Abstract

An optical beam waveguide consisting of a central aperture and a surrounding lens—a hybrid lens guide—possesses some important advantages of both iris guides and lens guides. Compared to an iris guide, it has a lower loss, and for small curvatures, is less sensitive to bends. Also it is less subject to breakdown than a lens guide. A theoretical study is presented. The hybrid lens guide can be specified by four parameters; the loss is investigated as a function of each. The effects of bending and a method to reduce such effects are also discussed.

© 1973 Optical Society of America

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References

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  1. G. Goubau, “Beam Waveguide” in Advances in Microwaves, L. Young, Ed. (Academic Press, New York, 1968), Vol. 3, p. 67.
  2. D. Gloge, Bell Syst. Tech. J. 46, 721 (1967).
  3. J. E. Degenford, M. D. Sirkis, W. H. Steier, IEEE Trans. MTT-12, 445 (1964).
  4. A. C. Beck, IEEE Trans. MTT-15, 433 (1967).
  5. J. W. Mink. Electron. Lett. 7, 527 (1971).
    [CrossRef]
  6. P. F. Checcacci, A. Consortini, A. M. Scheggi, Appl. Opt. 10, 1363 (1971).
    [CrossRef] [PubMed]
  7. P. F. Checcacci, R. Falciai, A. Scheggi, IEEE Trans. MTT-20, 608 (1972).
  8. M. De, J. W. Y. Lit, R. Tremblay, Appl. Opt. 7, 483 (1968).
    [CrossRef] [PubMed]
  9. R. Boulay, J. W. Y. Lit, R. Tremblay, Opt. Commun. 4, 163 (1971).
    [CrossRef]
  10. F. K. Schwering, IRE Trans. AP-10, 99 (1962).
  11. A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).
  12. J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965).
    [CrossRef]
  13. For a similar example of the application of the FFT, see A. L. Boyer, P. M. Hirsch, J. A. Jordan, L. B. Lesem, D. L. Van Rooy, “Reconstruction of Ultrasonic Images by Backward Propagation,” in Acoustical Holography, A. F. Metherell, Ed. (Plenum Press, New York, 1971), Vol. 3, p. 333.
    [CrossRef]
  14. W. T. Cochran et al.Proc. IEEE 55, 1664 (1967).
    [CrossRef]
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 21.
  16. P. A. Banger, R. Tremblay, Can. J. Phys. 49, 1290 (1971).
    [CrossRef]
  17. G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

1972 (1)

P. F. Checcacci, R. Falciai, A. Scheggi, IEEE Trans. MTT-20, 608 (1972).

1971 (4)

R. Boulay, J. W. Y. Lit, R. Tremblay, Opt. Commun. 4, 163 (1971).
[CrossRef]

J. W. Mink. Electron. Lett. 7, 527 (1971).
[CrossRef]

P. F. Checcacci, A. Consortini, A. M. Scheggi, Appl. Opt. 10, 1363 (1971).
[CrossRef] [PubMed]

P. A. Banger, R. Tremblay, Can. J. Phys. 49, 1290 (1971).
[CrossRef]

1968 (1)

1967 (3)

D. Gloge, Bell Syst. Tech. J. 46, 721 (1967).

A. C. Beck, IEEE Trans. MTT-15, 433 (1967).

W. T. Cochran et al.Proc. IEEE 55, 1664 (1967).
[CrossRef]

1965 (1)

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965).
[CrossRef]

1964 (1)

J. E. Degenford, M. D. Sirkis, W. H. Steier, IEEE Trans. MTT-12, 445 (1964).

1962 (1)

F. K. Schwering, IRE Trans. AP-10, 99 (1962).

1961 (2)

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

Banger, P. A.

P. A. Banger, R. Tremblay, Can. J. Phys. 49, 1290 (1971).
[CrossRef]

Beck, A. C.

A. C. Beck, IEEE Trans. MTT-15, 433 (1967).

Boulay, R.

R. Boulay, J. W. Y. Lit, R. Tremblay, Opt. Commun. 4, 163 (1971).
[CrossRef]

Boyd, G. D.

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

Boyer, A. L.

For a similar example of the application of the FFT, see A. L. Boyer, P. M. Hirsch, J. A. Jordan, L. B. Lesem, D. L. Van Rooy, “Reconstruction of Ultrasonic Images by Backward Propagation,” in Acoustical Holography, A. F. Metherell, Ed. (Plenum Press, New York, 1971), Vol. 3, p. 333.
[CrossRef]

Checcacci, P. F.

P. F. Checcacci, R. Falciai, A. Scheggi, IEEE Trans. MTT-20, 608 (1972).

P. F. Checcacci, A. Consortini, A. M. Scheggi, Appl. Opt. 10, 1363 (1971).
[CrossRef] [PubMed]

Cochran, W. T.

