Abstract

With reference to the analysis contained in J. Opt. Soc. Am. A 9, 765 (1992), certain explanations of and remarks on the entire series of nonparaxial corrections for the fundamental Gaussian beam are presented.

© 2002 Optical Society of America

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References

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  1. A. Wünsche, “Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams,” J. Opt. Soc. Am. A 9, 765–774 (1992).
    [CrossRef]
  2. R. Borghi, M. Santarsiero, M. A. Porras, “Nonparaxial Bessel–Gauss beams,” J. Opt. Soc. Am. A 18, 1618–1626 (2001).
    [CrossRef]
  3. T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation for light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
    [CrossRef]
  4. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), formula 7.421.2.
  5. Ref. 4, p. xliii.
  6. Ref. 4, formula 8.973.3.
  7. Ref. 4, formula 8.971.5

2001

1992

1985

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Equations (76)

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2r2+1rr+2z2+k02Fn,0(r, z)=0,
Fn,0(r, z)=fn,0(r, z)exp(ik0z),
2r2+1rr+2ik0z+2z2fn,0(r, z)=0.
fn,0(r, z)=m=0m=f 02mf n,0(m)(r, z),
2r2+1rr+2ik0zfn,0(0)(r, z)=0
2r2+1rr+2ik0zfn,0(m)(r, z)
=-f0-22z2fn,0(m-1)(r, z)form=1, 2,,
fn,0(0)(r, 0)=(-1)n22nn!Ln(u)exp(-u),
u=r2/w02
Ln(v)=exp(v)n!dndvn[vn exp(-v)].
fn,0(m)(r, z)=0dηηJ0(rη)f¯n,0(m)(η, z)
f¯n,0(m)(η, z)=0drrJ0(rη)fn,0(m)(r, z).
fn,0(0)(r, z)=0dηηJ0(rη)f¯n,0(0)(η, 0)exp-iη22k0z.
f¯n,0(0)(η, 0)=w02(n+1)2(-1)nη2n exp-η2w024.
fn,0(0)(r, z)=w02(n+1)2(-1)n0dηηJ0(rη)η2n×exp-η2w024(1-ζ),
ζ=-iz/l0.
fn,0(0)(r, z)=(-1)n22nn!(1-ζ)n+1Ln(v)exp(-v),
v=u/(1-ζ).
fn,0(0)(r, z)=(-1)n22nnζn0dηηJ0(rη)w022×exp-η2w024(1-ζ)=(-1)n22nnζnf0,0(0)(r, z).
f0,0(0)(r, z)=(1-ζ)-1 exp(-v)
f ¯0,0(0)(η, z)=w022exp-η2w024(1-ζ).
f¯0,0(m)(η, z)=f¯0,0(0)(η, z)g¯0,0(m)(η, z),
g¯0,0(0)(η, z)=1.
ζg¯0,0(m)(η, z)=2ζ2+η2w022ζ+η4w0416g¯0,0(m-1) (η, z)
form=1, 2,.
g¯0,0(1)(η, z)=η4w0416ζ
g¯0,0(2)(η, z)=η6w0632ζ+η8w08512ζ2.
g¯0,0(m)(η, z)=r=1r=mCr(m)ζrη2w024m+r,
C0(m)=0,Cr(m)=0,forr>m.
C1(1)=1,C1(2)=2,C2(2)=1/2.
Cr(m)=1rCr-1(m-1)+2Cr(m-1)+(r+1)Cr+1(m-1).
Cr(m)=(2m)!m1(r-1)!(m-r)!(m+r)!.
f0,0(m)(r, z)=r=1r=mCr(m)ζr0dηηJ0(rη)η2w024m+r×w022exp-η2w024(1-ζ).
f0,0(m)(r, z)=r=1r=mCr(m)ζrm+rζm+rf0,0(0)(r, z).
f0,0(r, z)=f0,0(0)(r, z)+m=1m=f02m(2m)!m
×r=1r=mζr(r-1)!(m-r)!(m+r)!m+rζm+r
