Abstract

Fourier decomposition of a given amplitude distribution into plane waves and the subsequent superposition of these waves after propagation is a powerful yet simple approach to diffraction problems. Many vector diffraction problems can be formulated in this way, and the classical results are usually the consequence of a stationary-phase approximation to the resulting integrals. For situations in which the approximation does not apply, a factorization technique is developed that substantially reduces the required computational resources. Numerical computations are based on the fast-Fourier-transform algorithm, and the practicality of this method is shown with several examples.

© 1989 Optical Society of America

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References

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  1. E. Wolf, “Electromagnetic diffraction in optical systems: an integral representation of the image field,” Proc. R. Soc. Ser. A 253, 349–357 (1959).
    [CrossRef]
  2. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems: structure of the image field in an aplanatic system,” Proc. R. Soc. Ser. A 253, 358–379 (1959).
    [CrossRef]
  3. D. R. Rhodes, “On the stored energy of planar apertures,”IEEE Trans. Antennas Propag. AP-14, 676–683 (1966).
    [CrossRef]
  4. G. Borgiotti, “Fourier transforms method of aperture antennas problem,” Alta Frequenza 32, 196–204 (1963).
  5. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  6. K. Miyamoto, E. Wolf, “Generalization of the Maggi–Rubinowitz theory of the boundary diffraction wave—part I,”J. Opt. Soc. Am. 52, 615–625 (1962).
    [CrossRef]
  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]
  10. M. Mansuripur, “Distribution of light at and near the focus of high-numerical-aperture objectives,” J. Opt. Soc. Am. A 3, 2086–2093 (1986).
    [CrossRef]
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  12. M. Mansuripur, C. Pons, “Diffraction modeling of optical path for magneto-optical disk systems,” in Optical Storage Technology and Applications, D. Carlin, Y. Tsunoda, A. Jamberdino, eds., Proc. Soc. Photo-Opt. Instrum. Eng.899, 56–60 (1988).
    [CrossRef]
  13. F. Jenkins, H. White, Fundamentals of Physical Optics, 1st ed. (McGraw-Hill, New York, 1937).

1986 (1)

1984 (1)

1983 (1)

1966 (1)

D. R. Rhodes, “On the stored energy of planar apertures,”IEEE Trans. Antennas Propag. AP-14, 676–683 (1966).
[CrossRef]

1963 (1)

G. Borgiotti, “Fourier transforms method of aperture antennas problem,” Alta Frequenza 32, 196–204 (1963).

1962 (1)

1959 (2)

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

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

1958 (1)

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

Borgiotti, G.

G. Borgiotti, “Fourier transforms method of aperture antennas problem,” Alta Frequenza 32, 196–204 (1963).

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).

Jenkins, F.

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

Li, Y.

Mahajan, V. N.

Mansuripur, M.

M. Mansuripur, “Distribution of light at and near the focus of high-numerical-aperture objectives,” J. Opt. Soc. Am. A 3, 2086–2093 (1986).
[CrossRef]

M. Mansuripur, C. Pons, “Diffraction modeling of optical path for magneto-optical disk systems,” in Optical Storage Technology and Applications, D. Carlin, Y. Tsunoda, A. Jamberdino, eds., Proc. Soc. Photo-Opt. Instrum. Eng.899, 56–60 (1988).
[CrossRef]

Miyamoto, K.

Pons, C.

M. Mansuripur, C. Pons, “Diffraction modeling of optical path for magneto-optical disk systems,” in Optical Storage Technology and Applications, D. Carlin, Y. Tsunoda, A. Jamberdino, eds., Proc. Soc. Photo-Opt. Instrum. Eng.899, 56–60 (1988).
[CrossRef]

Rhodes, D. R.

D. R. Rhodes, “On the stored energy of planar apertures,”IEEE Trans. Antennas Propag. AP-14, 676–683 (1966).
[CrossRef]

Richards, B.

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

White, H.

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

Wolf, E.

