Abstract

We consider the focusing of light by a high-aperture lens into a stratified medium. The solution is based on our previously obtained results [J. Opt. Soc. Am. A 12, 325 (1995)], where we represented the illumination incident upon a plane interface between media with mismatched refractive indices as a sum of plane waves. The present solution is obtained in terms of plane waves, and it satisfies Maxwell’s equations. The diffraction integrals are obtained in a form that is readily computable. We present numerical examples for some practical cases.

© 1997 Optical Society of America

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References

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  1. R. D. Vré, L. Hesselnik, “Analysis of photorefractive stratified volume holography optical elements,” J. Opt. Soc. Am. B 9, 1800–1808 (1994).
    [CrossRef]
  2. C. J. R. Sheppard, T. J. Connolly, J. Lee, C. J. Cogswell, “Confocal imaging of a stratified medium,” Appl. Opt. 33, 631–640 (1994).
    [CrossRef] [PubMed]
  3. P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
    [CrossRef]
  4. After submission of the present manuscript the authors were informed that a similarly rigorous theory appeared by V. Dhayalan, entitled “Focusing of electromagnetic waves,” Ph.D. dissertation (University of Bergen, Norway, 1996).
  5. E. Wolf, “Electromagnetic diffraction in optical systems, I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
    [CrossRef]
  6. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  7. P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755–776 (1909).
    [CrossRef]
  8. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970).
  9. P. Török, P. Varga, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
    [CrossRef]
  10. P. Török, P. Varga, A. Konkol, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. II,” J. Opt. Soc. Am. A 13, 2232–2238 (1996).
    [CrossRef]
  11. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannely, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

1996 (1)

P. Török, P. Varga, A. Konkol, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. II,” J. Opt. Soc. Am. A 13, 2232–2238 (1996).
[CrossRef]

1995 (2)

P. Török, P. Varga, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
[CrossRef]

P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
[CrossRef]

1994 (2)

R. D. Vré, L. Hesselnik, “Analysis of photorefractive stratified volume holography optical elements,” J. Opt. Soc. Am. B 9, 1800–1808 (1994).
[CrossRef]

C. J. R. Sheppard, T. J. Connolly, J. Lee, C. J. Cogswell, “Confocal imaging of a stratified medium,” Appl. Opt. 33, 631–640 (1994).
[CrossRef] [PubMed]

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems, I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

1909 (1)

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755–776 (1909).
[CrossRef]

Booker, G. R.

P. Török, P. Varga, A. Konkol, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. II,” J. Opt. Soc. Am. A 13, 2232–2238 (1996).
[CrossRef]

P. Török, P. Varga, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
[CrossRef]

P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970).

Cogswell, C. J.

Connolly, T. J.

Debye, P.

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755–776 (1909).
[CrossRef]

Dhayalan, V.

After submission of the present manuscript the authors were informed that a similarly rigorous theory appeared by V. Dhayalan, entitled “Focusing of electromagnetic waves,” Ph.D. dissertation (University of Bergen, Norway, 1996).

Flannely, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannely, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Hesselnik, L.

R. D. Vré, L. Hesselnik, “Analysis of photorefractive stratified volume holography optical elements,” J. Opt. Soc. Am. B 9, 1800–1808 (1994).
[CrossRef]

Konkol, A.

P. Török, P. Varga, A. Konkol, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. II,” J. Opt. Soc. Am. A 13, 2232–2238 (1996).
[CrossRef]

Laczik, Z.

P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
[CrossRef]

Lee, J.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannely, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Sheppard, C. J. R.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannely, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Török, P.

P. Török, P. Varga, A. Konkol, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. II,” J. Opt. Soc. Am. A 13, 2232–2238 (1996).
[CrossRef]

P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
[CrossRef]

P. Török, P. Varga, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
[CrossRef]

Varga, P.

P. Török, P. Varga, A. Konkol, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. II,” J. Opt. Soc. Am. A 13, 2232–2238 (1996).
[CrossRef]

P. Török, P. Varga, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
[CrossRef]

P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannely, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Vré, R. D.

