Abstract

The evolution of the amplitudes of two general, codirectional coupled waves is represented as a rotation of a generalized Poincaré sphere in analogy to polarized light. The method permits simple, pictorial representation of coupling effects in directional couplers and “monomode” optical fibers.

© 1977 Optical Society of America

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References

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  1. G. N. Ramachandran, S. Ramaseshan in Handbuch der Physik, S. Flügge, ed. (Springer, Berlin, 1961), Vol. 25/1, p. 1; M. Born, E. Wolf, Principles of Optics, (Pergamon, New York1975), p. 31.
    [CrossRef]
  2. A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973). The quantity A exp(iβaz) of that reference corresponds to the A1 used here, and xab there to ix12 here.
    [CrossRef]
  3. R. M. A. Azzam, N. M. Bashara, Appl. Opt. 12, 62 (1973).
    [CrossRef] [PubMed]
  4. V. Ramaswamy, R. D. Standley, in Digest of Topical Meeting on Optical Fiber Transmission II (Optical Society of America, Washington, D.C., 1977) paper TuD4.
  5. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).
  6. R. V. Schmidt, H. Kogelnik, Appl. Phys. Lett. 28, 503 (1976).
    [CrossRef]
  7. F. P. Kapron, N. F. Borrelli, D. B. Keck, IEEE J. Quantum Electron. QE-8, 222 (1972).
    [CrossRef]
  8. W. A. Gambling, D. N. Payne, H. Matsumara in Digest of Topical Meeting on Optical Fiber Transmission II (Optical Society of America, Washington, D.C., 1977) paper TuD5.
  9. A. Simon, R. Ulrich (to be published).
  10. R. M. A. Azzam, N. M. Bashara, Opt. Commun. 5, 319 (1972).
    [CrossRef]

1976

R. V. Schmidt, H. Kogelnik, Appl. Phys. Lett. 28, 503 (1976).
[CrossRef]

1973

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973). The quantity A exp(iβaz) of that reference corresponds to the A1 used here, and xab there to ix12 here.
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Appl. Opt. 12, 62 (1973).
[CrossRef] [PubMed]

1972

F. P. Kapron, N. F. Borrelli, D. B. Keck, IEEE J. Quantum Electron. QE-8, 222 (1972).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Opt. Commun. 5, 319 (1972).
[CrossRef]

1969

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Appl. Opt. 12, 62 (1973).
[CrossRef] [PubMed]

R. M. A. Azzam, N. M. Bashara, Opt. Commun. 5, 319 (1972).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Appl. Opt. 12, 62 (1973).
[CrossRef] [PubMed]

R. M. A. Azzam, N. M. Bashara, Opt. Commun. 5, 319 (1972).
[CrossRef]

Borrelli, N. F.

F. P. Kapron, N. F. Borrelli, D. B. Keck, IEEE J. Quantum Electron. QE-8, 222 (1972).
[CrossRef]

Gambling, W. A.

W. A. Gambling, D. N. Payne, H. Matsumara in Digest of Topical Meeting on Optical Fiber Transmission II (Optical Society of America, Washington, D.C., 1977) paper TuD5.

Kapron, F. P.

F. P. Kapron, N. F. Borrelli, D. B. Keck, IEEE J. Quantum Electron. QE-8, 222 (1972).
[CrossRef]

Keck, D. B.

F. P. Kapron, N. F. Borrelli, D. B. Keck, IEEE J. Quantum Electron. QE-8, 222 (1972).
[CrossRef]

Kogelnik, H.

R. V. Schmidt, H. Kogelnik, Appl. Phys. Lett. 28, 503 (1976).
[CrossRef]

Marcatili, E. A. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

Matsumara, H.

W. A. Gambling, D. N. Payne, H. Matsumara in Digest of Topical Meeting on Optical Fiber Transmission II (Optical Society of America, Washington, D.C., 1977) paper TuD5.

Payne, D. N.

W. A. Gambling, D. N. Payne, H. Matsumara in Digest of Topical Meeting on Optical Fiber Transmission II (Optical Society of America, Washington, D.C., 1977) paper TuD5.

Ramachandran, G. N.

G. N. Ramachandran, S. Ramaseshan in Handbuch der Physik, S. Flügge, ed. (Springer, Berlin, 1961), Vol. 25/1, p. 1; M. Born, E. Wolf, Principles of Optics, (Pergamon, New York1975), p. 31.
[CrossRef]

Ramaseshan, S.

