Abstract

A repeated FFT algorithm method is devised and calculation of the field distribution across a plane situated close to and far from the focal plane becomes possible. This overcomes the shortcomings of the fast Fourier transform (FFT) algorithm by which a substantial reduction of the computation time for the field distribution of focused light can be achieved only for the local field distribution adjacent to the focal plane.

© 1988 Optical Society of America

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References

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  1. V. B. Jipson, C. C. Williams, “Two-Dimensional Modeling of an Optical Disk Readout,” Appl. Opt. 22, 2202 (1983).
    [CrossRef] [PubMed]
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 30–56.
  3. H. Kubota, Wave Optics (Iwanami, Tokyo, 1971), pp. 242–244, in Japanese.
  4. E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, NJ, 1974).

1983 (1)

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, NJ, 1974).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 30–56.

Jipson, V. B.

Kubota, H.

H. Kubota, Wave Optics (Iwanami, Tokyo, 1971), pp. 242–244, in Japanese.

Williams, C. C.

Appl. Opt. (1)

Other (3)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 30–56.

H. Kubota, Wave Optics (Iwanami, Tokyo, 1971), pp. 242–244, in Japanese.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, NJ, 1974).

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Figures (5)

Fig. 1
Fig. 1

Light propagation from the aperture plane to the focal plane over the distance f. The observed plane is parallel to the focal plane at normal distance z.

Fig. 2
Fig. 2

Intensity distributions calculated by the FFT algorithm across the observed plane, parallel to the focal plane at distance z. Intensity values are plotted, magnified linearly with the square of z.

Fig. 3
Fig. 3

Intensity distributions calculated by the focal pivot method across the observed plane, parallel to the focal plane at distance z. Intensity values are plotted, magnified linearly with the square of z.

Fig. 4
Fig. 4

Intensity distributions across the plane immediately behind the objective plane with an amplitude transmittance gT(x,y).

Fig. 5
Fig. 5

Intensity distributions after further propagation over the distance z from the focal plane.

Equations (27)

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g P ( u , v ) = { 1 inside the lens aperture ( u 2 + v 2 a 2 ) , 0 other than above ( u 2 + v 2 > a 2 ) ,
g 1 ( u , v ) = g 0 ( u , v ) g P ( u , v ) exp [ i κ ( f - r 0 ) ] ,
g 0 ( u , v ) = exp [ - ( u 2 + v 2 ) ln 2 / 2 a 2 ] .
g F ( ξ , η ) = 1 / i λ · - + g 1 ( u , v ) exp ( i κ r ) r d u d v ,
g 2 ( x , y ) = 1 / i λ · - + g 1 ( u , v ) exp ( i κ ρ ) / ρ d u d v ,
r = ( f 2 + u 2 + v 2 ) 1 / 2 + ( ξ 2 + η 2 ) / 2 f - ( u ξ + v η ) / f , ρ = [ ( f - z ) 2 + u 2 + v 2 ] 1 / 2 + ( x 2 + y 2 ) / 2 ( f - z ) - ( u x + v y ) / ( f - z ) .
g F ( ξ , η ) = 1 / i λ f · exp [ i κ ( ξ 2 + η 2 ) / 2 f ] × F κ ξ / f , κ η / f { g 1 ( u , v ) exp [ i κ ( f 2 + u 2 + v 2 ) 1 / 2 ] } ,
F f x , f y { g ( x , y ) } = - + g ( x , y ) exp [ - i ( f x x + f y y ) ] d x d y ,
g ( x , y ) = 1 / ( 2 π ) 2 · - + F f x , f y { g ( x , y ) } × exp [ i ( f x x + f y y ) ] d f x d f y .
g 2 ( x , y ) = 1 / i λ ( f - z ) · exp [ i κ ( x 2 + y 2 ) / 2 ( f - z ) ] × F κ x f - z , κ y f - x ( g 1 ( u , v ) exp { i κ [ ( f - z ) 2 + u 2 + v 2 ] 1 / 2 } ) .
d = 2 z a / ( f 2 - a 2 ) 1 / 2 .
d = 2 K 0 z a / ( f 2 - a 2 ) 1 / 2 ,
z < f λ ( f 2 - a 2 ) 1 / 2 / ( λ ( f 2 - a 2 ) 1 / 2 + 2 K 0 a δ ) .
N δ = 2 K A a ,
z < f λ N ( f 2 - a 2 ) 1 / 2 / [ λ N ( f 2 - a 2 ) 1 / 2 + 4 K 0 K A a 2 ] .
d F = K F λ f / a ,
δ < a / K F .
g F ( ξ , η ) = 1 / i λ · - + g 2 ( x , y ) exp ( i κ ρ ) / ρ d x d y ,
g 2 ( x , y ) = i / λ z · exp [ - i κ ( z 2 + x 2 + y 2 ) 1 / 2 ] × F - κ x / z , - κ y / z { g F ( ξ , η ) exp [ - i κ ( ξ 2 + η 2 ) / 2 z ] } .
g 2 ( x , y ) = i / λ 2 z f · exp [ - i κ ( z 2 + x 2 + y 2 ) 1 / 2 ] × F - κ x / z , - κ y / z ( F κ ξ / f , κ η / f { g 1 ( u , v ) × exp [ i κ ( f 2 + u 2 + v 2 ) 1 / 2 ] } × exp [ i κ ( ξ 2 + η 2 ) ( 1 / f - 1 / z ) / 2 ] ) .
δ > 2 K 0 f a / N ( f 2 - a 2 ) 1 / 2 .
K A / K 0 > f / ( f 2 - a 2 ) 1 / 2 ,
K A K F < N / 2.
g T ( x , y ) = { 0 ( x - x 0 ) 2 + ( y - y 0 ) 2 d 0 2 / 4 , 1 ( x - x 0 ) 2 + ( y - y 0 ) 2 > d 0 2 / 4.
g F ( ξ , η ) = 1 / i λ · - + g 2 ( x , y ) g T ( x , y ) exp ( i κ ρ ) / ρ d x d y .
g 2 ( x , y ) = 1 / i λ · - + g F ( ξ , η ) exp ( i κ ρ ) ρ d ξ d η .
g 2 ( x , y ) = - 1 / ( λ z ) 2 · exp [ i κ ( z 2 + x 2 + y 2 ) 1 / 2 ] × F κ x / z , κ y / z ( F κ ξ / z , κ η / z { g 2 ( x , y ) g T ( x , y ) × exp [ i κ ( z 2 + x 2 + y 2 ) 1 / 2 ] } × exp [ i κ ( ξ 2 + η 2 ) / z ] ) .

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