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]

K. Miyamoto, E. Wolf, “Generalization of the Maggi–Rubinowitz theory of the boundary diffraction wave—part I,”J. Opt. Soc. Am. 52, 615–625 (1962).
[CrossRef]

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

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

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

Alta Frequenza (1)

G. Borgiotti, “Fourier transforms method of aperture antennas problem,” Alta Frequenza 32, 196–204 (1963).

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]

IEEE Trans. Antennas Propag. (1)

D. R. Rhodes, “On the stored energy of planar apertures,”IEEE Trans. Antennas Propag. AP-14, 676–683 (1966).
[CrossRef]

J. Opt. Soc. Am. (1)

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

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

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

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

Other (4)

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

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

M. Mansuripur, C. Pons, “Diffraction modeling of optical path for magneto-optical disk systems,” in Optical Storage Technology and Applications, D. Carlin, Y. Tsunoda, A. Jamberdino, eds., Proc. Soc. Photo-Opt. Instrum. Eng.899, 56–60 (1988).
[CrossRef]

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

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Figures (22)

Fig. 1
Fig. 1

Prism as a diffraction grating. A plane, linearly polarized incident beam is diffracted into a plane wave propagating along σ ^ = (σx, σy, σz). The relationship between the incident and the diffracted polarizations is summarized in Table 1.

Fig. 2
Fig. 2

Cross section of a ring lens. The central region (r < R1) is flat and does not affect the incident beam. The curved surface peaks at r = R0, and the aperture radius of the lens is R2. A collimated beam is brought to focus by this lens on a sharp ring of radius R0 at the focal plane. The amplitude distribution at the exit pupil is given by Eq. (3.41), and, in the absence of aberrations, the ring width is diffraction limited.

Fig. 3
Fig. 3

Circular aperture of radius R = 500λ illuminated with a plane uniform wave. (a) Intensity pattern at a distance of 83,333λ from the aperture; both direct and extended Fresnel techniques give the same result. (b) Intensity distribution at z = 125,000λ. Again both methods give the same result. (c) Intensity pattern at z = 1,000,000λ obtained by the extended Fresnel method. (d) Same as (c) but with the direct Fresnel method. In all cases Nmax = Nmay = 256 and Lmax = Lmay = 5000.

Fig. 4
Fig. 4

Circular aperture of radius R = 3λ illuminated with a plane uniform wave, polarized in the x direction. The intensity patterns are calculated at a distance of 20λ from the aperture. (a) and (b) show, respectively, the x and the z components of polarization obtained by the direct Fresnel method. (c) and (d) show the results of extended Fresnel calculations, and (e) and (f) show the results obtained from the Fraunhofer diffraction formula.

Fig. 5
Fig. 5

Circular aperture of radius R = 1000λ illuminated with a uniform wave of curvature C = 5 × 10−6 corresponding to a positive spherical lens of f = 200,000λ and NA = 0.005. The intensity patterns are calculated with the extended Fresnel technique, using Nmax, = Nmay = 512 and Lmax = Lmay = 10,000. (a) z = 100,000,(b) z = 125,000, (c) z = 150,000, (d) z = 160,000, (e) z = 200,000, and (f) z = 240,000.

Fig. 6
Fig. 6

Circular aperture of radius R = 1000λ illuminated with a uniform wave of curvature C = −5 × 10−6. Intensity on the optical axis is shown here as a function of distance from the aperture. The solid curve is calculated from the analytic expression derived in Ref. 9. The crosses are obtained by numerical computations using direct (or extended) Fresnel method.

Fig. 7
Fig. 7

Geometry of groove structure for a transmission phase diffraction grating. All parameters are in units of the wavelength λ.

Fig. 8
Fig. 8

Diffraction from a transmission phase grating. The grating parameters are α = 2, β = 5, γ = 7, δ = 10, ζ = 0.5, θ =−45°, and xc = yc = 1.06. The incident beam is Gaussian with an e−1 radius r0 = 20 and has no curvature (C = 0). The intensity patterns are calculated with either the direct or the extended Fresnel technique, using Nmax = Nmay = 512 and Lmay= Lmay = 350. (a) z = 200, (b) z = 300, (c) z = 500, (d) z = 1000.

