Abstract

The first Rayleigh–Sommerfeld diffraction formula is treated in an exact form as a three-dimensional (3D) convolution in the spatial domain. Therefore, a 3D Fourier transform can be employed to convert the 3D diffracted electromagnetic field to the reciprocal space without approximations, which we call the 3D angular spectrum (3D-AS) method. It is also demonstrated that if evanescent waves are neglected, the 3D-AS method can be readily implemented numerically, with the results in good agreement with theoretical predictions.

© 2013 Optical Society of America

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References

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2013 (1)

2011 (1)

2006 (1)

R. L. Lucke, Eur. J. Phys. 27, 193 (2006).
[CrossRef]

2005 (2)

1999 (1)

1993 (1)

1992 (1)

1989 (1)

1984 (1)

1964 (1)

1961 (1)

Alfieri, D.

Andrés, P.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2005).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, 2000).

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, 1966).

Coppola, G.

Cuche, E.

Dainty, J. C.

Depeursinge, C.

Ferraro, P.

Finizio, A.

Goodman, J. W.

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

Grilli, S.

Hrynevych, M.

Javidi, B.

Kou, S. S.

Li, Y.

Lin, J.

Lucke, R. L.

R. L. Lucke, Eur. J. Phys. 27, 193 (2006).
[CrossRef]

Marquet, P.

Martinez-Corral, M.

McCutchen, C. W.

Nicola, S. D.

Ojeda-Castañeda,

Oppenheim, A. V.

A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall, 1975).

Osterberg, H.

Pierattini, G.

Rodguez-Herrera, O. G.

Rusch, J. J.

Rutt, H. N.

Schafer, R. W.

A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall, 1975).

Sheppard, C. J. R.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Smith, L. W.

Steane, A. M.

Striano, V.

Urbach, H. P.

Veerman, J. A. C.

Wolf, E.

Y. Li and E. Wolf, J. Opt. Soc. Am. A 1, 801 (1984).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2005).

Yuan, X. C.

Appl. Opt. (1)

Eur. J. Phys. (1)

R. L. Lucke, Eur. J. Phys. 27, 193 (2006).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Opt. Express (1)

Opt. Lett. (2)

Other (6)

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, 1966).

R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, 2000).

A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall, 1975).

A. E. Siegman, Lasers (University Science, 1986).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2005).

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

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Figures (3)

Fig. 1.
Fig. 1.

Geometry for the diffraction calculation from an aperture illuminated from the left.

Fig. 2.
Fig. 2.

3D field distribution of a propagating Gaussian beam. (a) Transverse intensity and (b) phase when y=z=0; (c) axial intensity and (d) phase when x=y=0. The results from the 3D-AS method (blue circles) are compared with the theoretical prediction (red solid curve).

Fig. 3.
Fig. 3.

Intensity distributions along the (a) lateral and (c) axial directions for the 3D-AS method, (b) comparison of transverse line intensities (y=z=0) from the 3D-AS method (blue dots), the Debye model (continuous red line), and a paraxial model (continuous brown line). 3D-AS matches well with the more accurate Debye model. (d) Comparison of axial intensities (x=y=0) around the geometrical focus F from the 3D-AS method (blue dots), the Smith model (red line), and the Debye model (brown line). 3D-AS matches well with the more accurate Smith model.

Equations (10)

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U(x,y,z)=12πΣU0(ξ,η,0)K(ξ,η,0)dξdη,
K(ξ,η,ζ)=ζ[exp(ikρ)ρ].
U(x,y,z)=12πVV0(ξ,η,ζ)K(ξ,η,ζ)dξdηdζ,
U(x,y,z)=12πF31{F2{U0(ξ,η,0)}×F3{K(ξ,η,ζ)}}.
K^(fx,fy,fz)=i2πfzG^(fx,fy,fz),
G^(fr)=4π0exp(ikr)rsin(2πfrr)2πfrrr2dr,
G^A=i2fr[δ(fr1/λ)δ(fr+1/λ)]=iδ(fr21/λ2),G^B=12πfr[1(fr1/λ)+1(fr+1/λ)]=1π(fr21/λ2).
G^A=i2M[δ(fzM)+δ(fz+M)],G^B=12πM[1(fzM)1(fz+M)].
G^={iλδ(fr1/λ)fz>00fz<0.
K^=2πλfzδ(fr1/λ),fz0&z>0.

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