Abstract

A new method for computing ray-based approximations to optical wave fields is demonstrated through simple examples involving wave propagation in free space and in a gradient-index waveguide. The analytic solutions that exist for these cases make it easy to compare the new estimates with exact results. A particularly simple RMS error estimate is developed here, and corrections to the basic field estimate are also discussed and tested. A key step for any ray-based method is the choice of a family of rays to be associated with the initial wave field. We show that, for maximal accuracy, not only must the initial field be considered in choosing the rays, but so too must the medium that is to carry the wave.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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. G. W. Forbes, M. A. Alonso, “Asymptotic estimation of the optical wave propagator. II. relative validity,” J. Opt. Soc. Am. A 15, 1341–1354 (1998).
    [CrossRef]
  3. M. A. Alonso, G. W. Forbes, “Using rays better. II. Ray families for simple wave fields,” J. Opt. Soc. Am. A 18, 1146–1159 (2001).
    [CrossRef]
  4. If γ is changed after the integration, the change in θγ can be found by using the fact that round (θγ/π),where round(t) denotes the closest integer to t, is independent of γ. It follows that, if γ is changed from γato γb(where γbmay be complex) after the integration of Eq. (2.9), the new value of θγ can be found by using θγb=π round(θγa/π)+arctan{[Im(γb)X′+P′]/[Re(γb)X′]}, where |arctan(t)|⩽π/2 for all t.
  5. To verify this, consider the γ¯derivative of the right-hand side of Eq. (3.2). By using Eqs. (2.1) as well as Eqs. (3.3)–(3.6), it can be shown that this derivative vanishes identically.
  6. This definition follows upon averaging the relative error with a weight that is proportional to the squared modulus of the field. In the resulting expression the normalization factor is then the integral of |U|2,but this is replaced by |Uγ(j)|2in Eq. (3.7). Regardless, εγ(j)can be interpreted as an RMS relative error.
  7. With Eq. (2.5), the integral of |Uγ(0)(x, z)|2over all xcan be carried out in closed form, leaving ∫|Uγ(0)(x, z)|2dx=12 kπγ ∫a0(ξ)a0*(τ)Y′(ξ, z)Y′*(τ, z)H(ξ, z)H(τ, z)1/2×exp-kγ4[X(ξ, z)-X(τ, z)]2-k4γ[P(ξ, z)-P(τ, z)]2×exp-ik2[X(ξ, z)-X(τ, z)][P(ξ, z)+P(τ, z)]+ik[L(ξ, z)-L(τ, z)]dτdξ.Notice that this integral is most significant when ξ≈τ.The expression given in Eq. (3.9) results from approximating the integral over τ here by using the form of the saddle-point method given in an appendix in Ref. 3. [The saddle point is at τ=ξ,where Ω2=|Y′(ξ, z)|2/2γ≠0].
  8. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 109–127.
  9. There is an important exception, however, where the phase-space curve can coincide with a track of the medium. Consider, for example, a plane wave traveling in the z direction in free space. The ray family is described by Eqs. (5.13), although a0is now a constant. From Eq. (5.11) we see that a1vanishes identically in this case for any γ, and the estimate in Eq. (2.5) actually gives the exact solution. More generally, when the phase-space curve is chosen to coincide with a track of the medium and a0is chosen such that the relative weight associated with any segment of the track is conserved under propagation, Eq. (2.5) gives an accurate estimate of an eigenstate or mode of the medium (provided that γ is chosen appropriately). Of course, there is an extra restriction for closed tracks, which follows from requiring the agreement of the phase of the integrand in Eq. (2.5) at the (arbitrary) limits of integration: the phase-space area enclosed by the track must equal (m+1/2)λ,where m is an integer. This is consistent with the well-known quantization condition for the modes of a waveguide. The use of this method for solving eigenstate problems, where tunneling effects can be present, is the subject of further research.
  10. Y. A. Kravtsov, Y. I. Orlov, Caustics, Catastrophes and Wave Fields, 2nd ed. (Springer-Verlag, Berlin, 1999). See Chap. 6.
  11. This effect is not accounted for in Ref. 2, where the objective was to compare the errors of a variety of methods by using nothing more than the ray information. Note that this analysis considered a special case of the new method that was specific to the estimation of wave propagators for which the initial field is a delta function. The initial phase-space curve was then just a vertical line with constant A0and with P≡ξ.

2001 (2)

1998 (1)

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

Other (8)

If γ is changed after the integration, the change in θγ can be found by using the fact that round (θγ/π),where round(t) denotes the closest integer to t, is independent of γ. It follows that, if γ is changed from γato γb(where γbmay be complex) after the integration of Eq. (2.9), the new value of θγ can be found by using θγb=π round(θγa/π)+arctan{[Im(γb)X′+P′]/[Re(γb)X′]}, where |arctan(t)|⩽π/2 for all t.

