Abstract

The variations of the refractive index and the wavelength along the axis of an integrated-optics lens are taken into account in an improved set of equations of the beam-propagation method. It is shown that this implementation removes an inconsistency between the predictions of the standard beam-propagation method and those of geometrical optics.

© 1992 Optical Society of America

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References

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  1. J. M. Cowley, A. E. Moodie, Acta Crystallogr. 10, 609 (1957).
    [CrossRef]
  2. D. Van Dyck, Adv. Electron. Electron Phys. 65, 295 (1985).
    [CrossRef]
  3. M. D. Feit, J. A. Fleck, Appl. Opt. 17, 3990 (1978).
    [CrossRef] [PubMed]
  4. J. Van Roey, J. van der Donk, P. E. Lagasse, J. Opt. Soc. Am. 71, 803 (1981).
    [CrossRef]
  5. G. Pozzi, Ultramicroscopy 30, 417 (1989).
    [CrossRef]
  6. G. Pozzi, Optik (Stuttgart) 85, 15 (1990).
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  8. G. Pozzi, in Proceedings of International Symposium on Huygens’ Principle 1690-1990, Theory and Applications, H. Blok, H. A. Ferwerda, H. K. Kuiken, eds. (North-Holland, Amsterdam, 1992), p. 521.
  9. J. J. Gribble, J. M. Arnold, Opt. Lett. 13, 611 (1988).
    [CrossRef] [PubMed]

1990 (1)

G. Pozzi, Optik (Stuttgart) 85, 15 (1990).

1989 (1)

G. Pozzi, Ultramicroscopy 30, 417 (1989).
[CrossRef]

1988 (1)

1985 (1)

D. Van Dyck, Adv. Electron. Electron Phys. 65, 295 (1985).
[CrossRef]

1981 (1)

1978 (1)

1957 (1)

J. M. Cowley, A. E. Moodie, Acta Crystallogr. 10, 609 (1957).
[CrossRef]

Arnold, J. M.

Cowley, J. M.

J. M. Cowley, A. E. Moodie, Acta Crystallogr. 10, 609 (1957).
[CrossRef]

Feit, M. D.

Fleck, J. A.

Goodman, J. W.

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

Gribble, J. J.

Lagasse, P. E.

Moodie, A. E.

J. M. Cowley, A. E. Moodie, Acta Crystallogr. 10, 609 (1957).
[CrossRef]

Pozzi, G.

G. Pozzi, Optik (Stuttgart) 85, 15 (1990).

G. Pozzi, Ultramicroscopy 30, 417 (1989).
[CrossRef]

G. Pozzi, in Proceedings of International Symposium on Huygens’ Principle 1690-1990, Theory and Applications, H. Blok, H. A. Ferwerda, H. K. Kuiken, eds. (North-Holland, Amsterdam, 1992), p. 521.

van der Donk, J.

Van Dyck, D.

D. Van Dyck, Adv. Electron. Electron Phys. 65, 295 (1985).
[CrossRef]

Van Roey, J.

Acta Crystallogr. (1)

J. M. Cowley, A. E. Moodie, Acta Crystallogr. 10, 609 (1957).
[CrossRef]

Adv. Electron. Electron Phys. (1)

D. Van Dyck, Adv. Electron. Electron Phys. 65, 295 (1985).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Optik (Stuttgart) (1)

G. Pozzi, Optik (Stuttgart) 85, 15 (1990).

Ultramicroscopy (1)

G. Pozzi, Ultramicroscopy 30, 417 (1989).
[CrossRef]

Other (2)

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

G. Pozzi, in Proceedings of International Symposium on Huygens’ Principle 1690-1990, Theory and Applications, H. Blok, H. A. Ferwerda, H. K. Kuiken, eds. (North-Holland, Amsterdam, 1992), p. 521.

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Figures (2)

Fig. 1
Fig. 1

Contour map of the intensity of the wave front near the focal region.

Fig. 2
Fig. 2

Ray path for deriving the ray equation.

Equations (22)

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n 0 2 ( x , y , z ) = n 0 2 ( z ) - n 2 ( z ) ( x 2 + y 2 ) ,
T ( x , y , z i ) = n 0 ( z i ) n 0 ( z i + 1 ) exp [ - i π λ ( x 2 + y 2 ) × n 2 ( z i ) n 0 ( z i ) Δ z ] ,
ψ ( x , y , z i + 1 ) = ψ ( x , y , z i ) i λ i Δ z exp { i π λ i Δ z [ ( x - x ) 2 + ( y - y ) 2 ] } d x d y ,
ϕ ( x , y , z 0 ) = A ( z 0 ) exp [ 2 π i λ β ( z 0 ) ( x 2 + y 2 ) ] ,
ϕ ( x , y , z ) = A ( z 0 ) n 0 3 / 2 ( z 0 ) [ n 0 ( z 0 ) g ( z ) + 2 β ( z 0 ) h ( z ) ] n 0 ( z ) × exp { + i π n 0 ( z ) λ [ n 0 ( z 0 ) g ( z ) + 2 β ( z 0 ) h ( z ) n 0 ( z 0 ) g ( z ) + 2 β ( z 0 ) h ( z ) ] × ( x 2 + y 2 ) } ,
d d z [ n 0 ( z ) d r ( z ) d z ] + n 2 ( z ) n 0 ( z ) r ( z ) = 0
g ( z 0 ) = 1 = h ( z 0 ) ,             h ( z 0 ) = 0 = g ( z 0 ) .
n ( x , y , z ) = n ˜ / ( 1 + x 2 + y 2 + z 2 a 2 ) .
n 2 ( x , y , z ) = n ˜ 2 a 4 ( a 2 + z 2 ) 2 - 2 n ˜ 2 a 4 ( a 2 + z 2 ) 3 ( x 2 + y 2 ) ,
n 0 ( z ) = n ˜ a 2 a 2 + z 2
n 2 ( z ) = 2 n ˜ 2 a 4 ( a 2 + z 2 ) 3 .
( a 2 + z 2 ) d 2 r d z 2 - 2 z d r d z + 2 r = 0 ,
g ( z ) = ( 1 a 2 + z 0 2 ) z 2 + ( 2 z 0 2 a 2 + z 0 2 ) z + a 2 a 2 + z 0 2 ,
h ( z ) = ( z 0 a 2 + z 0 2 ) z 2 + ( a 2 - z 0 2 a 2 + z 0 2 ) z - a 2 z 0 2 a 2 + z 0 2 .
φ = 2 π λ n 0 ( z i - 1 ) Δ r sin θ ( z i - 1 ) - π λ ( r + Δ r ) 2 n 2 ( z i ) n 0 ( z i ) Δ z .
φ = 2 π λ n 0 ( z i ) Δ r sin θ ( z i ) - π λ r 2 n 2 ( z i ) n 0 ( z i ) Δ z .
n 0 ( z i ) sin θ ( z i ) - n 0 ( z i - 1 ) sin θ ( z i - 1 ) Δ z + 1 2 [ ( r + Δ r ) 2 - r 2 Δ r ] n 2 ( z i ) n 0 ( z i ) = 0 ,
n 0 ( z i ) sin θ ( z i ) - n 0 ( z i - 1 ) sin θ ( z i - 1 ) Δ z + n 2 ( z i ) n 0 ( z i ) = 0.
tan θ ( z ) = d r d z ,
d d z [ n 0 ( z ) sin θ ( z ) ] + n 2 ( z ) n 0 ( z ) r = 0.
sin θ ( z ) = tan θ ( z ) ,
d d z [ n 0 ( z ) d r d z ] + n 2 ( z ) n 0 ( z ) r = 0.

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