Abstract

The effects of curvature and anisotropy (leaky-mode) losses on the performance of geodesic lenses in Ti-diffused waveguides in LiNbO3 are examined. A model for the lenses is developed that is equivalent to the usual Fraunhofer diffraction theory, modified to account for the path-dependent losses. Numerical calculations show that the curvature losses are dominant and require fabrication of waveguides having well-confined fundamental modes. Higher-order modes are effectively attenuated by the lenses. The leaky-mode losses have the beneficial effect of reducing the side-lobe intensity in the focal plane.

© 1980 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. C. Hamilton, D. A. Wille, W. J. Miceli, Opt. Eng. 16 (5), 475 (1977).
    [CrossRef]
  2. C. M. Verber, D. W. Vahey, V. E. Wood, R. P. Kenan, N. F. Hartman, C. M. Chapman, in Technical Digest, International Conference on Integrated Optics and Optical Fiber Communication, Tokyo, 1977 (July1977), Talk B1–3.
  3. E. Spiller, J. S. Harper, Appl. Opt. 13, 2105 (1974).
    [CrossRef] [PubMed]
  4. C. M. Verber, D. W. Vahey, V. E. Wood, Appl. Phys. Lett. 28 (9), 514 (1976).
    [CrossRef]
  5. B. V. Chen, E. Marom, A. Lee, Appl. Phys. Lett. 31 (4), 263 (1977).
    [CrossRef]
  6. D. B. Anderson, R. L. Davis, J. T. Boyd, R. R. August, IEEE J. Quantum Electron. QE-13 (4), 275 (1977).
    [CrossRef]
  7. L. P. Boivin, Appl. Opt. 13, 391 (1974).
    [CrossRef] [PubMed]
  8. B. Chen, E. Marom, R. J. Morrison, Appl. Phys. Lett. 33 (6), 511 (1978).
    [CrossRef]
  9. G. E. Betts, J. C. Bradley, G. E. Marx, D. C. Schubert, H. A. Trenchard, Appl. Opt. 17, 2346 (1978).
    [CrossRef] [PubMed]
  10. D. W. Vahey, V. E. Wood, IEEE J. Quantum Electron. QE-13 (4), 275 (1977).
  11. D. W. Vahey, in Proceedings, Ultrasonics Symposium, J. deKlerk, B. R. McAvoy, Eds. (IEEE, Cherry Hill, N.J., 1978), pp. 70–73.
  12. G. E. Betts, G. E. Marx, Appl. Opt. 17, 3969 (1978).
    [CrossRef] [PubMed]
  13. S. K. Sheem, W. K. Burns, A. F. Milton, Opt. Lett. 3 (3), 76 (1978).
    [CrossRef] [PubMed]
  14. W. K. Burns, S. K. Sheem, A. F. Milton, to be published in IEEE J. Quantum ElectronQE-15 (11), (1979).
  15. D. W. Vahey, R. P. Kenan, V. E. Wood, C. M. Verber, N. F. Hartman, IEEE J. Quantum Electron. QE-13, (9), 6D (1977).
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5, pp. 77–100.
  17. G. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11 (6), 270 (1975).
    [CrossRef]
  18. G. B. Hocker, W. K. Burns, Appl. Opt. 16, 113 (1977).
    [CrossRef] [PubMed]
  19. D. Marcuse, Bell Syst. Tech. J. 50 (8), 2551 (1971).
  20. R. V. Schmidt, I. P. Kaminow, Appl. Phys. Lett. 25 (8), 458 (1974).
    [CrossRef]
  21. W. K. Burns, P. H. Klein, E. J. West, L. E. Plew, J. Appl. Phys. 50, 6175 (1979).
    [CrossRef]
  22. The data used here were obtained from the extraordinary mode measurements reported in Ref. 21.