To verify this, consider the γ¯derivative of the right-hand side of Eq. (3.2). By using Eqs. (2.1) as well as Eqs. (3.3)–(3.6), it can be shown that this derivative vanishes identically.

This definition follows upon averaging the relative error with a weight that is proportional to the squared modulus of the field. In the resulting expression the normalization factor is then the integral of |U|2,but this is replaced by |Uγ(j)|2in Eq. (3.7). Regardless, εγ(j)can be interpreted as an RMS relative error.

With Eq. (2.5), the integral of |Uγ(0)(x, z)|2over all xcan be carried out in closed form, leaving ∫|Uγ(0)(x, z)|2dx=12 kπγ ∫a0(ξ)a0*(τ)Y′(ξ, z)Y′*(τ, z)H(ξ, z)H(τ, z)1/2×exp-kγ4[X(ξ, z)-X(τ, z)]2-k4γ[P(ξ, z)-P(τ, z)]2×exp-ik2[X(ξ, z)-X(τ, z)][P(ξ, z)+P(τ, z)]+ik[L(ξ, z)-L(τ, z)]dτdξ.Notice that this integral is most significant when ξ≈τ.The expression given in Eq. (3.9) results from approximating the integral over τ here by using the form of the saddle-point method given in an appendix in Ref. 3. [The saddle point is at τ=ξ,where Ω2=|Y′(ξ, z)|2/2γ≠0].

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 109–127.

There is an important exception, however, where the phase-space curve can coincide with a track of the medium. Consider, for example, a plane wave traveling in the z direction in free space. The ray family is described by Eqs. (5.13), although a0is now a constant. From Eq. (5.11) we see that a1vanishes identically in this case for any γ, and the estimate in Eq. (2.5) actually gives the exact solution. More generally, when the phase-space curve is chosen to coincide with a track of the medium and a0is chosen such that the relative weight associated with any segment of the track is conserved under propagation, Eq. (2.5) gives an accurate estimate of an eigenstate or mode of the medium (provided that γ is chosen appropriately). Of course, there is an extra restriction for closed tracks, which follows from requiring the agreement of the phase of the integrand in Eq. (2.5) at the (arbitrary) limits of integration: the phase-space area enclosed by the track must equal (m+1/2)λ,where m is an integer. This is consistent with the well-known quantization condition for the modes of a waveguide. The use of this method for solving eigenstate problems, where tunneling effects can be present, is the subject of further research.

Y. A. Kravtsov, Y. I. Orlov, Caustics, Catastrophes and Wave Fields, 2nd ed. (Springer-Verlag, Berlin, 1999). See Chap. 6.

This effect is not accounted for in Ref. 2, where the objective was to compare the errors of a variety of methods by using nothing more than the ray information. Note that this analysis considered a special case of the new method that was specific to the estimation of wave propagators for which the initial field is a delta function. The initial phase-space curve was then just a vertical line with constant A0and with P≡ξ.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (16)

Fig. 1
Fig. 1

Amplitude of the Gaussian beam in free space specified by the initial condition in Eq. (4.12), for kw=5. The dotted lines represent phase contours separated by 2π, and the solid black lines are some of the rays described by Eqs. (4.14). Notice that these rays are approximately orthogonal to the phase fronts for zkw2 and that the spacing of the rays there is inversely proportional to the square of the field amplitude.

Fig. 2
Fig. 2

Plot of kw2z-1εˆγ(0), from Eq. (4.16), as a function of the real and imaginary parts of ln[zγ(z)]. Notice the zero at zγ=-i and the high values (which exceed the gray-scale maximum) near the singularity at zγ=0. The condition of insensitivity concerns the changes in the field estimate that result when γ moves on this plot within a circle of unit diameter centered at the point associated with the chosen value of γ. Moving about in such a disk corresponds to changing (in either direction) the modulus of γ by as much as a factor of e1/21.65 and its phase by up to half a radian, i.e., 30°.

Fig. 3
Fig. 3

Plots of the RMS errors εˆγ(0), [given in Eq. (4.16)] and εγ(0) as functions of z/w, with wγ=10 and for kw=10, 100, and 1000, for the field estimate in Eq. (4.15). Notice that the initial RMS error is below 1% in all cases.

Fig. 4
Fig. 4

Plots of εˆγ(0) [given in Eq. (4.16)] and εγ(1) with γ=10/w, as well as εγ(0)=εγ(1) for γ=-i/z, as functions of z/w, for kw=10, 100, and 1000, for the field estimates in Eqs. (4.15) and (4.17). Notice that the Rayleigh range corresponds to z/w<kw.

