Abstract

We study how a converging spherical wave gets distorted by a plane dielectric interface. The fields in the second medium are obtained by evaluating the m-theory diffraction integral on the interface. The loss of intensity and the form of the intensity distribution are investigated. Examples are presented for various refractive-index contrasts and depths of focus. In general the intensity gets spread out over a volume that is large compared with the case without refractive-index contrast. It was found that moving the focusing lens a distance d toward the interface does not result in an equal shift of the intensity profile. This latter point has important practical implications.

© 1996 Optical Society of America

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References

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  1. E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London A 253, 349–357 (1959).
    [Crossref]
  2. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
    [Crossref]
  3. F. Kottler, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. IV, pp. 283–314.
  4. F. Kottler, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. VI, pp. 333–377.
  5. B. Karczewski, E. Wolf, “Comparison of three theories of electromagnetic diffraction at an aperture. Part I: coherence matrices,” J. Opt. Soc. Am. 56, 1207–1214 (1966).
    [Crossref]
  6. B. Karczewski, E. Wolf, “Comparison of three theories of electromagnetic diffraction at an aperture. Part II: the far field,” J. Opt. Soc. Am. 56, 1214–1219 (1966).
    [Crossref]
  7. A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561–B1565 (1965).
    [Crossref]
  8. C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
    [Crossref]
  9. W. R. Smythe, “The double current sheet in diffraction,” Phys. Rev. 72, 1066–1070 (1947).
    [Crossref]
  10. H. Severin, “Zur Theorie der Beugung elektromagnetischer Wellen,” Z. Phys. 129, 426–439 (1951).
    [Crossref]
  11. G. Toraldo di Francia, Electromagnetic Waves (Interscience, New York, 1955).
  12. B. B. Baker, E. T. Copson, The Mathematical Theory of Huygen’s Principle, 2nd ed. (Clarendon, Oxford, 1950).
  13. T. D. Visser, S. H. Wiersma, “Diffraction of converging electromagnetic waves,” J. Opt. Soc. Am. A 9, 2034–2047 (1992).
    [Crossref]
  14. H. Ling, S-W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
    [Crossref]
  15. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986). This reference gives an alternative account of the work of Ling and Lee.
  16. S. Nemoto, “Waist shift of a Gaussian beam by plane dielectric interfaces,” Appl. Opt. 27, 1833–1839 (1988).
    [Crossref] [PubMed]
  17. J. Gasper, G. C. Sherman, J. J. Stamnes, “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface,” J. Opt. Soc. Am. 66, 955–961 (1976).
    [Crossref]
  18. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  19. J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
    [Crossref]
  20. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941). This book gives a clear derivation of the uniqueness theorem.
  21. T. D. Visser, J. L. Oud, “Volume measurements in 3-D microscopy,” Scanning 16, 198–200 (1994).
    [Crossref]

1994 (1)

T. D. Visser, J. L. Oud, “Volume measurements in 3-D microscopy,” Scanning 16, 198–200 (1994).
[Crossref]

1992 (1)

1988 (1)

1984 (1)

1976 (1)

1966 (2)

1965 (1)

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561–B1565 (1965).
[Crossref]

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London 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 A 253, 358–379 (1959).
[Crossref]

1954 (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[Crossref]

1951 (1)

H. Severin, “Zur Theorie der Beugung elektromagnetischer Wellen,” Z. Phys. 129, 426–439 (1951).
[Crossref]

1947 (1)

W. R. Smythe, “The double current sheet in diffraction,” Phys. Rev. 72, 1066–1070 (1947).
[Crossref]

1939 (1)

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

Baker, B. B.

B. B. Baker, E. T. Copson, The Mathematical Theory of Huygen’s Principle, 2nd ed. (Clarendon, Oxford, 1950).

Boivin, A.

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561–B1565 (1965).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Bouwkamp, C. J.

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[Crossref]

Chu, L. J.

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

Copson, E. T.

B. B. Baker, E. T. Copson, The Mathematical Theory of Huygen’s Principle, 2nd ed. (Clarendon, Oxford, 1950).

Gasper, J.

Karczewski, B.

Kottler, F.

F. Kottler, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. IV, pp. 283–314.

F. Kottler, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. VI, pp. 333–377.

Lee, S-W.

Ling, H.

Nemoto, S.

Oud, J. L.

T. D. Visser, J. L. Oud, “Volume measurements in 3-D microscopy,” Scanning 16, 198–200 (1994).
[Crossref]

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 A 253, 358–379 (1959).
[Crossref]

Severin, H.

H. Severin, “Zur Theorie der Beugung elektromagnetischer Wellen,” Z. Phys. 129, 426–439 (1951).
[Crossref]

Sherman, G. C.

