Abstract

The traditional design procedure for the waveguide Fresnel lens was carried over from those of bulk optics and micro-optics. In this design it is assumed that the lens thickness is negligibly small with respect to the focal length. This criterion does not hold for many integrated optic devices, in particular those with small mode-index modulations and long wavelengths. Under these conditions, the focal properties of the lens become unpredictable and the lens efficiency is reduced, both of which severely limit the usefulness of the lens as a waveguide-to-fiber coupler. To correct for this shortcoming, the standard Fresnel lens design procedure was modified to acocunt for the thickness of the lens explicitly. Both the standard and the modified Fresnel lens designs are outlined. A comparison of the limitations of the two lenses predicts better performance for the modified Fresnel lens. This is supported through computer-simulation results for a pair of test lenses.

© 1995 Optical Society of America

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References

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  1. J. R. Busch, V. E. Wood, R. P. Kenan, C. M. Verber, “Evaporated As2S3 Luneburg lenses for LiNbO3:Ti optical waveguides,” in Optical Information Processing for Aerospace Applications, NASA Conf. Pub. 2207 (NASA, Washington, D.C., 1981), pp. 251–261.
  2. V. E. Wood, J. R. Busch, D. T. Moore, C. B. Wooley, W. H. Southwell, “Rectangular Luneburg-type lenses for integrated optics,” Opt. Lett. 8, 226–228 (1983).
    [CrossRef] [PubMed]
  3. J. Sochacki, “New simplified method for designing the smooth-transition gradient-index and geodesic waveguide lenses of radial symmetry,” J. Lightwave Technol. 8, 667–672 (1990).
    [CrossRef]
  4. S. Forouhar, W. S. C. Chang, S.-K. Yao, “Performance and limitations of chirped grating lenses of Ti-indiffused LiNbO3 planar waveguides,” J. Lightwave Technol. 2, 503–511 (1984).
    [CrossRef]
  5. J.-M. P. Delavaux, W. S. C. Chang, “Effect of fabrication tolerances on the diffraction performance of chirped grating lenses on planar optical waveguides,” Appl. Opt. 24, 227–231 (1985).
    [CrossRef] [PubMed]
  6. Gen-ichi Hatakoshi, Shun-ichi Tanaka, “Grating lenses for integrated optics,” Opt. Lett. 2, 142–144 (1978).
    [CrossRef] [PubMed]
  7. G. C. Righini, G. Molesini, “Design of optical-waveguide homogeneous refracting lenses,” Appl. Opt. 27, 4193–4198 (1988).
    [CrossRef] [PubMed]
  8. P. J. R. Laybourn, G. Molesini, G. C. Righini, “Homogeneous refracting lenses for integrated optical circuits,” J. Mod. Opt. 35, 1029–1048 (1988).
    [CrossRef]
  9. Z. Nikolov, B. Pantchev, “Single aplanatic homogeneous refracting waveguide lenses without field curvature,” Opt. Lett. 17, 1429–1431 (1992).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  12. T. Q. Vu, J. A. Norris, C. S. Tsai, “Formation of negative- index-change waveguide lenses in LiNO3 by using ion milling,” Opt. Lett. 13, 1141–1143 (1988); T. Q. Vu, C. S. Tsai, “Ion-milled waveguide lenses and lens arrays in GaAs,” J. Lightwave Technol. 7, 1559–1566 (1989).
    [CrossRef] [PubMed]
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    [CrossRef]
  14. H. Nishihara, T. Suhara, “Micro Fresnel lenses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1987), Vol. 24, pp. 3–37.
    [CrossRef]
  15. C. Deshui, G. Xiurong, L. Shurong, H. Gongzhang, “Fresnel lenses in silicon nitride waveguides,” Chin. Phys. 10, 970–976 (1990).
  16. Z. Jaroszewicz, Z. Jaroszewicz, R. Staronski, J. Jochakiand, G. Righini, “Planar Fresnel lens with multiple phase jump,” Pure Appl. Opt. 3, 667–677 (1994).
    [CrossRef]
  17. M. A. Forastiere, G. C. Righini, “A family of curved Fresnel lenses for integrated optics: modelling by the beam propagation method,” Pure Appl. Opt. 3, 291–300 (1994).
    [CrossRef]
  18. C. W. Pitt, J. D. Skinner, G. R. Trotter, “Computer simulation of thin film lenses,” Opt. Commun. 53, 87–90 (1985).
    [CrossRef]
  19. J. V. Roey, J. V. der Donk, P. E. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am. 71, 803–810 (1981).
    [CrossRef]
  20. F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1993), Chap. 20, p. 410.