Fig. 9
Fig. 9

Cylindrical lens with a focal length f = 106λ and the aperture radius R = 2500λ. The incident beam is plane and uniform. The intensity patterns are obtained with the direct Fresnel method, using Nmax = Nmay = 512 and Lmax = Lmay = 40,000. (a) z = 0.75 × 106; (b) z = 106; (c) z = 1.25 × 106.

Fig. 10
Fig. 10

Spherical lens with a numerical aperture NA = 0.5 and a focal length f = 3500λ, illuminated with a plane uniform wave, linearly polarized along the x axis. (a) Intensity pattern for the x component at the focus; (b) intensity pattern for the y component; (c) intensity distribution for the z component. For these computations, Nmax = Nmay = 980 and Lmax = Lmay = 20,000.

Fig. 11
Fig. 11

Spherical lens with a numerical aperture NA = 0.5 and a focal length f = 3500λ, illuminated with a plane uniform wave, linearly polarized along the x axis. (a) Intensity pattern for the x component at a distance of 5λ from the focal plane; (b) intensity pattern for the x component at a distance of 10λ from the focal plane.

Fig. 12
Fig. 12

Astigmatic lens with fx = 20,000λ, fy = 20,200λ, and NA = 0.15. The incident beam is plane and uniform. For these calculations, Nmax = Nmay = 512 and Lmax = Lmay = 40,000. The intensity pattern is shown at various distances from the lens: (a) z = 20,000, (b) z = 20,100, and (c) z = 20,200.

Fig. 13
Fig. 13

Ring lens with R1 = 50λ, R0 = 100λ, R2 = 2500λ, and f = 25,000λ. The incident beam is plane and uniform. The intensity patterns shown are at (a) z = 24,750λ, (b) z = 25,000λ, and (c) z = 25,400λ. For these computations Nmax = Nmay = 512 and Lmax = Lmay = 25,000.

Fig. 14
Fig. 14

Various peaks of | ω ˙ η (r)| in the interval [r1, r2], plotted here as functions of η; the bold curve represents the largest peak. ηopt and ω ˙ max are the coordinates of the minimum point of the bold curve.

Fig. 15
Fig. 15

ηopt and ω ˙ max, as functions of θ2 for θ1= 0. Note that the scale for ω ˙ max is logarithmic. The particular value of θ1 chosen here corresponds to r1 = 0, while r2 = sin θ2.

Fig. 16
Fig. 16

Two plots of cos{2πf[(1 − r2)1/2 + ½ηr2]} versus r. In (a) the optimum value of η = 1.11395 is used, whereas in (b) η = 1 is used. In both cases f = 3500, r1 = 0, and r2 = 0.5.

Fig. 17
Fig. 17

Various peaks of | ω ˙ η (r)| in the interval [r1, r2], plotted here as functions of η; the bold curve represents the largest peak. ηopt and ω ˙ max are the coordinates of the minimum point of the bold curve.

Fig. 18
Fig. 18

ηopt and ω ˙ max as functions of θ2 for θ1 = 0. Note that the scale for ω ˙ max is logarithmic. The particular value of θ1 chosen here corresponds to r1 = 0, while r2 = tan θ2.

Fig. 19
Fig. 19

Two plots of cos{2πf[(1 + r2)1/2 − ½ηr2]} versus r. In (a) the optimum value of η = 0.8977 is used, whereas in (b) η = 1 is used. In both cases f = 3500, r1 = 0, and r2 = 0.577.

Fig. 20
Fig. 20

Various peaks of | ω ˙ η (r)| in the interval [r1, r2], plotted here as functions of η. The bold solid curve represents the largest peak to the right of R, whereas the bold dashed curve corresponds to the largest peak to the left of R. The larger of the two curves at any η thus gives the largest peak, and the optimum η is the point at which the largest peak is minimum.

Fig. 21
Fig. 21

ηopt and ω ˙ max as functions of R for r1 = 0.01 and r2 = 0.1. Note that the scale for ω ˙ max is logarithmic.

