadrado, J. M.

Friesem, A. A.

A. A. Friesem, U. Levy, Y. Silberberg, “Parallel Transmission of Images Through Single Optical Fibers,” Proc. IEEE 71, 208 (1983).
[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. A 3, 1604 (1986).
[CrossRef]

C. Gomez-Reino, J. Linares, J. Sochacki, “Image and Transform Transmission Through a GRIN Rod with Exponential Variation of the Equi-Index Surfaces,” Appl. Opt. 25, 1076 (1986).
[CrossRef] [PubMed]

C. Gomez-Reino, E. Larrea, M. V. Perez, J. M. Cuadrado, “Imaging, Transforming, and Modal Propagation in a Parabolic Gradient-Index Rod,” Appl. Opt. 24, 4379 (1985).
[CrossRef] [PubMed]

C. Gomez-Reino, M. V. Perez, J. Sochacki, M. Sochacka, “Image and Transform Transmission Through Divergent Conical GRIN Rods,” Opt. Commun. 55, 5 (1985).
[CrossRef]

C. Gomez-Reino, J. Sochacki, “Imaging and Transforming Capabilities of GRIN Rods with Noncylindrical Surfaces of Constant Index: a Family of Exact Solutions,” Appl. Opt. 24, 4375 (1985).
[CrossRef] [PubMed]

C. Gomez-Reino, M. V. Perez, J. Sochacki, M. Sochacka, “Image and Transform Transmission Through Divergent Parabolical GRIN Rods,” Opt. Commun. 55, 8 (1985).
[CrossRef]

C. Gomez-Reino, E. Larrea, M. V. Perez, J. M. Cuadrado, “Transmittance Function and Modal Propagation in a Conical Gradient-Index Rod,” Appl. Opt. 23, 1107 (1984).
[CrossRef] [PubMed]

C. Gomez-Reino, E. Larrea, “Pupil Effect in GRIN Material,” Appl. Opt. 22, 970 (1983).
[CrossRef] [PubMed]

C. Gomez-Reino, E. Larrea, “Imaging and Transforming Transmission Through a Medium with Nonrotation-Symmetric Gradient Index,” Appl. Opt. 22, 387 (1983).
[CrossRef] [PubMed]

C. Gomez-Reino, M. V. Perez, E. Larrea, “Imaging and Transforming Transmission Through Inhomogeneous Media with Revolution Symmetry,” Opt. Commun. 44, 8 (1982).
[CrossRef]

Iga, K.

K. Iga, “Theory for Gradient-Index Imaging,” Appl. Opt. 19, 1039 (1980).
[CrossRef] [PubMed]

K. Iga, Y. Kokubun, M. Oikawa, Fundamentals of Micro-Optics (Academic, New York, 1984).

Kokubun, Y.

K. Iga, Y. Kokubun, M. Oikawa, Fundamentals of Micro-Optics (Academic, New York, 1984).

Larrea, E.

Levy, U.

A. A. Friesem, U. Levy, Y. Silberberg, “Parallel Transmission of Images Through Single Optical Fibers,” Proc. IEEE 71, 208 (1983).
[CrossRef]

Linares, J.

Love, J. D.

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

Marchand, E. W.

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

Mikaelian, A. L.

A. L. Mikaelian, “Self-Focusing Media with Variable Index of Refraction,” Prog. Opt. 17, 279 (1980).
[CrossRef]

Milder, D. M.

D. M. Milder, “Ray and Wave Invariants for SOFAR Channel Propagation,” J. Acoust. Soc. Am. 46, 1259 (1969).
[CrossRef]

Oikawa, M.

K. Iga, Y. Kokubun, M. Oikawa, Fundamentals of Micro-Optics (Academic, New York, 1984).

Pask, C.

Perez, M. V.