Smythe, W. R.

W. R. Smythe, “The double current sheet in diffraction,” Phys. Rev. 72, 1066–1070 (1947).
[Crossref]

Stamnes, J. J.

J. Gasper, G. C. Sherman, J. J. Stamnes, “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface,” J. Opt. Soc. Am. 66, 955–961 (1976).
[Crossref]

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986). This reference gives an alternative account of the work of Ling and Lee.

Stratton, J. A.

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941). This book gives a clear derivation of the uniqueness theorem.

Toraldo di Francia, G.

G. Toraldo di Francia, Electromagnetic Waves (Interscience, New York, 1955).

Visser, T. D.

T. D. Visser, J. L. Oud, “Volume measurements in 3-D microscopy,” Scanning 16, 198–200 (1994).
[Crossref]

T. D. Visser, S. H. Wiersma, “Diffraction of converging electromagnetic waves,” J. Opt. Soc. Am. A 9, 2034–2047 (1992).
[Crossref]

Wiersma, S. H.

Wolf, E.

B. Karczewski, E. Wolf, “Comparison of three theories of electromagnetic diffraction at an aperture. Part II: the far field,” J. Opt. Soc. Am. 56, 1214–1219 (1966).
[Crossref]

B. Karczewski, E. Wolf, “Comparison of three theories of electromagnetic diffraction at an aperture. Part I: coherence matrices,” J. Opt. Soc. Am. 56, 1207–1214 (1966).
[Crossref]

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561–B1565 (1965).
[Crossref]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London 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 A 253, 349–357 (1959).
[Crossref]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

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

Phys. Rev. (3)

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561–B1565 (1965).
[Crossref]

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

W. R. Smythe, “The double current sheet in diffraction,” Phys. Rev. 72, 1066–1070 (1947).
[Crossref]

Proc. R. Soc. London A (2)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London 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 A 253, 358–379 (1959).
[Crossref]

Rep. Prog. Phys. (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[Crossref]

Scanning (1)

T. D. Visser, J. L. Oud, “Volume measurements in 3-D microscopy,” Scanning 16, 198–200 (1994).
[Crossref]

Z. Phys. (1)

H. Severin, “Zur Theorie der Beugung elektromagnetischer Wellen,” Z. Phys. 129, 426–439 (1951).
[Crossref]

Other (7)

G. Toraldo di Francia, Electromagnetic Waves (Interscience, New York, 1955).

B. B. Baker, E. T. Copson, The Mathematical Theory of Huygen’s Principle, 2nd ed. (Clarendon, Oxford, 1950).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

F. Kottler, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. IV, pp. 283–314.

F. Kottler, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. VI, pp. 333–377.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941). This book gives a clear derivation of the uniqueness theorem.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986). This reference gives an alternative account of the work of Ling and Lee.

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Figures (6)

Fig. 1
Fig. 1

Definition of the coordinate system. Shown at left are the unit wave vector k ^ and the electric vector Einc, both before refraction by an objective with semiaperture angle Ω. The incoming wave propagates perpendicular to the interface in the −z direction. The origin is placed at a distance f from the exit pupil. n ^is the unit wave vector after refraction by the lens.

Fig. 2
Fig. 2

Axial intensity distribution (in arbitrary units) for n1 = 1.51 and n2 = 1.33 (curve a). At center is shown the intensity profile without contrast, i.e., n1 = n2 = 1.51 (curve b). Curve c depicts the intensity for n1 = 1.33 and n2 = 1.51. (For all curves Ω = 60°, μ1 = μ2 = μ0, f = 10−2 m, fd = 50 μm, and λ = 632.8 nm.) As in all following examples both media are lossless.

Fig. 3
Fig. 3

Isointensity lines (a.u.) in the xy plane of maximum intensity (z = 7.54 nm). (Ω = 60°, μ1 = μ2 = μ0, f = 10−2 m, fd = 50 μm, and λ = 632.8 nm.)

Fig. 4
Fig. 4

Isointensity lines (a.u.) in the xz plane (y = 0). (Ω = 60°, μ1 = μ2 = μ0, f = 10−2 m, fd = 50 μm, and λ = 632.8 nm.) Note: the scale of the two axes is different.

Fig. 5
Fig. 5

Distance between the peak and the interface plotted versus the position of the lens. (Note: the distance between the lens and the interface is given by d = fzi.) Only if n1 = n2 (curve b) does the peak precisely follow the movement of the lens. If n1 > n2 (curve c) the peak position shifts less than that of the lens. For n1 < n2 (curve a) the opposite holds. In all cases Ω = 60°, μ1 = μ2 = μ0, f = 10−2 m, and λ = 632.8 nm. In curve a n1 = 1.33, n2 = 1.51, in curve b n1 = n2 = 1.33, and in curve c n1 = 1.51, n2 = 1.33.

