Abstract

Methods of geometric optics are used to characterize a multimode optical fiber. The discrete propagation modes are derived by applying the phase resonance constraint to equations of constant phase surfaces. This constraint provides a very clear geometrical interpretation of discrete propagation modes, and provides a link between the well known Wentzel, Kramers, Brillouin (WKB) method and geometric optics.

© 1990 Optical Society of America

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References

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  1. A. Ankiewicz, “Comparison of Wave and Ray Techniques for Solution of Graded Index Optical Waveguide Problems,” Optica Acta 25, 361–375 (1978).
    [CrossRef]
  2. J. P. Gordon, “Optics of General Guiding Media,” Bell Syst. Tech. J.321–331 (1966).
  3. D. Gloge, E. A. J. Marcatili, “Impulse Response of Fibers With Ring-Shaped Parabolic Index Distribution,” Bell Syst. Tech. J. 52, 1161–1168 (1973).
  4. D. Glóbge, E. A. J. Marcatili, “Multimode Theory of Graded- Core Fiber,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
  5. R. Olshansky, D. B. Keck, “Pulse Broadening in Graded- Index Optical Fibers,” Appl. Phys. 15, 483–491 (1976).
  6. R. Olshansky, “Propagation in Glass Optical Waveguides,” Rev. Mod. Phys. 51, 341–367 (1979).
    [CrossRef]
  7. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, Cincinnati, 1982).
  8. M. J. Adams, An Introduction to Optical Waveguides (Wiley, Chichester, New York, 1981).
  9. A. H. Cherin, An Introduction to Optical Fibers (McGraw-Hill, New York, St. Louis, 1983).
  10. D. L. LeeElectromagnetic Principles of Integrated Optics (Wiley, New York, Chichester, 1986).
  11. A. Ankiewicz, C. Pask, “Geometric Optics Approach to Light Acceptance and Propagation in Graded Index Fibers,” Opt. Quant. Electron. 9, 87–109 (1977).
    [CrossRef]
  12. S. Cornbleet, Microwave and Optical Ray Geometry (Wiley, Chichester, New York, 1984).
  13. K. Luneburg, Mathematical Theory of Optics (U. of California Press, Berkeley, 1964).
  14. E. W. Marchand, Gradient Index Optics (Academic, New York, San Francisco, 1978).

1979

R. Olshansky, “Propagation in Glass Optical Waveguides,” Rev. Mod. Phys. 51, 341–367 (1979).
[CrossRef]

1978

A. Ankiewicz, “Comparison of Wave and Ray Techniques for Solution of Graded Index Optical Waveguide Problems,” Optica Acta 25, 361–375 (1978).
[CrossRef]

1977

A. Ankiewicz, C. Pask, “Geometric Optics Approach to Light Acceptance and Propagation in Graded Index Fibers,” Opt. Quant. Electron. 9, 87–109 (1977).
[CrossRef]

1976

R. Olshansky, D. B. Keck, “Pulse Broadening in Graded- Index Optical Fibers,” Appl. Phys. 15, 483–491 (1976).

1973

D. Gloge, E. A. J. Marcatili, “Impulse Response of Fibers With Ring-Shaped Parabolic Index Distribution,” Bell Syst. Tech. J. 52, 1161–1168 (1973).

D. Glóbge, E. A. J. Marcatili, “Multimode Theory of Graded- Core Fiber,” Bell Syst. Tech. J. 52, 1563–1578 (1973).

1966

J. P. Gordon, “Optics of General Guiding Media,” Bell Syst. Tech. J.321–331 (1966).

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, Chichester, New York, 1981).

Ankiewicz, A.

A. Ankiewicz, “Comparison of Wave and Ray Techniques for Solution of Graded Index Optical Waveguide Problems,” Optica Acta 25, 361–375 (1978).
[CrossRef]

A. Ankiewicz, C. Pask, “Geometric Optics Approach to Light Acceptance and Propagation in Graded Index Fibers,” Opt. Quant. Electron. 9, 87–109 (1977).
[CrossRef]

Cherin, A. H.

A. H. Cherin, An Introduction to Optical Fibers (McGraw-Hill, New York, St. Louis, 1983).

Cornbleet, S.

