Abstract

We consider the problem of light focusing by a high-aperture lens through a plane interface between two media with different refractive indices. We compare two recently published diffraction theories and a new geometrical optics description. The two diffraction approaches exhibit axial distributions with little difference. The description based on geometrical optics is shown to agree well with the diffraction optics results. Also, some implications for three-dimensional imaging are discussed.

© 1997 Optical Society of America

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References

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  1. 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]
  2. H. Ling, S.-W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
    [Crossref]
  3. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Chap. 16, pp. 482–500.
  4. S. Chang, J. H. Jo, S. S. Lee, “Theoretical calculations of optical force exerted on a dielectric sphere in the evanescent field generated with a totally-reflected focused Gaussian beam,” Opt. Commun. 108, 133–143 (1994).
    [Crossref]
  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. (Leipzig) 30, 755–776 (1909).
    [Crossref]
  8. R. K. Luneburg, Mathematical Theory of Optics, 2nd ed. (U. of California Press, Berkeley, Calif., 1964), Chap. VI, pp. 319–323.
  9. 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]
  10. S. H. Wiersma, T. D. Visser, “Defocusing of a converging electromagnetic wave by a plane dielectric interface,” J. Opt. Soc. Am. A 13, 320–325 (1996).
    [Crossref]
  11. The name m theory was coined by Karczewski and Wolf in two papers: 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);“Comparison of three theories of electromagnetic diffraction at an aperture. Part II: The far field,” J. Opt. Soc. Am. 56, 1214–1219 (1966).
    [Crossref]
  12. W. R. Smythe, “The double current sheet in diffraction,” Phys. Rev. 72, 1066–1070 (1947).
    [Crossref]
  13. G. Toraldo di Francia, Electromagnetic Waves (Interscience, New York, 1955), Chap. 10, pp. 213–223.
  14. H. Severin, “Zur Theorie der Beugung electromagnetischer Wellen,” Z. Phys. 129, 426–439 (1951).
    [Crossref]
  15. P. Török, C. J. R. Sheppard, P. Varga, “Study of evanescent waves for transmission near-field optical microscopy,” J. Mod. Opt. 43, 1167–1183 (1996).
    [Crossref]
  16. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), pp. 486–488.
  17. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Chap. 3, pp. 125–127.
  18. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1991), Chap. 3, pp. 113–117.
  19. T. D. Visser, J. L. Oud, “Volume measurements in 3-D microscopy,” Scanning 16, 198–200 (1994).
    [Crossref]
  20. C. J. R. Sheppard, P. Török, “Effects of specimen refractive index on confocal imaging,” J. Microsc. 185, 366–374 (1997).
  21. P. Török, G. R. Booker, Z. Laczik, R. Falster, “A new confocal SIRM incorporating reflection, transmission and double-pass modes either with or without differential phase contrast imaging,” Inst. Phys. Conf. Ser. 134, 771–774 (1993).
  22. A. Sommerfeld, Optics (Academic, New York, 1954), Chap. V, p. 200.
  23. J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
    [Crossref]

1997 (1)

C. J. R. Sheppard, P. Török, “Effects of specimen refractive index on confocal imaging,” J. Microsc. 185, 366–374 (1997).

1996 (2)

P. Török, C. J. R. Sheppard, P. Varga, “Study of evanescent waves for transmission near-field optical microscopy,” J. Mod. Opt. 43, 1167–1183 (1996).
[Crossref]

S. H. Wiersma, T. D. Visser, “Defocusing of a converging electromagnetic wave by a plane dielectric interface,” J. Opt. Soc. Am. A 13, 320–325 (1996).
[Crossref]

1995 (1)

1994 (2)

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

S. Chang, J. H. Jo, S. S. Lee, “Theoretical calculations of optical force exerted on a dielectric sphere in the evanescent field generated with a totally-reflected focused Gaussian beam,” Opt. Commun. 108, 133–143 (1994).
[Crossref]

1993 (1)

P. Török, G. R. Booker, Z. Laczik, R. Falster, “A new confocal SIRM incorporating reflection, transmission and double-pass modes either with or without differential phase contrast imaging,” Inst. Phys. Conf. Ser. 134, 771–774 (1993).