W. T. Cochran et al.Proc. IEEE 55, 1664 (1967).
[CrossRef]

Consortini, A.

Cooley, J. W.

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965).
[CrossRef]

De, M.

Degenford, J. E.

J. E. Degenford, M. D. Sirkis, W. H. Steier, IEEE Trans. MTT-12, 445 (1964).

Falciai, R.

P. F. Checcacci, R. Falciai, A. Scheggi, IEEE Trans. MTT-20, 608 (1972).

Fox, A. G.

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Gloge, D.

D. Gloge, Bell Syst. Tech. J. 46, 721 (1967).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 21.

Gordon, J. P.

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

Goubau, G.

G. Goubau, “Beam Waveguide” in Advances in Microwaves, L. Young, Ed. (Academic Press, New York, 1968), Vol. 3, p. 67.

Hirsch, P. M.

For a similar example of the application of the FFT, see A. L. Boyer, P. M. Hirsch, J. A. Jordan, L. B. Lesem, D. L. Van Rooy, “Reconstruction of Ultrasonic Images by Backward Propagation,” in Acoustical Holography, A. F. Metherell, Ed. (Plenum Press, New York, 1971), Vol. 3, p. 333.
[CrossRef]

Jordan, J. A.

For a similar example of the application of the FFT, see A. L. Boyer, P. M. Hirsch, J. A. Jordan, L. B. Lesem, D. L. Van Rooy, “Reconstruction of Ultrasonic Images by Backward Propagation,” in Acoustical Holography, A. F. Metherell, Ed. (Plenum Press, New York, 1971), Vol. 3, p. 333.
[CrossRef]

Lesem, L. B.

For a similar example of the application of the FFT, see A. L. Boyer, P. M. Hirsch, J. A. Jordan, L. B. Lesem, D. L. Van Rooy, “Reconstruction of Ultrasonic Images by Backward Propagation,” in Acoustical Holography, A. F. Metherell, Ed. (Plenum Press, New York, 1971), Vol. 3, p. 333.
[CrossRef]

Li, T.

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Lit, J. W. Y.

R. Boulay, J. W. Y. Lit, R. Tremblay, Opt. Commun. 4, 163 (1971).
[CrossRef]

M. De, J. W. Y. Lit, R. Tremblay, Appl. Opt. 7, 483 (1968).
[CrossRef] [PubMed]

Mink, J. W.

J. W. Mink. Electron. Lett. 7, 527 (1971).
[CrossRef]

Scheggi, A.

P. F. Checcacci, R. Falciai, A. Scheggi, IEEE Trans. MTT-20, 608 (1972).

Scheggi, A. M.

Schwering, F. K.

F. K. Schwering, IRE Trans. AP-10, 99 (1962).

Sirkis, M. D.

J. E. Degenford, M. D. Sirkis, W. H. Steier, IEEE Trans. MTT-12, 445 (1964).

Steier, W. H.

J. E. Degenford, M. D. Sirkis, W. H. Steier, IEEE Trans. MTT-12, 445 (1964).

Tremblay, R.

R. Boulay, J. W. Y. Lit, R. Tremblay, Opt. Commun. 4, 163 (1971).
[CrossRef]

P. A. Banger, R. Tremblay, Can. J. Phys. 49, 1290 (1971).
[CrossRef]

M. De, J. W. Y. Lit, R. Tremblay, Appl. Opt. 7, 483 (1968).
[CrossRef] [PubMed]

Tukey, J. W.

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965).
[CrossRef]

Van Rooy, D. L.

For a similar example of the application of the FFT, see A. L. Boyer, P. M. Hirsch, J. A. Jordan, L. B. Lesem, D. L. Van Rooy, “Reconstruction of Ultrasonic Images by Backward Propagation,” in Acoustical Holography, A. F. Metherell, Ed. (Plenum Press, New York, 1971), Vol. 3, p. 333.
[CrossRef]

Appl. Opt. (2)

Bell Syst. Tech. J. (3)

D. Gloge, Bell Syst. Tech. J. 46, 721 (1967).

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

Can. J. Phys. (1)

P. A. Banger, R. Tremblay, Can. J. Phys. 49, 1290 (1971).
[CrossRef]

Electron. Lett. (1)

J. W. Mink. Electron. Lett. 7, 527 (1971).
[CrossRef]

IEEE Trans. (3)

J. E. Degenford, M. D. Sirkis, W. H. Steier, IEEE Trans. MTT-12, 445 (1964).

A. C. Beck, IEEE Trans. MTT-15, 433 (1967).

P. F. Checcacci, R. Falciai, A. Scheggi, IEEE Trans. MTT-20, 608 (1972).

IRE Trans. (1)

F. K. Schwering, IRE Trans. AP-10, 99 (1962).

Math. Comp. (1)

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965).
[CrossRef]

Opt. Commun. (1)

R. Boulay, J. W. Y. Lit, R. Tremblay, Opt. Commun. 4, 163 (1971).
[CrossRef]

Proc. IEEE (1)

W. T. Cochran et al.Proc. IEEE 55, 1664 (1967).
[CrossRef]

Other (3)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 21.