S. Cornbleet, Microwave and Optical Ray Geometry (Wiley, Chichester, New York, 1984).

Glóbge, D.

D. Glóbge, E. A. J. Marcatili, “Multimode Theory of Graded- Core Fiber,” Bell Syst. Tech. J. 52, 1563–1578 (1973).

Gloge, D.

D. Gloge, E. A. J. Marcatili, “Impulse Response of Fibers With Ring-Shaped Parabolic Index Distribution,” Bell Syst. Tech. J. 52, 1161–1168 (1973).

Gordon, J. P.

J. P. Gordon, “Optics of General Guiding Media,” Bell Syst. Tech. J.321–331 (1966).

Keck, D. B.

R. Olshansky, D. B. Keck, “Pulse Broadening in Graded- Index Optical Fibers,” Appl. Phys. 15, 483–491 (1976).

Lee, D. L.

D. L. LeeElectromagnetic Principles of Integrated Optics (Wiley, New York, Chichester, 1986).

Luneburg, K.

K. Luneburg, Mathematical Theory of Optics (U. of California Press, Berkeley, 1964).

Marcatili, E. A. J.

D. Glóbge, E. A. J. Marcatili, “Multimode Theory of Graded- Core Fiber,” Bell Syst. Tech. J. 52, 1563–1578 (1973).

D. Gloge, E. A. J. Marcatili, “Impulse Response of Fibers With Ring-Shaped Parabolic Index Distribution,” Bell Syst. Tech. J. 52, 1161–1168 (1973).

Marchand, E. W.

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

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, Cincinnati, 1982).

Olshansky, R.

R. Olshansky, “Propagation in Glass Optical Waveguides,” Rev. Mod. Phys. 51, 341–367 (1979).
[CrossRef]

R. Olshansky, D. B. Keck, “Pulse Broadening in Graded- Index Optical Fibers,” Appl. Phys. 15, 483–491 (1976).

Pask, C.

A. Ankiewicz, C. Pask, “Geometric Optics Approach to Light Acceptance and Propagation in Graded Index Fibers,” Opt. Quant. Electron. 9, 87–109 (1977).
[CrossRef]

Appl. Phys.

R. Olshansky, D. B. Keck, “Pulse Broadening in Graded- Index Optical Fibers,” Appl. Phys. 15, 483–491 (1976).

Bell Syst. Tech. J.

J. P. Gordon, “Optics of General Guiding Media,” Bell Syst. Tech. J.321–331 (1966).

D. Gloge, E. A. J. Marcatili, “Impulse Response of Fibers With Ring-Shaped Parabolic Index Distribution,” Bell Syst. Tech. J. 52, 1161–1168 (1973).

D. Glóbge, E. A. J. Marcatili, “Multimode Theory of Graded- Core Fiber,” Bell Syst. Tech. J. 52, 1563–1578 (1973).

Opt. Quant. Electron.

A. Ankiewicz, C. Pask, “Geometric Optics Approach to Light Acceptance and Propagation in Graded Index Fibers,” Opt. Quant. Electron. 9, 87–109 (1977).
[CrossRef]

Optica Acta

A. Ankiewicz, “Comparison of Wave and Ray Techniques for Solution of Graded Index Optical Waveguide Problems,” Optica Acta 25, 361–375 (1978).
[CrossRef]

Rev. Mod. Phys.

R. Olshansky, “Propagation in Glass Optical Waveguides,” Rev. Mod. Phys. 51, 341–367 (1979).
[CrossRef]

Other

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, Cincinnati, 1982).

M. J. Adams, An Introduction to Optical Waveguides (Wiley, Chichester, New York, 1981).

A. H. Cherin, An Introduction to Optical Fibers (McGraw-Hill, New York, St. Louis, 1983).

D. L. LeeElectromagnetic Principles of Integrated Optics (Wiley, New York, Chichester, 1986).

S. Cornbleet, Microwave and Optical Ray Geometry (Wiley, Chichester, New York, 1984).

K. Luneburg, Mathematical Theory of Optics (U. of California Press, Berkeley, 1964).

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

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Figures (11)

Fig. 1
Fig. 1

Skew rays.

Fig. 2
Fig. 2

Flight invariant definitions.

Fig. 3
Fig. 3

Phase velocities in 2-D structure.