Fig. 22
Fig. 22

Two plots of cos(2πf{[1 + (rR)2]1/2 − ½ηr2) versus fr. In (a) the optimum value of η = 0.272 is used, whereas in (b) η = 0 is used. In both cases f = 25000, r1 = 0.01, and r2 = 0.1.

Tables (1)

Tables Icon

Table 1 Components of the Polarization Vector of a Plane Wavea Propagating along the Unit Vector σ ^ = (σx, σy, σz)

Equations (123)

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T ( u , v ) = F { t ( x , y ) } = - t ( x , y ) exp [ - i 2 π ( x u + y v ) ] d x d y .
1 a b T ( u a , v b ) = F { t ( a x , b y ) } = - t ( a x , b y ) × exp [ - i 2 π ( x u + y v ) ] d x d y .
t ( x , y ) = F - 1 { T ( u , v ) } = - T ( u , v ) exp [ i 2 π ( x u + y v ) ] d u d v .
A ( x , y ) = exp { i 2 π λ C [ 1 + C 2 ( x 2 + y 2 ) ] 1 / 2 } .
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 β ( λ x , λ y , λ z ) F - 1 { λ - 2 T ( σ x λ , σ y λ ) Ψ α β ( σ x , σ y ) × exp [ i 2 π z ( 1 - σ x 2 - σ y 2 ) 1 / 2 ] } .
Δ min { R 10 , ( 1 + C 2 R 2 ) 1 / 2 2 C R } .
σ max = C R ( 1 + C 2 R 2 ) 1 / 2 + 1 2 R .
L max max { 10 R , 2 z σ max ( 1 - σ max 2 ) 1 / 2 } .
G ( σ x , δ y ) = λ - 2 T ( σ x / λ , σ y / λ ) Ψ α β ( σ x , σ y ) × exp { i 2 π z [ ( 1 - σ x 2 - σ y 2 ) 1 / 2 + ½ η ( σ x 2 + σ y 2 ) ] }
t ˜ α β ( λ x , λ y , λ z ) = F - 1 { G ( σ x , σ y ) exp [ - i π η z ( σ x 2 + σ y 2 ) } .
t ˜ α β ( λ x , λ y , λ z ) = - F - 1 { G ( σ x , σ y ) } ( - i η z ) × exp { i π η z [ ( x - x ) 2 + ( y - y ) 2 ] } d x d y .
H ( x , y ) = - i η z exp [ i π η z ( x 2 + y 2 ) ] F - 1 { G ( σ x , σ y ) } ;
t ˜ α β ( λ x , λ y , λ z ) = exp [ i π η z ( x 2 + y 2 ) ] F { H ( η z X , η z Y ) } .
L ma x max { 10 R , 2 z ω ˙ max } .
Δ η z 2 ( R + z ω ˙ max ) .
L ˜ ma x = η z N ma x / L ma x .
W ( σ x , σ y ) = x σ x + y σ y + z ( 1 - σ x 2 - σ y 2 ) 1 / 2 .
σ x 0 = x / ( x 2 + y 2 + z 2 ) 1 / 2 ,
σ y 0 = y / ( x 2 + y 2 + z 2 ) 1 / 2 .
W ( σ x , σ y ) = ( x 2 + y 2 + z 2 ) 1 / 2 { 1 - ½ [ 1 + ( x / z ) 2 ] ( σ x / σ x 0 ) 2 - ( xy / z ) 2 ( σ x - σ x 0 ) ( σ y - σ y 0 ) - ½ [ 1 + ( y / z ) 2 ] ( σ y - σ y 0 ) 2 } + .
