Abstract

A nonparaxial vector-field method is used to describe the behavior of low-f-number microlenses by use of ray propagation, Fresnel coefficients and the solution of Maxwell equations to determine the field propagating through the lens boundaries, followed by use of the Rayleigh-Sommerfeld method to find the diffracted field behind the lenses. This approach enables the phase, the amplitude, and the polarization of the diffracted fields to be determined. Numerical simulations for a convex-plano lens illustrate the effects of the radii of curvature, the lens apertures, the index of refraction, and the wavelength on the variations of the focal length, the focal plane field distribution, and the cross polarization of the field in the focal plane.

© 2004 Optical Society of America

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2002

J. W. M. Chon, X. Gan, M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81, 1576–1578 (2002).
[CrossRef]

1999

E. Park, M. Kim, Y. Kwon, “Microlens for efficient coupling between LED and optical fiber,” IEEE Photon. Technol. Lett. 11, 439–441 (1999).
[CrossRef]

S. Calixto, M. Ornelas-Rodriguez, “Mid-infrared microlenses fabricated by melting method,” IEEE Photon. Technol. Lett. 17, 1212–1214 (1999).

M. A. Alonso, A. A. Asatryan, G. W. Forbes, “Beyond the Fresnel approximation for focused waves,” J. Opt. Soc. Am. A 16, 1958–1969 (1999).
[CrossRef]

Q. Cao, “Correction to the paraxial approximation solutions in transversely nonuniform refractive-index media,” J. Opt. Soc. Am. A 16, 2494–2499 (1999).
[CrossRef]

1998

1995

1994

1992

1984

1981

W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. 71, 7–14 (1981).
[CrossRef]

Y. Li, E. Wolf, “Focal shift in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

1979

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

1975

M. Lax, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1965

A. I. Carswell, “Measurements of the longitudinal component of the electromagnetic field at the focus of a coherent beam,” Phys. Rev. Lett. 15, 647–649 (1965).
[CrossRef]

1959

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

Alonso, M. A.

Asatryan, A. A.

Barakat, R.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University, Cambridge, England, 1999).

Calixto, S.

S. Calixto, M. Ornelas-Rodriguez, “Mid-infrared microlenses fabricated by melting method,” IEEE Photon. Technol. Lett. 17, 1212–1214 (1999).

Cao, Q.

Carswell, A. I.

A. I. Carswell, “Measurements of the longitudinal component of the electromagnetic field at the focus of a coherent beam,” Phys. Rev. Lett. 15, 647–649 (1965).
[CrossRef]

Chon, J. W. M.

J. W. M. Chon, X. Gan, M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81, 1576–1578 (2002).
[CrossRef]

Deng, X.

Fainman, Y.

Forbes, G. W.

Gan, X.

J. W. M. Chon, X. Gan, M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81, 1576–1578 (2002).
[CrossRef]

Gu, M.

J. W. M. Chon, X. Gan, M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81, 1576–1578 (2002).
[CrossRef]

M. Gu, Advanced Optical Imaging Theory, Vol. 75 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 2000).
[CrossRef]

Harvey, J. E.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Hrynevych, M.

Hsu, W.

Kim, M.

E. Park, M. Kim, Y. Kwon, “Microlens for efficient coupling between LED and optical fiber,” IEEE Photon. Technol. Lett. 11, 439–441 (1999).
[CrossRef]

Kwon, Y.

E. Park, M. Kim, Y. Kwon, “Microlens for efficient coupling between LED and optical fiber,” IEEE Photon. Technol. Lett. 11, 439–441 (1999).
[CrossRef]

Lax, M.

M. Lax, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Li, Y.

Y. Li, E. Wolf, “Focal shift in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Ornelas-Rodriguez, M.

S. Calixto, M. Ornelas-Rodriguez, “Mid-infrared microlenses fabricated by melting method,” IEEE Photon. Technol. Lett. 17, 1212–1214 (1999).

Park, E.

E. Park, M. Kim, Y. Kwon, “Microlens for efficient coupling between LED and optical fiber,” IEEE Photon. Technol. Lett. 11, 439–441 (1999).
[CrossRef]

Prata, A.

Richards, B.

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

Shamir, J.

Sheppard, C. J. R.

Southwell, W. H.

Walther, A.

A. Walther, The Ray and Wave Theory of Lenses (Cambridge University, Cambridge, England, 1995).
[CrossRef]

Wang, A.

Wolf, E.

Y. Li, E. Wolf, “Focal shift in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

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

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University, Cambridge, England, 1999).

Wünsche, A.

Am. J. Phys.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

J. W. M. Chon, X. Gan, M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81, 1576–1578 (2002).
[CrossRef]

IEEE Photon. Technol. Lett.

E. Park, M. Kim, Y. Kwon, “Microlens for efficient coupling between LED and optical fiber,” IEEE Photon. Technol. Lett. 11, 439–441 (1999).
[CrossRef]

S. Calixto, M. Ornelas-Rodriguez, “Mid-infrared microlenses fabricated by melting method,” IEEE Photon. Technol. Lett. 17, 1212–1214 (1999).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

Y. Li, E. Wolf, “Focal shift in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Phys. Rev. A

M. Lax, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Phys. Rev. Lett.

A. I. Carswell, “Measurements of the longitudinal component of the electromagnetic field at the focus of a coherent beam,” Phys. Rev. Lett. 15, 647–649 (1965).
[CrossRef]

Proc. R. Soc. A

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

Other

M. Gu, Advanced Optical Imaging Theory, Vol. 75 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 2000).
[CrossRef]

A. Walther, The Ray and Wave Theory of Lenses (Cambridge University, Cambridge, England, 1995).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University, Cambridge, England, 1999).

