Abstract

The use of Poincare sphere visualization of two-coupled-mode problems is reviewed, and a new vector form of the usual integrated optics differential equations is presented. Unlike previous similar treatments involving propagation over discrete uniform intervals, we show the relationship of the differential form of the coupled mode equations to the Stokes-Mueller formalism, using the Feynman-Vernon-Hellwarth method for the two-level Schrodinger equation. A specific example is given for the tuned fiber coil optical isolator, for which the present method allows a solution by inspection.

© 1989 Optical Society of America

Full Article  |  PDF Article

Corrections

Stephen R. Chinn, "Differential coupled mode analysis and the Poincare sphere: erratum," Appl. Opt. 28, 3795-3795 (1989)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-28-18-3795

References

  • View by:
  • |
  • |
  • |

  1. W. A. Shurcliff, Polarized Light (Harvard U.P., Cambridge, 1962), Sec. 2.2.
  2. R. Wolfe, J. Hegarty, J. F. Dillon, L. C. Luther, G. K. Celler, L. E. Trimble, C. S. Dorsey, “Thin-Film Waveguide Magneto-optic Isolator,” Appl. Phys. Lett. 46, 817 (1985).
    [CrossRef]
  3. S. R. Korotky, “Three-Space Representation of Phase-Mismatch Switching in Coupled Two-State Optical Systems,” IEEE J. Quantum Electron. QE-22, 952 (1986).
    [CrossRef]
  4. See, for example, A. Hardy, W. Streifer, “Coupled Mode Theory of Parallel Waveguides,” IEEE/OSA J. Lightwave Technol.LT-3, 1135–1146 (1985).
    [CrossRef]
  5. R. P. Feynman, F. L. Vernon, R. W. Hellwarth, “Geometrical Representation of the Schrodinger Equation for Solving Maser Problems,” J. Appl. Phys. 28, 49 (1957).
    [CrossRef]
  6. G. W. Day, D. N. Payne, A. J. Barlow, J. J. Ramskov-Hansen, “Faraday Rotation in Coiled, Monomode Optical Fibers: Isolators, Filters, and Magnetic Sensors,” Opt. Lett. 7, 238–240 (1982).
    [CrossRef] [PubMed]
  7. G. W. Day, D. N. Payne, A. J. Barlow, J. J. Ramskov-Hansen, “Design and Performance of Tuned Fiber Coil Isolators,” IEEE/OSA J. Lightwave Technol. LT-2, 56–60 (1984).
    [CrossRef]
  8. E. P. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic, New York, 1959), Sec. 15.3.
  9. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1950), Sec. 4.5.
  10. R. W. Schmieder, “Stokes-Algebra Formalism,” J. Opt. Soc. Am. 59, 297 (1969).
    [CrossRef]
  11. H. Takenaka, “A Unified Formalism for Polarization Optics by Using Group Theory,” Nouv. Rev. Opt. 4, 37 (1973).
    [CrossRef]
  12. A. Yariv, Quantum Electronics (Wiley, New York, 1975).

1986 (1)

S. R. Korotky, “Three-Space Representation of Phase-Mismatch Switching in Coupled Two-State Optical Systems,” IEEE J. Quantum Electron. QE-22, 952 (1986).
[CrossRef]

1985 (1)

R. Wolfe, J. Hegarty, J. F. Dillon, L. C. Luther, G. K. Celler, L. E. Trimble, C. S. Dorsey, “Thin-Film Waveguide Magneto-optic Isolator,” Appl. Phys. Lett. 46, 817 (1985).
[CrossRef]

1984 (1)

G. W. Day, D. N. Payne, A. J. Barlow, J. J. Ramskov-Hansen, “Design and Performance of Tuned Fiber Coil Isolators,” IEEE/OSA J. Lightwave Technol. LT-2, 56–60 (1984).
[CrossRef]

1982 (1)

1973 (1)

H. Takenaka, “A Unified Formalism for Polarization Optics by Using Group Theory,” Nouv. Rev. Opt. 4, 37 (1973).
[CrossRef]

1969 (1)

1957 (1)

R. P. Feynman, F. L. Vernon, R. W. Hellwarth, “Geometrical Representation of the Schrodinger Equation for Solving Maser Problems,” J. Appl. Phys. 28, 49 (1957).
[CrossRef]

Barlow, A. J.

