Abstract

A key step in any ray-based method for propagating waves is the choice of a family of rays to be associated with the initial wave field. We develop some basic prescriptions for constructing initial ray families to match two particular types of waves. Various Gaussian and Bessel beams are separately given special treatment because of their general interest. These ideas are directly useful for a newly developed method for ray-based wave modeling. The new method expresses the wave as a superposition of ray contributions that is independent of the width of the field element associated with each ray. This insensitivity is investigated here even when the elemental width varies from ray to ray. The results increase the applicability of the new wave-modeling scheme.

© 2001 Optical Society of America

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References

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  1. G. W. Forbes, M. A. Alonso, “Using rays better. I. Theory for smoothly varying media,” J. Opt. Soc. Am. A 18, 1132–1145 (2001).
    [CrossRef]
  2. See, for example, the review presented in Section 5 of Yu. A. Kravtsov, G. W. Forbes, A. A. Asatryan, “Theory and applications of complex rays,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, Amsterdam1999), Vol. XXXIX, pp. 1–62.
  3. M. A. Alonso, G. W. Forbes, “Phase space distributions for high-frequency fields,” J. Opt. Soc. Am. A 17, 2288–2300 (2000).
    [CrossRef]
  4. N. G. de Bruijn, Asymptotic Methods in Analysis (Dover, New York, 1981), Chap. 5.
  5. N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986), Chap. 7.
  6. To see this, one must take into account Eq. (I-2.4).
  7. G. W. Forbes, M. A. Alonso, “Using rays better. III. Error estimates and illustrative applications in smooth media,” J. Opt. Soc. Am. A18 (to be published).
  8. Point sources are useful in this context because they corre-spond to Green’s functions. The propagator corresponds to the z0 derivative of the Green’s function and is used when considering plane-to-plane imaging. See, for example, M. A. Alonso, G. W. Forbes, “Semigeometrical estimation of Green’s functions and wave propagators in optics,” J. Opt. Soc. Am. A 14, 1076–1086 (1997).
    [CrossRef]
  9. This result also follows from using the saddle-point method in Eq. (1.1) with γ(ξ, z).
  10. As stated in Ref. 3, examples of these fields are the modes of z independent guiding smooth media.
  11. The arbitrary constant of integration in L is set to zero in Eq. (5.6), since it gives only a global phase that can be absorbed by c. The same is done in subsequent examples.
  12. Notice that we must now use the version of this method for integrals over two variables. See Ref. 4.
  13. R. L. Gordon, G. W. Forbes, “Optimal resolution with extreme depth of focus,” Opt. Commun. 150, 277–286 (1998).
    [CrossRef]

2001 (1)

2000 (1)

1998 (1)

R. L. Gordon, G. W. Forbes, “Optimal resolution with extreme depth of focus,” Opt. Commun. 150, 277–286 (1998).
[CrossRef]

1997 (1)

Alonso, M. A.

Asatryan, A. A.

See, for example, the review presented in Section 5 of Yu. A. Kravtsov, G. W. Forbes, A. A. Asatryan, “Theory and applications of complex rays,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, Amsterdam1999), Vol. XXXIX, pp. 1–62.

Bleistein, N.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986), Chap. 7.

de Bruijn, N. G.

N. G. de Bruijn, Asymptotic Methods in Analysis (Dover, New York, 1981), Chap. 5.

Forbes, G. W.

G. W. Forbes, M. A. Alonso, “Using rays better. I. Theory for smoothly varying media,” J. Opt. Soc. Am. A 18, 1132–1145 (2001).
[CrossRef]

M. A. Alonso, G. W. Forbes, “Phase space distributions for high-frequency fields,” J. Opt. Soc. Am. A 17, 2288–2300 (2000).
[CrossRef]

R. L. Gordon, G. W. Forbes, “Optimal resolution with extreme depth of focus,” Opt. Commun. 150, 277–286 (1998).
[CrossRef]

Point sources are useful in this context because they corre-spond to Green’s functions. The propagator corresponds to the z0 derivative of the Green’s function and is used when considering plane-to-plane imaging. See, for example, M. A. Alonso, G. W. Forbes, “Semigeometrical estimation of Green’s functions and wave propagators in optics,” J. Opt. Soc. Am. A 14, 1076–1086 (1997).
[CrossRef]

See, for example, the review presented in Section 5 of Yu. A. Kravtsov, G. W. Forbes, A. A. Asatryan, “Theory and applications of complex rays,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, Amsterdam1999), Vol. XXXIX, pp. 1–62.