1979 (1)

W. K. Burns, P. H. Klein, E. J. West, L. E. Plew, J. Appl. Phys. 50, 6175 (1979).
[CrossRef]

1978 (4)

1977 (6)

D. W. Vahey, V. E. Wood, IEEE J. Quantum Electron. QE-13 (4), 275 (1977).

D. W. Vahey, R. P. Kenan, V. E. Wood, C. M. Verber, N. F. Hartman, IEEE J. Quantum Electron. QE-13, (9), 6D (1977).

B. V. Chen, E. Marom, A. Lee, Appl. Phys. Lett. 31 (4), 263 (1977).
[CrossRef]

D. B. Anderson, R. L. Davis, J. T. Boyd, R. R. August, IEEE J. Quantum Electron. QE-13 (4), 275 (1977).
[CrossRef]

M. C. Hamilton, D. A. Wille, W. J. Miceli, Opt. Eng. 16 (5), 475 (1977).
[CrossRef]

G. B. Hocker, W. K. Burns, Appl. Opt. 16, 113 (1977).
[CrossRef] [PubMed]

1976 (1)

C. M. Verber, D. W. Vahey, V. E. Wood, Appl. Phys. Lett. 28 (9), 514 (1976).
[CrossRef]

1975 (1)

G. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11 (6), 270 (1975).
[CrossRef]

1974 (3)

1971 (1)

D. Marcuse, Bell Syst. Tech. J. 50 (8), 2551 (1971).

Anderson, D. B.

D. B. Anderson, R. L. Davis, J. T. Boyd, R. R. August, IEEE J. Quantum Electron. QE-13 (4), 275 (1977).
[CrossRef]

August, R. R.

D. B. Anderson, R. L. Davis, J. T. Boyd, R. R. August, IEEE J. Quantum Electron. QE-13 (4), 275 (1977).
[CrossRef]

Betts, G. E.

Boivin, L. P.

Boyd, J. T.

D. B. Anderson, R. L. Davis, J. T. Boyd, R. R. August, IEEE J. Quantum Electron. QE-13 (4), 275 (1977).
[CrossRef]

Bradley, J. C.

Burns, W. K.

W. K. Burns, P. H. Klein, E. J. West, L. E. Plew, J. Appl. Phys. 50, 6175 (1979).
[CrossRef]

S. K. Sheem, W. K. Burns, A. F. Milton, Opt. Lett. 3 (3), 76 (1978).
[CrossRef] [PubMed]

G. B. Hocker, W. K. Burns, Appl. Opt. 16, 113 (1977).
[CrossRef] [PubMed]

G. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11 (6), 270 (1975).
[CrossRef]

W. K. Burns, S. K. Sheem, A. F. Milton, to be published in IEEE J. Quantum ElectronQE-15 (11), (1979).

Chapman, C. M.

C. M. Verber, D. W. Vahey, V. E. Wood, R. P. Kenan, N. F. Hartman, C. M. Chapman, in Technical Digest, International Conference on Integrated Optics and Optical Fiber Communication, Tokyo, 1977 (July1977), Talk B1–3.

Chen, B.

B. Chen, E. Marom, R. J. Morrison, Appl. Phys. Lett. 33 (6), 511 (1978).
[CrossRef]

Chen, B. V.

B. V. Chen, E. Marom, A. Lee, Appl. Phys. Lett. 31 (4), 263 (1977).
[CrossRef]

Davis, R. L.

D. B. Anderson, R. L. Davis, J. T. Boyd, R. R. August, IEEE J. Quantum Electron. QE-13 (4), 275 (1977).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5, pp. 77–100.

Hamilton, M. C.

M. C. Hamilton, D. A. Wille, W. J. Miceli, Opt. Eng. 16 (5), 475 (1977).
[CrossRef]

Harper, J. S.

Hartman, N. F.

D. W. Vahey, R. P. Kenan, V. E. Wood, C. M. Verber, N. F. Hartman, IEEE J. Quantum Electron. QE-13, (9), 6D (1977).

C. M. Verber, D. W. Vahey, V. E. Wood, R. P. Kenan, N. F. Hartman, C. M. Chapman, in Technical Digest, International Conference on Integrated Optics and Optical Fiber Communication, Tokyo, 1977 (July1977), Talk B1–3.

Hocker, G. B.

G. B. Hocker, W. K. Burns, Appl. Opt. 16, 113 (1977).
[CrossRef] [PubMed]

G. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11 (6), 270 (1975).
[CrossRef]

Kaminow, I. P.

