Abstract

Although maximal localization is a basic notion in the consideration of phase-space representations of fields, it has not yet been pursued for general wave fields. We develop measures of spatial and directional spreads for nonparaxial waves in free space. These measures are invariant under translation and rotation and are shown to reduce to the conventional ones when applied to paraxial fields. The associated uncertainty relation sets limits to joint localization in coordinate and frequency space. This relation provides a basis for the definition of a joint localization measure that is analogous to the beam propagation factor (i.e., M2) of paraxial optics. The results are first developed for two-dimensional fields and then generalized to three dimensions.

© 2000 Optical Society of America

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References

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  1. L. Cohen, Time-Frequency Analysis (Prentice-Hall, N.J., 1995), Chap. 3.
  2. G. Folland, A. Sitaram, “The uncertainty principle: a mathematical survey,” J. Fourier Anal. Appl. 3, 207–238 (1997).
    [CrossRef]
  3. See Ref. 1, p. 46.
  4. M. W. Sasnett, “Propagation of multimode laser beams—the M2 factor,” in The Physics and Technology of Laser Resonators, D. R. Hall, P. E. Jackson, eds. (Hilger, Bristol, UK, 1989), pp. 132–142.
  5. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
    [CrossRef]
  6. T. F. Johnston, “M2 concept characterises beam quality,” Laser Focus World 26(5), 173–183 (1990).
  7. The usefulness of M2 as a measure of beam quality has been questioned as a result of the sensitivity to noise of the second moments. See, for example, G. N. Lawrence, “Proposed international standard for laser-beam quality falls short,” Laser Focus World 30(7), 109–114 (1994).
  8. M. A. Porras, “The best quality optical beam beyond the paraxial approximation,” Opt. Commun. 111, 338–349 (1994).
    [CrossRef]
  9. G. W. Forbes, M. A. Alonso, “Measures of spread for periodic distributions and the associated uncertainty relations,” Am. J. Phys. (to be published).
  10. Since we are considering wave fields generated by distant sources, we ignore for now any evanescent components, which correspond to complex values of θ.
  11. A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), pp. 51–57.
  12. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 128–133.
  13. P. González-Casanova, K. B. Wolf, “Interpolation for solutions of the Helmholtz equation,” Numer. Methods Partial Diff. Eqns. 11, 77–91 (1995).
    [CrossRef]
  14. M. A. Alonso, “Measurement of Helmholtz wave fields,” J. Opt. Soc. Am. A 17, 1256–1264 (2000).
    [CrossRef]
  15. This form follows simply upon consideration of the process of wrapping a Gaussian distribution around a ring and summing the multiple values at each location. The Fourier coefficients of the result are then given as an integral over all θ of a Gaussian times exp(imθ), which is once again a Gaussian. The form of Eq. (9.4) then follows.
  16. This form is completely analogous to the standard form of the angular momentum operator used in elementary quantum mechanics. See, for example, R. L. Liboff, Quantum Mechanics (Addison-Wesley, Reading, Mass., 1980), p. 327.
  17. See the reference given in Note 7.
  18. E. Wigner, “On the correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  19. H. W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
    [CrossRef]
  20. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  21. M. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  22. A. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 42, 32–37 (1980).
  23. K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
    [CrossRef]

2000 (1)

1999 (1)

1997 (1)

G. Folland, A. Sitaram, “The uncertainty principle: a mathematical survey,” J. Fourier Anal. Appl. 3, 207–238 (1997).
[CrossRef]

1995 (2)

H. W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[CrossRef]

P. González-Casanova, K. B. Wolf, “Interpolation for solutions of the Helmholtz equation,” Numer. Methods Partial Diff. Eqns. 11, 77–91 (1995).
[CrossRef]

1994 (2)

The usefulness of M2 as a measure of beam quality has been questioned as a result of the sensitivity to noise of the second moments. See, for example, G. N. Lawrence, “Proposed international standard for laser-beam quality falls short,” Laser Focus World 30(7), 109–114 (1994).

M. A. Porras, “The best quality optical beam beyond the paraxial approximation,” Opt. Commun. 111, 338–349 (1994).
[CrossRef]

1990 (1)

T. F. Johnston, “M2 concept characterises beam quality,” Laser Focus World 26(5), 173–183 (1990).

1980 (1)

A. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 42, 32–37 (1980).

1979 (1)

1968 (1)

1932 (1)

E. Wigner, “On the correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Alonso, M. A.

Bastiaans, M.

Cohen, L.