1994 (2)

Z. Jaroszewicz, Z. Jaroszewicz, R. Staronski, J. Jochakiand, G. Righini, “Planar Fresnel lens with multiple phase jump,” Pure Appl. Opt. 3, 667–677 (1994).
[CrossRef]

M. A. Forastiere, G. C. Righini, “A family of curved Fresnel lenses for integrated optics: modelling by the beam propagation method,” Pure Appl. Opt. 3, 291–300 (1994).
[CrossRef]

1993 (1)

A. L. Belostotsky, A. S. Leonov, “Design of aplanatic waveguide Fresnel lenses and aberration-free planar optical systems,” J. Lightwave Technol. 11, 1314–1319 (1993).
[CrossRef]

1992 (1)

1990 (2)

J. Sochacki, “New simplified method for designing the smooth-transition gradient-index and geodesic waveguide lenses of radial symmetry,” J. Lightwave Technol. 8, 667–672 (1990).
[CrossRef]

C. Deshui, G. Xiurong, L. Shurong, H. Gongzhang, “Fresnel lenses in silicon nitride waveguides,” Chin. Phys. 10, 970–976 (1990).

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).
[CrossRef]

1988 (4)

1985 (2)

1984 (1)

S. Forouhar, W. S. C. Chang, S.-K. Yao, “Performance and limitations of chirped grating lenses of Ti-indiffused LiNbO3 planar waveguides,” J. Lightwave Technol. 2, 503–511 (1984).
[CrossRef]

1983 (1)

1981 (1)

1978 (1)

Belostotsky, A. L.

A. L. Belostotsky, A. S. Leonov, “Design of aplanatic waveguide Fresnel lenses and aberration-free planar optical systems,” J. Lightwave Technol. 11, 1314–1319 (1993).
[CrossRef]

Busch, J. R.

V. E. Wood, J. R. Busch, D. T. Moore, C. B. Wooley, W. H. Southwell, “Rectangular Luneburg-type lenses for integrated optics,” Opt. Lett. 8, 226–228 (1983).
[CrossRef] [PubMed]

J. R. Busch, V. E. Wood, R. P. Kenan, C. M. Verber, “Evaporated As2S3 Luneburg lenses for LiNbO3:Ti optical waveguides,” in Optical Information Processing for Aerospace Applications, NASA Conf. Pub. 2207 (NASA, Washington, D.C., 1981), pp. 251–261.

Chang, W. S. C.

J.-M. P. Delavaux, W. S. C. Chang, “Effect of fabrication tolerances on the diffraction performance of chirped grating lenses on planar optical waveguides,” Appl. Opt. 24, 227–231 (1985).
[CrossRef] [PubMed]

S. Forouhar, W. S. C. Chang, S.-K. Yao, “Performance and limitations of chirped grating lenses of Ti-indiffused LiNbO3 planar waveguides,” J. Lightwave Technol. 2, 503–511 (1984).
[CrossRef]

Delavaux, J.-M. P.

der Donk, J. V.

Deshui, C.

C. Deshui, G. Xiurong, L. Shurong, H. Gongzhang, “Fresnel lenses in silicon nitride waveguides,” Chin. Phys. 10, 970–976 (1990).

Forastiere, M. A.

M. A. Forastiere, G. C. Righini, “A family of curved Fresnel lenses for integrated optics: modelling by the beam propagation method,” Pure Appl. Opt. 3, 291–300 (1994).
[CrossRef]

Forouhar, S.

S. Forouhar, W. S. C. Chang, S.-K. Yao, “Performance and limitations of chirped grating lenses of Ti-indiffused LiNbO3 planar waveguides,” J. Lightwave Technol. 2, 503–511 (1984).
[CrossRef]

Gongzhang, H.

C. Deshui, G. Xiurong, L. Shurong, H. Gongzhang, “Fresnel lenses in silicon nitride waveguides,” Chin. Phys. 10, 970–976 (1990).

Hatakoshi, Gen-ichi

Jaroszewicz, Z.

