Abstract

This paper deals with Fraunhofer diffraction by an ensemble of independent randomly oriented ice crystals of assorted shapes, like those of cirrus clouds. There is no restriction on the shape of each crystal. It is shown that light flux density in the Fourier plane is azimuth-invariant and varies as 1/sin4θ, θ being the angle of diffraction. The analytical formula proposed is exact. The key point of this study is conservation of electromagnetic energy.

© 2012 Optical Society of America

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References

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  1. G. L. Stephens, S. C. Tsay, P. W. J. Stackhouse, and P. J. Flateau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climatic feedback,” J. Atmos. Sci. 47, 1742–1754 (1990).
    [CrossRef]
  2. L. M. Miloshevich and A. J. Heymsfield, “A balloon-borne continuous cloud particle replicator for measuring vertical profiles of cloud microphysical properties: instrument design, performance and collection efficiency analysis,” J. Atmos. Ocean. Technol. 14, 753–768 (1997).
    [CrossRef]
  3. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  4. P. Wendling, R. Wendling, and H. Weickmann, “Scattering of solar radiation by hexagonal ice crystals,” Appl. Opt. 18, 2663–2671 (1979).
    [CrossRef]
  5. Y. Takano and S. Asano, “Fraunhofer diffraction by ice crystals suspended in the atmosphere,” J. Meteorol. Soc. Jpn. 61, 289–299 (1983).
  6. J. P. Pérez, Optique, Fondements et Applications, 7th ed.(Dunod, 2011).
  7. S. Ganci, “Simple derivation of formulas for Fraunhofer diffraction at polygonal apertures from Maggi–Rubinowicz transformation,” J. Opt. Soc. Am. 1, 559–561 (1984).
    [CrossRef]
  8. K. Miyamoto and E. Wolf, “Generalization of the Maggi–Rubinowicz theory of the boundary diffraction wave. Part I,” J. Opt. Soc. Am. 52, 615–625 (1962).
    [CrossRef]
  9. K. Miyamoto and E. Wolf, “Generalization of the Maggi–Rubinowicz theory of the boundary diffraction wave. Part II,” J. Opt. Soc. Am. 52, 626–637 (1962).
    [CrossRef]
  10. W. Sillitto, “Fraunhofer diffraction at straight-edged apertures,” J. Opt. Soc. Am. 69, 765–770 (1979).
    [CrossRef]
  11. R. C. Smith and J. S. Marsh, “Diffraction patterns of simple apertures,” J. Opt. Soc. Am. 64, 798–803 (1974).
    [CrossRef]
  12. F. M. Schwerd, Die Beugungserscheinungen aus den Fundamentalgesetzen der Undulationstheorie Analytisch Entwickelt und In Bildern Dargestellt (Mannheim, 1835).
  13. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

1997 (1)

L. M. Miloshevich and A. J. Heymsfield, “A balloon-borne continuous cloud particle replicator for measuring vertical profiles of cloud microphysical properties: instrument design, performance and collection efficiency analysis,” J. Atmos. Ocean. Technol. 14, 753–768 (1997).
[CrossRef]

1990 (1)

G. L. Stephens, S. C. Tsay, P. W. J. Stackhouse, and P. J. Flateau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climatic feedback,” J. Atmos. Sci. 47, 1742–1754 (1990).
[CrossRef]

1984 (1)

S. Ganci, “Simple derivation of formulas for Fraunhofer diffraction at polygonal apertures from Maggi–Rubinowicz transformation,” J. Opt. Soc. Am. 1, 559–561 (1984).
[CrossRef]

1983 (1)

Y. Takano and S. Asano, “Fraunhofer diffraction by ice crystals suspended in the atmosphere,” J. Meteorol. Soc. Jpn. 61, 289–299 (1983).

1979 (2)

1974 (1)

1962 (2)

Asano, S.

Y. Takano and S. Asano, “Fraunhofer diffraction by ice crystals suspended in the atmosphere,” J. Meteorol. Soc. Jpn. 61, 289–299 (1983).

Born, M.

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

Flateau, P. J.

G. L. Stephens, S. C. Tsay, P. W. J. Stackhouse, and P. J. Flateau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climatic feedback,” J. Atmos. Sci. 47, 1742–1754 (1990).
[CrossRef]

Ganci, S.

S. Ganci, “Simple derivation of formulas for Fraunhofer diffraction at polygonal apertures from Maggi–Rubinowicz transformation,” J. Opt. Soc. Am. 1, 559–561 (1984).
[CrossRef]

Heymsfield, A. J.

L. M. Miloshevich and A. J. Heymsfield, “A balloon-borne continuous cloud particle replicator for measuring vertical profiles of cloud microphysical properties: instrument design, performance and collection efficiency analysis,” J. Atmos. Ocean. Technol. 14, 753–768 (1997).
[CrossRef]

Marsh, J. S.

Miloshevich, L. M.

L. M. Miloshevich and A. J. Heymsfield, “A balloon-borne continuous cloud particle replicator for measuring vertical profiles of cloud microphysical properties: instrument design, performance and collection efficiency analysis,” J. Atmos. Ocean. Technol. 14, 753–768 (1997).
[CrossRef]

Miyamoto, K.

Pérez, J. P.

J. P. Pérez, Optique, Fondements et Applications, 7th ed.(Dunod, 2011).

Schwerd, F. M.

F. M. Schwerd, Die Beugungserscheinungen aus den Fundamentalgesetzen der Undulationstheorie Analytisch Entwickelt und In Bildern Dargestellt (Mannheim, 1835).

Sillitto, W.

Smith, R. C.

