Abstract

I present a formulation for treating diffraction effects on total irradiance in the case of a Planck source; earlier work generally depended on calculating diffraction effects on spectral irradiance followed by summation over spectral components. The formulation is derived and demonstrated for Fraunhofer diffraction by circular apertures, rectangular apertures and slits, and Fresnel diffraction by circular apertures. The prospects for treating other sources and optical systems are also discussed.

© 2004 Optical Society of America

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References

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  1. See, for example, J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), pp. 427ff.
  2. All of these examples are discussed in M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 8.
  3. See, for example, J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), pp. 66–67.
  4. See, for example, W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974), p. 96.
  5. K. Miyamoto, E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave—part I,” J. Opt. Soc. Am. 52, 612–625 (1962).
  6. K. Miyamoto, E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave—part II,” J. Opt. Soc. Am. 52, 626–637 (1962).
    [CrossRef]
  7. K. Knopp, Theory and Application of Infinite Series, 2nd English ed. (Blackie and Son, London, 1946), p. 419.

1962 (2)

Born, M.

All of these examples are discussed in M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 8.

Goodman, J. W.

See, for example, J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), pp. 66–67.

Jackson, J. D.

See, for example, J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), pp. 427ff.

Knopp, K.

K. Knopp, Theory and Application of Infinite Series, 2nd English ed. (Blackie and Son, London, 1946), p. 419.

Miyamoto, K.

Welford, W. T.

See, for example, W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974), p. 96.

Wolf, E.

J. Opt. Soc. Am. (2)

Other (5)

K. Knopp, Theory and Application of Infinite Series, 2nd English ed. (Blackie and Son, London, 1946), p. 419.

See, for example, J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), pp. 427ff.

All of these examples are discussed in M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 8.

See, for example, J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), pp. 66–67.

See, for example, W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974), p. 96.

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Figures (6)

Fig. 1
Fig. 1

Canonical optical arrangement considered in this work. Light emitted from an extended source passes by N optical elements before reaching a detector. The optical axis is assumed to be the z axis. Ap., aperture.

Fig. 2
Fig. 2

Rectangular aperture with several geometrical parameters indicated.

Fig. 3
Fig. 3

(dE/dA s )/Ξ0 for case of Fraunhofer diffraction by a rectangular aperture along three lines in the (θ x , θ y ) plane as discussed in the text.

Fig. 4
Fig. 4

Universal function h(z) relevant for the case of Fraunhofer diffraction by a rectangular slit, plotted as h(z)/z 2 versus z.

Fig. 5
Fig. 5

Universal behavior of η(α) for the case of Fraunhofer diffraction by a circular aperture, as a function of angular parameter α (solid curve), and asymptotic result (dashed curve).

Fig. 6
Fig. 6

Diffraction effects on total irradiance because of a circular aperture for a geometry discussed in the text versus distance from the optical axis, r d . The solid curve shows results of numerical calculations, and the dashed line shows what is expected from geometrical optics.

Tables (2)

Tables Icon

Table 1 Function h(z) Versus z

Tables Icon

Table 2 Function η(α) = (dE/dA s )/Ξ0 Versus α

Equations (87)

