Abstract

By using the Rayleigh–Sommerfeld diffraction formula of the first kind for the two-dimensional case, the impulse response of a thin planar lens is given and the influence of primary aberrations is considered. Respective degrading factors are recognized and a two-dimensional diffraction theory of aberrations in the Seidel approximation is presented. The influence of the material parameters and the geometry of the thin planar lens on the respective aberrations is discussed. The authors show that in several cases, not only spherical aberration but also field curvature is of concern. The thin planar lens as an imaging system and as a Fourier transformer is considered and two apertures—rectangular and Gaussian—are compared.

© 1980 Optical Society of America

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References

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  1. G. C. Righini, V. Russo, S. Sottini, and G. Toraldo di Francia, “Geodesic Lenses for Guided Optical Waves,” Appl. Opt. 12, 1477–1481 (1973).
    [Crossref] [PubMed]
  2. E. Spiller and J. S. Harper, “High Resolution Lenses for Optical Waveguides,” Appl. Opt. 13, 2105–2108 (1974).
    [Crossref] [PubMed]
  3. C. M. Verber, D. W. Vahey, and V. E. Wood, “Focal properties of geodesic waveguide lenses,” Appl. Phys. Lett. 28, 514–516 (1976).
    [Crossref]
  4. Van E. Wood, “Effect of edge-rounding on geodesic lenses,” Appl.Opt. 15, 2817–2820 (1976).
    [Crossref]
  5. D. W. Vahey and V. E. Wood, “Focal Characteristics of Spheroidal Geodesic Lenses for Integrated Optical Processing,” IEEE J. Quantum Electron. QE-13, 129–133 (1977).
    [Crossref]
  6. W. H. Southwell, “Inhomogeneous optical waveguide lens analysis,” J. Opt. Soc. Am. 67, 1010–1014 (1977).
    [Crossref]
  7. W. H. Southwell, “Index profiles for generalized Luneburg lenses and their use in planar optical waveguides,” J. Opt. Soc. Am. 67, 1004–1009 (1977),
    [Crossref]
  8. D. B. Anderson, R. L. Davis, J. T. Boyd, and R. R. August, “Comparison of Optical-Waveguide Lens Technologies,” IEEE J. Quantum Electron. QE-13, 275–281 (1977).
    [Crossref]
  9. W. H. Southwell, “Geodesic optical waveguide lens analysis,” J. Opt. Soc. Am. 67, 1293–1299 (1977).
    [Crossref]
  10. R. Shubert and J. H. Harris, “Optical guided-wave focusing and diffraction,” J. Opt. Soc. Am. 61, 154–161 (1971).
    [Crossref]
  11. P. K. Tien, “Integrated optics and new wave phenomena in optical waveguides,” Rev. Mod. Phys. 49, 361–420 (1977).
    [Crossref]
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5, Sec. 3.
  13. See, for example, D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), pp. 8–9. Note that our case is slightly more general and leads to the one considered in Ref. 13, if β·r→ βz.
  14. The relation given in Ref. 10 is erroneous. This form is correct.
  15. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), Chaps. 5 and 9.
  16. The problem of field curvature for geodesic lenses was pointed out by G. C. Righini, V. Russo, and S. Sottini, “The family of perfect aspherical geodesic lenses for integrated optical circuits,” IEEE J. Quantum Electron. QE-14, 1–3 (1978).
  17. J. R. Shewell and E. Wolf, “Inverse Diffraction and a New Reciprocity Theorem,” J. Opt. Soc. Am. 58, 1596–1603 (1968).
    [Crossref]
  18. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Chap. 5, Sec. 4.

1978 (1)

The problem of field curvature for geodesic lenses was pointed out by G. C. Righini, V. Russo, and S. Sottini, “The family of perfect aspherical geodesic lenses for integrated optical circuits,” IEEE J. Quantum Electron. QE-14, 1–3 (1978).

1977 (6)

D. W. Vahey and V. E. Wood, “Focal Characteristics of Spheroidal Geodesic Lenses for Integrated Optical Processing,” IEEE J. Quantum Electron. QE-13, 129–133 (1977).
[Crossref]

D. B. Anderson, R. L. Davis, J. T. Boyd, and R. R. August, “Comparison of Optical-Waveguide Lens Technologies,” IEEE J. Quantum Electron. QE-13, 275–281 (1977).
[Crossref]

P. K. Tien, “Integrated optics and new wave phenomena in optical waveguides,” Rev. Mod. Phys. 49, 361–420 (1977).
[Crossref]