Z. Jaroszewicz, Z. Jaroszewicz, R. Staronski, J. Jochakiand, G. Righini, “Planar Fresnel lens with multiple phase jump,” Pure Appl. Opt. 3, 667–677 (1994).
[CrossRef]

Z. Jaroszewicz, Z. Jaroszewicz, R. Staronski, J. Jochakiand, G. Righini, “Planar Fresnel lens with multiple phase jump,” Pure Appl. Opt. 3, 667–677 (1994).
[CrossRef]

Jochakiand, J.

Z. Jaroszewicz, Z. Jaroszewicz, R. Staronski, J. Jochakiand, G. Righini, “Planar Fresnel lens with multiple phase jump,” Pure Appl. Opt. 3, 667–677 (1994).
[CrossRef]

Kenan, R. P.

J. R. Busch, V. E. Wood, R. P. Kenan, C. M. Verber, “Evaporated As2S3 Luneburg lenses for LiNbO3:Ti optical waveguides,” in Optical Information Processing for Aerospace Applications, NASA Conf. Pub. 2207 (NASA, Washington, D.C., 1981), pp. 251–261.

Lagasse, P. E.

Laybourn, P. J. R.

P. J. R. Laybourn, G. Molesini, G. C. Righini, “Homogeneous refracting lenses for integrated optical circuits,” J. Mod. Opt. 35, 1029–1048 (1988).
[CrossRef]

Leonov, A. S.

A. L. Belostotsky, A. S. Leonov, “Design of aplanatic waveguide Fresnel lenses and aberration-free planar optical systems,” J. Lightwave Technol. 11, 1314–1319 (1993).
[CrossRef]

Molesini, G.

P. J. R. Laybourn, G. Molesini, G. C. Righini, “Homogeneous refracting lenses for integrated optical circuits,” J. Mod. Opt. 35, 1029–1048 (1988).
[CrossRef]

G. C. Righini, G. Molesini, “Design of optical-waveguide homogeneous refracting lenses,” Appl. Opt. 27, 4193–4198 (1988).
[CrossRef] [PubMed]

Moore, D. T.

Nikolov, Z.

Nishihara, H.

H. Nishihara, T. Suhara, “Micro Fresnel lenses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1987), Vol. 24, pp. 3–37.
[CrossRef]

Norris, J. A.

Pantchev, B.

Pedrotti, F. L.

F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1993), Chap. 20, p. 410.

Pedrotti, L. S.

F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1993), Chap. 20, p. 410.

Pitt, C. W.

C. W. Pitt, S. Reid, S. Reynolds, J. Skinner, “Waveguiding Fresnel lenses: modelling and fabrication,” J. Mod. Opt. 35, 1079–1111 (1988).
[CrossRef]

C. W. Pitt, J. D. Skinner, G. R. Trotter, “Computer simulation of thin film lenses,” Opt. Commun. 53, 87–90 (1985).
[CrossRef]

Reid, S.

C. W. Pitt, S. Reid, S. Reynolds, J. Skinner, “Waveguiding Fresnel lenses: modelling and fabrication,” J. Mod. Opt. 35, 1079–1111 (1988).
[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).
[CrossRef]

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).
[CrossRef]

C. W. Pitt, S. Reid, S. Reynolds, J. Skinner, “Waveguiding Fresnel lenses: modelling and fabrication,” J. Mod. Opt. 35, 1079–1111 (1988).
[CrossRef]

Righini, G.

Z. Jaroszewicz, Z. Jaroszewicz, R. Staronski, J. Jochakiand, G. Righini, “Planar Fresnel lens with multiple phase jump,” Pure Appl. Opt. 3, 667–677 (1994).
[CrossRef]

Righini, G. C.

M. A. Forastiere, G. C. Righini, “A family of curved Fresnel lenses for integrated optics: modelling by the beam propagation method,” Pure Appl. Opt. 3, 291–300 (1994).
[CrossRef]

P. J. R. Laybourn, G. Molesini, G. C. Righini, “Homogeneous refracting lenses for integrated optical circuits,” J. Mod. Opt. 35, 1029–1048 (1988).
[CrossRef]

G. C. Righini, G. Molesini, “Design of optical-waveguide homogeneous refracting lenses,” Appl. Opt. 27, 4193–4198 (1988).
[CrossRef] [PubMed]

Roey, J. V.

Shurong, L.

C. Deshui, G. Xiurong, L. Shurong, H. Gongzhang, “Fresnel lenses in silicon nitride waveguides,” Chin. Phys. 10, 970–976 (1990).