t ˜ α β ( λ x , λ y , λ z ) = - ( i / z ) exp [ i 2 π z ( 1 + x 2 + y 2 z 2 ) 1 / 2 ] × Ψ α β ( σ x 0 , σ y 0 ) ( 1 - σ x 0 2 - σ y 0 2 ) [ λ - 2 T ( σ x 0 / λ , σ y 0 / λ ) ] ,
L max 10 R .
L ˜ ma x = z · N ma x L ma x .
t ( x , y ) = τ ˜ 0 ( x , y ) exp { - i 2 π λ [ f 2 + ( x - x c ) 2 + ( y - y c ) 2 ] 1 / 2 } .
τ ˜ 0 ( x , y ) = τ 0 { x - x c [ 1 + ( x - x c f ) 2 + ( y - y c f ) 2 ] 1 / 2 + x c , y - y c [ 1 + ( x - x c f ) 2 + ( y - y c f ) 2 ] 1 / 2 + y c } × [ 1 + ( x - x c f ) 2 + ( y - y c f ) 2 ] - 1 .
τ ( x , y ) = { τ ˜ 0 ( x , y ) exp ( - i 2 π f λ { [ 1 + ( x - x c ) 2 + ( y - y c ) 2 f 2 ] 1 / 2 - 1 2 η ( x - x c ) 2 + ( y - y c ) 2 f 2 } ) 0             [ ( x - x c ) 2 + ( y - y c ) 2 ] 1 / 2 > f tan [ arcsin ( NA ) ] .
t ( x , t ) = τ ( x , y ) exp { - i π η λ f [ ( x - x c ) 2 + ( y - y c ) 2 ] } .
τ ( f η X , f η Y ) = τ ˜ 0 ( f η X , f η Y ) exp ( - i 2 π f × { [ 1 + ( X - X c ) 2 + ( Y - Y c ) 2 η 2 ] 1 / 2 + 1 2 ( X - X c ) 2 + ( Y - Y c ) 2 η } ) ,
t ( f η X , f η Y ) = τ ( f η X , f η Y ) × exp { - i π f η [ ( X - X c ) 2 + ( Y - Y c ) 2 ] } .
( η f ) 2 T ( η u f , η v f ) = - i η f - F { τ ( f η X , f η Y ) } × exp { - i 2 π [ X c ( u - u ) + Y c ( v - v ) ] } × exp { i π η f [ ( u - u ) 2 + ( v - v ) 2 ] } d u d v .
λ - 2 T ( σ x λ , σ y λ ) = - i f η exp [ i π f η ( σ x 2 + σ y 2 ) - i 2 π λ ( x c σ x + y c σ y ) ] × - F { τ ( f η X , f η Y ) } × exp { i π η f [ ( u + x c λ ) 2 + ( v + y c λ ) 2 ] } . × exp [ - i 2 π ( u σ x + v σ y ) ] d u d v .
h ( u , v ) = - i f η exp { i π η f [ ( u + x x ) 2 + ( v + y c ) 2 ] } × F { τ ( λ f η X , λ f η Y ) } ;
λ - 2 T ( σ x λ , σ y λ ) = exp { i 2 π [ f 2 η ( σ x 2 + σ y 2 ) - x c σ c - y c σ y ] } × F { h ( u , v ) } .
g ( σ x , σ y ) = exp [ i 2 π ( z - f ) ( 1 - σ x 2 - σ y 2 ) 1 / 2 × exp { i 2 π f [ ( 1 - σ x 2 - σ y 2 ) 1 / 2 + 1 2 η ( σ x 2 + σ y 2 ) ] } × exp [ - i 2 π ( x c σ x + y c σ y ) ] F { h ( u , v ) } ,
t ˜ α β ( λ x , λ y , λ z ) = F - 1 { Ψ α β ( σ x , σ y ) g ( σ x , σ y ) } .
Δ min { R 10 , 1 2 ω ˙ max } .
σ max = f η ( ω ˙ max + 1 2 R ) .
