Abstract

It is reported an approximation to the scalar and vectorial Rayleigh-Sommerfeld diffraction integrals performed with a Fourier transform operation, as done in Fresnel and Fraunhoffer approximations, but suitable for paraxial, non-paraxial and off-axis regimes. High accuracy is obtained even for waves with low f numbers. The approximation becomes exact on-axis.

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References

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  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
  2. J. J. Stamnes, Waves in Focal Regions (IOP Publishing Limited, 1986).
  3. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  4. C. J. R. Sheppard, “Limitations of the paraxial Debye approximation,” Opt. Lett. 38(7), 1074–1076 (2013).
    [Crossref] [PubMed]
  5. A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202(1-3), 17–20 (2002).
    [Crossref]
  6. D. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B 23(6), 1228–1234 (2006).
    [Crossref]
  7. H. Ye, C.-W. Qiu, K. Huang, J. Teng, B. Luk’yanchuk, and S. P. Yeo, “Creation of a longitudinally polarized subwavelength hotspot with an utra-thin planar lens: vectorial Rayleigh-Sommerfeld method,” Laser Phys. Lett. 10(6), 1–8 (2013).
    [Crossref]
  8. X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12(11), 115707 (2010).
    [Crossref]
  9. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. 253(1274), 358–379 (1959).
    [Crossref]
  10. L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. 212(4-6), 343–352 (2002).
    [Crossref]
  11. E. Lalor, “Conditions for the validity of the angular spectrum of plane waves,” J. Opt. Soc. Am. 58(9), 1235–1237 (1968).
    [Crossref]
  12. W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. 71(1), 7–14 (1981).
    [Crossref]

2013 (2)

C. J. R. Sheppard, “Limitations of the paraxial Debye approximation,” Opt. Lett. 38(7), 1074–1076 (2013).
[Crossref] [PubMed]

H. Ye, C.-W. Qiu, K. Huang, J. Teng, B. Luk’yanchuk, and S. P. Yeo, “Creation of a longitudinally polarized subwavelength hotspot with an utra-thin planar lens: vectorial Rayleigh-Sommerfeld method,” Laser Phys. Lett. 10(6), 1–8 (2013).
[Crossref]

2010 (1)

X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12(11), 115707 (2010).
[Crossref]

2006 (1)

2002 (2)

L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. 212(4-6), 343–352 (2002).
[Crossref]

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202(1-3), 17–20 (2002).
[Crossref]

1981 (1)

1968 (1)

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. 253(1274), 358–379 (1959).
[Crossref]

Ciattoni, A.

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202(1-3), 17–20 (2002).
[Crossref]

Crosignani, B.

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202(1-3), 17–20 (2002).
[Crossref]

Deng, D.

Di Porto, P.

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202(1-3), 17–20 (2002).
[Crossref]

Hao, X.

X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12(11), 115707 (2010).
[Crossref]

Helseth, L. E.

L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. 212(4-6), 343–352 (2002).
[Crossref]

Huang, K.

H. Ye, C.-W. Qiu, K. Huang, J. Teng, B. Luk’yanchuk, and S. P. Yeo, “Creation of a longitudinally polarized subwavelength hotspot with an utra-thin planar lens: vectorial Rayleigh-Sommerfeld method,” Laser Phys. Lett. 10(6), 1–8 (2013).
[Crossref]

Kuang, C.

X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12(11), 115707 (2010).
[Crossref]

Lalor, E.

Liu, X.

X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12(11), 115707 (2010).
[Crossref]

Luk’yanchuk, B.

H. Ye, C.-W. Qiu, K. Huang, J. Teng, B. Luk’yanchuk, and S. P. Yeo, “Creation of a longitudinally polarized subwavelength hotspot with an utra-thin planar lens: vectorial Rayleigh-Sommerfeld method,” Laser Phys. Lett. 10(6), 1–8 (2013).
[Crossref]

Qiu, C.-W.

H. Ye, C.-W. Qiu, K. Huang, J. Teng, B. Luk’yanchuk, and S. P. Yeo, “Creation of a longitudinally polarized subwavelength hotspot with an utra-thin planar lens: vectorial Rayleigh-Sommerfeld method,” Laser Phys. Lett. 10(6), 1–8 (2013).
[Crossref]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. 253(1274), 358–379 (1959).
[Crossref]

Sheppard, C. J. R.