Skinner, J.

C. W. Pitt, S. Reid, S. Reynolds, J. Skinner, “Waveguiding Fresnel lenses: modelling and fabrication,” J. Mod. Opt. 35, 1079–1111 (1988).
[CrossRef]

Skinner, J. D.

C. W. Pitt, J. D. Skinner, G. R. Trotter, “Computer simulation of thin film lenses,” Opt. Commun. 53, 87–90 (1985).
[CrossRef]

Sochacki, J.

J. Sochacki, “New simplified method for designing the smooth-transition gradient-index and geodesic waveguide lenses of radial symmetry,” J. Lightwave Technol. 8, 667–672 (1990).
[CrossRef]

Southwell, W. H.

Staronski, R.

Z. Jaroszewicz, Z. Jaroszewicz, R. Staronski, J. Jochakiand, G. Righini, “Planar Fresnel lens with multiple phase jump,” Pure Appl. Opt. 3, 667–677 (1994).
[CrossRef]

Suhara, T.

H. Nishihara, T. Suhara, “Micro Fresnel lenses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1987), Vol. 24, pp. 3–37.
[CrossRef]

Tanaka, Shun-ichi

Trotter, G. R.

C. W. Pitt, J. D. Skinner, G. R. Trotter, “Computer simulation of thin film lenses,” Opt. Commun. 53, 87–90 (1985).
[CrossRef]

Tsai, C. S.

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).
[CrossRef]

Verber, C. M.

J. R. Busch, V. E. Wood, R. P. Kenan, C. M. Verber, “Evaporated As2S3 Luneburg lenses for LiNbO3:Ti optical waveguides,” in Optical Information Processing for Aerospace Applications, NASA Conf. Pub. 2207 (NASA, Washington, D.C., 1981), pp. 251–261.

Vu, T. Q.

Wood, V. E.

V. E. Wood, J. R. Busch, D. T. Moore, C. B. Wooley, W. H. Southwell, “Rectangular Luneburg-type lenses for integrated optics,” Opt. Lett. 8, 226–228 (1983).
[CrossRef] [PubMed]

J. R. Busch, V. E. Wood, R. P. Kenan, C. M. Verber, “Evaporated As2S3 Luneburg lenses for LiNbO3:Ti optical waveguides,” in Optical Information Processing for Aerospace Applications, NASA Conf. Pub. 2207 (NASA, Washington, D.C., 1981), pp. 251–261.

Wooley, C. B.

Xiurong, G.

C. Deshui, G. Xiurong, L. Shurong, H. Gongzhang, “Fresnel lenses in silicon nitride waveguides,” Chin. Phys. 10, 970–976 (1990).

Yao, S.-K.

S. Forouhar, W. S. C. Chang, S.-K. Yao, “Performance and limitations of chirped grating lenses of Ti-indiffused LiNbO3 planar waveguides,” J. Lightwave Technol. 2, 503–511 (1984).
[CrossRef]

Appl. Opt. (2)

Chin. Phys. (1)

C. Deshui, G. Xiurong, L. Shurong, H. Gongzhang, “Fresnel lenses in silicon nitride waveguides,” Chin. Phys. 10, 970–976 (1990).

J. Lightwave Technol. (3)

A. L. Belostotsky, A. S. Leonov, “Design of aplanatic waveguide Fresnel lenses and aberration-free planar optical systems,” J. Lightwave Technol. 11, 1314–1319 (1993).
[CrossRef]

J. Sochacki, “New simplified method for designing the smooth-transition gradient-index and geodesic waveguide lenses of radial symmetry,” J. Lightwave Technol. 8, 667–672 (1990).
[CrossRef]

S. Forouhar, W. S. C. Chang, S.-K. Yao, “Performance and limitations of chirped grating lenses of Ti-indiffused LiNbO3 planar waveguides,” J. Lightwave Technol. 2, 503–511 (1984).
[CrossRef]

J. Mod. Opt. (2)

P. J. R. Laybourn, G. Molesini, G. C. Righini, “Homogeneous refracting lenses for integrated optical circuits,” J. Mod. Opt. 35, 1029–1048 (1988).
[CrossRef]

C. W. Pitt, S. Reid, S. Reynolds, J. Skinner, “Waveguiding Fresnel lenses: modelling and fabrication,” J. Mod. Opt. 35, 1079–1111 (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).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