Fig. 4
Fig. 4

Meridional ray wavefronts; parabolic fiber, n0 = 1.4, n1 = 1.2, ρ = 30 μm, λ = 1300 nm, and m = 40.

Fig. 5
Fig. 5

Meridional rays satisfying transverse phase shift constraint; sech profile, n0 = 1.4, n1 = 1.385, ρ = 30 μm, and λ = 1300 nm.

Fig. 6
Fig. 6

Ray reflecting from both outer caustic surfaces in ring fiber. a) propagation, b) rotation.

Fig. 7
Fig. 7

Ray reflecting from outer caustic surfaces in ring fiber. a) propagation, b) rotation.

Fig. 8
Fig. 8

Phase velocities in cylindrical structure.

Fig. 9
Fig. 9

Skew ray wavefronts. a) propagation, b) rotation; parabolic profile, n0 = 1.4, n1 = 1.2, ρ = 30 μ, λ = 1300 nm, m = 32, and q = 32.

Fig. 10
Fig. 10

Wavefronts in the cylindrical fiber.

Fig. 11
Fig. 11

Discrete propagation modes in AB space; parabolic profile, n0 = 1.4, n1 = 1.385, ρ = 30 μm, and λ = 1300 nm.

Equations (44)

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A = n ( r ) cos a ,
B = r n ( r ) cos β sin α .
d z d r = A n ( r ) 2 - A 2 - B 2 / r 2 ,
d θ d r = B n ( r ) 2 - A 2 - B 2 / r 2 .
d z d r = A n ( r ) 2 - A 2 .
r 1 r 2 k 2 n ( r ) 2 - β 2 - ν 2 / r 2 d r = ( μ + 1 / 2 ) π ,
0 r 2 k 2 n ( r ) 2 - β 2 d r = ( μ + 1 / 2 ) π / 2.
d f 1 d z d f 2 d z = - 1.
( d r d z ) flight path · ( d r d z ) constant phase curve = - 1 ,
( d r d z ) constant phase curve = - ( d z d r ) flight path .
d r d z = - A n ( r ) 2 - A 2 ,
n ( r ) 2 - A 2 d r = - A d z .
r s r n ( r ) 2 - A 2 d r = - A ( z - z s ) .
v l = c n ( r ) cos α = c A , v t = c n ( r ) sin α .
v l T = c T A = λ A ,
P s = ( z z , r s ) = ( m λ / 4 A 0 ) , P = ( z , r ) = ( 0 , r c ) ,
0 r c n ( r ) 2 - A 2 d r = m λ / 4.
0 r c k 2 n ( r ) 2 - β 2 d r = m π / 2 ,             where β = 2 π λ A .
ψ = s π 2 = m π 2 + p π 2 .
s = m + p .
p = z a λ / 4 A = 4 z a A λ .
s = 4 z a n 0 λ .
A = n 0 - m λ 4 z a .
n ( r ) = n r 2 - δ 1 r 2 - δ 2 / r 2 .
λ B q = 2 π ,
B = q λ 2 π .
n ( r ) 2 - A 2 - B 2 / r 2 d r = - A d z .
r s r n ( r ) 2 - A 2 - B 2 / r 2 d r = - A ( z - z s ) .
r 1 r 2 n ( r ) 2 - A 2 - B 2 / r 2 d r = m λ / 2.
r d θ d r = B r n ( r ) 2 - A 2 - B 2 / r 2 .
( d r r d θ ) constant phase curve = - ( r d θ d r ) flight path .
d r d θ = - B n ( r ) 2 - A 2 - B 2 / r 2 ,
n ( r ) 2 - A 2 - B 2 / r 2 d r = - B d θ .
r s r n ( r ) 2 - A 2 - B 2 / r 2 d r = - B ( θ - θ s ) .
r 1 r n ( r ) 2 - A 2 - B 2 / r 2 d r = m λ / 2.
r 1 r 2 k 2 n ( r ) 2 - β 2 - ν 2 / r 2 d r = m π .
β = 2 π λ A             and             ν = 2 π λ B .
ψ = s 2 π = m 2 π + p 2 π + w 2 π .
p = 4 z a A λ ,
w = 4 θ r B λ .
w = 2 q θ r π .
s = m + p + w = constant .
m + w = constant .
m + q = constant .

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