R. V. Schmidt, I. P. Kaminow, Appl. Phys. Lett. 25 (8), 458 (1974).
[CrossRef]

Kenan, R. P.

D. W. Vahey, R. P. Kenan, V. E. Wood, C. M. Verber, N. F. Hartman, IEEE J. Quantum Electron. QE-13, (9), 6D (1977).

C. M. Verber, D. W. Vahey, V. E. Wood, R. P. Kenan, N. F. Hartman, C. M. Chapman, in Technical Digest, International Conference on Integrated Optics and Optical Fiber Communication, Tokyo, 1977 (July1977), Talk B1–3.

Klein, P. H.

W. K. Burns, P. H. Klein, E. J. West, L. E. Plew, J. Appl. Phys. 50, 6175 (1979).
[CrossRef]

Lee, A.

B. V. Chen, E. Marom, A. Lee, Appl. Phys. Lett. 31 (4), 263 (1977).
[CrossRef]

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 50 (8), 2551 (1971).

Marom, E.

B. Chen, E. Marom, R. J. Morrison, Appl. Phys. Lett. 33 (6), 511 (1978).
[CrossRef]

B. V. Chen, E. Marom, A. Lee, Appl. Phys. Lett. 31 (4), 263 (1977).
[CrossRef]

Marx, G. E.

Miceli, W. J.

M. C. Hamilton, D. A. Wille, W. J. Miceli, Opt. Eng. 16 (5), 475 (1977).
[CrossRef]

Milton, A. F.

S. K. Sheem, W. K. Burns, A. F. Milton, Opt. Lett. 3 (3), 76 (1978).
[CrossRef] [PubMed]

W. K. Burns, S. K. Sheem, A. F. Milton, to be published in IEEE J. Quantum ElectronQE-15 (11), (1979).

Morrison, R. J.

B. Chen, E. Marom, R. J. Morrison, Appl. Phys. Lett. 33 (6), 511 (1978).
[CrossRef]

Plew, L. E.

W. K. Burns, P. H. Klein, E. J. West, L. E. Plew, J. Appl. Phys. 50, 6175 (1979).
[CrossRef]

Schmidt, R. V.

R. V. Schmidt, I. P. Kaminow, Appl. Phys. Lett. 25 (8), 458 (1974).
[CrossRef]

Schubert, D. C.

Sheem, S. K.

S. K. Sheem, W. K. Burns, A. F. Milton, Opt. Lett. 3 (3), 76 (1978).
[CrossRef] [PubMed]

W. K. Burns, S. K. Sheem, A. F. Milton, to be published in IEEE J. Quantum ElectronQE-15 (11), (1979).

Spiller, E.

Trenchard, H. A.

Vahey, D. W.

D. W. Vahey, R. P. Kenan, V. E. Wood, C. M. Verber, N. F. Hartman, IEEE J. Quantum Electron. QE-13, (9), 6D (1977).

D. W. Vahey, V. E. Wood, IEEE J. Quantum Electron. QE-13 (4), 275 (1977).

C. M. Verber, D. W. Vahey, V. E. Wood, Appl. Phys. Lett. 28 (9), 514 (1976).
[CrossRef]

C. M. Verber, D. W. Vahey, V. E. Wood, R. P. Kenan, N. F. Hartman, C. M. Chapman, in Technical Digest, International Conference on Integrated Optics and Optical Fiber Communication, Tokyo, 1977 (July1977), Talk B1–3.

D. W. Vahey, in Proceedings, Ultrasonics Symposium, J. deKlerk, B. R. McAvoy, Eds. (IEEE, Cherry Hill, N.J., 1978), pp. 70–73.

Verber, C. M.

D. W. Vahey, R. P. Kenan, V. E. Wood, C. M. Verber, N. F. Hartman, IEEE J. Quantum Electron. QE-13, (9), 6D (1977).

C. M. Verber, D. W. Vahey, V. E. Wood, Appl. Phys. Lett. 28 (9), 514 (1976).
[CrossRef]

C. M. Verber, D. W. Vahey, V. E. Wood, R. P. Kenan, N. F. Hartman, C. M. Chapman, in Technical Digest, International Conference on Integrated Optics and Optical Fiber Communication, Tokyo, 1977 (July1977), Talk B1–3.