1984 (1)

1976 (1)

1966 (1)

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]

1951 (1)

H. Severin, “Zur Theorie der Beugung electromagnetischer 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]

1909 (1)

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

Booker, G. R.

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, G. R. Booker, Z. Laczik, R. Falster, “A new confocal SIRM incorporating reflection, transmission and double-pass modes either with or without differential phase contrast imaging,” Inst. Phys. Conf. Ser. 134, 771–774 (1993).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1991), Chap. 3, pp. 113–117.

Chang, S.

S. Chang, J. H. Jo, S. S. Lee, “Theoretical calculations of optical force exerted on a dielectric sphere in the evanescent field generated with a totally-reflected focused Gaussian beam,” Opt. Commun. 108, 133–143 (1994).
[Crossref]

Chu, L. J.

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

Debye, P.

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

Falster, R.

P. Török, G. R. Booker, Z. Laczik, R. Falster, “A new confocal SIRM incorporating reflection, transmission and double-pass modes either with or without differential phase contrast imaging,” Inst. Phys. Conf. Ser. 134, 771–774 (1993).

Gasper, J.

Jo, J. H.

S. Chang, J. H. Jo, S. S. Lee, “Theoretical calculations of optical force exerted on a dielectric sphere in the evanescent field generated with a totally-reflected focused Gaussian beam,” Opt. Commun. 108, 133–143 (1994).
[Crossref]

Karczewski, B.

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]

P. Török, G. R. Booker, Z. Laczik, R. Falster, “A new confocal SIRM incorporating reflection, transmission and double-pass modes either with or without differential phase contrast imaging,” Inst. Phys. Conf. Ser. 134, 771–774 (1993).

Lee, S. S.

S. Chang, J. H. Jo, S. S. Lee, “Theoretical calculations of optical force exerted on a dielectric sphere in the evanescent field generated with a totally-reflected focused Gaussian beam,” Opt. Commun. 108, 133–143 (1994).
[Crossref]

Lee, S.-W.

Ling, H.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics, 2nd ed. (U. of California Press, Berkeley, Calif., 1964), Chap. VI, pp. 319–323.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Chap. 3, pp. 125–127.

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, Ser. A 253, 358–379 (1959).
[Crossref]

Severin, H.

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

Sheppard, C. J. R.

C. J. R. Sheppard, P. Török, “Effects of specimen refractive index on confocal imaging,” J. Microsc. 185, 366–374 (1997).

P. Török, C. J. R. Sheppard, P. Varga, “Study of evanescent waves for transmission near-field optical microscopy,” J. Mod. Opt. 43, 1167–1183 (1996).
[Crossref]

Sherman, G. C.

Smythe, W. R.

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

Sommerfeld, A.

A. Sommerfeld, Optics (Academic, New York, 1954), Chap. V, p. 200.

Stamnes, J. J.

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), pp. 486–488.

Toraldo di Francia, G.

G. Toraldo di Francia, Electromagnetic Waves (Interscience, New York, 1955), Chap. 10, pp. 213–223.

Török, P.

C. J. R. Sheppard, P. Török, “Effects of specimen refractive index on confocal imaging,” J. Microsc. 185, 366–374 (1997).

P. Török, C. J. R. Sheppard, P. Varga, “Study of evanescent waves for transmission near-field optical microscopy,” J. Mod. Opt. 43, 1167–1183 (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, G. R. Booker, Z. Laczik, R. Falster, “A new confocal SIRM incorporating reflection, transmission and double-pass modes either with or without differential phase contrast imaging,” Inst. Phys. Conf. Ser. 134, 771–774 (1993).

Varga, P.

Visser, T. D.

Wiersma, S. H.

Wolf, E.

The name m theory was coined by Karczewski and Wolf in two papers: 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);“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. 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]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Chap. 3, pp. 125–127.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1991), Chap. 3, pp. 113–117.

Ann. Phys. (Leipzig) (1)

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

Inst. Phys. Conf. Ser. (1)

P. Török, G. R. Booker, Z. Laczik, R. Falster, “A new confocal SIRM incorporating reflection, transmission and double-pass modes either with or without differential phase contrast imaging,” Inst. Phys. Conf. Ser. 134, 771–774 (1993).