G. W. Forbes, M. A. Alonso, “Using rays better. III. Error estimates and illustrative applications in smooth media,” J. Opt. Soc. Am. A18 (to be published).

Gordon, R. L.

R. L. Gordon, G. W. Forbes, “Optimal resolution with extreme depth of focus,” Opt. Commun. 150, 277–286 (1998).
[CrossRef]

Handelsman, R. A.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986), Chap. 7.

Kravtsov, Yu. A.

See, for example, the review presented in Section 5 of Yu. A. Kravtsov, G. W. Forbes, A. A. Asatryan, “Theory and applications of complex rays,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, Amsterdam1999), Vol. XXXIX, pp. 1–62.

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

R. L. Gordon, G. W. Forbes, “Optimal resolution with extreme depth of focus,” Opt. Commun. 150, 277–286 (1998).
[CrossRef]

Other (9)

See, for example, the review presented in Section 5 of Yu. A. Kravtsov, G. W. Forbes, A. A. Asatryan, “Theory and applications of complex rays,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, Amsterdam1999), Vol. XXXIX, pp. 1–62.

N. G. de Bruijn, Asymptotic Methods in Analysis (Dover, New York, 1981), Chap. 5.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986), Chap. 7.

To see this, one must take into account Eq. (I-2.4).

G. W. Forbes, M. A. Alonso, “Using rays better. III. Error estimates and illustrative applications in smooth media,” J. Opt. Soc. Am. A18 (to be published).

This result also follows from using the saddle-point method in Eq. (1.1) with γ(ξ, z).

As stated in Ref. 3, examples of these fields are the modes of z independent guiding smooth media.

The arbitrary constant of integration in L is set to zero in Eq. (5.6), since it gives only a global phase that can be absorbed by c. The same is done in subsequent examples.

Notice that we must now use the version of this method for integrals over two variables. See Ref. 4.

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Figures (7)

Fig. 1
Fig. 1

Errors in the estimate of a Gaussian field resulting from using γ=100(kw2)-1 (solid curve) and γ=100(kw2)-1×exp[-ξ2/(2w2)] (dashed curve). Notice that the accuracy for the second case is consistently better only within the interval |x|<w. Near x=0, the plot gives the relative error because the raw difference is scaled by the maximum value of the Gaussian field at its center.

Fig. 2
Fig. 2

Ray family associated with a Hermite–Gaussian beam propagating from its waist in free space. It is no surprise to see the basic hyperbolic structure of the Gaussian beam emerge so clearly from the rays. Further, there are two rays at each point inside this hyperbolic region, and interference between them leads to the multiple lobes of the Hermite–Gaussians of nonzero order.

Fig. 3
Fig. 3

Plots of (2m+1)|g(ξ, z0; γ¯, 2βγ¯)/(ka0)| as functions of ξ for β=1 (solid curve) and β=-1 (dashed curve). Notice that the two curves are identical other than for a displacement and that both have maxima of the order of unity.

Fig. 4
Fig. 4

Plots of estimates of a Hermite–Gaussian field of orders (a) m=0 and (b) m=7. In both cases three values of γ were used: γ¯/2, γ¯ (which corresponds to the exact solution), and 2γ¯. Notice that the estimates in (b) are considerably less sensitive to changes in γ.

Fig. 5
Fig. 5

Initial conditions for the ray families associated with (a) a Hermite–Gaussian field, (b) a Laguerre–Gaussian field, and (c) a Bessel field. The positions X of a sample of these rays at z=z0 (uniformly spaced in ξ and η in each case) are represented by gray dots, and their associated initial directions P are represented by arrows. The length of the arrows is proportional to the sine of the angle between the ray and the z axis. While the families in (a) and (b) occupy only the finite regions of the (x, y) plane shown in pale gray, the ray family in (c) actually extends over the whole plane except for the central circle of radius ρ. Also in (c), notice that, for m0 as shown, the arrows circulate about the z axis, hence the field’s nonzero angular momentum.The rays in any plane containing the z axis for (b) have precisely the form shown in Fig. 2 (see also the ray tangents there in the plane z=0). Also represented as gray ellipses in (c) are the effective dimensions of the contributions associated with some of these rays, resulting from the use of Eq. (6.15). The minor and major semiaxes of these ellipses correspond to (kγ1)-1/2 and (kγ2)-1/2, respectively.