W. H. Southwell, “Index profiles for generalized Luneburg lenses and their use in planar optical waveguides,” J. Opt. Soc. Am. 67, 1004–1009 (1977),
[Crossref]

W. H. Southwell, “Inhomogeneous optical waveguide lens analysis,” J. Opt. Soc. Am. 67, 1010–1014 (1977).
[Crossref]

W. H. Southwell, “Geodesic optical waveguide lens analysis,” J. Opt. Soc. Am. 67, 1293–1299 (1977).
[Crossref]

1976 (2)

C. M. Verber, D. W. Vahey, and V. E. Wood, “Focal properties of geodesic waveguide lenses,” Appl. Phys. Lett. 28, 514–516 (1976).
[Crossref]

Van E. Wood, “Effect of edge-rounding on geodesic lenses,” Appl.Opt. 15, 2817–2820 (1976).
[Crossref]

1974 (1)

1973 (1)

1971 (1)

1968 (1)

Anderson, D. B.

D. B. Anderson, R. L. Davis, J. T. Boyd, and R. R. August, “Comparison of Optical-Waveguide Lens Technologies,” IEEE J. Quantum Electron. QE-13, 275–281 (1977).
[Crossref]

August, R. R.

D. B. Anderson, R. L. Davis, J. T. Boyd, and R. R. August, “Comparison of Optical-Waveguide Lens Technologies,” IEEE J. Quantum Electron. QE-13, 275–281 (1977).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), Chaps. 5 and 9.

Boyd, J. T.

D. B. Anderson, R. L. Davis, J. T. Boyd, and R. R. August, “Comparison of Optical-Waveguide Lens Technologies,” IEEE J. Quantum Electron. QE-13, 275–281 (1977).
[Crossref]

Davis, R. L.

D. B. Anderson, R. L. Davis, J. T. Boyd, and R. R. August, “Comparison of Optical-Waveguide Lens Technologies,” IEEE J. Quantum Electron. QE-13, 275–281 (1977).
[Crossref]

Felsen, L. B.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Chap. 5, Sec. 4.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5, Sec. 3.

Harper, J. S.

Harris, J. H.

Marcuse, D.

See, for example, D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), pp. 8–9. Note that our case is slightly more general and leads to the one considered in Ref. 13, if β·r→ βz.

Marcuvitz, N.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Chap. 5, Sec. 4.

Righini, G. C.

The problem of field curvature for geodesic lenses was pointed out by G. C. Righini, V. Russo, and S. Sottini, “The family of perfect aspherical geodesic lenses for integrated optical circuits,” IEEE J. Quantum Electron. QE-14, 1–3 (1978).

G. C. Righini, V. Russo, S. Sottini, and G. Toraldo di Francia, “Geodesic Lenses for Guided Optical Waves,” Appl. Opt. 12, 1477–1481 (1973).
[Crossref] [PubMed]

Russo, V.

The problem of field curvature for geodesic lenses was pointed out by G. C. Righini, V. Russo, and S. Sottini, “The family of perfect aspherical geodesic lenses for integrated optical circuits,” IEEE J. Quantum Electron. QE-14, 1–3 (1978).

G. C. Righini, V. Russo, S. Sottini, and G. Toraldo di Francia, “Geodesic Lenses for Guided Optical Waves,” Appl. Opt. 12, 1477–1481 (1973).
[Crossref] [PubMed]

Shewell, J. R.

Shubert, R.

Sottini, S.

The problem of field curvature for geodesic lenses was pointed out by G. C. Righini, V. Russo, and S. Sottini, “The family of perfect aspherical geodesic lenses for integrated optical circuits,” IEEE J. Quantum Electron. QE-14, 1–3 (1978).

G. C. Righini, V. Russo, S. Sottini, and G. Toraldo di Francia, “Geodesic Lenses for Guided Optical Waves,” Appl. Opt. 12, 1477–1481 (1973).
[Crossref] [PubMed]

Southwell, W. H.

Spiller, E.

Tien, P. K.

P. K. Tien, “Integrated optics and new wave phenomena in optical waveguides,” Rev. Mod. Phys. 49, 361–420 (1977).
[Crossref]

Toraldo di Francia, G.

Vahey, D. W.

D. W. Vahey and V. E. Wood, “Focal Characteristics of Spheroidal Geodesic Lenses for Integrated Optical Processing,” IEEE J. Quantum Electron. QE-13, 129–133 (1977).
[Crossref]

C. M. Verber, D. W. Vahey, and V. E. Wood, “Focal properties of geodesic waveguide lenses,” Appl. Phys. Lett. 28, 514–516 (1976).
[Crossref]

Verber, C. M.