L. Cohen, Time-Frequency Analysis (Prentice-Hall, N.J., 1995), Chap. 3.

Erdélyi, A.

A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), pp. 51–57.

Folland, G.

G. Folland, A. Sitaram, “The uncertainty principle: a mathematical survey,” J. Fourier Anal. Appl. 3, 207–238 (1997).
[CrossRef]

Forbes, G. W.

K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
[CrossRef]

G. W. Forbes, M. A. Alonso, “Measures of spread for periodic distributions and the associated uncertainty relations,” Am. J. Phys. (to be published).

González-Casanova, P.

P. González-Casanova, K. B. Wolf, “Interpolation for solutions of the Helmholtz equation,” Numer. Methods Partial Diff. Eqns. 11, 77–91 (1995).
[CrossRef]

Johnston, T. F.

T. F. Johnston, “M2 concept characterises beam quality,” Laser Focus World 26(5), 173–183 (1990).

Lawrence, G. N.

The usefulness of M2 as a measure of beam quality has been questioned as a result of the sensitivity to noise of the second moments. See, for example, G. N. Lawrence, “Proposed international standard for laser-beam quality falls short,” Laser Focus World 30(7), 109–114 (1994).

Lee, H. W.

H. W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[CrossRef]

Liboff, R. L.

This form is completely analogous to the standard form of the angular momentum operator used in elementary quantum mechanics. See, for example, R. L. Liboff, Quantum Mechanics (Addison-Wesley, Reading, Mass., 1980), p. 327.

Lohmann, A.

A. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 42, 32–37 (1980).

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 128–133.

Porras, M. A.

M. A. Porras, “The best quality optical beam beyond the paraxial approximation,” Opt. Commun. 111, 338–349 (1994).
[CrossRef]

Sasnett, M. W.

M. W. Sasnett, “Propagation of multimode laser beams—the M2 factor,” in The Physics and Technology of Laser Resonators, D. R. Hall, P. E. Jackson, eds. (Hilger, Bristol, UK, 1989), pp. 132–142.

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

Sitaram, A.

G. Folland, A. Sitaram, “The uncertainty principle: a mathematical survey,” J. Fourier Anal. Appl. 3, 207–238 (1997).
[CrossRef]

Walther, A.

Wigner, E.

E. Wigner, “On the correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 128–133.

Wolf, K. B.

K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
[CrossRef]

P. González-Casanova, K. B. Wolf, “Interpolation for solutions of the Helmholtz equation,” Numer. Methods Partial Diff. Eqns. 11, 77–91 (1995).
[CrossRef]

J. Fourier Anal. Appl. (1)

G. Folland, A. Sitaram, “The uncertainty principle: a mathematical survey,” J. Fourier Anal. Appl. 3, 207–238 (1997).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Laser Focus World (2)

T. F. Johnston, “M2 concept characterises beam quality,” Laser Focus World 26(5), 173–183 (1990).

The usefulness of M2 as a measure of beam quality has been questioned as a result of the sensitivity to noise of the second moments. See, for example, G. N. Lawrence, “Proposed international standard for laser-beam quality falls short,” Laser Focus World 30(7), 109–114 (1994).

Numer. Methods Partial Diff. Eqns. (1)

P. González-Casanova, K. B. Wolf, “Interpolation for solutions of the Helmholtz equation,” Numer. Methods Partial Diff. Eqns. 11, 77–91 (1995).
[CrossRef]

Opt. Commun. (2)

M. A. Porras, “The best quality optical beam beyond the paraxial approximation,” Opt. Commun. 111, 338–349 (1994).
[CrossRef]

A. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 42, 32–37 (1980).

Phys. Rep. (1)

H. W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[CrossRef]

Phys. Rev. (1)

E. Wigner, “On the correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Other (11)

L. Cohen, Time-Frequency Analysis (Prentice-Hall, N.J., 1995), Chap. 3.

This form follows simply upon consideration of the process of wrapping a Gaussian distribution around a ring and summing the multiple values at each location. The Fourier coefficients of the result are then given as an integral over all θ of a Gaussian times exp(imθ), which is once again a Gaussian. The form of Eq. (9.4) then follows.

This form is completely analogous to the standard form of the angular momentum operator used in elementary quantum mechanics. See, for example, R. L. Liboff, Quantum Mechanics (Addison-Wesley, Reading, Mass., 1980), p. 327.

See the reference given in Note 7.

G. W. Forbes, M. A. Alonso, “Measures of spread for periodic distributions and the associated uncertainty relations,” Am. J. Phys. (to be published).

