Abstract

Classical diffraction theory is used to investigate the effects of high numerical aperture on the focusing of coherent light. By expanding the diffracted beam in plane waves, we show that the lens action can be expressed as a succession of three Fourier transforms. Furthermore, polarization effects are included in the model in a natural way. Some numerical results of the theory are also presented.

© 1986 Optical Society of America

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References

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  1. H. H. Hopkins, “The Airy disk formula for systems of high relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943).
    [CrossRef]
  2. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  3. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  5. M. V. Klein, Optics (Wiley, New York, 1970).
  6. F. Jenkins, H. White, Fundamentals of Physical Optics, 1st ed. (McGraw-Hill, New York, 1937).
  7. G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
    [CrossRef]
  8. Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
    [CrossRef]
  9. V. N. Mahajan, “Axial irradiance and optimum focusing of laser beams,” Appl. Opt. 22, 3042–3053 (1983).
    [CrossRef] [PubMed]

1984 (1)

1983 (1)

1958 (1)

G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
[CrossRef]

1943 (1)

H. H. Hopkins, “The Airy disk formula for systems of high relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943).
[CrossRef]

Born, M.

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

Farnell, G. W.

G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
[CrossRef]

Goodman, J. W.

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

Hopkins, H. H.

H. H. Hopkins, “The Airy disk formula for systems of high relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

Jenkins, F.

F. Jenkins, H. White, Fundamentals of Physical Optics, 1st ed. (McGraw-Hill, New York, 1937).

Klein, M. V.

M. V. Klein, Optics (Wiley, New York, 1970).

Li, Y.

Mahajan, V. N.

White, H.

F. Jenkins, H. White, Fundamentals of Physical Optics, 1st ed. (McGraw-Hill, New York, 1937).

Wolf, E.

Appl. Opt. (1)

Can. J. Phys. (1)

G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
[CrossRef]

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

Proc. Phys. Soc. London (1)

H. H. Hopkins, “The Airy disk formula for systems of high relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943).
[CrossRef]

Other (5)

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

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

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

M. V. Klein, Optics (Wiley, New York, 1970).

F. Jenkins, H. White, Fundamentals of Physical Optics, 1st ed. (McGraw-Hill, New York, 1937).

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Figures (7)

Fig. 1
Fig. 1

Prism as a diffraction grating: A plane, linearly polarized incident beam is diffracted into a plane wave propagating in the direction σ ˆ = ( σ x , σ y , σ z ). The polarization of the outgoing beam is related to the polarization of the incoming beam and the orientation of the prism as described in the text.

Fig. 2
Fig. 2

Power-transmission coefficient for a circular aperture versus the normalized aperture radius. The inset shows the transmission coefficient obtained from the vector Smythe–Kirchhoff approximation according to the following equation: η = 1 ( λ / 4 π R ) 0 4 π R / λ J 0 ( x ) d x.

Fig. 3
Fig. 3

Fresnel diffraction from a circular aperture of radius R = 5λ at a distance z0 = 25λ. The X and Y axes are normalized by the wavelength λ, whereas the vertical axis has units of maximum intensity, Imax. The incident beam is plane, propagating in the Z direction with unit incident power on the aperture. The incident polarization is linear in the X direction. (a) Intensity distribution for the X component of polarization, Imax = 0.50 × 10−1. (b) Intensity distribution for the Z component of polarization, Imax = 0.18 × 10−3.

Fig. 4
Fig. 4

Fresnel diffraction from a circular aperture of radius R = 5λ at a distance z0 = 12λ. The incident beam is plane with unit intensity at the aperture. The polarization effects are ignored. The intensity at the observation plane has circular symmetry; thus only the radial distribution is shown. Notice that the central spot is dark and the peak intensity is twice the intensity of the incident beam at the aperture.