C. M. Verber, D. W. Vahey, and V. E. Wood, “Focal properties of geodesic waveguide lenses,” Appl. Phys. Lett. 28, 514–516 (1976).
[Crossref]

Wolf, E.

Wood, V. E.

D. W. Vahey and V. E. Wood, “Focal Characteristics of Spheroidal Geodesic Lenses for Integrated Optical Processing,” IEEE J. Quantum Electron. QE-13, 129–133 (1977).
[Crossref]

C. M. Verber, D. W. Vahey, and V. E. Wood, “Focal properties of geodesic waveguide lenses,” Appl. Phys. Lett. 28, 514–516 (1976).
[Crossref]

Wood, Van E.

Van E. Wood, “Effect of edge-rounding on geodesic lenses,” Appl.Opt. 15, 2817–2820 (1976).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

C. M. Verber, D. W. Vahey, and V. E. Wood, “Focal properties of geodesic waveguide lenses,” Appl. Phys. Lett. 28, 514–516 (1976).
[Crossref]

Appl.Opt. (1)

Van E. Wood, “Effect of edge-rounding on geodesic lenses,” Appl.Opt. 15, 2817–2820 (1976).
[Crossref]

IEEE J. Quantum Electron. (3)

D. W. Vahey and V. E. Wood, “Focal Characteristics of Spheroidal Geodesic Lenses for Integrated Optical Processing,” IEEE J. Quantum Electron. QE-13, 129–133 (1977).
[Crossref]

The problem of field curvature for geodesic lenses was pointed out by G. C. Righini, V. Russo, and S. Sottini, “The family of perfect aspherical geodesic lenses for integrated optical circuits,” IEEE J. Quantum Electron. QE-14, 1–3 (1978).

D. B. Anderson, R. L. Davis, J. T. Boyd, and R. R. August, “Comparison of Optical-Waveguide Lens Technologies,” IEEE J. Quantum Electron. QE-13, 275–281 (1977).
[Crossref]

J. Opt. Soc. Am. (5)

Rev. Mod. Phys. (1)

P. K. Tien, “Integrated optics and new wave phenomena in optical waveguides,” Rev. Mod. Phys. 49, 361–420 (1977).
[Crossref]

Other (5)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5, Sec. 3.

See, for example, D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), pp. 8–9. Note that our case is slightly more general and leads to the one considered in Ref. 13, if β·r→ βz.

The relation given in Ref. 10 is erroneous. This form is correct.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), Chaps. 5 and 9.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Chap. 5, Sec. 4.

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Figures (7)

FIG. 1
FIG. 1

Geometry of the planar waveguide. n0 eff is the constant effective refractive index of the thin film.

FIG. 2
FIG. 2

Geometry of a planar system including a thin planar lens. I represents the one-dimensional aperture of the lens, δ is the thickness of the lens, do denotes the distance between the object line and the lens, di the distance between the observation line and the lens.

FIG. 3
FIG. 3

Normalized impulse response |h|/|hG(0)| versus fl. hG is the impulse response without aberrations, hG(0) is the value of the impulse response in the Gaussian image plane, f is a spatial frequency. Curve (1) is the sinc function, curves (2), and (3), and (4) are impulse responses including aberrations for the cases y0 = 0, y0 = I/4, and y0 = I/2, respectively (y0 is the coordinate of the point source). Values of the parameters assumed: I = 6 mm, n0 eff = 1.55, λ0 = 0.6328 μm, and F = 40 n 0 eff 1 / 3 mm.

FIG. 4
FIG. 4

Maximum phase error versus fl for the following kinds of aberrations: spherical aberration–ΔΦA, coma–ΔΦB, field curvature–ΔΦc, and distortion–ΔΦD. The values of the parameters are as in Fig. 3 and y = l/2 = 3 mm.

FIG. 5
FIG. 5

Maximum phase error ΔΦA induced by spherical aberration versus the distance between the object line and the lens (do) for the following assumed values of the parameters: λ0 = 0.6328 μm, l = 6 mm, F = 40 n 0 eff 1 / 3 mm, y = l/2 mm.

FIG. 6
FIG. 6

Response |H| of a lens working as a Fourier transformer with a rectangular aperture versus fl. Curve (1) is a sinc function, curve (2) denotes the response for α = 0 (a is the angle of incidence of the guided wave on the lens), and curve (3) is the response for α = tan 1 [ ( 3 / 20 ) n 0 eff 1 / 3 ].