R. D. Vré, L. Hesselnik, “Analysis of photorefractive stratified volume holography optical elements,” J. Opt. Soc. Am. B 9, 1800–1808 (1994).
[CrossRef]

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems, I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970).

Ann. Phys. (1)

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755–776 (1909).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. A (2)

P. Török, P. Varga, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
[CrossRef]

P. Török, P. Varga, A. Konkol, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. II,” J. Opt. Soc. Am. A 13, 2232–2238 (1996).
[CrossRef]

J. Opt. Soc. Am. B (1)

R. D. Vré, L. Hesselnik, “Analysis of photorefractive stratified volume holography optical elements,” J. Opt. Soc. Am. B 9, 1800–1808 (1994).
[CrossRef]

J. Opt. Soc. Am. A (1)

P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
[CrossRef]

Proc. R. Soc. London Ser. A (2)

E. Wolf, “Electromagnetic diffraction in optical systems, I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Other (3)

After submission of the present manuscript the authors were informed that a similarly rigorous theory appeared by V. Dhayalan, entitled “Focusing of electromagnetic waves,” Ph.D. dissertation (University of Bergen, Norway, 1996).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannely, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970).

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Figures (5)

Fig. 1
Fig. 1

Diagram showing light focused by a lens into a single medium.

Fig. 2
Fig. 2

Diagram showing light focused by a lens into two media separated by a planar interface.

Fig. 3
Fig. 3

Time-averaged electric-energy density distributions as functions of the scan position for a lens numerical aperture of 0.9 and for focusing from air (n1 = 1) through a cover glass (n2 = 1.54) into an aqueous medium (n3 = 1.33). The thicknesses of the cover glasses are 120 (solid curve), 170 (dashed curve), and 220 µm (dashed-dotted curve) and the depth in the aqueous medium is 50 µm. The lens is not corrected for the cover-glass thickness.

Fig. 4
Fig. 4

Time-averaged electric-energy density distributions as functions of the scan position for a lens numerical aperture of 0.9 and for focusing from air (n1 = 1) through a cover glass (n2 = 1.54) into an aqueous medium (n3 = 1.33). The thicknesses of the cover glasses are 120 (solid curve), 170 (dashed curve), and 220 µm (dashed-dotted curve) and the depth in the aqueous medium is 50 µm. The lens is corrected for a 170-µm cover glass.

Fig. 5
Fig. 5

Time-averaged electric-energy density distributions as functions of scan position for a lens numerical aperture of 0.9 and for focusing from air (n1 = 1) through a cover glass (n2 = 1.54) into an aqueous medium (n3 = 1.33). The lens is corrected for a 170-µm cover glass. The solid curve corresponds to a 0-µm depth and a 170-µm cover-glass thickness, whereas the dashed curve corresponds to a 50-µm depth and a 120-µm cover-glass thickness.

Equations (48)

