Abstract

Focusing characteristics of optical waveguide lenses are analyzed by the beam propagation method (BPM) instead of the ray tracing method. By use of the BPM, we can observe field distributions of a converging or diverging light beam after it passes through a waveguide lens. Variations of the spot width and magnitude of diffraction can immediately be evaluated with this calculation. The BPM calculations are used for a mode-index, Luneburg, and geodesic lenses. For the application of the method to the geodesic lens, the surface deformation is converted into an equivalent index.

© 1990 Optical Society of America

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References

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  1. C. S. Tsai, “Guided-Wave Acoustooptic Bragg Modulators for Wide Band Integrated Optic Communications and Signal Processing,” IEEE Trans. Circuits Syst. CAS-26, 1072–1098 (1979).
    [Crossref]
  2. M. K. Barnoski, B. Chen, T. R. Joseph, J. Y. Lee, O. G. Ramer, “Integrated-Optic Spectrum Analyzer,” IEEE Trans. Circuits Syst. CAS-26, 1113–1124 (1979).
    [Crossref]
  3. C. M. Verber, R. P. Kenan, J. R. Busch, “Correlator Based on an Integrated Optical Spatial Light Modulator,” Appl. Opt. 20, 1626–1629 (1981).
    [Crossref] [PubMed]
  4. D. Y. Zang, C. S. Tsai, “Titanium-Indiffused Proton-Exchanged Waveguide Lenses in LiNbO3 for Optical Information Processing,” Appl. Opt. 25, 2264–2271 (1986).
    [Crossref] [PubMed]
  5. T. Suhara, S. Fujiwara, H. Nishihara, “Proton-Exchanged Fresnel Lenses in Ti:LiNbO3 Waveguides,” Appl. Opt. 25, 3379–3383 (1986).
    [Crossref] [PubMed]
  6. S. A. Reid, M. Varasi, S. Reynolds, “Double Dilute Melt Proton Exchange Fresnel Lenses for LiNbO3 Optical Waveguides,” J. Opt. Commun. 10, 67–73 (1989).
  7. G. C. Righini, G. Molesini, “Design of Optical-Waveguide Homogeneous Refracting Lenses,” Appl. Opt. 27, 4193–4199 (1988).
    [Crossref] [PubMed]
  8. W. H. Southwell, “Planar Optical Waveguide Lens Design,” Appl. Opt. 21, 1985–1988 (1982).
    [Crossref] [PubMed]
  9. J. Van Roey, J. van der Donk, P. E. Lagasse, “Beam-Propagation Method: Analysis and Assessment,” J. Opt. Soc. Am. 71, 803–810 (1981).
    [Crossref]
  10. G. P. Bava, P. Rosina, I. Montrosset, “Numerical Analysis of Planar Fresnel Lenses,” J. Mod. Opt. 35, 863–869 (1988).
    [Crossref]
  11. A. L. Dawar, S. M. Al-Shukri, R. M. De La Rue, A. C. G. Nutt, G. Stewart, “Fabrication and Characterization of Titanium Indiffused Proton Exchanged Optical Waveguides in Z-Cut LiNbO3,” Opt. Commun. 61, 100–104 (1987).
    [Crossref]
  12. W. H. Southwell, “Index Profiles for Generalized Luneburg Lenses and Their Use in Planar Optical Waveguides,” J. Opt. Soc. Am. 67, 1010–1014 (1977).
    [Crossref]

1989 (1)

S. A. Reid, M. Varasi, S. Reynolds, “Double Dilute Melt Proton Exchange Fresnel Lenses for LiNbO3 Optical Waveguides,” J. Opt. Commun. 10, 67–73 (1989).

1988 (2)

G. P. Bava, P. Rosina, I. Montrosset, “Numerical Analysis of Planar Fresnel Lenses,” J. Mod. Opt. 35, 863–869 (1988).
[Crossref]

G. C. Righini, G. Molesini, “Design of Optical-Waveguide Homogeneous Refracting Lenses,” Appl. Opt. 27, 4193–4199 (1988).
[Crossref] [PubMed]

1987 (1)

A. L. Dawar, S. M. Al-Shukri, R. M. De La Rue, A. C. G. Nutt, G. Stewart, “Fabrication and Characterization of Titanium Indiffused Proton Exchanged Optical Waveguides in Z-Cut LiNbO3,” Opt. Commun. 61, 100–104 (1987).
[Crossref]

1986 (2)

1982 (1)

1981 (2)

1979 (2)

C. S. Tsai, “Guided-Wave Acoustooptic Bragg Modulators for Wide Band Integrated Optic Communications and Signal Processing,” IEEE Trans. Circuits Syst. CAS-26, 1072–1098 (1979).
[Crossref]

M. K. Barnoski, B. Chen, T. R. Joseph, J. Y. Lee, O. G. Ramer, “Integrated-Optic Spectrum Analyzer,” IEEE Trans. Circuits Syst. CAS-26, 1113–1124 (1979).
[Crossref]

1977 (1)

Al-Shukri, S. M.