FIG. 7
FIG. 7

Response |HM|of a lens working as a Fourier transformer with a Gaussian aperture versus fl. Curve (1) is a sinc function, curve (2) denotes a response for α = 0, and a = l/2 = 3 mm [a described in Eq. (16)], and curve (3) is a response for α = tan 1 [ ( 3 / 20 ) n 0 eff 1 / 3 ], a = l/2 = 3 mm. Values for the other parameters as in Fig. 3.

Equations (31)

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H x ( x , y , z ) = e i β · r ( A e ibx + B e ibx ) ,
H x ( x , y , z ) = U ( y , z ) ( A e ibx + B e ibx ) ,
( 2 y 2 + 2 z 2 + β 2 ) U ( y , z ) = 0 ,
e × Re { ( 1 / 2 ) E × H * } = 0 ,
H = ( ω μ 0 ) 1 β × E = ( / μ 0 ) 1 / 2 e × E ,
S = e ( / μ 0 ) | E | 2 = e υ w ,
U ( y , z ) = 1 2 π + Ũ 0 ( β y ) e i ( y β y + z β z ) d β y ,
β z = { ( β 2 β y 2 ) 1 / 2 for | β y | β ( homogeneous ) . i ( β y 2 β 2 ) 1 / 2 for | β y | > β ( evanescent ) .
U ( y ) = i 2 + U ( y 0 ) z { H 0 ( 1 ) ( β r ) } d y 0 ,
U ( y ) = ( β / 2 π z ) 1 / 2 e i 3 π / 4 + U ( y 0 ) e i β r d y 0 .
[ 2 y 2 + 2 z 2 + k 0 2 n eff 2 ( y , z ) ] U ( y , z ) = 0 ,
n eff ( y , z ) = n 0 eff + Δ n max [ 1 y 2 / ( l / 2 ) 2 ] × exp [ 2 z 2 / ( δ / 3 ) 2 ] ,
F = 3 8 π n 0 eff l 2 Δ n max δ .
U 2 ( y , z ) = exp [ i k 0 z 1 z 2 n eff ( y , z ) d z ] U 1 ( y , z ) ,
t ( y ) = exp ( i β Δ 0 ) exp ( i β y 2 / F ) ,
U 2 ( y ) = P ( y ) t ( y ) U 1 ( y ) ,
P ( y ) = exp ( y 2 / a 2 ) rect ( y / l )
U 1 ( y ) = exp ( i π / 4 ) λ d 0 exp [ i β ( 1 2 d 0 ( y y 0 ) 2 1 8 d 0 3 ( y y 0 ) 4 ) ] ,
h ( y i , y 0 ) = exp [ i 3 π / 4 ] λ d i + U 2 ( y ) × exp [ i β ( 1 2 d i ( y i y ) 2 1 8 d i 3 ( y i y ) 4 ) ] d y ,
h ( y i , y 0 ) = exp [ i π / 2 ] λ d i d 0 × exp { i β [ Δ 0 + 1 2 ( y 0 2 d 0 + y i 2 d i ) 1 8 ( y 0 4 d 0 3 + y i 4 d i 3 ) ] } × + P ( y ) P A B ( y ; y 0 , y i ) exp [ i β 2 ( 1 d 0 + 1 d i 1 F ) y 2 ] × exp ( 2 π ify ) d y ,
P A B ( y ; y 0 , y i ) = exp ( i Δ Φ ) ,
Δ Φ = β 8 ( 1 d 0 3 + 1 d i 3 ) y 4 + β 2 ( y 0 d 0 3 + y i d i 3 ) y 3 3 β 4 ( y 0 2 d 0 3 + y i 2 d i 3 ) y 2 + β 2 ( y 0 3 d 0 3 + y i 3 d i 3 ) y .
F = 40 n 0 eff 1 / 3 m m .
U ( y , z ) = 1 2 π + Ũ ( β y , z ) exp ( i y β y ) d β y ,
Ũ ( β y , z ) = F ̂ { U ( y , z ) } = + U ( y , z ) exp ( i y β y ) d y .
Ũ ( β y , z ) = exp [ i z β z ] Ũ 0 ( β y )
Ũ ( β y , z 2 ) = exp [ i β z ( z 2 z 1 ) ] Ũ ( β y , z 1 ) .
U ( y 2 ) = + g ( y 2 y 1 ) U ( y 1 ) d y 1
g ( y 2 y 1 ) = 1 2 π + exp { i [ ( y 2 y 1 ) β y + ( z 2 z 1 ) β z ] } d β y .
H 0 ( 1 ) ( β r ) = 1 π + exp { i [ ( y 2 y 1 ) β y + ( z 2 z 1 ) β z ] } β z d β y ,
g ( y 2 y 1 ) = i 2 z 2 [ H 0 ( 1 ) ( β r ) ] .