J. Microsc. (1)

C. J. R. Sheppard, P. Török, “Effects of specimen refractive index on confocal imaging,” J. Microsc. 185, 366–374 (1997).

J. Mod. Opt. (1)

P. Török, C. J. R. Sheppard, P. Varga, “Study of evanescent waves for transmission near-field optical microscopy,” J. Mod. Opt. 43, 1167–1183 (1996).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

S. Chang, J. H. Jo, S. S. Lee, “Theoretical calculations of optical force exerted on a dielectric sphere in the evanescent field generated with a totally-reflected focused Gaussian beam,” Opt. Commun. 108, 133–143 (1994).
[Crossref]

Phys. Rev. (2)

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

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

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 electromagnetischer Wellen,” Z. Phys. 129, 426–439 (1951).
[Crossref]

Other (7)

G. Toraldo di Francia, Electromagnetic Waves (Interscience, New York, 1955), Chap. 10, pp. 213–223.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), pp. 486–488.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Chap. 3, pp. 125–127.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1991), Chap. 3, pp. 113–117.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Chap. 16, pp. 482–500.

R. K. Luneburg, Mathematical Theory of Optics, 2nd ed. (U. of California Press, Berkeley, Calif., 1964), Chap. VI, pp. 319–323.

A. Sommerfeld, Optics (Academic, New York, 1954), Chap. V, p. 200.

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Figures (7)

Fig. 1
Fig. 1

Geometry of the system.

Fig. 2
Fig. 2

Ray tracing for focusing through an interface. A lens with focal length f and semiaperture angle Ω1 is placed at a distance d in front of an interface.

Fig. 3
Fig. 3

Comparison of theories 1 and 2. The lens numerical aperture was NA=1.4, and computations were performed for a glass/water interface (n1=1.54, n2=1.33) with focusing depths f-d of (a) 10, (b) 50, and (c) 100 μm. Individual images are normalized to the intensity obtained for the 10-μm focusing depth.

Fig. 4
Fig. 4

Results of theory 2 for a lens with semiaperture angle 60° and for a wavelength of 632.8 nm (HeNe laser). The focusing depth f-d=50 μm, the focal length f=1×10-2 m, and μ1=μ2=μ0. The axial intensity distribution is shown for (curve a) n1=1.51 and n2=1.33, (curve b) n1=n2=1.51, and (curve c) n1=1.33 and n2=1.51.

Fig. 5
Fig. 5

Intensity distributions according to geometrical optics. Curves a and b correspond to curves a and c, respectively, of Fig. 4. The detector radius =1×10-8 m.

Fig. 6
Fig. 6

Distance between the peak and the interface plotted versus the position of the lens. 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 cm, and λ=632.8 nm. In curve a, n1=1.33 and n2=1.51; in curve b, n1=n2=1.33; and in curve c, n1=1.51 and n2=1.33.

Fig. 7
Fig. 7

Geometry used for the construction of the Green’s functions G±.

Equations (64)