C. W. Pitt, J. D. Skinner, G. R. Trotter, “Computer simulation of thin film lenses,” Opt. Commun. 53, 87–90 (1985).
[CrossRef]

Opt. Lett. (4)

Pure Appl. Opt. (2)

Z. Jaroszewicz, Z. Jaroszewicz, R. Staronski, J. Jochakiand, G. Righini, “Planar Fresnel lens with multiple phase jump,” Pure Appl. Opt. 3, 667–677 (1994).
[CrossRef]

M. A. Forastiere, G. C. Righini, “A family of curved Fresnel lenses for integrated optics: modelling by the beam propagation method,” Pure Appl. Opt. 3, 291–300 (1994).
[CrossRef]

Other (3)

H. Nishihara, T. Suhara, “Micro Fresnel lenses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1987), Vol. 24, pp. 3–37.
[CrossRef]

J. R. Busch, V. E. Wood, R. P. Kenan, C. M. Verber, “Evaporated As2S3 Luneburg lenses for LiNbO3:Ti optical waveguides,” in Optical Information Processing for Aerospace Applications, NASA Conf. Pub. 2207 (NASA, Washington, D.C., 1981), pp. 251–261.

F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1993), Chap. 20, p. 410.

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Figures (9)

Fig. 1
Fig. 1

Geometry for the design of a SFL. Phase modulation due to the lens is assumed to occur discontinuously at the line y = f.

Fig. 2
Fig. 2

Profile of a SFL for λ0 = 1.53 μm, Δn = 0.005, n eff = 2.148, and f = 4.158 mm. The shaded area represents the region of higher effective mode index.

Fig. 3
Fig. 3

Geometry for modeling the SFL as a fictitious thick lens and designing the MFL. The SFL design procedure counts the phase along the path from (x, y) to (0, 0) via (x, f), and the MFL design procedure counts the phase along the path from (x, y) directly to (0, 0).

Fig. 4
Fig. 4

Family of curves generated by MFL equations, where m is the order of the lens zone.

Fig. 5
Fig. 5

Profile of MFL for λ0 = 1.53 μm, Δn = 0.005, n eff = 2.148, and f = 4.158 mm. The shaded area represents the region of higher effective mode index.

Fig. 6
Fig. 6

Reflectance that an incoming on-axis ray experiences at the lens boundary as a function of distance from the lens axis for SFL and MFL lenses.

Fig. 7
Fig. 7

E-field magnitude in the designed-for focal plane (y = 0) for both the SFL and the MFL.

Fig. 8
Fig. 8

E-field phase in the designed-for focal plane (y = 0) for both the SFL and the MFL.

Fig. 9
Fig. 9

Irradiance profile in the designed-for focal plane (y = 0) for both the SFL and the MFL.

Tables (1)

Tables Icon

Table 1 Lens and Waveguide Parameters for Test Lenses (MFL and SFL)

Equations (13)

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

Δ ϕ ( x ) = ϕ converging ( x ) - ϕ plane ( x ) = 2 π n eff λ 0 [ ( x 2 + f 2 ) 1 / 2 - f ] - 2 m π ,
L ( x ) = L max [ 1 - Δ ϕ ( x ) 2 π ] ,
L max = λ 0 / Δ n
n eff ( y - f ) + n eff ( x 2 + f 2 ) 1 / 2 = n eff ( y - f ) + n eff f + m λ 0 ,
y = f + n eff Δ n [ ( x 2 + f 2 ) 1 / 2 - f - m λ eff ] ,
n eff ( x 2 + y 2 ) 1 / 2 = n eff ( y - f ) + n eff f + m λ 0 .
( x 2 + y 2 ) 1 / 2 = n r ( y - f ) + f + m λ eff ,
y = - b ± ( b 2 - 4 a c ) 1 / 2 2 a ,
a = 1 - n r 2 , b = 2 n r [ f ( n r - 1 ) - m λ eff ] , c = x 2 - [ f ( n r - 1 ) - m λ eff ] . 2
η = | E s ( x ) E f * ( x ) d x | 2 E s 2 d x E f 2 d x ,
r = n eff cos ( θ i ) - n eff cos ( θ t ) n eff cos ( θ i ) + n eff cos ( θ t ) ,
E ( r ) = 1 r E i exp ( j k r ) F ( θ ) ,
F ( θ ) = 1 + cos ( θ ) 2 ,

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