A. L. Dawar, S. M. Al-Shukri, R. M. De La Rue, A. C. G. Nutt, G. Stewart, “Fabrication and Characterization of Titanium Indiffused Proton Exchanged Optical Waveguides in Z-Cut LiNbO3,” Opt. Commun. 61, 100–104 (1987).
[Crossref]

Barnoski, M. K.

M. K. Barnoski, B. Chen, T. R. Joseph, J. Y. Lee, O. G. Ramer, “Integrated-Optic Spectrum Analyzer,” IEEE Trans. Circuits Syst. CAS-26, 1113–1124 (1979).
[Crossref]

Bava, G. P.

G. P. Bava, P. Rosina, I. Montrosset, “Numerical Analysis of Planar Fresnel Lenses,” J. Mod. Opt. 35, 863–869 (1988).
[Crossref]

Busch, J. R.

Chen, B.

M. K. Barnoski, B. Chen, T. R. Joseph, J. Y. Lee, O. G. Ramer, “Integrated-Optic Spectrum Analyzer,” IEEE Trans. Circuits Syst. CAS-26, 1113–1124 (1979).
[Crossref]

Dawar, A. L.

A. L. Dawar, S. M. Al-Shukri, R. M. De La Rue, A. C. G. Nutt, G. Stewart, “Fabrication and Characterization of Titanium Indiffused Proton Exchanged Optical Waveguides in Z-Cut LiNbO3,” Opt. Commun. 61, 100–104 (1987).
[Crossref]

De La Rue, R. M.

A. L. Dawar, S. M. Al-Shukri, R. M. De La Rue, A. C. G. Nutt, G. Stewart, “Fabrication and Characterization of Titanium Indiffused Proton Exchanged Optical Waveguides in Z-Cut LiNbO3,” Opt. Commun. 61, 100–104 (1987).
[Crossref]

Fujiwara, S.

Joseph, T. R.

M. K. Barnoski, B. Chen, T. R. Joseph, J. Y. Lee, O. G. Ramer, “Integrated-Optic Spectrum Analyzer,” IEEE Trans. Circuits Syst. CAS-26, 1113–1124 (1979).
[Crossref]

Kenan, R. P.

Lagasse, P. E.

Lee, J. Y.

M. K. Barnoski, B. Chen, T. R. Joseph, J. Y. Lee, O. G. Ramer, “Integrated-Optic Spectrum Analyzer,” IEEE Trans. Circuits Syst. CAS-26, 1113–1124 (1979).
[Crossref]

Molesini, G.

Montrosset, I.

G. P. Bava, P. Rosina, I. Montrosset, “Numerical Analysis of Planar Fresnel Lenses,” J. Mod. Opt. 35, 863–869 (1988).
[Crossref]

Nishihara, H.

Nutt, A. C. G.

A. L. Dawar, S. M. Al-Shukri, R. M. De La Rue, A. C. G. Nutt, G. Stewart, “Fabrication and Characterization of Titanium Indiffused Proton Exchanged Optical Waveguides in Z-Cut LiNbO3,” Opt. Commun. 61, 100–104 (1987).
[Crossref]

Ramer, O. G.

M. K. Barnoski, B. Chen, T. R. Joseph, J. Y. Lee, O. G. Ramer, “Integrated-Optic Spectrum Analyzer,” IEEE Trans. Circuits Syst. CAS-26, 1113–1124 (1979).
[Crossref]

Reid, S. A.

S. A. Reid, M. Varasi, S. Reynolds, “Double Dilute Melt Proton Exchange Fresnel Lenses for LiNbO3 Optical Waveguides,” J. Opt. Commun. 10, 67–73 (1989).

Reynolds, S.

S. A. Reid, M. Varasi, S. Reynolds, “Double Dilute Melt Proton Exchange Fresnel Lenses for LiNbO3 Optical Waveguides,” J. Opt. Commun. 10, 67–73 (1989).

Righini, G. C.

Rosina, P.

G. P. Bava, P. Rosina, I. Montrosset, “Numerical Analysis of Planar Fresnel Lenses,” J. Mod. Opt. 35, 863–869 (1988).
[Crossref]

Southwell, W. H.

Stewart, G.

A. L. Dawar, S. M. Al-Shukri, R. M. De La Rue, A. C. G. Nutt, G. Stewart, “Fabrication and Characterization of Titanium Indiffused Proton Exchanged Optical Waveguides in Z-Cut LiNbO3,” Opt. Commun. 61, 100–104 (1987).
[Crossref]

Suhara, T.

Tsai, C. S.