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Lλλ, Ts, rs, Ωˆ=f1rs, Ωˆf2λ, Ts.
uk, rs, rdu0iλNAp1ApNd2r1d2rN×Gk, rs, r1Gk, rN, rd×expikδLrμ,
Gk, rμ, rν=expik|rμ-rν||rμ-rν|1zν-zμexpikzν-zμ+xν-xμ2+yν-yμ22zν-zμ
expikδLrμ=expik μ=1N-xμ2+yμ22fμ
Lrμ=zd-zs+x1-xs2+y1-ys22z1-zs++xd-xN2+yd-yN22zd-zN+δLrμ,
Δ=z1-zsz2-z1zd-zN.
fl, rs, rd=Ap1d2r1 ApNd2rNδl-L|rμ},
uk, rs, rd=u0ΔiλN-dlfl, rs, rdexpikl,
|uk, rs, rd|2=|u0|2Δ2λ2N-dl -dlfl, rs, rd×fl, rs, rdexpikl-l,
Tk, rs, rd=|uk, rs, rd/u0|2=k2N2π2NΔ2-dl -dlfl, rs, rd×fl, rs, rdexpikl-l=k2N-22π2NΔ2-dl -dldfl, rs, rddl×dfl, rs, rddlexpikl-l.
T0rs, rd=limk Tk, rs, rd|Illum.
T0rs, rd=1ds2dd21ds+1dd-1f-2.
Erd, Ts=Sourced2rsdErd, TsdAs=Sourced2rs0dλ dEλλ, rd, TsdAs.
dEλλ, rd, TsdAs=ρTk, rs, rd|u0|2=Tk, rs, rdLλλ, Ts.
dErd, TsdAs=0dλTk, rs, rdLλλ, Ts=c1π0dλλ5Tk, rs, rdexpc2/λTs-1,
dErd, TsdAs=c116π50dkk3Tk, rs, rdexpβk-1,
β=c22πTs=ckBTs.
dE0rd, TsdAs=3ζ4c18π5β4T0rs, rd.
dErd, TsdAs=c116π52π2NΔ2×-dl -dlfl, rs, rdfl, rs, rd×0dkk3+2Nexpikl-lexpβk-1=3+2N!c116π52π2NΔ2×-dl -dlfl, rs, rdfl, rs, rd×n=11nβ-il-l2N+4.
dErd, TsdAs=c116π52π2NΔ2×-dl -dl dfl, rs, rddl×dfl, rs, rddl0dkk1+2N×expikl-lexpβk-1=1+2N!c116π52π2NΔ2×-dl -dl dfl, rs, rddl×dfl, rs, rddl×n=11nβ-il-l2N+2.
ddlfl, rs, rdl-l0N-1=gδl-l0+bl, rs, rd,
dErd, TsdAs=3c164π7ds2dd2-dlSβ, l×-dsdfs, rs, rdds×dfs+l, rs, rdds,
dErd, TsdAs=3c1g2Sβ, 064π7ds2dd2+3c1g32π7ds2dd2×-dlSβ, lbl0+l, rs, rd+3c164π7ds2dd2-dlSβ, l×-dsbs, rs, rdbs+l, rs, rd=dE0dAsIG+IXβ, rs, rd+IBβ, rs, rd.
Lrs, r1, rdds+dd-θxx1-θyy1l0-θxx1-θyy1.
-dlfl, rs, rd=AAp1,
dErd, TsdAs=Ξ0=15ζ6c1AAp128π7ds2dd2β6.
dErd, TsdAs=3c1B232π7ds2dd2-l+-l-dl -l+-l-dl--l+-l-dl +l-+l+dl-+l-+l+dl -l+-l-dl++l-+l+dl +l-+l+dl×n=11nβ-il-l4=c1B264π7ds2dd2({ n=11[nβi(l-l')] | l=-l+l=-l-}| l=-l+l=-l-) .
1a-ib2+1a+ib2=a+ib2+a-ib2a2+b22=2a2-b2a2+b22=2a2+b2-4b2a2+b22,