Fig. 6
Fig. 6

Radial plots of estimates of a Laguerre–Gaussian field of orders (a) m=0 and (b) m=7. In both cases, three values of G were used: γ¯I/2, γ¯I (which corresponds to the exact solution), and 2γ¯I. Notice that the estimates in (b) are considerably less sensitive to changes in G.

Fig. 7
Fig. 7

Radial plots of the magnitude of the estimates of a Bessel field of order m=0, resulting from the use of γ1=γ2=kP02 (dotted curve) for which G is constant; 10γ1=0.1γ2=kP02 (dashed curve) for which G depends on ξ; and γ1=0 and γ2= (gray curve), which gives the exact solution.

Equations (144)

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P(ξ, z)n[X(ξ, z), z]X˙(ξ, z)/[1+X˙(ξ, z)]1/2,
Uγ(0)(x, z)
=k2πa0(ξ)Y(ξ, z)H(ξ, z) exp{-12kγ(z)[x-X(ξ, z)]2}×exp(ik{L(ξ, z)+[x-X(ξ, z)]P(ξ, z)})dξ,
Uγ(1)(x, z)=k2π a0(ξ)+a1[ξ, z,γ(z)]ik×Y(ξ, z)H(ξ, z) exp{-12kγ(z)[x-X(ξ, z)]2}×exp(ik{L(ξ, z)+[x-X(ξ, z)]P(ξ, z)})dξ.
a1(ξ, z, γb)=a1(ξ, z, γa)+g(ξ, z; γa, γb),
g(ξ, z;γa, γb)
γaγb a1γ dγ=iH γb-γa2 1YaYb a0H+a024H 51YaYb-YaYb+YaYb(YaYb)2.
k|a0(ξ)||g(ξ, z; γ, 2±1γ)|,
Gω(x, p)k2π ω1/4U(x, z0)×exp-kω (x-x)22-ikpx-x2dx,
Uγ(0)(x, z)=a0[ξS(x)]{H[ξS(x), z]X[ξS(x), z]}1/2+O(k-1)×exp{ikL[ξS(x), z]}.
L(ξ, z0)=arg[U(ξ, z0)]/k.
P(ξ, z0)=L(ξ, z0),
a0(ξ)=|U(ξ, z0)|H(ξ, z0)=|U(ξ, z0)|[n2(ξ, z0)-P2(ξ, z0)]1/4.
Uγ(1)(x, z0)exp(ikL)A0+a1ikH+12kY0×A0-i A0 PY0-i A0 P4Y0-5A0 P212Y02+O(k-2),
a1(ξ, z0, γ)=-iH2(γ+iP)A0-i A0P(γ+iP)-i A0P4(γ+iP)-5A0P212(γ+iP)2=limγa g(ξ, z0; γa, γ),
a1(ξ, z0, γ)=-iH2γ a0H.
U(x, z0)=U0 exp-x22w2.
a1=-i2γw2 ξ2w2-1a0.
12k|γ|w2 ξ2w2-11.
|γ|1kw2.
U(x, z0)=U0 sechπ2 xw.
a1=-iπ4γw2tanh2π2 ξw-sech2π2 ξwa0.
U˜γ(0)(p, z)k2π Uγ(0)(x, z)exp(-ikpx)dx=ik2π a0(ξ)
× P(ξ, z)/γ(z)-iX(ξ, z)H(ξ, z)1/2
×exp-k [p-P(ξ, z)]22γ(z)