C. Gomez-Reino, M. V. Perez, J. Sochacki, M. Sochacka, “Image and Transform Transmission Through Divergent Parabolical GRIN Rods,” Opt. Commun. 55, 8 (1985).
[CrossRef]

C. Gomez-Reino, M. V. Perez, J. Sochacki, M. Sochacka, “Image and Transform Transmission Through Divergent Conical GRIN Rods,” Opt. Commun. 55, 5 (1985).
[CrossRef]

C. Gomez-Reino, E. Larrea, M. V. Perez, J. M. Cuadrado, “Imaging, Transforming, and Modal Propagation in a Parabolic Gradient-Index Rod,” Appl. Opt. 24, 4379 (1985).
[CrossRef] [PubMed]

C. Gomez-Reino, E. Larrea, M. V. Perez, J. M. Cuadrado, “Transmittance Function and Modal Propagation in a Conical Gradient-Index Rod,” Appl. Opt. 23, 1107 (1984).
[CrossRef] [PubMed]

C. Gomez-Reino, M. V. Perez, E. Larrea, “Imaging and Transforming Transmission Through Inhomogeneous Media with Revolution Symmetry,” Opt. Commun. 44, 8 (1982).
[CrossRef]

Silberberg, Y.

A. A. Friesem, U. Levy, Y. Silberberg, “Parallel Transmission of Images Through Single Optical Fibers,” Proc. IEEE 71, 208 (1983).
[CrossRef]

Snyder, A. W.

Sochacka, M.

C. Gomez-Reino, M. V. Perez, J. Sochacki, M. Sochacka, “Image and Transform Transmission Through Divergent Parabolical GRIN Rods,” Opt. Commun. 55, 8 (1985).
[CrossRef]

C. Gomez-Reino, M. V. Perez, J. Sochacki, M. Sochacka, “Image and Transform Transmission Through Divergent Conical GRIN Rods,” Opt. Commun. 55, 5 (1985).
[CrossRef]

Sochacki, J.

C. Gomez-Reino, J. Linares, J. Sochacki, “Image and Transform Transmission Through a GRIN Rod with Exponential Variation of the Equi-Index Surfaces,” Appl. Opt. 25, 1076 (1986).
[CrossRef] [PubMed]

C. Gomez-Reino, M. V. Perez, J. Sochacki, M. Sochacka, “Image and Transform Transmission Through Divergent Conical GRIN Rods,” Opt. Commun. 55, 5 (1985).
[CrossRef]

C. Gomez-Reino, M. V. Perez, J. Sochacki, M. Sochacka, “Image and Transform Transmission Through Divergent Parabolical GRIN Rods,” Opt. Commun. 55, 8 (1985).
[CrossRef]

C. Gomez-Reino, J. Sochacki, “Imaging and Transforming Capabilities of GRIN Rods with Noncylindrical Surfaces of Constant Index: a Family of Exact Solutions,” Appl. Opt. 24, 4375 (1985).
[CrossRef] [PubMed]

Synder, A. W.

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

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

Yariv, A.

Appl. Opt.

J. Acoust. Soc. Am.

D. M. Milder, “Ray and Wave Invariants for SOFAR Channel Propagation,” J. Acoust. Soc. Am. 46, 1259 (1969).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

C. Gomez-Reino, M. V. Perez, J. Sochacki, M. Sochacka, “Image and Transform Transmission Through Divergent Conical GRIN Rods,” Opt. Commun. 55, 5 (1985).
[CrossRef]

C. Gomez-Reino, M. V. Perez, J. Sochacki, M. Sochacka, “Image and Transform Transmission Through Divergent Parabolical GRIN Rods,” Opt. Commun. 55, 8 (1985).
[CrossRef]

C. Gomez-Reino, M. V. Perez, E. Larrea, “Imaging and Transforming Transmission Through Inhomogeneous Media with Revolution Symmetry,” Opt. Commun. 44, 8 (1982).
[CrossRef]

Proc. IEEE

A. A. Friesem, U. Levy, Y. Silberberg, “Parallel Transmission of Images Through Single Optical Fibers,” Proc. IEEE 71, 208 (1983).
[CrossRef]

Prog. Opt.