dErd, TsdAs=c1B216π7ds2dd2β2ζ2-4π2×g2πl+-l-/β-4π2×g2πl++l-/β+2π2g4πl+/β+2π2g4πl-/β.
dfl, rs, rddl=AAp12l¯δl+l¯-AAp12l¯δl-l¯,
dErd, TsdAs=3c1AAp12128π7ds2dd2l¯2n=12nβ4-1nβ-2il¯4-1nβ+2il¯4=3c1AAp12128π7ds2dd2l¯2β42ζ4-S1, 2l¯/β.
Lrs, r1, rdl0-θxx1+12ds+12ddy12,
uk, rs, rd=u0iλdsdd-dlf1l, rs, rdexpikl×-dy1 expik12ds+12ddy12,
f1l, rs, rd=-rx+rxdx1δl--θxx1=Θ|θxrx|-lΘl+|θxrx|/|θx|.
Tk, rs, rd=k2πdsddds+dd×-dl -dlf1l, rs, rdf1l, rs, rd×expikl-l.
dErd, TsdAs=c132π6dsddds+dd×-dl -dlf1l, rs, rdf1l, rs, rd×0dkk4 expikl-lexpβk-1=3c14π6dsddds+dd×-dl -dlf1l, rs, rdf1l, rs, rd×n=1nβ-il-l-5=c116π6dsddds+dd×-dl -dl df1l, rs, rddl×df1l, rs, rddln=1nβ-il-l-3.
df1l, rs, rddl=δl+|θxrx|-δl-|θxrx|/|θx|.
dErd, TsdAs=c1hz16π6dsddds+ddθx2β3,
hz=2ζ3-n=11n-iz3-n=11n+iz3
Lrd, r1, rs=l0-x1θx-y1θy,
fl, rs, rd=2θ2R2θ2-l21/2
bl, rs, rd=-2lθ2R2θ2-l21/2.
dEdAs=3c18π7ds2dd2θ402RθdlSβ, l-Rθ+l/2Rθ-l/2dss-l/2s+l/2Rθ-s-l/2Rθ+s+l/2Rθ-s+l/2Rθ+s-l/21/2=3c18π7ds2dd2θ402RθdlSβ, lRθ-l/2-11dxRθ-l/22x2-l2/4(1-x2Rθ+l/2/Rθ-l/22-x2)1/2=3c18π7ds2dd2θ402RθdlSβ, lRθ+l/2-11dxRθ-l/22x2-l2/4(1-x21-Rθ-l/2/Rθ+l/22x2)1/2=3c18π7ds2dd2θ402RθdlSβ, l2Rθ+l01dx2Rθ-l2x2-l2(1-x21-2Rθ-l/2Rθ+l2x2)1/2=3c1R22π7ds2dd2θ201dySβ, 2Rθy1+y01dx1-y2x2-y2(1-x21-1-y/1+y2x2)1/2.
1-y2x2-y21+y=1+2y1+y-1+y×1-1-y1+y2x2,
dEdAs=3c1R22π7ds2dd2θ201dySβ, 2Rθy×1+2y1+yK1-y1+y2-1+yE1-y1+y2,
Ku=01dt1-t21-ut21/2,
Eu=01dt1-ut21-t21/2,
dEdAs=3c1R24π7ds2dd2θ2β401dyS1, αyK1-y2-2E1-y2=3c1AAp12π9ds2dd2α2β601dyS1, αyK1-y2-2E1-y2.
Am=01dyy2mK1-y2=π4Γm+1/2Γm+12,
Bm=01dyy2mE1-y2=π4Γm+1/2Γm+3/2Γm+1Γm+2,
dEdAs=3c1AAp12π9ds2dd2α2β6m=02π2msmAm-2Bmα2m,
ηα504ζ3π7α3+378 logeαπ7α5+636γ+12 loge2-11π7α5+.
Lrs, r1, rdds+dd+xd2+yd22ds+dd+x1-xi2+y1-yi212ds+12dd=l0+Cx1-xi2+y1-yi2,
Lrs, r1, rd=Cx1-xi2+y1-yi2=Cρ2,
θ=cos-1ρ2+τ2-R22τρ.
fl, rs, rd=π/C,
fl, rs, rd=θ/C.