Fig. 6
Fig. 6

Peak intensity (a.u.) versus n1. The index of refraction n2 is fixed at 1.33. (Ω = 60°, μ1 = μ2 = μ0, f = 10−2 m, fd = 50 μm, and λ = 632.8 nm.)

Equations (34)

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E = E inc exp [ i ( k 1 k ^ · r + ω t ) ] ,             ( z > f )
E inc = ( cos α , sin α , 0 ) ,
k ^ = ( 0 0 - 1 ) ,             n ^ = - ( sin θ cos ϕ sin θ sin ϕ cos θ ) .
E s = E inc · ( k ^ × n ^ ) k ^ × n ^ 2 ( k ^ × n ^ ) ,
E p = E inc · [ k ^ × ( k ^ × n ^ ) ] k ^ × ( k ^ × n ^ ) n ^ × ( k ^ × n ^ ) [ n ^ × ( k ^ × n ^ ) ] .
k ^ × n ^ k ^ × n ^ = ( - sin ϕ cos ϕ 0 ) ,
n ^ × ( k ^ × n ^ ) n ^ × ( k ^ × n ^ ) = ( cos θ cos ϕ cos θ sin ϕ - sin θ ) ,
k ^ × ( k ^ × n ^ ) k ^ × ( k ^ × n ^ ) = ( cos ϕ sin ϕ 0 ) .
E s = cos 1 / 2 θ sin ϕ ( sin ϕ - cos ϕ 0 ) ,
E p = cos 1 / 2 θ cos ϕ ( cos θ cos ϕ cos θ sin ϕ - sin θ ) .
E = E S 1 exp [ i k 1 n ^ · r ] ,             ( in exit pupil )
E S 1 = E s + E p .
t ( θ ) = ( f - d ) / cos θ ,
F ( θ ) = exp [ i k 1 ( f - f - d cos θ ) ] .
k i ( ω 2 i μ i ) 1 / 2 ,             ( i = 1 , 2 )
K ( θ ) = f cos 3 / 2 θ f - d .
E δ 0 ( θ , ϕ , z i + δ ) = K ( θ ) F ( θ ) E S 1 ( θ , ϕ ) .
q ^ = - ( sin θ cos ϕ sin θ sin ϕ cos θ ) ,
E p ; δ 0 ( z i - δ ) = η p E p ; δ 0 ( z i + δ ) q ^ × ( q ^ × n ^ ) q ^ × ( q ^ × n ^ ) .
E s ; δ 0 ( z i - δ ) = η s E s ; δ 0 ( z i + δ ) ,
E δ 0 ( z i - δ ) = E s ; δ 0 ( z i - δ ) + E p ; δ 0 ( z i - δ ) , = K ( θ ) F ( θ ) cos 1 / 2 θ [ η s sin ϕ ( sin ϕ - cos ϕ 0 ) [ + η p cos ϕ ( cos θ cos ϕ cos θ sin ϕ - sin θ ) ] ,
m ^ × E ( θ , ϕ , z i ) = K ( θ ) F ( θ ) cos 1 / 2 θ [ η s sin ϕ ( cos ϕ sin ϕ 0 ) [ + η p cos ϕ ( - cos θ sin ϕ cos θ cos ϕ 0 ) ] .
E ( x ) = 2 S ( m ^ × E ) × G d σ .
G ( p , x ) = exp ( i k 2 x - p ) 4 π x - p ,
G = ( 1 x - p - i k 2 ) G e ^ G .
e ^ G = x - p x - p .
d σ = t 2 ( θ ) tan θ d θ d ϕ             ( 0 θ Ω ) ,
e ^ G = - f - d s ( θ , z ) ( tan θ cos ϕ tan θ sin ϕ 1 - z / ( f - d ) ) ,
s ( θ , z ) = [ t 2 ( θ ) + z 2 - 2 z ( f - d ) ] 1 / 2 ,
E x ( 0 , 0 , z ) = C 0 Ω exp [ i ( k 2 s - k 1 t ) ] g ( θ , z ) d θ ,
C ( z ) = f 2 ( f - d ) 2 ( z f - d - 1 ) exp [ i k 1 f ] ,
g ( θ , z ) = ( 1 s 3 - i k 2 s 2 ) ( η s + η p cos θ ) tan θ .
h ( θ 1 ) = ( f - d ) tan θ 1 tan θ 2 = ( f - d ) n 2 n 1 cos θ 2 cos θ 1 .
Δ peak Δ lens = - h ( θ = 0 ) d = n 2 n 1 .

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