G. W. Day, D. N. Payne, A. J. Barlow, J. J. Ramskov-Hansen, “Design and Performance of Tuned Fiber Coil Isolators,” IEEE/OSA J. Lightwave Technol. LT-2, 56–60 (1984).
[CrossRef]

G. W. Day, D. N. Payne, A. J. Barlow, J. J. Ramskov-Hansen, “Faraday Rotation in Coiled, Monomode Optical Fibers: Isolators, Filters, and Magnetic Sensors,” Opt. Lett. 7, 238–240 (1982).
[CrossRef] [PubMed]

Celler, G. K.

R. Wolfe, J. Hegarty, J. F. Dillon, L. C. Luther, G. K. Celler, L. E. Trimble, C. S. Dorsey, “Thin-Film Waveguide Magneto-optic Isolator,” Appl. Phys. Lett. 46, 817 (1985).
[CrossRef]

Day, G. W.

G. W. Day, D. N. Payne, A. J. Barlow, J. J. Ramskov-Hansen, “Design and Performance of Tuned Fiber Coil Isolators,” IEEE/OSA J. Lightwave Technol. LT-2, 56–60 (1984).
[CrossRef]

G. W. Day, D. N. Payne, A. J. Barlow, J. J. Ramskov-Hansen, “Faraday Rotation in Coiled, Monomode Optical Fibers: Isolators, Filters, and Magnetic Sensors,” Opt. Lett. 7, 238–240 (1982).
[CrossRef] [PubMed]

Dillon, J. F.

R. Wolfe, J. Hegarty, J. F. Dillon, L. C. Luther, G. K. Celler, L. E. Trimble, C. S. Dorsey, “Thin-Film Waveguide Magneto-optic Isolator,” Appl. Phys. Lett. 46, 817 (1985).
[CrossRef]

Dorsey, C. S.

R. Wolfe, J. Hegarty, J. F. Dillon, L. C. Luther, G. K. Celler, L. E. Trimble, C. S. Dorsey, “Thin-Film Waveguide Magneto-optic Isolator,” Appl. Phys. Lett. 46, 817 (1985).
[CrossRef]

Feynman, R. P.

R. P. Feynman, F. L. Vernon, R. W. Hellwarth, “Geometrical Representation of the Schrodinger Equation for Solving Maser Problems,” J. Appl. Phys. 28, 49 (1957).
[CrossRef]

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1950), Sec. 4.5.

Hardy, A.

See, for example, A. Hardy, W. Streifer, “Coupled Mode Theory of Parallel Waveguides,” IEEE/OSA J. Lightwave Technol.LT-3, 1135–1146 (1985).
[CrossRef]

Hegarty, J.

R. Wolfe, J. Hegarty, J. F. Dillon, L. C. Luther, G. K. Celler, L. E. Trimble, C. S. Dorsey, “Thin-Film Waveguide Magneto-optic Isolator,” Appl. Phys. Lett. 46, 817 (1985).
[CrossRef]

Hellwarth, R. W.

R. P. Feynman, F. L. Vernon, R. W. Hellwarth, “Geometrical Representation of the Schrodinger Equation for Solving Maser Problems,” J. Appl. Phys. 28, 49 (1957).
[CrossRef]

Korotky, S. R.

S. R. Korotky, “Three-Space Representation of Phase-Mismatch Switching in Coupled Two-State Optical Systems,” IEEE J. Quantum Electron. QE-22, 952 (1986).
[CrossRef]

Luther, L. C.

R. Wolfe, J. Hegarty, J. F. Dillon, L. C. Luther, G. K. Celler, L. E. Trimble, C. S. Dorsey, “Thin-Film Waveguide Magneto-optic Isolator,” Appl. Phys. Lett. 46, 817 (1985).
[CrossRef]

Payne, D. N.