A. L. Mikaelian, “Self-Focusing Media with Variable Index of Refraction,” Prog. Opt. 17, 279 (1980).
[CrossRef]

Other

K. Iga, Y. Kokubun, M. Oikawa, Fundamentals of Micro-Optics (Academic, New York, 1984).

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

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

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

A. Erdelyi, Ed., Higher Transcendental Functions, Vol. 2, Bateman Manuscript Project (McGraw-Hill, New York, 1953).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

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Figures (2)

Fig. 1
Fig. 1

Coordinate system. An opaque screen with an aperture on the z = 0 plane separates the graded medium (z > 0 region) from the homogeneous medium (z < 0 region). Plane–polar coordinates (r,θ) are defined on the aperture plane.

Fig. 2
Fig. 2

Contours of constant refractive index for the quadratic taper [Eq. (3)]. Index values are labeled n0 > n1 > n2 > n3 and X denotes an axis perpendicular to the optical axis.

Tables (3)

Tables Icon

Table I Comparison of Adiabatic and Nonadiabatic Theories for a Linear Taper

Tables Icon

Table II Comparison of Adiabatic and Nonadiabatic Theories for a Quadratic Taper

Tables Icon

Table III Comparison of Adiabatic and Nonadiabatic Theories for an Exponential Taper

Equations (56)

Equations on this page are rendered with MathJax. Learn more.