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E2=R(-1)[P(2)]-1IP(1)LRE(0).
R=cos ϕ-sin ϕ0sin ϕcos ϕ0001,
L=cos θ10-sin θ1010sin θ10cos θ1,
P(n)=cos θn0sin θn010-sin θn0cos θn,
I=τp000τs000τp,
E(2)=cos θ1τp cos θ2 cos2 ϕ+τs sin2 ϕτp cos θ2 sin ϕ cos ϕ-τs sin ϕ cos ϕ-τp sin θ2 cos ϕ.
E1(x, y, zi)=-ik12π Ω1 a(s1x, s1y)s1z×exp[ik1(s1xx+s1yy+s1zzi)]ds1xds1y.
E2(x, y, zi)=-ik12π Ω1 M a(s1x,s1y)s1z×exp[ik1(s1xx+s1yy+s1zzi)]ds1xds1y,
E2(x, y, z)=-ik22π Ω2 F(s^2)×exp[ik2(s2xx+s2yy-s2zz)]ds2xds2y.
E2x(z)=ik1fl02 0Ω1 cos θ1 sin θ1×exp[i(f-d)(k1 cos θ1-k2 cos θ2)]×(τs+τp cos θ2)exp(-ik2z cos θ2)dθ1,
E(Q)=2 Σ[m×E(P)]×G(P, Q)dΣ,
G(P, Q)=exp(ik2s)4πs;
G=1s-ik2G Q-P|Q-P|,
s=[t2(θ1)+z2-2z(f-d)]1/2,
t(θ1)=(f-d)/cos θ1.
F(θ1)=exp{ik1[f-t(θ1)]},
A(θ1)=f cos3/2 θ1f-d.
E2x(z)=f(f-d)2 [z-(f-d)]0Ω1(τs+τp cos θ2)×exp{i[k2s+k1(f-t)]}×1s3-ik2s2tan θ1 dθ1.
ERS1(x, y, z)=-12π A E0(xP, yP, f-d)×z exp(ik2R)RdA,
R=[(x-xP)2+(y-yP)2+(z-f+d)2]1/2.
z exp(ik2R)R=(z-f+d)ik2R2-1R3exp(ik2R),
ERS1(x, y, z)=-z-f+d2π AEx(P)Ey(P)Ez(P)×ik2R2-1R3exp(ik2R)dA.
P-Q=(xP-x, yP-y, f-d-z),
s=R=|P-Q|=[(xP-x)2+(yP-y)2+(z-f+d)2]1/2
(m×E)×G
=-14π1R3-ik2R2exp(ik2R)×(z-f+d)Ex(P)(z-f+d)Ey(P)(y-yP)Ey(P)+(x-xP)Ex(P).
Em theory(0, 0, z)
=-z-f+d2π
×AEx(P)Ey(P)[yPEy(P)+xPEx(P)]/(z-f+d)×ik2R2-1R3exp(ik2R)dA.
tan θ1=ρf-d.
tan θ2=ρ/h(θ1).
h(θ1)=(f-d) tan θ1tan θ2=(f-d) n2 cos θ2n1 cos θ1,
0<θ1Ω1.
zm=f-d-h(Ω1),
zp=f-d-limθ10 h(θ1)=(f-d)(1-n2/n1).
h2(1-sin2 θ1)=(f-d)2(n2/n1)2×[1-(n1/n2)2 sin2 θ1].
sin2[θ1(h)]=[(f-d)2(n2/n1)2-h2]/[(f-d)2-h2].
θ1=arcsin(f-d)2[(n2/n1)2-1]2z(f-d)-z2+11/2,
n1n2.
Δh/tan θ2.
r=f sin θ1,
I(h; )=π|rI2-rII2|cos[θ2(h)].
apparentsize=n1n2 (truesize+peakwidth).
2f+k2f=h(Q),
f(Q)=-VG(P, Q)h(P)dP,
2G(P, Q)+k2G(P, Q)=-δ(Q-P).
G±(P, Q)=G(P, Q)±G(P, Q).
P2G±(P, Q)+k2G±(P, Q)
=-δ(Q-P)δ(Q-P).
Γ(P, Q)=c1G-(P, Q)+c2G+(P, Q).
P2Γ(P, Q)+k2Γ(P, Q)=-(c1+c2)δ(Q-P)+(c1-c2)δ(Q-P).
V [A×(×B)]dV=S[A×(×B)]m dS,
V[×A×B-A×(×B)]dV
=S[A×(×B)]m dS.
V[B×(×A)-A×(×B)]dV=S[A×(×B)-B×(×A)]m dS.
V[ΓP×(P×E)-EP×(P×Γ)]dP.
-V E[P(PΓ)]dP-E(Q)c=-E(Q)c,
Σ[E×(P×Γ)-Γ×(P×E)]m dΣ
=Σ E(P×Γ)×m dΣ,
(P×Γ)×m=(P×Γ)×m,
(×Γ)=2PG×c.
2 Σ[-(EG)(cm)+(Ec)(Gm)]dΣ
=2 Σ c(m×E)×G dΣ.
E(Q)=2 Σ (m×E)×G dΣ,

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