×exp(ik{L¯(ξ, z)-[p-P(ξ, z)]
×X(ξ, z)})dξ,
L¯(ξ, z0)=arg[U˜(ξ, z0)]/k,
X(ξ, z0)=-L¯(ξ, z0),
a0(ξ)=|U˜(ξ, z0)|H(ξ, z0)=|U˜(ξ, z0)|{n2[X(ξ, z0), z0]-ξ2}1/4.
U˜γ(1)(p, z)=ik2π a0(ξ)+a1[ξ, z, γ(z)]ik×Y(ξ, z)γ(z)H(ξ, z) exp-k [p-P(ξ, z)]22γ(z)×exp(ik{L¯(ξ, z)-[p-P(ξ, z)]X(ξ, z)})dξ.
a1(ξ, z0, γ)=-iH2(γ-1-iX) A0+i A0X(γ-1-iX)+i A0X4(γ-1-iX)-5A0X224(γ-1-iX)2=limγa0g[ξ, z0; γa, γ(z0)],
a1(ξ, z0, γ)=-iγH2 a0H.
U˜(p, z0)=U0kq exp-p22q2,
|γ|kq2=1kw2.
U˜(p, z0)=U0kq sechπ2 pq.
X(ξ, z0)=x0,L¯(ξ, z0)=-x0ξ,L(ξ, z0)=0.
a1(ξ, z0, γ)=iγa08H2 3-5 n2(x0, z0)H2,
ka1(ξ, z0, γ)a0=|γ|4n2(x0, z0).
Uγ+δγ-Uγ=0.
Uγ(x, z)=k2π A[ξ, z, γ(ξ, z), γ(ξ, z),]×Λ(ξ, z)g[x, z; γ(ξ, z), ξ]dξ,
g[x, z; γ ,ξ]exp{-(kγ/2)[x-X(ξ, z)]2}×exp(ik{L(ξ, z)+[x-X(ξ, z)]P(ξ, z)}),
Λ(ξ, z)γ(ξ, z)X(ξ, z)+iP(ξ, z).
n=0 Aγ(n)δγ(n)+XΛ-kΔ2A δγ2+O(δγ2)
×Λgdξ=0,
g=kΛΔ-γ Δ22g.
Δg=gkΛ+γ2ΛΔ2g=gkΛ m=0Δγ2Λm,
F2f(ξ)Δ2(x, ξ, z)g[x, z; γ(ξ, z), ξ]dξ=fXkΛ+Cˆ2 fk2+O(k-3)gdξ,
Cˆ2 f=1Λ fΛ-2fΛ γXΛ2+fΛ2 54 γ 2X2Λ2-32XγΛ-γXΛ.
n=0 Aγ (n)δγ (n)+12kΛCˆ2(AΛδγ)+O(δγ2)Λgdξ=0.
n=0A0γ (n)δγ(n)=0,
n=0 Amγ (n)δγ (n)=i2ΛCˆ2(Am-1Λδγ),
G(ξ, z; γa, γa, γa, γb, γb, γb)
A1(ξ, z, γb, γb, γb)-A1(ξ, z, γa, γa, γa)
=(ΓA0)+ΘA0
+38Γ1/3(Γ2/3)+Φ1Γ+Φ0+Φ-1ΓA0,
Γγa-γb2iΛaΛb,Θ12i γaΛa2-γbΛb2,
Φ1ΛaΛb-γaγbX212ΛaΛb,
Φ0(Λa+Λb)(Λaγb-Λbγa)24iΛa2Λb2+16i ΛaγaΛa3-ΛbγbΛb3+34Θ,
Φ-1112ΛaΛb γaΛa-γbΛb2+512Θ2.
g(ξ, z; γa, γa, γa, γb, γb, γb)
a1(ξ, z, γb, γb, γb)-a1(ξ, z, γa, γa, γa)
=H(ξ, z)G(ξ, z ; γa, γa, γa, γb, γb, γb).
k|a0||g[ξ, z0; γ, γ, γ, φγ, (φγ), (φγ)]|,
a1[ξ, z, γ(ξ, z)]
=a1[ξ, z0, γ(ξ, z0)]+g[ξ, z0; γ(ξ, z0), γ¯]+z0za˙1(ξ, z, γ¯)dz+g[ξ, z; γ¯, γ(ξ, z)],
a1(ξ, z0, γ, γ, γ)=limγa g(ξ, z0; γa, 0, 0, γ, γ, γ).
a1(ξ, z0, γ, γ, γ)=-iH2 a0γH,
a1(ξ, z0, γ, γ, γ)=limγa0 g(ξ, z0; γa, 0, 0, γ, γ, γ),
a1(ξ, z0, γ, γ, γ)=-iH2 γa0H.