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Figures (9)

Fig. 1
Fig. 1

Incident beam in a homogeneous medium with a refractive index of n 1 is introduced into a lens of diameter D made from material with a refractive index of n 2 and radii of curvature R 1 and R 2. The diffracted beam propagates in the homogeneous medium with a refractive index of n 3.

Fig. 2
Fig. 2

Representation of unit vectors on the lens boundaries.

Fig. 3
Fig. 3

Focal plane position as a function of the lens aperture for a lens with R 1/λ = 50, R 2/λ = ∞, and n 2 = 1.5.

Fig. 4
Fig. 4

Three-dimensional plot showing the lens aperture effect on the intensity distribution along the optical axis of a convex-plano lens. R 1/λ = -50, R 2/λ = ∞, n 2 = 1.5.

Fig. 5
Fig. 5

Performance of the lens changes as the radius of curvature varies for the rigorous solution (solid curve) and for the Fresnel approach (dashed curve). (D/λ = 60 for all cases). (a) R/λ = 40, (b) R/λ = 60, (c) R/λ = 80, (d) R/λ = 100.

Fig. 6
Fig. 6

Four lenses with the same f/#, but different radii of curvature R/λ (40, 60, 80, and 100), have slightly different performances. (a) Intensity without the paraxial approximation (solid curves) and the Fresnel approximation (dashed curves). (b) Error of the focal position of the paraxial approximation.

Fig. 7
Fig. 7

Lenses with the same f/# produce the same spot size at their focal planes.

Fig. 8
Fig. 8

Difference between the rigorous approach (solid curve) and the Fresnel approach (dashed curve) is greatly reduced when the wavelength is short. (a) R = 50 μm, D = 60 μm, λ = 1 μm; (b) R = 50 μm, D = 60 μm, λ = 0.1 μm.

Fig. 9
Fig. 9

Maximum cross polarization for (a) the variation of the curvature, (b) the variation of the aperture, and (c) the variation of the lens index.

Equations (35)

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2Ei+ki2Ei=0, i=1, 2, or 3.
E1rf=E0 exp-jk1 · rfx2,
e1s=-y1ρ1x1+x1ρ1x2,
e1p=-x1ρ1x1-y1ρ1x2,
e1t=R12-ρ121/2R1ρ1x1x1+y1x2+ρ1R1x3,
E1r1=E1sr1e1s-E1pr1cos α1e1t+E1pr1sin α1e1,
E1sr1=E0 exp-jk1d1x1ρ1,
E1pr1=-E0 exp-jk1d1y1ρ1,
d1=R1-R12-ρ12.
k2e2k=k2 sin Ω1e1t+k2 cos Ω1e1,
E2r2=T1sE2sr2e1s-T1pE2pr2cos Ω1e1t+T1pE2pr2sin Ω1e1,
E2mr1=exp-jk2d2E1 · e1m,
d2=D1-R1+R12-ρ12cos α2,
D1=R1-R12-D241/2,D2=R2-R22-D241/2.
E3r2=T2sE3sr2e2s-T2pE3pr2cos Ω2e2t+T2pE3pr2sin Ω2e2,
E3mr2=E2 · e2m exp-jk3d3,
d3=D2-r2 · x3/e3k · x3.
e2s=e1s, e2t=-x1ρ1x1-y1ρ1x2,e2=-x3, d3=0.
EIrI=E1xI, yI, zIx1+E2xI, yI, zIx2+E3xI, yI, zIx3,
Er=SIdsErIn GrI, r,
GrI, r=-exp-jr-rIk3/2π|r-rI|.
Er=-SIdsErI/2π|r-rI|2jk3+1/|r-rI|×r-rIeIexp-jk3r-rI,
Er=-jk3z-z1/2πr2exp-jk3r×SIdsErIexpjk3r · rI-rI2/2/r.
Er=-jk3z-z1/2πr2exp-jk3r×SIdsErIexpjk3r · rI/r.
Ixy=20 logExEy, Izy=20 logEzEy
z1=R12-D2/41/2-R12-x12-y121/2
e1=-x1x1+y1x2-R12-x12-y121/2x3/R1.
z2=R22-x22-y221/2-R22-D2/41/2
e2=-x2x1+y2x2+R22-x22-y221/2x3/R2.
αiri=cos-1ki · ei,Ωiri=sin-1ni sin αi/ni+1,
Tis=2ni cos αini cos αi+ni+1 cos Ωi,Tip=2ni cos αini+1 cos αi+ni cos Ωi.
r2=ρ2+z-zI2, ρ2=x2+y2, rI2=xI2+yI2
|r-r1|=r1+rI2-2r · rI/r21/2.
1|r-rI|1r, exp-jk|r-rI|exp-jkr-r · rI/r+rI2/2r, when r>rI.
1|r-rI|1r, exp-jk|r-rI|exp-jkr-r · rI/r, when r  rI.

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