Since we are considering wave fields generated by distant sources, we ignore for now any evanescent components, which correspond to complex values of θ.

A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), pp. 51–57.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 128–133.

See Ref. 1, p. 46.

M. W. Sasnett, “Propagation of multimode laser beams—the M2 factor,” in The Physics and Technology of Laser Resonators, D. R. Hall, P. E. Jackson, eds. (Hilger, Bristol, UK, 1989), pp. 132–142.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Modulus of the fields described by Eq. (9.2) for (a) m=0, (b) m=1, and (c) m=4. For these fields, M2=1, 5, and 65, respectively.

Fig. 2
Fig. 2

Modulus of the fields described by the spectral function in Eq. (9.4) for (a) kw=π/4 for which M21.22, (b) kw=π/2 for which M21.09, and (c) kw=π for which M21.02. Notice that the field in (c) resembles a paraxial Gaussian beam. It turns out that the plots for all kwπ/8 are effectively indistinguishable from Fig. 1(a). (For kw=π/8, it turns out that M21.01.)

Fig. 3
Fig. 3

M2 as a function of kw for the field described by the spectral function in Eq. (9.4). Notice that, away from its maximum at kwπ/4, M2 goes to unity both for small and large kw. Also notice the rapid falloff between π/4 and π/8 that was mentioned in Fig. 2.

Fig. 4
Fig. 4

Modulus of the fields described by the spectral function in Eq. (9.9) for α=π/2 and (a) kw=π/4 for which M21.20, (b) kw=π/2 for which M21.72, and (c) kw=π for which M210.6. Notice that the individual Gaussian components of each of these composite fields are plotted in Fig. 2.

Fig. 5
Fig. 5

M2 as a function of kw for the field described by the spectral function in Eq. (9.9) for α=0, π/8, 2π/8, 3π/8,…π. For π<α<2π, the curves in the continuation of this sequence overlie this set, but in reverse order.

Fig. 6
Fig. 6

Modulus of the fields described by the spectral function in Eq. (9.10) for kw=π and (a) n=2 for which M220.74, (b) n=3 for which M220.72, and (c) n=4 for which M219.65. M2 falls rapidly for larger values of n in this case: It is 8.13 for n=6, and 1.78 for n=8 (as can be seen from Fig. 7). Notice the standing waves generated by the counterpropagating components for n=2 and 4.

Fig. 7
Fig. 7

M2 as a function of kw/nπ for the field described by the spectral function in Eq. (9.10) for n=2, 3, 4,…10, 20, and 40.

Equations (135)