Uγ(0)(x, z)=k2π ξ0ξ0+Ξa0(ξ)Λ(ξ, z)H(ξ, z)×exp-12kγ(ξ, z)[x-X(ξ, z)]2×exp(ik{L(ξ, z)+[x-X(ξ, z)]P(ξ, z)})dξ.
[Λ(ξ+Ξ, z)]1/2=[Λ(ξ, z)]1/2 exp[±iπ],
ϕ=L(ξ+Ξ, z)-L(ξ, z).
ϕ=(2m+1)π/k,
X(ξ, z0)=X0 cos ξ,
P(ξ, z0)=P0 sin ξ,
A0(ξ, z0)=a0(ξ)/H(ξ, z0)=c.
L(ξ, z0)=X0P02(sin ξ cos ξ-ξ).
Uγ¯(0)(x, z0)=U0 exp-kγ¯2x2Hm(kγ¯x),
a1(ξ, z, γ, γ, γ)
=g(ξ, z; γ¯, 0, 0, γ, γ, γ)=-ka024(2m+1)(γ ¯cos ξ+i sin ξ)×{(γ-γ¯)[-9i exp(iξ)+6(γ-γ¯)×sin ξ+2i(γ-γ¯)2 exp(iξ)sin2 ξ]+24γ[exp(iξ)+γ sin ξ]+12γ[i exp(iξ)+γ sin ξ]},
γ(ξ, z0)=γτ(ξ)γ¯τ cos ξ-i sin ξcos ξ-iτ sin ξ,
g(ξ, z; γ¯, 0, 0, γτ, γτ, γτ)=ik(1-6τ+5τ3)a024(2m+1).
(2m+1)|g(ξ, z; γτ, γτ, γτ, 2±1γτ, 2±1γτ, 2±1γτ)/(ka0)|
Lξj(ξ, z)=P(ξ, z)·Xξj(ξ, z),
L(ξ, z0)=arg[U(ξ, z0)]/k,
P(ξ, z0)=Lξ(ξ, z0),
a0(ξ)=|U(ξ, z0)|H(ξ, z0),
U(x, z0)=U0 exp-x2+y22w2.
UγI(0)(x, z0)=U0[1+(kγw2)-1]×exp-x2+y22w2[1+(kγw2)-1].
L¯(ξ, z0)=arg[U˜(ξ, z0)]/k,
X(ξ, z0)=-L¯ξ(ξ, z0),
a0(ξ)=|U˜(ξ, z0)|H(ξ, z0),
U˜(p, z0)=U0kq2 exp-px2+py22q2,
UγI(0)(x, z0)=U0 exp-(1-kγw2)(x2+y2)2w2.
U(x, y, z0)=U0 exp-k γ¯xx2+γ¯yy22×Hm(kγ¯xx)Hm(kγ¯yy).
X(ξ, η, z0)=(X0 cos ξ, Y0 cos η),
P(ξ, η, z0)=(P0 sin ξ, Q0 sin η),
A0(ξ, η, z0)=a0(ξ, η)/[H(ξ, η, z0)]1/2=c.
L(ξ, η, z0)=X0P02(sin ξ cos ξ-ξ)
+Y0Q02(sin η cos η-η),
X(ξ, η, z0)=X0 cos ξ(cos η, sin η),
P(ξ, η, z0)=P0 sin ξ(cos η, sin η),
A0(ξ, η, z0)=a0(ξ, η)/[H(ξ, η, z0)]1/2=c.
(G·X+iP)(ξ, η)=iP02 exp(2iξ).
Uγ¯I(0)(x, z0)=U0 exp-kγ¯2|x|2Lm(kγ¯|x|2),
U0=2πkP0ci[-(2m+1)/2]m exp[-(2m+1)/2]/m!.
X(ξ, η, z0)=ρ(cos ξ, sin ξ)+η(sin ξ,-cos ξ),
P(ξ, η, z0)=P0(sin ξ,-cos ξ),
A0(ξ, η, z0)=a0(ξ, η)/[H(ξ, η, z0)]1/2=c.
L(ξ, η, z0)=P0ρξ+P0η.
G(ξ, z0)=cos ξ-sin ξsin ξcos ξγ100γ2cos ξsin ξ-sin ξcos ξ=γ1 cos2 ξ+γ2 sin2 ξ(γ1-γ2)sin ξ cos ξ(γ1-γ2)sin ξ cos ξγ1 sin2 ξ+γ2 cos2 ξ.
UG(0)(x, z0)