L ma x max { 10 R , 2 f η ( ω ˙ max + 1 2 R ) + 2 r c } .
Δ 1 2 ( ω ˙ max + η r c f ) .
L ma x N ma x min { R 10 , 1 2 ( ω ˙ max + η r c f ) } .
L ˜ ma x = f N ma x η L ma x .
λ - 2 T ( σ x λ , σ y λ ) = f 2 - τ ˜ 0 ( f X , f Y ) × exp [ - i 2 π f W ( X , Y ) ] d X d Y ,
W ( X , Y ) = [ 1 + ( X - X c ) 2 + ( Y - Y c ) 2 ] 1 / 2 + σ x X + σ y Y .
X 0 = X c - σ x ( 1 - σ x 2 - σ y 2 ) 1 / 2 ,
Y 0 = Y c - σ y ( 1 - σ x 2 - σ y 2 ) 1 / 2 .
W ( X , Y ) = σ x X c + σ y Y c + ( 1 - σ x 2 - σ y 2 ) 1 / 2 × [ 1 + ½ ( 1 - σ x 2 ) ( X - X 0 ) 2 - σ x σ y ( X - X 0 ) ( Y - Y 0 ) + ½ ( 1 - σ y 2 ) ( Y - Y 0 ) 2 ] + .
λ - 1 T ( σ x λ , σ y λ ) = τ ˜ ˜ 0 ( σ x , σ y ) × exp { - i 2 π f [ ( 1 - σ x 2 - σ y 2 ) 1 / 2 + σ x X c + σ y Y c ] } ,
τ ˜ ˜ 0 ( σ x , σ y ) = ( - i f 1 - σ x 2 - σ y 2 ) × τ ˜ 0 [ x c - f σ x ( 1 - σ x 2 - σ y 2 ) 1 / 2 , y c - f σ y ( 1 - σ x 2 - σ y 2 ) 1 / 2 ] .
t ˜ α β ( λ x , λ y , λ z ) = F - 1 { exp { i 2 π [ ( z - f ) ( 1 - σ x 2 - σ y 2 ) 1 / 2 - x c σ x - y c σ y ] } Ψ α β ( σ x , σ y ) τ ˜ ˜ 0 ( σ x , σ y ) } .
Δ f min { NA 10 , 1 2 { r c + z - f NA [ 1 - ( NA ) 2 ] 1 / 2 } - 1 } .
L ma x 10 f NA .
L ˜ ma x = f N ma x L ma x .
t ( x , y ) = { τ 0 ( x , y ) exp ( - i 2 π λ ( 2 f x f y f x + f y ) { 1 + ( f x + f y 2 f x f y ) [ ( x - x c ) 2 f x + ( y - y c ) 2 f y ] } 1 / 2 ) 0             { f x + f y 2 f x f y [ ( x - x c ) 2 f x + ( y - y c ) 2 f y ] } 1 / 2 > tan [ arcsin ( NA ) ]
t ( x , y ) = τ 0 ( x , y ) exp { - i π λ [ ( x - x c ) 2 f x + ( y - y c ) 2 f y ] } ,
τ ( x , y ) = τ 0 ( x , y ) exp [ - i 2 π ( 2 f x f y f x + f y ) × ( { 1 + ( f x + f y 2 f x f y ) [ ( x - x c ) 2 f x + ( y - y c ) 2 f y ] } 1 / 2 - 1 2 η ( f x + f y 2 f x f y ) [ ( x - x c ) 2 f x + ( y - y c ) 2 f y ] ) ] .
Δ x min { R x 10 , ( 1 2 + f x 2 f y ) 2 ω ˙ max } ,
Δ y min { R y 10 , ( 1 2 + f y 2 f x ) 1 / 2 2 ω ˙ max } .
h ( u , v ) = - i ( f x f y ) 1 / 2 η exp { i π η [ ( u + x c ) 2 f x + ( v + y c ) 2 f y ] } × F { τ ( λ f x η X , λ f y η Y ) } .
L max max { 10 R x , 2 f x η [ ω ˙ max ( 1 2 + f x 2 f y ) 1 / 2 + 1 2 R x ] + 2 x c } .