West, E. J.

W. K. Burns, P. H. Klein, E. J. West, L. E. Plew, J. Appl. Phys. 50, 6175 (1979).
[CrossRef]

Wille, D. A.

M. C. Hamilton, D. A. Wille, W. J. Miceli, Opt. Eng. 16 (5), 475 (1977).
[CrossRef]

Wood, V. E.

D. W. Vahey, R. P. Kenan, V. E. Wood, C. M. Verber, N. F. Hartman, IEEE J. Quantum Electron. QE-13, (9), 6D (1977).

D. W. Vahey, V. E. Wood, IEEE J. Quantum Electron. QE-13 (4), 275 (1977).

C. M. Verber, D. W. Vahey, V. E. Wood, Appl. Phys. Lett. 28 (9), 514 (1976).
[CrossRef]

C. M. Verber, D. W. Vahey, V. E. Wood, R. P. Kenan, N. F. Hartman, C. M. Chapman, in Technical Digest, International Conference on Integrated Optics and Optical Fiber Communication, Tokyo, 1977 (July1977), Talk B1–3.

Appl. Opt. (5)

Appl. Phys. Lett. (4)

R. V. Schmidt, I. P. Kaminow, Appl. Phys. Lett. 25 (8), 458 (1974).
[CrossRef]

C. M. Verber, D. W. Vahey, V. E. Wood, Appl. Phys. Lett. 28 (9), 514 (1976).
[CrossRef]

B. V. Chen, E. Marom, A. Lee, Appl. Phys. Lett. 31 (4), 263 (1977).
[CrossRef]

B. Chen, E. Marom, R. J. Morrison, Appl. Phys. Lett. 33 (6), 511 (1978).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Marcuse, Bell Syst. Tech. J. 50 (8), 2551 (1971).

IEEE J. Quantum Electron. (4)

G. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11 (6), 270 (1975).
[CrossRef]

D. W. Vahey, R. P. Kenan, V. E. Wood, C. M. Verber, N. F. Hartman, IEEE J. Quantum Electron. QE-13, (9), 6D (1977).

D. B. Anderson, R. L. Davis, J. T. Boyd, R. R. August, IEEE J. Quantum Electron. QE-13 (4), 275 (1977).
[CrossRef]

D. W. Vahey, V. E. Wood, IEEE J. Quantum Electron. QE-13 (4), 275 (1977).

J. Appl. Phys. (1)

W. K. Burns, P. H. Klein, E. J. West, L. E. Plew, J. Appl. Phys. 50, 6175 (1979).
[CrossRef]

Opt. Eng. (1)

M. C. Hamilton, D. A. Wille, W. J. Miceli, Opt. Eng. 16 (5), 475 (1977).
[CrossRef]

Opt. Lett. (1)

Other (5)

D. W. Vahey, in Proceedings, Ultrasonics Symposium, J. deKlerk, B. R. McAvoy, Eds. (IEEE, Cherry Hill, N.J., 1978), pp. 70–73.

C. M. Verber, D. W. Vahey, V. E. Wood, R. P. Kenan, N. F. Hartman, C. M. Chapman, in Technical Digest, International Conference on Integrated Optics and Optical Fiber Communication, Tokyo, 1977 (July1977), Talk B1–3.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5, pp. 77–100.

W. K. Burns, S. K. Sheem, A. F. Milton, to be published in IEEE J. Quantum ElectronQE-15 (11), (1979).

The data used here were obtained from the extraordinary mode measurements reported in Ref. 21.

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

Geometry for calculating waveguide-curvature losses and leaky-mode propagation losses on geodesic-lens performance.

Fig. 2
Fig. 2

Geometry for calculating ray attenuations and path lengths.

Fig. 3
Fig. 3

Normalized ray transmission T(h)/T(0) as a function of the normalized input ray height 2h/d.