D. Y. Zang, C. S. Tsai, “Titanium-Indiffused Proton-Exchanged Waveguide Lenses in LiNbO3 for Optical Information Processing,” Appl. Opt. 25, 2264–2271 (1986).
[Crossref] [PubMed]

C. S. Tsai, “Guided-Wave Acoustooptic Bragg Modulators for Wide Band Integrated Optic Communications and Signal Processing,” IEEE Trans. Circuits Syst. CAS-26, 1072–1098 (1979).
[Crossref]

van der Donk, J.

Van Roey, J.

Varasi, M.

S. A. Reid, M. Varasi, S. Reynolds, “Double Dilute Melt Proton Exchange Fresnel Lenses for LiNbO3 Optical Waveguides,” J. Opt. Commun. 10, 67–73 (1989).

Verber, C. M.

Zang, D. Y.

Appl. Opt. (5)

IEEE Trans. Circuits Syst. (2)

C. S. Tsai, “Guided-Wave Acoustooptic Bragg Modulators for Wide Band Integrated Optic Communications and Signal Processing,” IEEE Trans. Circuits Syst. CAS-26, 1072–1098 (1979).
[Crossref]

M. K. Barnoski, B. Chen, T. R. Joseph, J. Y. Lee, O. G. Ramer, “Integrated-Optic Spectrum Analyzer,” IEEE Trans. Circuits Syst. CAS-26, 1113–1124 (1979).
[Crossref]

J. Mod. Opt. (1)

G. P. Bava, P. Rosina, I. Montrosset, “Numerical Analysis of Planar Fresnel Lenses,” J. Mod. Opt. 35, 863–869 (1988).
[Crossref]

J. Opt. Commun. (1)

S. A. Reid, M. Varasi, S. Reynolds, “Double Dilute Melt Proton Exchange Fresnel Lenses for LiNbO3 Optical Waveguides,” J. Opt. Commun. 10, 67–73 (1989).

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

A. L. Dawar, S. M. Al-Shukri, R. M. De La Rue, A. C. G. Nutt, G. Stewart, “Fabrication and Characterization of Titanium Indiffused Proton Exchanged Optical Waveguides in Z-Cut LiNbO3,” Opt. Commun. 61, 100–104 (1987).
[Crossref]

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Figures (11)

Fig. 1
Fig. 1

Model of a TIPE mode-index lens for BPM calculation. The effective refractive indices of a waveguide and mode-index lens are that n g = 2.21 and n1 = 2.32, respectively. The radius of the circular lens is assumed to be 150 μm.

Fig. 2
Fig. 2

Focusing characteristic (a) and contour map (b) of a mode-index lens calculated by using the BPM. The input light is a plane Gaussian beam having a spot size of 40 μm and 633-nm wavelength.

Fig. 3
Fig. 3

Diffraction pattern on the focal plane for a mode-index lens.

Fig. 4
Fig. 4

Schematic diagram of a Luneburg lens. Lens parameters for the BPM calculation are as follows: bulk indices of SiO2 substrate and Corning 7059 glass waveguide are n1 = 1.47 and n2 = 1.57, respectively, and it is assumed that the radius of the overlay lens is 150 μm and the image distance s = 9. The thickness of the waveguide is 1 μm.

Fig. 5
Fig. 5

Focusing characteristic (a) and contour map (b) of a Luneburg lens calculated by using the BPM. The input light is a plane Gaussian beam having a spot size of 40 μm and 633-nm wavelength.

Fig. 6
Fig. 6

Diffraction pattern on a focal plane for a Luneburg lens.

Fig. 7
Fig. 7

Schematic diagram of a geodesic lens.

Fig. 8
Fig. 8

Top (a) and cross-sectional (b) views of a depression for a spherical geodesic lens.

Fig. 9
Fig. 9

Calculated equivalent-index profile for a spherical geodesic lens. The lens radius is 500 μm, and the maximum depth of the depression is 81 μm. A Corning 7059 glass (n2 = 1.57) waveguide 1 μm thick is constructed on a SiO2 (n1 = 1.47) substrate.

Fig. 10
Fig. 10

Focusing characteristic (a) and contour map (b) of a geodesic lens of which an equivalent-index profile is indicated in Fig. 9. Input light is a plane Gaussian beam having a spot size of 100 μm and 633-nm wavelength.

Fig. 11
Fig. 11

Diffraction pattern on a focal plane for a geodesic lens.

Equations (4)

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E ( x , z + δ z ) = E 0 ( x , z + δ z ) exp [ - j k 0 δ n 2 δ z / ( 2 n 0 ) ] ,
n ( r ) = exp [ w ( σ , s ) ] ,
w ( σ , s ) = 1 π σ 1 arcsin ( u / s ) ( u 2 - σ 2 ) 1 / 2 d u .
n e = n g / cos α ,

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