g ( σ x , σ y ) = exp { i 2 π z [ ( 1 - σ x 2 - σ y 2 ) 1 / 2 + f x 2 η z σ x 2 + f y 2 η z σ y 2 ] } × exp [ - i 2 π ( x c σ x + y c σ y ) ] F { h ( u , v ) } .
L ma x N ma x min { R x 10 , ( 1 2 + f x 2 f y ) 1 / 2 2 ( ω ˙ max + η x c f x ) } .
t ˜ α β ( λ x , λ y , λ z ) = F - 1 { Ψ α β ( σ x , σ y ) g ( σ x , σ y ) } .
L ˜ ma x = f x N ma x η L ma x ,
L ˜ ma y = f y N ma y η L ma y .
λ - 2 T ( σ x λ , σ y λ ) = τ ˜ 0 ( σ x , σ y ) exp ( - i 2 π { ( 2 f x f y f x + f y ) × [ 1 - ( 1 2 + f x 2 f y ) σ x 2 - ( 1 2 + f y 2 f x ) σ y 2 ] 1 / 2 + x c σ x + y c σ y } ) ,
τ ˜ 0 ( σ x , σ y ) = - i ( f x f y ) 1 / 2 1 - ( 1 2 + f x 2 f y ) σ x 2 - ( 1 2 + f y 2 f x ) σ y 2 × τ 0 { x c - f x σ x [ 1 - ( 1 2 + f x 2 f y ) σ x 2 - ( 1 2 + f y 2 f x ) σ y 2 ] 1 / 2 , y c - f y σ y [ 1 - ( 1 2 + f x 2 f y ) σ x 2 - ( 1 2 + f y 2 f x ) σ y 2 ] 1 / 2 } .
t ˜ α β ( λ x , λ y , λ z ) = F - 1 { exp ( i 2 π { z ( 1 - σ x 2 - σ y 2 ) 1 / 2 - ( 2 f x f y f x + f y ) × [ 1 - ( 1 2 + f x 2 f y ) σ x 2 - ( 1 2 + f y 2 f x ) σ y 2 ] 1 / 2 - x c σ x - y c σ y } ) Ψ α β ( σ x , σ y ) τ ˜ 0 ( σ x , σ y ) } .
Δ x f x min { NA 10 ( 1 2 + f x 2 f y ) 1 / 2 , 1 2 ( x c + | f x NA { ( 1 2 + f x 2 f y ) [ 1 - ( NA ) 2 ] } 1 / 2 - z NA [ 1 2 + f x 2 f y - ( NA ) 2 ] 1 / 2 | ) - 1 } .
L ma x 10 f x NA ( 1 2 + f x 2 f y ) 1 / 2 ,
L ˜ ma x = f x N ma x L ma x ,
L ˜ ma y = f y N ma y L ma y .
t ( x , y ) = τ 0 ( x , y ) exp [ - i 2 π λ ( f 2 + { [ ( x - x c ) 2 + ( y - y c ) 2 ] 1 / 2 - R 0 } 2 ) 12 ] .
τ ( x , y ) = { τ 0 ( x , y ) exp { - i 2 π f λ [ ( 1 + { [ ( x - x c ) 2 + ( y - y c ) 2 f 2 ] 1 / 2 - R 0 f } 2 ) 1 / 2 - 1 2 η ( x - x c ) 2 + ( y - y c ) 2 f 2 ] } 1 [ ( x - x c ) 2 + ( y - y c ) 2 ] 1 / 2 R 1 0 [ ( x - x c ) 2 + ( y - y c ) 2 ] 1 / 2 R 2 .
z = 1 N R 2 - N 4 .
Δ < [ 10 δ + 2 C R ( 1 + C 2 R 2 ) 1 / 2 ] - 1 .
σ max = C R ( 1 + C 2 R 2 ) 1 / 2 + 1 δ ,
t ( x , y ) = τ 0 ( x , y ) exp ( - i 2 π λ { f 2 + [ ( x - x c ) sin θ - ( y - y c ) cos θ ] 2 } 1 / 2 ) = 0             [ ( x - x c ) 2 + ( y - y c ) 2 ] > R 2 .