×f0,0(0)(r, z)
=T1f0,0(0)(r, z),
T1=1+m=1m=f02m(2m)!m×r=1r=mζr(r-1)!(m-r)!(m+r)!m+rζm+r.
f0,0(m)(r, z)=f02m(1-ζ)1+m(2m)!m×r=1r=mζr(1-ζ)rLm+r(v)exp(-v)(r-1)!(m-r)!.
T1f0,0(0)(r, z)=f0,0(0)(r, z)+m=1m=f02mζm!2mζ2m[ζm-1f0,0(0)(r, z)].
2mζ2m[ζm-1f0,0(0)(r, z)]
=q=0q=2m(2m)!q!(2m-q)!2m-qζ2m-q(ζm-1)qζq[f0,0(0)(r, z)].
2mζ2m[ζm-1f0,0(0)(r, z)]
=r=1r=m(2m)!(m-r)!(m+r)!m-rζm-r(ζm-1)m+rζm+r×[f0,0(0) (r, z)].
m-rζm-r(ζm-1)=(m-1)!(r-1)!ζr-1,
T1f0,0(0)(r, z)=f0,0(0)(r, z)+m-1m=f02m(2m)!m×r=1r=mζr(r-1)!(m-r)!(m+r)!m+rζm+r×[f0,0(0)(r, z)],
m+rζm+r[f0,0(0)(r, z)]
=1(1-ζ)1+m+rn=0n=(-1)nvn(n+1)m+rn!,
(n+1)m+r=(n+m+r)(n+m+r-1)(n+2)×(n+1)
m+rvm+r[vm+r exp(-v)]=n=0n=(-1)nvn(n+1)m+rn!.
m+rζm+r[f0,0(0)(r, z)]=(m+r)!(1-ζ)1+m+rLm+r(v)exp(-v).
T1f0,0(0)(r, z)=f0,0(0)(r, z)1+m=1m=f02m(1-ζ)m(2m)!m×r=1r=mζr(1-ζ)rLm+r(v)(r-1)!(m-r)!.
T1f0,0(0)(0, z)=1(1-ζ)1+m=1m=f02m(2m)!m!ζ(1-ζ)m+1×r=1r=m(m-1)!(r-1)!(m-r)!ζ1-ζr-1.
T1f0,0(0)(0, z)=1(1-ζ)1+m=1m=f02m(2m)!m!ζ(1-ζ)2m.
f0,0(m)(r, z)=f02m(1-ζ)1+m(2m)!m×s=1s=mζs(1-ζ)mLm+s-(s-1)(v)exp(-v)(s-1)!(m-s)!.
f0,0(m)(0, z)=s=1s=mf02m(1-ζ)1+2m(2m)!mζsLm+s-(s-1)(0)(s-1)!(m-s)!.
f0,0(m)(0, z)=f02m(1-ζ)1+2m(2m)!m!ζ.
Lm+s-(s-1)(0)=m+1m+s=(m+1)(m)(m-1)  (m+1-m-s+2)(m+1-m-s+1)(m+s)(m+s-1)  3.2.1.
f0,0(1)(r, z)ζf2,0(0)(r, z).
f0,0(2)(r, z)ζf3,0(0)(r, z)+ζ2f4,0(0)(r, z).
f0,0(3)(r, z)ζf4,0(0)(r, z)+ζ2f5,0(0)(r, z)+ζ3f6,0(0)(r, z).
f0,0(m)(r, z)r=1r=mζrfm+r,0(0)(r, z).
Lnα(v)=Lnα+1(v)-Ln-1α+1(v).
Lm+s-s+1(v)=t=1t=s(-1)t-1(s-1)!(t-1)!(s-t)!Lm+s-t+1(v).
f0,0(m)(r, z)=s=1s=mf02m(1-ζ)1+2m(2m)!mζs exp(-v)(s-1)!(m-s)!×t=1t=s(-1)t-1(s-1)!(t-1)!(s-t)!Lm+s-t+1(v).
f0,0(m)(r, z)=f02m(1-ζ)1+m(2m)!m×r=1r=mζr(1-ζ)mLm+r(v)exp(-v)(r-1)!×s=rs=m(-ζ)s-r(s-r)!(m-s)!
f0,0(m)(r, z)=f02m(1-ζ)1+m(2m)!m×r=1r=mζr(1-ζ)rLm+r(v)exp(-v)(r-1)!(m-r)!,
T2f0,0(0)(r, z)=f0,0(0)(r, z)+m=1m=f02m1m!2mζ2m[ζmf0,0(0)(r, z)].
T2f0,0(0)(r, z)=f0,0(0)(r, z)+m=1m=f02m(2m)!×r=0r=mζrr!(m-r)!(m+r)!m+rζm+r×[f0,0(0) (r, z)],
T2f0,0(0)(r, z)=f0,0(0)(r, z)1+m=1m=f02m(1-ζ)m(2m)!×r=0r=mζr(1-ζ)rLm+r(v)r!(m-r)!.
T2f0,0(0)(0, z)=1(1-ζ)1+m=1m=f02m(2m)!m!1(1-ζ)2m.
f0,0(m)(r, z)=f02m(1-ζ)1+m(2m)!×k=0k=mζk(1-ζ)mLm+k-k(v)exp(-v)k!(m-k)!
f0,0(m)(r, z)=f02m(1-ζ)1+m(2m)!×r=0r=mζr(1-ζ)rLm+r(v)exp(-v)r!(m-r)!.
Lm+k-k(v)=r=0r=k(-1)k-rk!r!(k-r)!Lm+r(v).
f0,0(m)(r, z)=f02m(1-ζ)1+m(2m)!×r=0r=mζr(1-ζ)mLm+r(v)exp(-v)r!×k=rk=m(-ζ)(k-r)(k-r)!(m-k)!.

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