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E P = - ik 1 2 π Ω a s 1 x ,   s 1 y s 1 z exp ik 1 s ˆ 1 · r p d s 1 x d s 1 y ,
E 1 1 x ,   y ,   - h 1 = - ik 1 2 π Ω 1 a s 1 x ,   s 1 y s 1 z × exp ik 1 s 1 x x + s 1 y y - s 1 z h 1 d s 1 x d s 1 y .
E N x ,   y ,   - h N - 1 = - ik 1 2 π Ω 1 Λ W s ˆ 1 × exp ik 1 s 1 x x + s 1 y y - s 1 z h N - 1 × d s 1 x d s 1 y .
E N x ,   y ,   z = - ik N 2 π Ω N F × exp ik N s Nx x + s Ny y - s Nz z d s Nx d s Ny .
k 2 s ˆ 2 - k 1 s ˆ 1 = k 2   cos   θ 2 - k 1   cos   θ 1 u ˆ ,
k N s Nx = = k 2 s 2 x = k 1 s 1 x ,   k N s Ny = = k 2 s 2 y = k 1 s 1 y .
E N x ,   y ,   z = - ik 1 2 π Ω 1 Λ W s ˆ 1 × exp ik 0 n N h N - 1 s Nz - n 1 h 1 s 1 z × exp ik N s Nz z exp ik 1 s 1 x x + s 1 y y d s 1 x d s 1 y .
s lz = 1 - k 1 2 k l 2 s 1 x 2 + s 1 y 2 1 / 2 .
s ˆ N = sin   θ N   cos   ϕ i ˆ + sin   θ N   sin   ϕ j ˆ + cos   θ N k ˆ ,
r p = r p ( sin   θ p   cos   ϕ p   i ˆ + sin   θ p   sin   ϕ p j ˆ + cos   θ p k ˆ .
e p , s , ζ = A θ 1 P 1 LRe 0 ,
R = cos   ϕ sin   ϕ 0 - sin   ϕ cos   ϕ 0 0 0 1 ,
L = cos   θ 1 0 sin   θ 1 0 1 0 - sin   θ 1 0 cos   θ 1 ,
P l = cos   θ l 0 - sin   θ l 0 1 0 sin   θ l 0 cos   θ l .
e N = R - 1 P N - 1 I N - 1 e p , s , ζ ,
I N - 1 = T p N - 1 0 0 0 T s N - 1 0 0 0 T p N - 1 .
T m N - 1 = t m N - 1 j = 1 N - 2 t m j exp i β j + 1 D m N - 1 ,
t s l = 2 n l   cos   θ l n l   cos   θ l + n n + 1   cos   θ l + 1 , t p l = 2 n l   cos   θ l n l + 1   cos   θ l + n l   cos   θ l + 1 ,
r s l = n l   cos   θ l - n l + 1   cos   θ l + 1 n l   cos   θ l + n l + 1   cos   θ l + 1 , r p l = n l + 1   cos   θ l - n l   cos   θ l + 1 n l + 1   cos   θ l + n l   cos   θ l + 1 .
e N = A θ 1 × T p N - 1   cos   θ N   cos 2   ϕ + T s N - 1   sin 2   ϕ T p N - 1   cos   θ N   sin   ϕ   cos   ϕ - T s N - 1   sin   ϕ   cos   ϕ - T p N - 1   sin   θ N   cos   ϕ .
A θ 1 = fl 0   cos 1 / 2   θ 1 ,
Λ W s ˆ 1 = e N .
κ = k 1   sin   θ 1   sin   θ p   cos ϕ - ϕ p ,
Ψ i = h N - 1 n N s Nz - n 1 h 1 s 1 z .
E N r p = - ik 1 2 π Ω 1 e N   exp ir p κ exp ik 0 Ψ i × exp ik N r p   cos   θ p   cos   θ N sin   θ 1 d θ 1 d ϕ .
E Nx = - iK I 0 N + I 2 N   cos   2 ϕ p ,   E Ny = - iK I 2 N   sin   2 ϕ p ,   E Nz = - 2 K I 1 N   cos   ϕ p ,
K = k 1 fl 0 2 ,