Fig. 5
Fig. 5

Diffraction from aberration-free, spherical lens with f = 4000λ and NA = 0.3. The incident beam is plane with unit amplitude, and the polarization effects are ignored. (a) The real part of the complex function τ(x, y) versus the normalized distance from the lens center. (b) The real and imaginary parts of the complex function g(σx, σy) versus (σx2 + σy2)1/2 at z0 = f. (c) Amplitude and phase of the distribution at the focal plane. The amplitude is normalized by the aperture area of the lens. (d) Amplitude and phase of the distribution at the focal plane obtained with standard approximations.

Fig. 6
Fig. 6

Diffraction from aberration-free, spherical lens with f = 4000λ and NA = 0.3. The incident beam is plane, is linearly polarized in the X direction, and has unit power in the aperture. (a) Intensity distribution for the X component of polarization at the focal plane. Imax = 0.276λ−2. (b) Intensity distribution for the Y component of polarization in the focal plane. Imax = 0.740 × 10−5 λ−2. (c) Intensity distribution for the Z component of polarization in the focal plane. Imax = 0.335 × 10−2 λ−2.

Fig. 7
Fig. 7

Intensity distribution for the X component of polarization at z0 = f ± 15λ (same lens as Fig. 6). Imax = 0.4 × 10−1 λ−2.

Equations (31)