G. W. Day, D. N. Payne, A. J. Barlow, J. J. Ramskov-Hansen, “Design and Performance of Tuned Fiber Coil Isolators,” IEEE/OSA J. Lightwave Technol. LT-2, 56–60 (1984).
[CrossRef]

G. W. Day, D. N. Payne, A. J. Barlow, J. J. Ramskov-Hansen, “Faraday Rotation in Coiled, Monomode Optical Fibers: Isolators, Filters, and Magnetic Sensors,” Opt. Lett. 7, 238–240 (1982).
[CrossRef] [PubMed]

Ramskov-Hansen, J. J.

G. W. Day, D. N. Payne, A. J. Barlow, J. J. Ramskov-Hansen, “Design and Performance of Tuned Fiber Coil Isolators,” IEEE/OSA J. Lightwave Technol. LT-2, 56–60 (1984).
[CrossRef]

G. W. Day, D. N. Payne, A. J. Barlow, J. J. Ramskov-Hansen, “Faraday Rotation in Coiled, Monomode Optical Fibers: Isolators, Filters, and Magnetic Sensors,” Opt. Lett. 7, 238–240 (1982).
[CrossRef] [PubMed]

Schmieder, R. W.

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U.P., Cambridge, 1962), Sec. 2.2.

Streifer, W.

See, for example, A. Hardy, W. Streifer, “Coupled Mode Theory of Parallel Waveguides,” IEEE/OSA J. Lightwave Technol.LT-3, 1135–1146 (1985).
[CrossRef]

Takenaka, H.

H. Takenaka, “A Unified Formalism for Polarization Optics by Using Group Theory,” Nouv. Rev. Opt. 4, 37 (1973).
[CrossRef]

Trimble, L. E.

R. Wolfe, J. Hegarty, J. F. Dillon, L. C. Luther, G. K. Celler, L. E. Trimble, C. S. Dorsey, “Thin-Film Waveguide Magneto-optic Isolator,” Appl. Phys. Lett. 46, 817 (1985).
[CrossRef]

Vernon, F. L.

R. P. Feynman, F. L. Vernon, R. W. Hellwarth, “Geometrical Representation of the Schrodinger Equation for Solving Maser Problems,” J. Appl. Phys. 28, 49 (1957).
[CrossRef]

Wigner, E. P.

E. P. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic, New York, 1959), Sec. 15.3.

Wolfe, R.

R. Wolfe, J. Hegarty, J. F. Dillon, L. C. Luther, G. K. Celler, L. E. Trimble, C. S. Dorsey, “Thin-Film Waveguide Magneto-optic Isolator,” Appl. Phys. Lett. 46, 817 (1985).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1975).

Appl. Phys. Lett. (1)

R. Wolfe, J. Hegarty, J. F. Dillon, L. C. Luther, G. K. Celler, L. E. Trimble, C. S. Dorsey, “Thin-Film Waveguide Magneto-optic Isolator,” Appl. Phys. Lett. 46, 817 (1985).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. R. Korotky, “Three-Space Representation of Phase-Mismatch Switching in Coupled Two-State Optical Systems,” IEEE J. Quantum Electron. QE-22, 952 (1986).
[CrossRef]

IEEE/OSA J. Lightwave Technol. (1)

G. W. Day, D. N. Payne, A. J. Barlow, J. J. Ramskov-Hansen, “Design and Performance of Tuned Fiber Coil Isolators,” IEEE/OSA J. Lightwave Technol. LT-2, 56–60 (1984).
[CrossRef]

J. Appl. Phys. (1)

R. P. Feynman, F. L. Vernon, R. W. Hellwarth, “Geometrical Representation of the Schrodinger Equation for Solving Maser Problems,” J. Appl. Phys. 28, 49 (1957).
[CrossRef]

J. Opt. Soc. Am. (1)

Nouv. Rev. Opt. (1)