fl, rs, rd=0.
fl, rs, rd=0
fl, rs, rd=θ/C
bl=1Cdθdl=1Cdθdρdρdl=12ρC2dθdρ.
bl=A-Blll-l1l2-l1/2,
A=τ2-R22, B=12C, l1=C|τ-R|2, l2=Cτ+R2.
IX=β4Cπζ4l1l2dlSβ, lA-Blll-l1l2-l1/2.
IB=β4C2π2ζ40l2-l1dlSβ, ll1l2-lds A-BsA-Bs-Blss+ls-l1s+l-l1l2-sl2-s-l1/2.
IB=β4C2π2ζ4a0l2-l1dlSβ, l×-1+1dx A/a-Bx-Bb/aA/a-Bx-Bb/a-Bl/ax+b/ax+b/a+l/a1-x21+l/a2-x21/2=β4B2C2κπ2ζ4a0l2-l1dlSβ, l-1+1dxn1-xn2-xx+d1x+d21-x21-κ2x21/2,
n1=ABa-ba, n2=ABa-ba-la, d1=ba, d2=ba+la, κ=11+l/a.
n1-xn2-xx+d1x+d2=1+1d2-d1ξ1x+d1-ξ2x+d2,
ξ1=d1+n1d1+n2,ξ2=d2+n1d2+n2.
1+1d2-d1ξ1x+d1-ξ2x+d21+1d2-d1d1ξ1d12-x2-d2ξ2d22-x2
IB=2β4B2C2κπ2ζ4a0l2-l1dlSβ, l01dx1-x21-κ2x21/2+ξ1/d1d2-d101dx1-d1-2x21-x21-κ2x21/2-ξ2/d2d2-d101dx1-d2-2x21-x21-κ2x21/2=2β4B2C2κπ2ζ4a0l2-l1dlSβ, lKκ2+ξ1/d1d2-d1Πd1-2, κ2-ξ2/d2d2-d1Πd2-2, κ2,
Πd2, κ2=01dt1-d2t21-t21-κ2t21/2
Sβ, l=n=1nβ+il-4+nβ-il-4
Sβ, l=l-4Sβ/l, 1=β-4S1, l/β,
b+il-4+b-il-4=b-il4+b+il4b2+l24=2b4-6b2l2+l4b2+l24=2b2+l22-8l2b2+l2+8l4b2+l24=2/b2+l22-16l2b2+l23+16l4b2+l24,
Sβ, l=2 n=1nβ2+l2-2-16l2n=1nβ2+l2-3+16l4n=1nβ2+l2-4.
Sβ, l=22π/β4S2z-16l22π/β6S3z+16l42π/β8S4z=22π/β4S2z-8z2S3z+8z4S4z,
Smz=n=14π2n2+z2-m.
S1z=12z11-exp-z-1z-12,
Sm+1z=-12mzddz Smz.
f=df/dz=expz/expz-1-exp2z/expz-12=f-f2,f=d2f/dz2=f-f21-2f=f-3f2+2f3,f=d3f/dz3=f-f21-6f+6f2=f-7f2+12f3-6f4.
S2z=-14z2 f+14z3 f-12z4-18z3, S3z=116z3 f-316z4 f+316z5 f-12z6-332z5, S4z=-196z4 f+116z5 f-532z6 f+532z7 f-12z8-564z7.
Sβ, l=-22π/β412z4+f12=-l-4+Oe-z.
4π2n2+z2-m=14π2mk=0×m-1+k!-z2/4π2kk!m-1!n2m+2k,
Smz=14π2mk=0×m-1+k!ζ2m+2k-z2/4π2kk!m-1! =k=0 sm,kz2k,
S1z=124-z21440+z460480-,S2z=11440-z230240+z4806400-,S3z=160480-z2806400+z415966720-,S4z=12419200-z223950080+691z4261534873600-,Sβ, l=32π4β411440-z26048+z469120-,,
Sβ, l=32π4β4s2,0+z2s2,1-8s3,0+k=2 z2ks2,k-8s3,k-1+8s4,k-2=k=0 skz2k,

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