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A ( x , y , z , t ) = A 0 exp [ i ( 2 π / λ ) ( x σ x + y σ y + z σ z ) ] exp [ i ω t ] .
A ( x , y ) = A 0 exp [ i ( 2 π / λ ) ( x σ x + y σ y ) ] ,
T ( S x , S y ) = + t ( x , y ) exp [ i 2 π ( x S x + y S y ) ] d x d y .
t ( x , y ) = λ 2 + T ( σ x / λ , σ y / λ ) × exp [ i ( 2 π / λ ) ( x σ x + y σ y ) ] d σ x d σ y .
s component = | σ y | ( σ x 2 + σ y 2 ) 1 / 2 , p component = | σ x | ( σ x 2 + σ y 2 ) 1 / 2 .
Ψ x = [ σ x 2 ( 1 σ x 2 σ y 2 ) 1 / 2 + σ y 2 ] σ x 2 + σ y 2 ,
Ψ y = σ x σ y [ 1 ( 1 σ x 2 σ y 2 ) 1 / 2 ] σ x 2 + σ y 2 ,
Ψ z = σ x ,
t a ( x , y , z 0 ) = λ 2 σ x 2 + σ y 2 1 Ψ a ( σ x , σ y ) T ( σ x / λ , σ y / λ ) × exp { i ( 2 π / λ ) [ x σ x + y σ y + z 0 ( 1 σ x 2 σ y 2 ) 1 / 2 ] } × d σ x d σ y ,
η = σ x 2 + σ y 2 1 | T ( σ x / λ , σ y / λ ) | 2 d σ x d σ y + | T ( σ x / λ , σ y / λ ) | 2 d σ x d σ y .
η = 0 1 ( 2 / x ) J 1 2 ( 2 π R x / λ ) d x = 1 J 0 2 ( 2 π R / λ ) J 1 2 ( 2 π R / λ ) .
W ( σ x , σ y ) = x σ x + y σ y + z o ( 1 σ x 2 σ y 2 ) 1 / 2 ,
σ x 0 = x / ( x 2 + y 2 + z 0 2 ) 1 / 2 , σ y 0 = y / ( x 2 + y 2 + z 0 2 ) 1 / 2 .
W ( σ x , σ y ) / ( x 2 + y 2 + z 0 2 ) 1 / 2 = 1 1 2 [ 1 + ( x / z 0 ) 2 ] ( σ x σ x o ) 2 ( x y / z 0 2 ) ( σ x σ x o ) ( σ y σ y o ) 1 2 [ 1 + ( y / z 0 ) 2 ] ( σ y σ y o ) 2 .
t α ( x , y , z 0 ) = ( i / λ ) exp [ i ( 2 π / λ ) ( x 2 + y 2 + z 0 2 ) 1 / 2 ] z 0 [ 1 + ( x 2 + y 2 ) / z 0 2 ] × Ψ α ( σ x 0 , σ y 0 ) T ( σ x 0 / λ , σ y 0 / λ ) .
t α ( λ x , λ y , λ z 0 ) = i exp ( i 2 π z 0 ) z 0 × exp ( i 2 π z 0 { [ 1 + ( x 2 + y 2 ) / z 0 2 ] 1 / 2 1 } ) × ( 1 σ x o 2 σ y 0 2 ) Ψ α ( σ x o , σ y 0 ) × [ λ 2 T ( σ x o / λ , σ y 0 / λ ) ] ,
λ 2 T ( σ x / λ , σ y / λ ) = { F { t ( λ x , λ y ) } , σ x 2 + σ y 2 1 0 , otherwise .
t α ( λ x , λ y , λ z 0 ) = exp ( i 2 π z 0 ) × F 1 { λ 2 T ( σ x / λ , σ y / λ ) Ψ α ( σ x , σ y ) × exp { i 2 π z 0 [ ( 1 σ x 2 σ y 2 ) 1 / 2 1 ] } } ,
z 0 / λ = ( 1 / N ) ( R / λ ) 2 ( N / 4 ) .
r = ( x 2 + y 2 ) 1 / 2 = 0.61 ( R / λ ) / N .
t ( x , y ) = τ 0 ( x , y ) exp { i ( 2 π / λ ) [ f ( f 2 + x 2 + y 2 ) 1 / 2 ] } .
t ( f x , f y ) = τ ( f x , f y ) exp [ i π f ( x 2 + y 2 ) ] .
τ ( f x , f y ) = τ 0 ( f x , f y ) exp { i 2 π f [ ( 1 + x 2 + y 2 ) 1 / 2 1 1 2 ( x 2 + y 2 ) ] } = τ 0 ( f x , f y ) exp { i 2 π f n = 2 ( 1 ) n ( 2 n 3 ) ! ! 2 n n ! ( x 2 + y 2 ) n } .
F { t ( f x , f y ) } = F { τ ( f x , f y ) } * ( i / f ) exp { i ( π / f ) ( u 2 + υ 2 ) } ,
f 2 T ( u / f , υ / f ) = ( i / f ) exp [ i ( π / f ) ( u 2 + υ 2 ) ] × + h ( u , υ ) exp [ i ( 2 π / f ) ( u u + υ υ ) ] d u d υ ,
h ( u , υ ) = exp [ i ( π / f ) ( u 2 + υ 2 ) ] F { τ ( f x , f y ) } .
λ 2 T ( σ x / λ , σ y / λ ) = i f exp [ i π f ( σ x 2 + σ y 2 ) ] F { h ( u , υ ) } .
t α ( λ x , λ y , λ z 0 ) = i f exp ( i 2 π z 0 ) F 1 { Ψ α ( σ x , σ y ) F { h ( u , υ ) } × exp { i 2 π z 0 [ 1 1 2 ( f / z 0 ) ( σ x 2 + σ y 2 ) ( 1 σ x 2 σ y 2 ) 1 / 2 ] } } .
t α ( λ x , λ y , λ z 0 ) = i f exp ( i 2 π z 0 ) F 1 { Ψ α ( σ x , σ y ) g ( σ x , σ y ) } ,
g ( σ x , σ y ) = { exp { i π [ ( f z 0 ) ( σ x 2 + σ y 2 ) 2 z 0 n = 2 ( 2 n 3 ) ! ! 2 n n ! ( σ x 2 + σ y 2 ) n ] } F { h ( u , υ ) } 0 , σ x 2 + σ y 2 > 1 .
t ( λ x , λ y , λ f ) = i f exp ( i 2 π f ) F 1 { F { h ( u , υ ) } } = i ( f / λ ) exp ( i 2 π f / λ ) exp [ i ( π λ / f ) × ( x 2 + y 2 ) 1 / 2 ] F { τ 0 ( f x , f y ) } .

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