G. N. Ramachandran, S. Ramaseshan in Handbuch der Physik, S. Flügge, ed. (Springer, Berlin, 1961), Vol. 25/1, p. 1; M. Born, E. Wolf, Principles of Optics, (Pergamon, New York1975), p. 31.
[CrossRef]

Ramaswamy, V.

V. Ramaswamy, R. D. Standley, in Digest of Topical Meeting on Optical Fiber Transmission II (Optical Society of America, Washington, D.C., 1977) paper TuD4.

Schmidt, R. V.

R. V. Schmidt, H. Kogelnik, Appl. Phys. Lett. 28, 503 (1976).
[CrossRef]

Simon, A.

A. Simon, R. Ulrich (to be published).

Standley, R. D.

V. Ramaswamy, R. D. Standley, in Digest of Topical Meeting on Optical Fiber Transmission II (Optical Society of America, Washington, D.C., 1977) paper TuD4.

Ulrich, R.

A. Simon, R. Ulrich (to be published).

Yariv, A.

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973). The quantity A exp(iβaz) of that reference corresponds to the A1 used here, and xab there to ix12 here.
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

R. V. Schmidt, H. Kogelnik, Appl. Phys. Lett. 28, 503 (1976).
[CrossRef]

Bell Syst. Tech. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

IEEE J. Quantum Electron.

F. P. Kapron, N. F. Borrelli, D. B. Keck, IEEE J. Quantum Electron. QE-8, 222 (1972).
[CrossRef]

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973). The quantity A exp(iβaz) of that reference corresponds to the A1 used here, and xab there to ix12 here.
[CrossRef]

Opt. Commun.

R. M. A. Azzam, N. M. Bashara, Opt. Commun. 5, 319 (1972).
[CrossRef]

Other

G. N. Ramachandran, S. Ramaseshan in Handbuch der Physik, S. Flügge, ed. (Springer, Berlin, 1961), Vol. 25/1, p. 1; M. Born, E. Wolf, Principles of Optics, (Pergamon, New York1975), p. 31.
[CrossRef]

V. Ramaswamy, R. D. Standley, in Digest of Topical Meeting on Optical Fiber Transmission II (Optical Society of America, Washington, D.C., 1977) paper TuD4.

W. A. Gambling, D. N. Payne, H. Matsumara in Digest of Topical Meeting on Optical Fiber Transmission II (Optical Society of America, Washington, D.C., 1977) paper TuD5.

A. Simon, R. Ulrich (to be published).

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Figures (3)

Fig. 1
Fig. 1

Any combination of two waves, F1, F2, can be represented by a point C on the generalized Poincaré sphere.

Fig. 2
Fig. 2

The evolution of states can be described by indicating on the sphere (a) the trajectories C(s) or (b) the position of the local axis R(s) of rotation, together with the local angle of rotation θ(s) = s∫Ωds.

Fig. 3
Fig. 3

(a) Evolution of the input state F1 in directional couplers. (b) In a monomode fiber, the combined effects of biréfringent β, optical activity α, and twist τ result in cycloidal trajectories C(s).

Equations (12)

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A 1 = i k 1 A 1 + i x 12 A 2 , A 2 = i x 21 A 1 + i k 2 A 2 .
x 12 = x 21 * = x exp ( i γ ) .
A 1 = C sin ( π / 4 + ψ ) exp ( i Φ + i ϕ ) , A 2 = C sin ( π / 4 ψ ) exp ( i Φ i ϕ ) .
C = ( | A 1 | 2 + | A 2 | 2 ) 1 / 2 ,
Φ = 1 2 [ arg ( A 1 ) + arg ( A 2 ) ] ,
ψ = arctan [ ( | A 1 | | A 2 | ) / ( | A 1 | + | A 2 | ) ] ,
ϕ = 1 2 [ arg ( A 1 ) arg ( A 2 ) ] .
ψ = x sin ( 2 ϕ γ ) ,
ϕ = δ x cos ( 2 ϕ γ ) tan 2 ψ .
Ω = 2 ( δ 2 + x 2 ) 1 / 2
2 ψ R = arctan ( δ / x ) ,
2 ϕ R = γ .

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