Fig. 4
Fig. 4

Focal-plane intensity distributions for four waveguides in the case of lens illumination by a uniform beam. (a) b = 0.1, 〈T〉 = −372 dB; (b) b = 0.2, 〈T〉 = −41.2 dB; (c) b = 0.3, 〈T〉 = −1.4 dB; (d) b = 0.4, 〈T〉 = −2.0 dB.

Fig. 5
Fig. 5

Focal-plane intensity distribution for four waveguides in the case of lens illumination by a truncated-Gaussian beam. (a) b = 0.1, 〈T〉 = −377 dB; (b) b = 0.2, 〈T〉 = −41.6 dB; (c) b = 0.3, 〈T〉 = −1.0 dB; (d) b = 0.4, 〈T〉 = −1.5 dB.

Fig. 6
Fig. 6

Focal-plane intensity distributions in the absence of losses: (a) uniform illumination; (b) truncated-Gaussian illumination.

Tables (2)

Tables Icon

Table I Parameters for Waveguides Used in Lens Calculations (λ0 = 0.83 μm)

Tables Icon

Table II Maximum and Minimum Ray Transmissions Associated with Leaky-mode (LM) and Curvature (C) Losses

Equations (19)

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

E n ( x ) = ( i f λ ) - 1 / 2 d n - Δ / 2 d n + Δ / 2 T ( h ) A ( h ) exp ( - i 2 π h x / f λ ) d h ,
E ( x ) = n = - N N E n ( x ) .
E ( x ) = 2 ( i f λ ) - 1 / 2 0 d / 2 T ( h ) A ( h ) cos ( 2 π h x / f λ ) d h .
( f λ ) 1 / 2 T ( h ) / T ( h ) min .
E ( x ) = 2 Δ ( i f λ ) - 1 / 2 sinc X [ A 0 T 0 / 2 + n = 1 N A n T n cos ( 2 π n X ) ] ,
T ( h ) = T 1 ( h ) T 2 ( h ) T 3 ( h ) ,
T i ( h ) = exp [ - 0.5 c α i ( h , s ) d s ] ,
T i ( h ) exp [ - 0.5 α i ( h ) l i ( h ) ] , l i ( h ) = c d s , c = c out ,             i = 1 , c = c in ,             i = 2 , 3.
R = 2 f [ R c / ( 4 f - R c ) ] 1 / 2 ,
l 2 l 3 = 2 R sin - 1 [ ( R c / R ) cos ( ϕ i - ψ / 2 ) ] , ϕ i = sin - 1 ( h / R c ) , ψ = sin - 1 ( h / f ) . } .
α 2 ( h ) = [ 4 ( n o 2 - n e 2 ) 1 / 2 sin 2 ψ / 2 n o 2 cos ψ / 2 ] [ Δ n ( 1 - b ) D ( - l n b ) 1 / 2 + k 0 - 1 ( 2 Δ n n e b ) - 1 / 2 ] , ψ = sin - 1 h / f ,
l 1 ( h ) = [ f - R c cos ( ϕ i - ψ ) ] / cos ψ .
α 3 = 2 ( 2 Δ n b / n e ) 1 / 2 ( 1 - b ) D ( - l n b ) 1 / 2 + k 0 - 1 ( 2 Δ n n e b ) - 1 / 2 exp [ g ( R ) ] , g ( R ) = k 0 ( 2 Δ n n e b ) 1 / 2 [ D ( - l n b ) 1 / 2 - 4 Δ n b R / 3 n e ] .
Δ n e 1.08 ( τ / D ) ,
Δ n e 5.04 × 10 - 5 τ             ( Å ) .
V k 0 D ( 2 n e Δ n ) 1 / 2 ,
T L M max = 20 log 10 T 1 ( 0 ) T 2 ( 0 ) = 0 , T L M min = 20 log 10 T 1 ( d / 2 ) T 2 ( d / 2 ) , T C min = 20 log 10 T 3 ( 0 ) , T C max = 20 log 10 T 3 ( d / 2 ) ,
I ( x ) = 20 log 10 [ E ( x ) / E ( 0 ) ] ,
T = 20 log 10 [ E ( 0 ) / E 0 ( 0 ) ] ,

Metrics