Δ min { R 10 , [ 2 NA + 2 C R ( 1 + C 2 R 2 ) 1 / 2 ] - 1 } .
σ max = NA + C R ( 1 + C 2 R 2 ) 1 / 2 + 1 2 R ,
ω η ( r ) = ( 1 - r 2 ) 1 / 2 + ½ η r 2 ,             r 1 r r 2 ,
ω ˙ η ( r ) = - ( r - 2 - 1 ) - 1 / 2 + η r ,             r 1 r r 2 .
ω ¨ η ( r ) = - ( 1 - r 2 ) - 3 / 2 + η ,             r 1 r r 2 ,
r 0 = ( 1 - η - 2 / 3 ) 1 / 2 .
( 1 - r 1 2 ) - 3 / 2 η ( 1 - r 2 2 ) - 3 / 2 .
ω ˙ η ( r 0 ) = ( η 2 / 3 - 1 ) 3 / 2 .
sin θ 1 = r 1 , sin θ 2 = r 2 ;
ω ˙ η ( r 1 ) = η sin θ 1 - tan θ 1 ,
ω ˙ η ( r 2 ) = η sin θ 2 - tan θ 2 .
1 cos 3 θ 1 η 1 cos 3 θ 2 .
η 0 = tan θ 1 + tan θ 2 sin θ 1 + sin θ 2 .
ω η ( r ) = ( 1 + r 2 ) 1 / 2 - ½ η r 2 ,             r 1 r r 2 ,
ω ˙ η ( r ) = ( 1 + r - 2 ) - 1 / 2 - η r ,             r 1 r r 2 .
ω ¨ η ( r ) = ( 1 + r 2 ) - 3 / 2 - η ,             r 1 r r 2 ,
r 0 = ( η - 2 / 3 - 1 ) 1 / 2 .
( 1 + r 2 2 ) - 3 / 2 η ( 1 + r 1 2 ) - 3 / 2 .
ω ˙ η ( r 0 ) = ( 1 - η 2 / 3 ) 3 / 2 .
tan θ 1 = r 1 , tan θ 2 = r 2 ;
ω ˙ η ( r 1 ) = - η tan θ 1 + sin θ 1 ,
ω ˙ η ( r 2 ) = - η tan θ 2 + sin θ 2 .
cos 3 θ 2 η cos 3 θ 1 .
η 0 = sin θ 1 + sin θ 2 tan θ 1 + tan θ 2 .
ω ˙ max ( A ) = tan θ 2 - η opt ( A ) sin θ 2 ,
ω ˙ max ( B ) = η opt ( B ) tan θ 2 - sin θ 2 .
ω ˙ max ( A ) / η opt ( A ) = ω ˙ max ( B ) ,
ω ˙ max ( B ) / η opt ( B ) = ω ˙ max ( A ) .
ω η ( r ) = [ 1 + ( r - R ) 2 ] 1 / 2 - ½ η r 2 ,
ω ˙ η ( r ) = ( r - R ) [ 1 + ( r - R ) 2 ] - 1 / 2 - η r ,             r 1 r r 2 .
ω ¨ η ( r ) = [ 1 + ( r - R ) 2 ] - 3 / 2 - η ,             r 1 r r 2 ,
r 0 ± = R ± ( η - 2 / 3 - 1 ) 1 / 2 .
[ 1 + ( r 2 - R ) 2 ] - 3 / 2 η 1 ,
[ 1 + ( R - r 1 ) 2 ] - 3 / 2 η 1.
ω ˙ η ( r 0 ± ) = ± ( 1 - η 2 / 3 ) 3 / 2 - η R .
tan θ 1 = r 1 ,
tan θ 2 = r 2 ,
tan ϕ 1 = R - r 1 ,
tan ϕ 2 = r 2 - R ;
ω ˙ η ( r 1 ) = - sin ϕ 1 - η tan θ 1 ,
ω ˙ η ( r 2 ) = sin ϕ 2 - η tan θ 2 .
cos 3 ϕ 2 η 1             ( for r 0 + ) ,
cos 3 ϕ 1 η 1             ( for r 0 - ) .
cos 3 ϕ 2 < sin ϕ 2 tan θ 2 .
η opt = sin ϕ 2 - sin ϕ 1 tan θ 2 + tan θ 1 .

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