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U˜(p, z)k2π -U(x, z)exp(-ikxp)dx.
U(x, z)=k2π -U˜(p, z)exp(ikxp)dp.
U˜(p, z)=U˜(p, 0)exp[ikz(1-p2/2)].
Δx2(z)N-1-|U(x, z)|2(x-x¯)2dx,
N-|U(x, z)|2dx=-|U˜(p, z)|2dp,
x¯(z)N-1-|U(x, z)|2xdx.
p¯N-1-|U˜(p, z)|2pdp=(kN)-1- ImU(x, z)U*x(x, z)dx,
Δp2N-1-|U˜(p, z)|2(p-p¯)2dp=k-2N-1-Ux(x, z)2dx-p¯2.
Δx2Δp2-(x¯p¯-x¯p¯)214k2,
xp¯(z)1Nk-x ImU(x, z)U*x(x, z)dx=1Nk-p ImU˜*(p, z)U˜p(p, z)dp.
M24k2[Δk2Δp2-(xp¯-xp¯¯)2]1.
(2+k2)U(r)=0,
U(r)=k2π Sφ(θ)exp[ikr·u(θ)]dθ,
φ(θ)=limr exp-ikr+iπ4r21-ikrU[ru(θ)].
φ(θ)=12sign(cos θ)k2π×- U(x, 0)cos θ-ikUz(x, 0)×exp(-ikx sin θ)dx.
φ(θ)=cos θ+|cos θ|2k2π ×-U(x, 0)exp(-ikx sin θ)dx=cos θ+|cos θ|2U˜(sin θ, 0),
φd|φ=Sφd*(θ)φ(θ)dθ,
Nφφ|φ=S|φ(θ)|2dθ,
ONφ-1φ|Oˆφ=Nφ-1Sφ*(θ)Oˆφ(θ)dθ.
u=Nφ-1S|φ(θ)|2(sin θ, cos θ)dθ.
Δθ21-|u|2.
lˆ=-ikθ.
l=-ikNφSφ*(θ)φ(θ)dθ=1kNφS Im[φ*(θ)φ(θ)]dθ,
l2=1k2NφS|φ(θ)|2dθ.
φ(θ)=|φ(θ)|exp[iΦ(θ)].
kl=-1NφS|φ(θ)|2Φ(θ)dθ,
k2l2=1NφS|φ(θ)|2dθ+1NφS|φ(θ)|2[Φ(θ)]2dθ.
φ(θ)=12πmcm exp(imθ).
kl=Nφ-1mm|cm|2,
k2l2=Nφ-1mm2|cm|2.
U[ru(θ)]=k2π mcmJm(kr)exp[im(θ+π/2)].
Δl2l2-l2.
f|fg|g|f |g|2,
f(θ)=Nφ-1/2[u(θ)-u]φ(θ),
g(θ)=vkNφ-1/2(lˆ-l)φ(θ),
f|f=1-u·u=Δθ2.
g|g=k2|v|2Δl2.
f|g=kNφ-1Sφ*(θ)v·[u(θ)-u](lˆ-l)φ(θ)dθ
=v·Nφ-1S Im[φ*(θ)φ(θ)]u(θ)dθ-i2u-kul,
k2Δθ2Δl214|u|2+κ2,
κku·lˆu+ulˆ2|u|=Nφ-1u|u|·S Im[φ*(θ)φ(θ)]udθ=Nφ-1S Im[φ*(θ)φ(θ)]sin(θ-ϕ)dθ.
Δθ2k2Δl2+1414+κ2.
RˆαU(r)=U(r·α),Rˆαφ(θ)=φ(θ-α),
Tˆr0U(r)=U(r-r0),
Tˆr0φ(θ)=φ(θ)exp[-ikr0·u(θ)],
lrTˆrl=l-r·u,
l2rTˆrl2=l2-r·lˆu+ulˆ+r·uu·r,
κrTˆrκ=κ-kr·uu·u|u|=κ+kr·uu·u|u|,
uu=cos2 θ,-sin θ cos θ-sin θ cos θ,sin2 θ.
Δθ2k2(Δl)r2+1414+κr2,
(Δl)r2l2r-lr2=Δl2-r·(lˆu+ulˆ-2ulˆ)+r·(uu-uu)·r.
rC=12uu-1·lˆu+ulˆ=uu·lˆu+ulˆ2 det(uu),
uu-1=1det(uu)sin2 θ,sin θ cos θsin θ cos θ,cos2 θ=uudet(uu).
lrC=l-u·uu·lˆu+ulˆ2 det(uu),
l2rC=l2-lˆu+ulˆ·uu·lˆu+ulˆ4 det(uu),
κrC=0.
Δθ2l2rC+14k2-lrC214k2.
M2Δθ2(4k2l2rC+1),
M21+4k2lrC21.
φ(arcsin p)=(1-p2)1/2U˜(p, 0),
Nφ=S|φ(θ)|2dθ=1-p2¯2+O(pM4)N,
u=Nφ-1|U˜(p, 0)|2(p(1-p2)1/2, 1-p2)dp
=p¯+O(pM3), 1-p2¯2+O(pM4),
Δθ2=1-|u|2=(p2¯-p¯2)[1+O(pM2)]=[1+O(pM2)]Δp2.
l2=[1+O(pM2)]k2NU˜p(p, 0)2dp
=[1+O(pM2)]N-1-|xU(x, 0)|2dx=[1+O(pM2)]x2¯.