Stackhouse, P. W. J.

G. L. Stephens, S. C. Tsay, P. W. J. Stackhouse, and P. J. Flateau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climatic feedback,” J. Atmos. Sci. 47, 1742–1754 (1990).
[CrossRef]

Stephens, G. L.

G. L. Stephens, S. C. Tsay, P. W. J. Stackhouse, and P. J. Flateau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climatic feedback,” J. Atmos. Sci. 47, 1742–1754 (1990).
[CrossRef]

Takano, Y.

Y. Takano and S. Asano, “Fraunhofer diffraction by ice crystals suspended in the atmosphere,” J. Meteorol. Soc. Jpn. 61, 289–299 (1983).

Tsay, S. C.

G. L. Stephens, S. C. Tsay, P. W. J. Stackhouse, and P. J. Flateau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climatic feedback,” J. Atmos. Sci. 47, 1742–1754 (1990).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

Weickmann, H.

Wendling, P.

Wendling, R.

Wolf, E.

Appl. Opt. (1)

J. Atmos. Ocean. Technol. (1)

L. M. Miloshevich and A. J. Heymsfield, “A balloon-borne continuous cloud particle replicator for measuring vertical profiles of cloud microphysical properties: instrument design, performance and collection efficiency analysis,” J. Atmos. Ocean. Technol. 14, 753–768 (1997).
[CrossRef]

J. Atmos. Sci. (1)

G. L. Stephens, S. C. Tsay, P. W. J. Stackhouse, and P. J. Flateau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climatic feedback,” J. Atmos. Sci. 47, 1742–1754 (1990).
[CrossRef]

J. Meteorol. Soc. Jpn. (1)

Y. Takano and S. Asano, “Fraunhofer diffraction by ice crystals suspended in the atmosphere,” J. Meteorol. Soc. Jpn. 61, 289–299 (1983).

J. Opt. Soc. Am. (5)

Other (4)

F. M. Schwerd, Die Beugungserscheinungen aus den Fundamentalgesetzen der Undulationstheorie Analytisch Entwickelt und In Bildern Dargestellt (Mannheim, 1835).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

J. P. Pérez, Optique, Fondements et Applications, 7th ed.(Dunod, 2011).

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

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Figures (1)

Fig. 1.
Fig. 1.

Diffraction by an aperture in a screen D. ki and kd are incident and diffracted wavevectors. Oz is the optical axis, EXY is the Fourier plane, and P is a point of this plane of polar coordinate (ρ,φ).

Equations (28)

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ΦD|t̲(x,y)|2dS=D^|t̲^(u,v)|2dS^,
t̲(x,y)=ψ̲+(x,y)ψ̲(x,y),
Φ=12ε0c|ψ̲|2D|t̲(x,y)|2dS,
ΦDr|t̲r(x,y)|2dSr.
Φ=i=1NΦii=1N[Di|t̲i(x,y)|2dSi]=i=1N[D^i|ti̲^(u,v)|2dS^i],
{D=i=1NDit̲i0ifDiDand0otherwise
i=1N[Di|t̲i(x,y)|2dSi]=i=1N[D|t̲i(x,y)|2dS]=D[i=1N|t̲i(x,y)|2]dS.
{D^=i=1ND^iti̲^0ifD^iD^and0otherwise
i=1N[D^i|ti̲^(u,v)|2dS^i]=i=1N[D^|ti̲^(u,v)|2dS^]=D^[i=1N|ti̲^(u,v)|2]dS^.
D[i=1N|t̲i(x,y)|2]dS=D^[i=1N|ti̲^(u,v)|2]dS^
D[i=1NÉi(x,y)]dS=D^[i=1NÉ^i(u,v)]dS^,
ΦD[i=1N|t̲i(x,y)|2]dS=Dr[i=1N|t̲r,i(x,y)|2]dSr,
ΦD^[i=1N|ti̲^(x,y)|2]dS^=D^r[i=1N|t̲^r,i(u,v)|2]dS^r.
|t̲^r,i(ξ)|2=(πDi24)2[2J1(πξDi)πξDi]2=14ξ2Di2[J1(πξDi)]2,
|t̲^r,i(0)|2=(πDi24)2=Si2
i=1N|t̲^r,i(0)|2=NSi2.
Φ0{i=1NDi2[J1(πξDi)]2}14ξ2dS^r.
i=1NDi2[J1(πξDi)]2=NDi2[J1(πξDi)]2=N(πξ)2Zi2[J1(Zi)]2
Zi2[J1(Zi)]2=1ZMZmZmZMg(Z)dZ=1DMDmDmDMg(Zπξ)dD=1DMDmDmDMf(D)dD=f(Di)
Φf(Da,i)4π2N01ξ4dS^r.
É^(ξ)f(Da,i)4π2N1ξ4whereξ0.
É=dΦdSr=dΦdS^rdS^rdSr=É^(λR)2.
É^(θ)f(Da,i)4π2Nλ4sin4θwithθ0.
É^(θ)=ΦSbf(Da,i)4π2Nλ4sin4θandÉ^(0)=ΦSbNSi2.
Φ=S^É^dS^=ΦSbS^(i=1N|t̲^i(ξ)|2)dS^=ΦSbi=1N{0Si2[2J1(πξDi)πξDi]22πξdξ}=ΦSbi=1N8Si2πDi2[0J12(X)XdX]
0J12(X)XdX=12,
Φ=S^É^dS^=ΦSbi=1N4πDi2Si2=ΦSbi=1NSi=ΦSbSt,
hcλPdcosθdSrR2=É^λ2dSrR2and thenPd=É^hcλcosθ.

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