n 2 = n 0 2 ( 1 r 2 / ρ 0 2 ) .
n 2 = n 0 2 [ 1 r 2 / ρ 2 ( z ) ]
ρ ( z ) = ρ 0 ( 1 + z / L ) 2
[ 2 + ( n 2 / n 0 2 ) ] E = 0 .
Ψ untap , j m = r j exp ( r 2 / 2 ρ 0 ) L m j ( r 2 / ρ 0 ) ( j θ ) sin cos exp ( i β j m z ) j , m = 0 , 1 , 2 , ,
β j m = [ 1 2 ( j + 2 m + 1 ) / ρ 0 ]
Ψ local , j m = r j exp [ r 2 / 2 ρ ( z ) ] L m j [ r 2 / ρ ( z ) ] ( j θ ) sin cos × exp [ i Φ j m ( z ) ] ,
Φ j m ( z ) = 0 z β j m ( s ) d s .
Ψ j m = [ 1 / ρ ( z ) ] j + 1 r j exp [ r 2 / 2 ρ ( z ) ] L m j [ r 2 / ρ ( z ) ] × ( j θ ) sin cos exp [ i Φ j m ( z ) ] .
2 ( j + 2 m + 1 ) / ρ ( z ) 1 ,
β j m ( z ) = [ 1 2 ( j + 2 m + 1 ) / ρ ( z ) ] 1 ( j + 2 m + 1 ) / ρ ( z ) .
Φ j m ( z ) = z ( j + 2 m + 1 ) γ ( z ) ,
γ ( z ) = 0 z d s / ρ ( s ) .
| z B d ρ / d z | ρ ( z ) ,
| [ π z / γ ( z ) ] d ρ / d z | ρ ( z ) .
E ( r , z = 0 ) = { ϕ ( r ) , r < a , 0 , r > a
E ( r , z ) = m = 0 a m Ψ 0 m ( r , z ) .
a m = ( 2 / ρ 0 ) 0 a s ϕ ( s ) exp ( s 2 / 2 ρ 0 ) L m ( s 2 / ρ 0 ) d s .
E ( r , z ) = [ 1 / i σ ¯ ( z ) ζ ( z ) ] exp { i [ z + r 2 / 2 τ ¯ ( z ) ] } × 0 a ζ ( z ) s ϕ [ s / ζ ( z ) ] exp [ i s 2 / 2 τ ¯ ( z ) ] J 0 [ r s / σ ¯ ( z ) ] d s ,
σ ¯ ( z ) = ρ ( z ) sin γ ( z ) ,
τ ¯ ( z ) = ρ ( z ) tan γ ( z ) ,
ζ ( z ) = [ ρ ( z ) / ρ 0 ] .
ϕ [ s / ζ ( z ) ] 1
I ( r , z ) = [ 1 / ζ ( z ) cos γ ( z ) ] 2 [ U 1 2 ( p ¯ , q ¯ ) + U 2 2 ( p ¯ , q ¯ ) ] ,
p ¯ = a 2 ζ 2 ( z ) / τ ¯ ( z ) ,
q ¯ = a r ζ ( z ) / σ ¯ ( z ) .
I ( r = 0 , z ) = [ a 2 ζ ( z ) / 2 σ ¯ ( z ) ] 2 sinc 2 [ a 2 ζ 2 ( z ) / 4 τ ¯ ( z ) ] .
γ ( z ˜ m ) = ( 2 m + 1 ) π / 2 .
r = r i ζ ( z ) cos 0 z d s / [ ρ 2 ( s ) r i 2 ζ 2 ( s ) ] .
z = z ˜ m + z
I [ a 2 ζ ( z ˜ m ) / 2 ρ ( z ˜ m ) ] 2 sinc 2 [ a 2 z / 4 ρ 0 ρ ( z ˜ m ) ] .
Ω T = 4 π ρ 0 ρ ( z ˜ m ) / a 2 .
Ω T / Ω P = ζ 2 ( z ˜ m ) ,
Ω T / ρ ( z ˜ m ) = Ω P / ρ 0
I ( r , z = z ˜ m ) = [ a 2 ζ ( z ˜ m ) / 2 ρ ( z ˜ m ) ] 2 [ 2 J 1 ( u ) / u ] 2 ,
u = ζ ( z ˜ m ) a r / ρ ( z ˜ m ) .
ω T = 1 . 22 π ρ ( z ˜ m ) / a ζ ( z ˜ m ) .
ω T / ω p = ζ ( z ˜ m ) .
γ ( z ˜ image , m ) = m π ,
M m = ζ ( z ˜ image , m ) .
D L / ρ 0
D exp ( z / L ) π .
J ν ( x ) ( 2 / π x ) cos ( x ν π / 2 π / 4 ) ( | x | ) ,
Ψ j m basis = [ 1 / ρ ( z ) ] j + 1 / 2 r j exp [ r 2 / 2 ρ ( z ) ] L m j [ r 2 / ρ ( z ) ] × ( j θ ) sin cos H μ ( 1 ) , ( 2 ) ( L + z ) ,
μ = ( 1 / 2 ) [ 1 + 8 ( j + 2 m + 1 ) L 2 / ρ 0 ] .
{ H μ ( 1 ) , ( 2 ) ( L + z ) for basis fields , exp [ ± i Φ j m ( z ) ] / ( L + z ) 1 / 2 for adiabatic modes .
Φ j m ( z ) = C 1 + [ ( L + z ) 2 ν 2 ] ν sec 1 [ ( L + z ) / ν ] ,
μ 1 ,
( L + z ) μ .
H μ ( 1 ) , ( 2 ) ( μ sec δ ) ( 2 / π μ tan δ ) exp [ ± i ( μ tan δ μ δ π / 4 ) ] ( μ 1 ) .
[ ( π / 2 ) ( L + z ) ] H μ ( 1 ) , ( 2 ) ( L + z ) exp { ± i [ [ ( L + z ) 2 ν 2 ] ν sec 1 [ ( L + z ) / ν ] π / 4 ] } .
tanh ( 2 / ρ 0 ) ( 2 / ρ 0 ) a 2 / 2 min { [ coth ( 2 / ρ 0 ) ] / ( 2 / ρ 0 ) , 2 / ( 2 / ρ 0 ) 2 .
[ ( 2 m + 1 ) 2 / 2 ] 0 z d s / ρ 2 ( s ) π .
Δ A { 1 / [ 2 tanh 1 ( a 2 / ρ 0 ) ] , a 2 / ρ 0 1 , 1 / [ 2 coth 1 ( a 2 / ρ 0 ) ] , a 2 / ρ 0 > 1 .
0 z d s / ρ 2 ( s ) 2 π T 2 ( a 2 / ρ 0 ) ,
T ( x ) = { tanh 1 ( x ) / [ 1 + tanh 1 ( x ) ] , x 1 , coth 1 ( x ) / [ 1 + coth 1 ( x ) ] , x > 1 .

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