Fig. 5
Fig. 5

Amplitude of the Gaussian beam in free space specified by the initial condition in Eq. (4.19), for q=(kw)-1=0.2. The dotted lines represent phase contours separated by 2π, and the solid black lines are some of the rays described by Eqs. (4.21). Notice that these rays are approximately orthogonal to the phase fronts for zkw2 and that the spacing of the rays there is inversely proportional to the square of the field amplitude.

Fig. 6
Fig. 6

Plot of f [Re(γ)z, Im(γ)z]=kzεˆγ(0), expressed now as a function of the real and imaginary parts of ln[zγ(z)], for the field estimate in Subsection 4.B. Notice the zero at zγ=0 and the high values (which exceed the gray-scale maximum) near the line of singularities at zγ=-is for s(0, 1] (corresponding to the left half of the bottom edge of the plot).

Fig. 7
Fig. 7

Generic form for z0 of the phase-space curve corresponding to the ray family described by Eqs. (4.21). Notice that the slope of this plot at the origin is unity.

Fig. 8
Fig. 8

Amplitude of the field that satisfies the initial condition in Eq. (4.12) and propagates in the positive z direction in the medium described by Eq. (5.1), for kn0/v=100 and vw=0.25. The dotted lines represent phase contours separated by 2π. The solid black lines represent some of the rays described by Eqs. (5.5) together with Eqs. (4.13a), (4.13b), and (5.10). Notice in (b) that, once vz has reached 100, the rays have become so entangled that, certainly with this coarse ray sampling, the link to the wave field is not clear at all. Nevertheless, in the examples considered below, we go out to vz=500.

Fig. 9
Fig. 9

Phase-space curves for the example in Section 5, corresponding to the parametric plots of Eqs. (5.5) [with Eqs. (4.13a), (4.13b), and (5.10)] as functions of ξ, for vz=(a) 0, (b) 10, and (c) 500. The number of loops described by the segment corresponding to ξ[-0.99,0.99] is given by 2vzn0[1/H(0.99)-1/H(0)]=18vz.

Fig. 10
Fig. 10

Plot of k¯εˆγ(0) as a function of the real and imaginary parts of γ(z)/γ¯ for the field estimate of Section 5, with vz=10. The minimum is located at γ/γ¯0.8-0.4i, where k¯εˆγ(0)5.64. At γ/γ¯=1, k¯εˆγ(0)8.64.

Fig. 11
Fig. 11

Plots of the amplitude of the field considered in Section 5 and its estimates Uγ(0) and Uγ(1) as functions of the transverse variable vx, with vw=0.25, γ=γ¯, k¯=400, and for vz=(a)0, (c) 10, and (e) 500. The local errors of Uγ(0) and Uγ(1) for the same values are shown, respectively, in (b), (d), and (f). Since the field and its estimates are even functions of x, the plots are shown for vx0.

Fig. 12
Fig. 12

Plots of the RMS errors εˆγ(0), εγ(0), and εγ(1), as functions of vz for vw=0.25, γ=γ¯, and k¯=400. Notice that, for large z, εˆγ(0)z3/4.

Fig. 13
Fig. 13

Plot of ε¯γ(0)limz(vz)-3/4k¯εˆγ(0) as a function of vw, where the rays were chosen according to the first prescription for the initial condition given in Eq. (4.12). When, instead, the rays are chosen according to the second prescription, this plot still describes ε¯γ(0), but now as a function of q/n0=(k¯vw)-1.

Fig. 14
Fig. 14

Plots of ε¯γ(0)limz(vz)-3/4k¯ε¯γ(0) as a function of r2/w2, where the rays were chosen according to the first prescription for the initial condition given in Eq. (5.14). These curves correspond, from top to bottom, to log10(vw)=-3, -2.5, -2, -1.5, -1, -0.5, and 0. Notice that, with the exception of the bottom one, these curves are approximately parallel straight lines.

Fig. 15
Fig. 15

Plots of ε¯γ(0)limz(vz)-3/4k¯εˆγ(0) as a function of v2r2 where the rays were chosen according to the second prescription for the initial condition in Eq. (5.14). These curves correspond, from top to bottom, to log10(q/n0)=-3, -2.5, -2, -1.5, -1, -0.5, and 0. Again, with the exception of the bottom one, these curves are approximately parallel straight lines.

Fig. 16
Fig. 16

For the displaced Gaussian initial field given in Eq. (5.14), the field estimate based on the rays given by the first prescription is more accurate than the one found by using the second prescription whenever r>τ, where τ corresponds to the value of r of the crossing of the curves w/(1-v2r2) (shown in black) and w0 exp[-r2/(2w2)] (shown in gray).

Equations (96)

Equations on this page are rendered with MathJax. Learn more.