For a similar example of the application of the FFT, see A. L. Boyer, P. M. Hirsch, J. A. Jordan, L. B. Lesem, D. L. Van Rooy, “Reconstruction of Ultrasonic Images by Backward Propagation,” in Acoustical Holography, A. F. Metherell, Ed. (Plenum Press, New York, 1971), Vol. 3, p. 333.
[CrossRef]

G. Goubau, “Beam Waveguide” in Advances in Microwaves, L. Young, Ed. (Academic Press, New York, 1968), Vol. 3, p. 67.

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Figures (12)

Fig. 1
Fig. 1

Hybrid lens waveguide in one dimension.

Fig. 2
Fig. 2

Geometry used for calculating fields of a HGL.

Fig. 3
Fig. 3

Curved HLG.

Fig. 4
Fig. 4

Equivalent resonator for the guide in Fig. 2.

Fig. 5
Fig. 5

Power loss vs ϕj of a HLG.

Fig. 6
Fig. 6

Power loss vs N of a HLG for A: g = 0.5, A/B = 0.56; B: g = 0., A/B = 0.56; C: g = 1, A/B = 1 (planar iris).

Fig. 7
Fig. 7

Power loss vs g for a HLG (N = 1.22) for A: A/B = 0.88; B: A/B = 0.7; C: A/B = 0.64; D: A/B = 0.6; E: A/B = 0.56

Fig. 8
Fig. 8

Power loss vs A/B for a HLG (N = 1.22) for A: g = 0.8, B: g = 0.5, C: g = 0, C: g = (3rd mode).

Fig. 9
Fig. 9

Amplitude of the 0 order mode for A: N = 1.22, g = 0, ϕj = 0,A/B = 0.56, B: N = 1.22, g = 0, ϕj = 0, A/B = 0.3.

Fig. 10
Fig. 10

Power loss vs 1/ R ¯ for a curved guide (N = 1.22) A: Planar iris; B: HLG with A/B = 0.56, g = 0.5; C: HLG as in B with a prism (θ = α).

Fig. 11
Fig. 11

Power loss vs ϕj for a curved HLG-plus-prism (θ = α).

Fig. 12
Fig. 12

Power loss vs θ for a curved HLG-plus-prism (ϕj = 0).

Equations (7)

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U 2 ( x ¯ 2 ) = exp ( - π / 4 ) / ( 2 π ) 1 2 [ - A c B A c B P ( x ¯ 1 , x ¯ 2 ) U 1 ( x ¯ 1 ) d x ¯ 1 + V L ( x ¯ 1 ) P ( x ¯ 1 , x ¯ 2 ) U 1 ( x ¯ 1 ) d x ¯ 1 ] ,
P ( x ¯ 1 , x ¯ 2 ) = exp i ( x ¯ 1 - x ¯ 2 ) 2 / 2 , L ( x ¯ 1 ) = exp i { ϕ j + [ ( x ¯ 1 - 2 - ( c A / B ) 2 ] ( g - 1 ) } ,
U n m ( x ¯ n ) = γ m U n - 1 m ( x ¯ n ) ,
U 2 ( x ¯ 2 ) = exp ( - i π / 4 ) / ( 2 π ) 1 2 [ - A c B A c B P ( x ¯ 1 , x ¯ 2 ) × S ( x ¯ 1 , x ¯ 2 ) U 1 ( x ¯ 1 ) d x ¯ 1 + V P ( x ¯ 1 , x ¯ 2 ) L ( x ¯ 1 ) S ( x ¯ 1 , x ¯ 2 ) T ( x ¯ 1 ) U ( x ¯ 1 ) d x ¯ 1 ] ,
T ( x ¯ 1 ) = exp [ - i x ¯ 1 ( 1 - A ¯ / x ¯ 1 ) a / R ¯ ]
Δ x = ( λ z / M ) 1 2 ,
D ^ ( l Δ x ) = p = - ( M / 2 ) p = ( M / 2 ) - 1 D ( p Δ x ) exp ( ± 2 π i p l / M ) for l = - ( M / 2 ) , - ( M / 2 ) + 1 , ( M / 2 ) - 1.

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