Southwell, W. H.

Teng, J.

H. Ye, C.-W. Qiu, K. Huang, J. Teng, B. Luk’yanchuk, and S. P. Yeo, “Creation of a longitudinally polarized subwavelength hotspot with an utra-thin planar lens: vectorial Rayleigh-Sommerfeld method,” Laser Phys. Lett. 10(6), 1–8 (2013).
[Crossref]

Wang, T.

X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12(11), 115707 (2010).
[Crossref]

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. 253(1274), 358–379 (1959).
[Crossref]

Ye, H.

H. Ye, C.-W. Qiu, K. Huang, J. Teng, B. Luk’yanchuk, and S. P. Yeo, “Creation of a longitudinally polarized subwavelength hotspot with an utra-thin planar lens: vectorial Rayleigh-Sommerfeld method,” Laser Phys. Lett. 10(6), 1–8 (2013).
[Crossref]

Yeo, S. P.

H. Ye, C.-W. Qiu, K. Huang, J. Teng, B. Luk’yanchuk, and S. P. Yeo, “Creation of a longitudinally polarized subwavelength hotspot with an utra-thin planar lens: vectorial Rayleigh-Sommerfeld method,” Laser Phys. Lett. 10(6), 1–8 (2013).
[Crossref]

J. Opt. (1)

X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12(11), 115707 (2010).
[Crossref]

J. Opt. Soc. Am. (2)

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

Laser Phys. Lett. (1)

H. Ye, C.-W. Qiu, K. Huang, J. Teng, B. Luk’yanchuk, and S. P. Yeo, “Creation of a longitudinally polarized subwavelength hotspot with an utra-thin planar lens: vectorial Rayleigh-Sommerfeld method,” Laser Phys. Lett. 10(6), 1–8 (2013).
[Crossref]

Opt. Commun. (2)

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202(1-3), 17–20 (2002).
[Crossref]

L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. 212(4-6), 343–352 (2002).
[Crossref]

Opt. Lett. (1)

Proc. R. Soc. Lond. (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. 253(1274), 358–379 (1959).
[Crossref]

Other (3)

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

J. J. Stamnes, Waves in Focal Regions (IOP Publishing Limited, 1986).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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Figures (8)

Fig. 1
Fig. 1 Geometry for the Rayleigh-Sommerfeld diffraction formula.
Fig. 2
Fig. 2 Diffraction patterns I RS , I F and I N of a circular aperture of semi-diameter a = 5mm. Various curves of I N are given to highlight that A z is only a scale factor.
Fig. 3
Fig. 3 Percentage difference between normalized intensities of I RS , and I N , corresponding to a circular aperture of semi-diameter a = 5mm and converging spherical waves of various f numbers.
Fig. 4
Fig. 4 Percentage difference between normalized intensities of I RS , and I N , corresponding to a circular aperture of variable diameter D as function of three f numbers. Continuous, dashed and dotted-dashed lines corresponds to f/0.6, f/1 and f/2, respectively.
Fig. 5
Fig. 5 Diffraction intensities at the focus plane obtained with vectorial diffraction equations and azimuthal polarization. (a)-(c) corresponds to | U x | 2 +| U y | 2 ,| U z | 2 and its sum, computed from our approximation Eq. (18). (d) are the plots of the azimuthal polarization calculated with Eq. (18) and with the exact Eqs. (14,16). The two intensities are indistinguishable on the scale in the figure.
Fig. 6
Fig. 6 Vectorial diffraction intensities | U x | 2 +| U y | 2 +| U z | 2 at the focus plane, corresponding to an azimuthal polarization state. Continuous, dotted-continuous and dashed lines were obtained with Eqs. (14,16), (18) and (19), respectively.
Fig. 7
Fig. 7 Vectorial diffraction intensities as obtained in Fig. 6 but with a defocus of f/60 mm towards the aperture. Continuous, dotted-continuous and dashed lines were obtained with Eqs. (14,16), (18) and (19), respectively.
Fig. 8
Fig. 8 Vectorial diffraction intensities as obtained in Fig. 7 but using A z =1.29z to obtain I_N. The other curves are equal to those shown in Fig. 7.

