Abstract

An angular spectrum representation in three dimensions is used to develop three-dimensional Fourier forms of the first and second Rayleigh–Sommerfeld diffraction formulae and the Kirchhoff diffraction formula. For forward-propagating waves, these reduce to three-dimensional Fourier representations for diffraction in the forward half-space.

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References

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  1. H. Weyl, “Ausbreitung elektromagnetische Wellen über einem ebenen Leiter,” Ann. Phys. 365, 481–500 (1919).
    [CrossRef]
  2. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
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  3. C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).
  4. P. Andrés, M. Martinez-Corral, and J. Ojeda-Castañeda, “Off-axis focal shift for rotationally nonsymmetric screens,” Opt. Lett. 18, 1290–1293 (1993).
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  5. J. Lin, X. C. Yuan, S. S. Kou, C. J. R. Sheppard, O. G. Rodríguez-Herrera, and J. C. Dainty, “Direct calculation of a three-dimensional diffraction field,” Opt. Lett. 36, 1341–1343 (2011).
    [CrossRef]
  6. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
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    [CrossRef]
  8. H. Arnoldus, “Representation of the near-field, middle-field, and far-field electromagnetic Green’s functions in reciprocal space,” J. Opt. Soc. Am. B 18, 547–555 (2001).
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    [CrossRef]
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  18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  19. G. C. Sherman, “Application of the convolution theorem to Rayleigh’s integral formulas,” J. Opt. Soc. Am. 57, 546–547 (1967).
    [CrossRef]
  20. C. L. Andrews, “Diffraction pattern of a circular aperture at short distances,” Phys. Rev. 71, 777–786 (1947).
    [CrossRef]
  21. M. Totzeck, “Validity of the scalar Kirchhoff and Rayleigh–Sommerfeld diffraction theories in the near field of small phase objects,” J. Opt. Soc. Am. A 8, 27–32 (1991).
    [CrossRef]
  22. S. S. Kou, C. J. R. Sheppard, and J. Lin, “Exact evaluation of a volumetric diffracted field with 3D convolution: the 3D angular spectrum method,” Opt. Lett. (submitted).

2011 (2)

2001 (1)

1997 (1)

C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).

1993 (1)

1991 (1)

1970 (1)

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

1967 (1)

1964 (1)

1947 (1)

C. L. Andrews, “Diffraction pattern of a circular aperture at short distances,” Phys. Rev. 71, 777–786 (1947).
[CrossRef]

1919 (1)

H. Weyl, “Ausbreitung elektromagnetische Wellen über einem ebenen Leiter,” Ann. Phys. 365, 481–500 (1919).
[CrossRef]

1912 (1)

A. Sommerfeld, “Die Greensche Funktion der Schwingungsgleichung,” Jahresbericht der Deutschen Mathematiker-Vereinigung 21, 309–353 (1912).

Andrés, P.

Andrews, C. L.

C. L. Andrews, “Diffraction pattern of a circular aperture at short distances,” Phys. Rev. 71, 777–786 (1947).
[CrossRef]

Arnoldus, H.

Born, M.

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

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications, McGraw-Hill Electrical and Electronic Engineering Series (McGraw-Hill, 1978).

Carter, W. H.

Clemmow, P. C.

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

Dainty, J. C.

Dirac, P. A. M.

P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford University, 1981).

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1978).

Goodman, J. W.

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

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1962).

Kou, S. S.

J. Lin, X. C. Yuan, S. S. Kou, C. J. R. Sheppard, O. G. Rodríguez-Herrera, and J. C. Dainty, “Direct calculation of a three-dimensional diffraction field,” Opt. Lett. 36, 1341–1343 (2011).
[CrossRef]

S. S. Kou, C. J. R. Sheppard, and J. Lin, “Exact evaluation of a volumetric diffracted field with 3D convolution: the 3D angular spectrum method,” Opt. Lett. (submitted).

Larkin, K. G.

C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).

Lin, J.

J. Lin, X. C. Yuan, S. S. Kou, C. J. R. Sheppard, O. G. Rodríguez-Herrera, and J. C. Dainty, “Direct calculation of a three-dimensional diffraction field,” Opt. Lett. 36, 1341–1343 (2011).
[CrossRef]

S. S. Kou, C. J. R. Sheppard, and J. Lin, “Exact evaluation of a volumetric diffracted field with 3D convolution: the 3D angular spectrum method,” Opt. Lett. (submitted).

Martinez-Corral, M.

McCutchen, C. W.

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1978).

Ojeda-Castañeda, J.

Rodríguez-Herrera, O. G.

Sheppard, C. J. R.

J. Lin, X. C. Yuan, S. S. Kou, C. J. R. Sheppard, O. G. Rodríguez-Herrera, and J. C. Dainty, “Direct calculation of a three-dimensional diffraction field,” Opt. Lett. 36, 1341–1343 (2011).
[CrossRef]

C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).

S. S. Kou, C. J. R. Sheppard, and J. Lin, “Exact evaluation of a volumetric diffracted field with 3D convolution: the 3D angular spectrum method,” Opt. Lett. (submitted).

Sherman, G. C.

Sommerfeld, A.

A. Sommerfeld, “Die Greensche Funktion der Schwingungsgleichung,” Jahresbericht der Deutschen Mathematiker-Vereinigung 21, 309–353 (1912).

A. Sommerfeld, Optics, Lectures on Theoretical Physics (Academic, 1964).

Totzeck, M.