=kc2π 02π exp{ikP0[ρξ+r sin(ξ-θ)]}×exp-kγ12[ρ-r cos(ξ-θ)]2×-[-γ2(γ1η+iP0)]1/2×exp-kγ22[η-r sin(ξ-θ)]2dηdξ,
UG(0)(x, z0)
=kc2π exp(imθ)02π exp[i(mα+kP0r sin α)]×exp-kγ12 mkP0-r cos α2×-[-γ2(γ1η+iP0)]1/2×exp-kγ22(η-r sin α)2dηdα.
UG(0)(x, z0)=U0 exp(imθ)Jm(kP0r),
A0m(ξ, z0)=A0(ξ, z0)M[X(ξ, z0)].
A0m(ξ, η, z0)=c exp-ρ2+η22w2,
I=α(τ)exp[-kΩ(τ)]dτ.
I=α0+α1σ+α22σ2+×exp-kΩ0+Ω22σ2+dσ,
I=exp(-kΩ0)α0+α1σ+α22σ2+×1-kΩ36σ3+Ω424σ4++×exp-k Ω22σ2dσ.
I2πkΩ2 exp(-kΩ0)α0+1k 524 α0Ω32Ω23-α0Ω48Ω22-α1Ω32Ω22+α22Ω2+O(k-2).
Uγ¯(0)(x, z0)
=cikP02π 02π expi ξ2-kγ¯2(x-X0 cos ξ)2-ik X0P02(ξ+sin ξ cos ξ)+ikxP0 sin ξdξ=cikP02π exp-kγ¯2x202π expi 1-kX0P02ξ×expkX0P0xX0-cos ξ2exp(iξ)dξ. 
Uγ¯(0)(x, z0)=cikP02π exp-kγ¯2x2hm(kγ¯x),
hm(t)=02π exp-imξ+(2m+1)×t(2m+1)1/2-cos ξ2exp(iξ)dξ.
hm(t)=-i  z-m-1×exp(2m+1)zt(2m+1)1/2-z2+14dz.
hm(n)(t)=-i(2m+1)n/2× z-m-1+n×exp(2m+1)zt(2m+1)1/2-z2+14dz.
hm(t)-2thm(t)
=-i  z-m[(2m+1)z-2t2m+1]×exp(2m+1)×zt(2m+1)1/2-z2+14dz=-i  z-m-2 zexp(2m+1)zt(2m+1)1/2-z2+14dz=-i(-2m)  z-m-1×exp(2m+1)zt(2m+1)1/2-z2+14dz=-2mhm(t),
h0(t)=-i  z-1 expzt-z2+14dz=2π exp-14.
hm(m)(t)=-i(2m+1)m/2  z-1×exp(2m+1)zt(2m+1)1/2-z2+14dz=2π(2m+1)m/2 exp-2m+14.
hm(t)=2π[(2m+1)/4]m/2×exp[-(2m+1)/4]Hm(t)/m!,
Uγ¯(0)(x, z0)=c (2πikP0)1/2m! 2m+14m/2×exp-2m+14×exp-kγ¯2x2Hm(kγ¯x).
Uγ¯I(0)(x, z0)
=kP0c2π i exp-kγ¯2r202π02π expi1-kX0P02ξ×expkX0P0r cos(η-θ)X0-cos ξ2exp(iξ)dξdη.
Uγ¯I(0)(x, z0)=kP0c2π i exp-kγ¯2r2lm(kγ¯r),
lm(t)=02πfm[t cos(η-θ)]dη=02πfm(t cos η)dη,
fm(t)=02π exp-imξ+(2m+2)
×t(2m+2)1/2-cos ξ2exp(iξ)dξ.
fm(t)=2π[(m+1)/2]m/2 exp[-(m+1)/2]Hm(t)/m!.
lm(t)=2πm! m+12m/2 exp-m+12×02πHm(t cos η)dη=(2π)2(m/2)! 1+(-1)m2 -m+12m/2×exp-m+12Lm/2(t2),
Uγ¯I(0)(x, z0)=i 2πkP0cm! -2m+12m×exp-2m+12×exp-kγ¯2r2Lm(kγr¯2).

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