 l=[1+O(pM2)]x¯,
u=[1+O(pM2)](1,-p¯),
uu=[1+O(pM2)]p2¯,p¯p¯,1,
lˆu+ulˆ=[2+O(pM2)](x¯,-xp¯).
lrC=O(pM2),
l2rC=[1+O(pM2)][Δx2-(xp¯-x ¯p¯)2/Δp2].
M2=[1+O(pM2)](M2+Δp2)=[1+O(pM2)]M2.
φ(θ)=12πexp(imθ),
U[ru(α)]=kJm(kr)exp[im(α+π/2)].
Δθ2=1,
l=lrC=m/k,
l2=l2rC=m2/k2.
φ(θ)=γ(θ, kw) 12πm exp-m22k2w2+imθ.
Nφ=m exp-m2k2w2.
Δθ2=1-m exp-m2k2w2-2×m exp-m2+m+1/2k2w22.
lrC=0,
l2rC=m exp-m2k2w2-1×mm2k2exp-m2k2w2.
φ(θ)=γ(θ+α/2, kw)+γ(θ-α/2, kw),
φ(θ)=j=1nγ(θ+2πj/n, kw).
U(r)=k2πS0π sin θφ(θ, ϕ)exp[ikr·u(θ, ϕ)]dθdϕ,
u(θ, ϕ)(sin θ cos ϕ, sin θ sin ϕ, cos θ).
φ1|φ2=S0π sin θ φ1*(θ, ϕ)φ2(θ, ϕ)dθdϕ.
ΔΩ21-|u|2,
u1NφS0π sin θ|φ(θ, ϕ)|2u(θ, ϕ)dθdϕ,
Lˆ-iksin ϕθ+cos ϕtan θϕ,-cos ϕθ
+sin ϕtan θϕ,-ϕ.
uvLv=Lvuv
uvLv±=Lv±uv+ikuv=Lv±uv+uvLv±+i2kuv,
f(θ, φ)=Nφ-1/2[u(θ, ϕ)-u]φ(θ, ϕ),
g(θ, ϕ)=kNφ-1/2V·(Lˆ-L)φ(θ, ϕ),
f|f=ΔΩ2,
g|g=k2(L2-L·L),
f|g=k(u·V·Lˆ-u·V·Lˆ).
V0+=010001100,Vx+=01000-1100,
Vy+=010001-100,Vz+=0-10001100.
f|g=k(u·Vμ+·Lˆ-u·Vμ+·Lˆ)=k[bx, μ(uyLˆz-uyLˆz)+by, μ(uzLˆx-uzLˆx)+bz, μ(uxLˆy-uxLˆy)],
k2ΔΩ2(L2-L·L)
k(|uyLˆz-uyLˆz|2+|uzLˆx-uzLˆx|2
+|uxLˆy-uxLˆy|2).
k2ΔΩ2(L2-L·(L))|u|24+|κ+|2,
κv+kLˆv-uv++uv+Lˆv-2-uv+Lˆv-.
ΔΩ214+k2(L2-L·L)14+|κ+|2.
Vμ-=(Vμ+)T=(Vμ+)-1.
ΔΩ214+k2(L2-L·L)14+|κ-|2,
κv-kLˆv+uv-+uv-Lˆv+2-uv-Lˆv+.
T˜rφ(θ, ϕ)=φ(θ, ϕ)exp[-ikr·u(θ, ϕ)],
LrTˆrL=L-r×u,
L2rTˆrL2=L2-Lˆ·r×u-r×u·Lˆ+(r×u)·(r×u)=L2+r·u×Lˆ-Lˆ×u+r·(I-uu)·r,
rC=12(I-uu)-1·u×Lˆ-Lˆ×u.
L2rC=L2+14u×Lˆ-Lˆ×u·(I-uu)-1·u×Lˆ-Lˆ×u,
ΔΩ2(k2L2rC+14)14.
M2ΔΩ2(4k2L2rC+1).
ΔΩ2=1-πNφ0π sin(2θ)|φ(θ)|2dθ2,
L2=2πNφ0π sin θ|φ(θ)|2dθ,
Nφ=2π0π sin θ|φ(θ)|2dθ.
M(l, θ) k2π-ππφθ+α2φ*θ-α2exp2ikl sinα2dα.
l=x cos θ-z sin θ=r·u(θ).
S M(x cos θ-z sin θ, θ)dθ=|U(x, z)|2.
-M(l, θ)dl=|φ(θ)|2.
S-M(l, θ)dldθ=Nφ.
Mr0(l, θ)Tˆr0M(l, θ)=M[l+r0·u(θ), θ].
|φ1|φ2|2=2πkS-M1(l, θ)BˆM2(l, θ)dldθ,
Bˆf(l)=-B(l-l)f(l)dl,B(l) J1(2kl)2l.
BˆmM(l, θ)k2π-ππφθ+α2φ*θ-α2×exp2ikl sinα2cosmα2dα.
u=Nφ-1S-u(θ)BˆmM(l, θ)dldθ
l=Nφ-1S-lBˆmM(l, θ)dldθ
Nφ-1S-l2BˆmM(l, θ)dldθ=l2+m-14k2.
lm=Nφ-1S-(lBˆ)mM(l, θ)dldθ,
lr=Nφ-1SlBˆMr(l, θ)dldθ=Nφ-1SlBˆM(l+r·u, θ)dldθ=Nφ-1S(l-r·u)BˆM(l, θ)dldθ=l-r·u.

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