Tyc, T.

T. Tyc and X. Zhang, “Perfect lenses in focus,” Nature 480, 42–43 (2011).
[CrossRef]

Weyl, H.

H. Weyl, “Ausbreitung elektromagnetische Wellen über einem ebenen Leiter,” Ann. Phys. 365, 481–500 (1919).
[CrossRef]

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

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

Yuan, X. C.

Zhang, X.

T. Tyc and X. Zhang, “Perfect lenses in focus,” Nature 480, 42–43 (2011).
[CrossRef]

Ann. Phys. (1)

H. Weyl, “Ausbreitung elektromagnetische Wellen über einem ebenen Leiter,” Ann. Phys. 365, 481–500 (1919).
[CrossRef]

J. Opt. Soc. Am. (3)

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

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

Jahresbericht der Deutschen Mathematiker-Vereinigung (1)

A. Sommerfeld, “Die Greensche Funktion der Schwingungsgleichung,” Jahresbericht der Deutschen Mathematiker-Vereinigung 21, 309–353 (1912).

Nature (1)

T. Tyc and X. Zhang, “Perfect lenses in focus,” Nature 480, 42–43 (2011).
[CrossRef]

Opt. Commun. (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Opt. Lett. (2)

Optik (1)

C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).

Phys. Rev. (1)

C. L. Andrews, “Diffraction pattern of a circular aperture at short distances,” Phys. Rev. 71, 777–786 (1947).
[CrossRef]

Other (9)

J. D. Jackson, Classical Electrodynamics (Wiley, 1962).

P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford University, 1981).

A. Sommerfeld, Optics, Lectures on Theoretical Physics (Academic, 1964).

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

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

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

R. N. Bracewell, The Fourier Transform and its Applications, McGraw-Hill Electrical and Electronic Engineering Series (McGraw-Hill, 1978).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1978).

S. S. Kou, C. J. R. Sheppard, and J. Lin, “Exact evaluation of a volumetric diffracted field with 3D convolution: the 3D angular spectrum method,” Opt. Lett. (submitted).

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Equations (17)

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G^(fr)=G^A(fr)+G^B(fr),
δ(fr21/λ2)=λ2[δ(fr1/λ)+δ(fr+1/λ)]=δ[fz(1/λ2fx2fy2)1/2]2|fz|+δ[fz+(1/λ2fx2fy2)1/2]2|fz|,
G^A(f)=iλ2[H(fz)δ(fr1/λ)+H(fz)δ(fr1/λ)],
G^A(f)=i2fzδ[fz(1λ2fx2fy2)1/2]i2fzδ[fz+(1λ2fx2fy2)1/2],G^B(f)=12πfz[fz(1λ2fx2fy2)1/2]+12πfz[fz+(1λ2fx2fy2)1/2].
K^RS1(f)=2πifzG^=K^ARS1(f)+K^BRS1(f)=πλfzδ(fr1/λ)2ifzfr21/λ2.
A(fx,fy)=Aout(fx,fy)+Aev(fx,fy)=[Uout(x,y)z=0++Uev(x,y)z=0+]×exp[i2π(fxx+fyy)]dxdy.
U^(f)=12πA(fx,fy)[K^ARS1(f)+K^BRS1(f)]=12πA(fx,fy)K^RS1(f).
K^ARS1(f)=πδ[fz(1λ2fx2fy2)1/2]πδ[fz+(1λ2fx2fy2)1/2],K^BRS1(f)=ifz(1λ2fx2fy2)1/2ifz+(1λ2fx2fy2)1/2.
K^AeffRS1(f)=2πδ[fz(1λ2fx2fy2)1/2]=4πifzH(fz)G^A,K^BeffRS1(f)=2ifzfr21λ2=2πifzG^B,(fx2+fy2)>1/λ2,=0,(fx2+fy2)<1/λ2.
U(r)=12πF31[Aev(fx,fy)K^AeffRS1(f)+Aout(fx,fy)K^BeffRS1(f)],
B(fx,fy)=Bout(fx,fy)+Bev(fx,fy)=[Uoutz(x,y)z=0++Uevz(x,y)z=0+]×exp[i2π(fxx+fyy)]dxdy.
U^(f)=12πB(fx,fy)[K^ARS2(f)+K^BRS2(f)].
U(r)=12πF31[Bout(fx,fy)K^AeffRS2(f)+Bev(fx,fy)K^BeffRS2(f)],z0,
K^AeffRS2(f)=2i(1λ2fx2fy2)1/2δ[fz(1λ2fx2fy2)1/2],K^BeffRS2(f)=1π(fr21λ2),(fx2+fy2)>1/λ,=0,(fx2+fy2)<1/λ.
B(fx,fy)=i2π(1/λ2fx2fy2)1/2×[UIIIout(x,y)z=0++UIIIev(x,y)z=0+]×exp[i2π(fxx+fyy)]dxdy=i2π(1/λ2fx2fy2)1/2A(fx,fy).
U(r)=14πF31[Aout(fx,fy)K^AeffRS1(f)+Aev(fx,fy)K^BeffRS1(f)Bout(fx,fy)K^AeffRS2(f)Bev(fx,fy)K^BeffRS2(f)],
U(r)=14πF31{[Bout(fx,fy)+i2πfzAout(fx,fy)]2H(fz)G^A(f)+[Bev(fx,fy)+i2πfzAev(fx,fy)]G^B(f)}.

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