Equations (26)

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G RS ( x 0 , y 0 )= z 2π Σ g(x,y)(jk 1 R ) exp(jkR) R 2 dxdy ,
R= z 2 + (x x 0 ) 2 + (y y 0 ) 2 ;
Rz+ (x x 0 ) 2 + (y y 0 ) 2 2z ,
G F ( y 0 , y 0 )=A Σ g(x,y)exp(jk r 2 /2z)exp[j2π(x x 0 +y y 0 )/λz]dxdy ,
A= jexp(jkz) λz exp[jπ( x 0 2 + y 0 2 )/λz]andr= x 2 + y 2 .
R R n (x x 0 +y y 0 ) R n + ( y 2 + z 2 ) x 0 2 +( x 2 + z 2 ) y 0 2 2xy x 0 y 0 R n 3 ,
R n = z 2 + x 2 + y 2 .
R R n (x x 0 +y y 0 ) A z ,
G N ( x 0 , y 0 )= z 2π Σ g(x,y)(jk 1 R n ) exp(jk R n ) R n 2 exp[ j2π(x x 0 +y y 0 ) λ A z ]dxdy ,
A z =[ Σ x 2 dx ]/[ Σ ( x 2 / z 2 + x 2 )dx ],
A z = 3 a 4 4 ( a 2 + z 2 ) 1/2 12 z 2 ( a 2 + z 2 ) 1/2 +8 z 3 .
R R n x 0 [ x z x 3 2 z 3 +o (x/z) 5 ],
g I (x,y)= exp(jk R n ) R n .
U x,y ( x 0 , y 0 )= z 2π Σ E x,y (x,y)g(x,y)(jkn 1 R ) exp(jknR) R 2 dxdy ,
U z ( x 0 , y 0 )= 1 2π Σ E z (x,y, x 0 , y 0 )g(x,y)(jkn 1 R ) exp(jknR) R 2 dxdy ,
U z ( x 0 , y 0 )= U xy ( x 0 , y 0 )+ 1 2π Σ E xy (x,y)g(x,y)(jkn 1 R ) exp(jknR) R 2 dxdy ,
U xy ( x 0 , y 0 )= x 0 z U x ( x 0 , y 0 )+ y 0 z U y ( x 0 , y 0 )and E xy (x,y)=[x E x (x,y)+y E y (x,y)].
(jkn 1 R ) exp(jknR) R 2 (jkn 1 R n ) exp(jkn R n ) R n 2 exp[ j2π(xn x 0 +yn y 0 ) λ A z ].
U ( x 0 , y 0 )= jc λ 0 α 0 2π P sinθ cosθ exp[jknH(z, x 0 , y 0 ,θ,φ)dθdφ ,
  U ( x 0 , y 0 )=( U x , U y , U z )( x 0 , y 0 ), P =( P 1 , P 2 , P 3 ), H(z, x 0 , y 0 ,θ,φ)=zcosθ+ x 0 sinθcosφ+ y 0 sinθsinφ, P 1 =[1+(cosθ1)cos φ 2 ] E x +(cosθ1)cosφsinφ E y , P 2 =(cosθ1)cosφsinφ E x +[1+(cosθ1)sin φ 2 ] E y , P 3 =sinθcosφ E x sinθsinφ E y ,
G N ( r 0 )=z 0 a g(r)(jk 1 R r ) exp(jk R r ) R r 2 J 0 [2πr r 0 /(λ A z )]rdr ,
G N ( x n0 , y n0 )=B Σ g(x,y)(jk 1 R n0 ) exp(jk R n0 ) R n0 2 exp[ j2π(x x 0 +y y 0 ) λ B z ]dxdy ,
{ ( jk 1 R n ) z 2π R n 2 exp(jk R n ) }=exp[ jkz 1 (λu) 2 (λv) 2 ],
G N ( x n0 , y n0 )=exp{ jk(z/ B z ) B z 2 ( x 0 2 + y 0 2 ) },
G N ( x 0 , y 0 )= jkz 2π Σ f(x,y) R n 3 exp[ j2π(x x 0 +y y 0 ) λ A z ]dxdy .
G N ( r 0 )=jkz 0 a J 0 [2πr r 0 /(λ A z )] R r 3 rdr ,

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