H. Takenaka, “A Unified Formalism for Polarization Optics by Using Group Theory,” Nouv. Rev. Opt. 4, 37 (1973).
[CrossRef]

Opt. Lett. (1)

Other (5)

See, for example, A. Hardy, W. Streifer, “Coupled Mode Theory of Parallel Waveguides,” IEEE/OSA J. Lightwave Technol.LT-3, 1135–1146 (1985).
[CrossRef]

A. Yariv, Quantum Electronics (Wiley, New York, 1975).

E. P. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic, New York, 1959), Sec. 15.3.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1950), Sec. 4.5.

W. A. Shurcliff, Polarized Light (Harvard U.P., Cambridge, 1962), Sec. 2.2.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Diagram of Poincare sphere coordinate system with the axes defined by Eq. (7). The mode or polarization amplitudes E x and E y are defined to be real when removed from the parenthetical expressions of the real and imaginary terms.

Fig. 2
Fig. 2

Numerical solution for the modal Stokes vector in the fiber coil with δ = 1.00 and F0 = 0.02. The phase matched tuning condition Δβ = 1/R has been used, and the propagation distance is 25π = 78.54.

Fig. 3
Fig. 3

Results of Fig. 2 shown in a reference frame rotating at constant rate 2δ about the S3 axis. On this scale, the analytic solution would be nearly indistinguishable from the numerical result shown.

Fig. 4
Fig. 4

Difference between the numerical and analytic result for S3 as a function of distance. The normalized constant total intensity S0 = 1.

Fig. 5
Fig. 5

Difference between the numerical and analytic result for S2 in the rotating reference frame as a function of distance. The normalized constant total intensity S0 = 1.

Fig. 6
Fig. 6

Rotating frame numerical results for the phase mismatched case with δ = 1.00, F0 = 0.02, and Λ = π/0.99 giving 2δ − (1/R) = 0.02. The propagation distance is π/(0.02√2) = 111.07.

Equations (15)

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

[ E x ( z ) E y ( z ) ] = [ cos ϕ z 2 j Δ β ϕ sin ϕ z 2 2 F ϕ sin ϕ z 2 2 F * ϕ sin ϕ z 2 cos ϕ z 2 + j Δ β ϕ sin ϕ z 2 ] [ E x ( 0 ) E y ( 0 ) ] ,
d E x d z = j δ E x F E y , d E y d z = F * E x + j δ E y ,
j ћ d a d t = a ( ћ ω 0 2 + V a a ) + b V a b
r 1 = a b * + b a * , r 2 = j ( a b * b a * ) , r 3 = a a * b b * .
d r d t = ω × r ,
ω 1 = ( V a b + V b a ) / ћ , ω 2 = j ( V a b V b a ) / ћ , ω 3 = ω 0 .
S 1 = E x E y * + E x * E y = 2 Re ( E x E y * ) ; S 2 = j ( E x E y * E x * E y ) = 2 Im ( E x E y * ) ; S 3 = E x E x * E y E y * .
d S d z = Ω × S , Ω = [ 2 Im ( F ) , 2 Re ( F ) , 2 δ ] .
| E x | 2 | E y | 2 = cos ( 2 θ ) = 1 tan 2 θ 1 + tan 2 θ = δ 2 | F | 2 δ 2 + | F | 2 .
| E y | 2 = | F | 2 δ 2 + | F | 2 .
F = F 0 cos z R F 0 cos 2 π z Λ ,
d S r d z = ( Ω r Δ ) × S r ,
Ω r Δ = F 0 { sin 4 δ z , 2 cos 2 2 δ z , 0 } r = F 0 { 0 , 1 , 0 } r + F 0 { sin 4 δ z , cos 4 δ z , 0 } r .
| E x ( z ) | 2 | E y ( z ) | 2 = cos ( F 0 z ) ,
| E y ( z ) | 2 / | E x ( 0 ) | 2 = sin 2 ( F 0 z / 2 ) .

Metrics