I 0 N = 0 α cos   θ 1   sin   θ 1   exp ik 0 Ψ i × T s N - 1 + T p N - 1   cos   θ N × J 0 k 1   sin   θ 1 r p   sin   θ p exp ik N r p   cos   θ p   cos   θ N d θ 1 , I 1 N = 0 α cos   θ 1   sin   θ 1   exp ik 0 Ψ i T p N - 1 sin   θ N × J 1 k 1   sin   θ 1 r p   sin   θ p exp ik N r p   cos   θ p   cos   θ N d θ 1 , I 2 N = 0 α cos   θ 1   sin   θ 1   exp ik 0 Ψ i × T s N - 1 - T p N - 1   cos   θ N × J 2 k 1   sin   θ 1 r p   sin   θ p exp ik N r p   cos   θ p   cos   θ N d θ 1 ,
T m N - 1 = t m N - 1 j = 1 N - 2 t m j exp i β j + 1 D m N - 1 ,
Ψ a = h 1 n 2   cos   θ 2 + h 2 n 3   cos   θ 3 - n 2   cos   θ 2 + + h N - 2 n N - 1   cos   θ N - 1 - n N - 2   cos   θ N - 2 - h N - 1 n N - 1   cos   θ N - 1 .
Ψ = j = 1 N - 1 h j n j + 1   cos   θ j + 1 - n j   cos   θ j ,
Ψ = - h 1 n 1   cos   θ 1 + h N - 1 n N   cos   θ N + l = 2 N - 1 h l - 1 - h l n l   cos   θ l .
Ψ = - h 1 n 1 1 - ρ 2   sin 2   α + h N - 1 n N 1 - n 1 n N 2   ρ 2   sin 2   α 1 / 2 + j = 2 N - 1 h j - 1 - h j n j 1 - n 1 n j 2 ρ 2   sin 2   α 1 / 2 ,
Ψ = - h 1 n 1 l = 0 1 l ! a l ρ 2 l   sin 2 l   α + h m l = 0 1 l ! a l n 1 n 1 n N 2 l - 1 ρ 2 l   sin 2 l   α + j = 2 N - 1 h j - 1 - h j l = 0 1 l ! a l n 1 n 1 n j 2 l - 1 ρ 2 l   sin 2 l   α ,
T s , p N - 1 = τ s , p = t 12 s , p t 23 s , p exp i β 1 + r 12 s , p r 23 s , p exp 2 i β ,
w e r p ,   z ,   θ p = 1 16 π E N · E N * .
T m N - 1 = B m N - 1 A m 1 ,
T m N - 1 = 2 p m 1 K 11 N - 1 + K 12 N - 1 p m N p m 1 + K 21 N - 1 + K 22 N - 1 p m N ,
p s j = n j   cos   θ j ,     p p j = 1 n j cos   θ j ,     j = 1 ,   2 ,   ,   N ,
K N - 1 = K N - 2 · M N - 1 h N - 1 - h N - 2 ,
M N - 1 = cos   β N - 1 - i p m N - 1 sin   β N - 1 - ip m N - 1   sin   β N - 1 cos   β N - 1 .
2 p m 1 T m N - 1 = K 1 j N - 2 M j 1 N - 1 + K 1 j N - 2 M j 2 N - 1 p m N p m 1 + K 2 j N - 2 M j 1 N - 1 + K 2 j N - 2 M j 2 N - 1 p m N ,
T m N - 1 = t m N - 1 T m N - 2   exp i β N - 1 D N - 2 ,
t m N - 1 = 2 p m N - 1 p m N - 1 + p m N
D n - 2 = 1 + 2 p m 1 r m N - 1 T N - 2 K 11 N - 2 - K 12 N - 2 p m N - 1 p m 1 + K 21 N - 2 - K 22 N - 2 p m N - 1 exp 2 i β N - 1 ,
r m N - 1 = p m N - 1 - p m N p m N - 1 + p m N ,
T m N - 1 = t m N - 1 j = 1 N - 2 t m j   exp i β j + 1 D m N - 1 ,
D m 2 = 1 + r m 1 r m 2   exp 2 i β 2 , D m 3 = 1 + r m 1 r m 2   exp 2 i β 2 , + r m 3   exp 2 i β 2 + β 3 + r m 2 r m 3   exp 2 i β 3 , D m 4 = 1 + r m 1 r m 2   exp 2 i β 2 + r m 3   exp 2 i β 2 + β 3 + r m 4   exp 2 i β 2 + β 3 + β 4 + r m 2 r m 3   exp 2 i β 3 + r m 4   exp 2 i β 3 + β 4 + r m 3   r m 4   exp 2 i β 4 + r m 1 r m 2 r m 3 r m 4   exp 2 i β 2 + β 4 .

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