[2+k2n2(x, z)]U(x, z)=0,
X˙(ξ, z)=P(ξ, z)H(ξ, z),
P˙(ξ, z)=12H(ξ, z) n2x[X(ξ, z), z],
H(ξ, z){n2[X(ξ, z), z]-P2(ξ, z)}1/2.
H˙(ξ, z)=12H(ξ, z) n2z[X(ξ, z), z].
L(ξ, z)=P(ξ, z)X(ξ, z),
L˙(ξ, z)=n2[X(ξ, z), z]H(ξ, z),
Uγ(0)(x, z)=k2π  a0(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.
Y(ξ, z)γ(z)X(ξ, z)+iP(ξ, z).
L(ξ, z)=L(ξ, z0)+z0z n2[X(ξ, z), z]{n2[X(ξ, z), z]-P2(ξ, z)}1/2 dz.
tan[θγ(ξ, z)]=P(ξ, z)γX(ξ, z),
θ˙γ=γ(XP˙-X˙P)γ2X2+P2=-γ(γ2X2+P2)H×P2-X22 2n2x2(X, z)+1H2 PP-X2 n2x(X, z)2.
Y=(γ 2X2+P2)1/4 exp(iθγ/2).
Uγ(1)(x, z)=k2π a0(ξ)+a1(ξ, z)ikY(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)=a1(ξ, z0)+g[ξ, z0; γ(z0), γ¯]+Δa¯1(ξ, z0, z, γ¯)+g[ξ, z; γ¯, γ(z)],
g(ξ, z; γa, γb)i2 Hγaγb 1Yξ 1Y ξ a0HYdγ=iH γb-γa2 1YaYb a0H+a024H 51YaYb-YaYb+YaYb(YaYb)2.
Δa¯1(ξ, z0, z, γ¯)z0za¯˙1(ξ, z, γ¯)dz,
a¯˙1=12HY¯ Fa0HY¯+Ga0HY¯+Ka02HY¯,
F=-12Y¯ 2n2x2(X, z)+2(γ¯2+Y¯˙2),
G=1HY¯[HY¯¨Y¯-HY¯˙Y¯˙-H˙Y¯˙Y¯],
K=Y¯¨-2Y¯˙1/2 ξ H˙Y¯˙1/2H-2Y¯˙1/4 ξ Y¯˙Y¯˙3/4Y¯+Y¯H1/2 z H˙H3/2-112Y¯ 3 2ξ2 2n2x2(X, z)+Y¯4 3n2x3(X, z) ξ XY¯4.
εγ(j)(z)|U(x, z)-Uγ(j)(x, z)|2dx|Uγ(j)(x, z)|2dx1/2.
U(x, z)-Uγ(0)(x, z)=k2π a1ik+O(k-2)×YH exp{-kγ(x-X)2/2+ik[L+(x-X)P]}dξ.
|Uγ(0)(x, z)|2dx= |a0(ξ)|2H(ξ, z) dξ+O(k-1),
|U(x, z)-Uγ(0)(x, z)|2dx=1k2  |a1(ξ, z)|2H(ξ, z) dξ+O(k-3).
εˆγ(0)(z)1k H-1(ξ, z)|a1(ξ, z)|2dξH-1(ξ, z)|a0(ξ)|2dξ1/2.
L(ξ, z0)=arg[U(ξ, z0)]/k,
P(ξ, z0)=L(ξ, z0),
a0(ξ)=|U(ξ, z0)|[n2(ξ, z0)-P2(ξ, z0)]1/4.
a1(ξ, z0)=-iH2(γ+iP) A0-i A0P(γ+iP)-i A0P4(γ+iP)-5A0P212(γ+iP)2=limγa g[ξ, z0; γa, γ(z0)],
U˜(p, z0)k2π U(x, z0)exp(-ikxp)dx.
X(ξ, z0)=-L¯(ξ, z0),
a0(ξ)=|U˜(ξ, z0)|{n2[X(ξ, z0), z0]-ξ2}1/4,
L(ξ, z0)=L¯(ξ, z0)+ξ X(ξ, z0).
a1(ξ, z0)=iH2(γ-1-iX) A0+i A0X(γ-1-iX)+i A0 X4(γ-1-iX)-5A0X212(γ-1-iX)2=limγa0 g[ξ, z0; γa, γ(z0)],
n(x, z)1.
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(ikz1-p2),|p|1U˜(p, 0)exp(-kzp2-1),|p|>1.
P(ξ, z)=P(ξ, 0),
X(ξ, z)=X(ξ, 0)+P(ξ, 0)H(ξ)z,
L(ξ, z)=L(ξ, 0)+zH(ξ),
H(ξ)=[1-P2(ξ, 0)]1/2.
a¯˙1(ξ, z)=γ¯2B1Y¯2+B2Y¯Y¯3+B3Y¯Y¯3+B4Y¯2Y¯4,
B1-a02H3-2PPH5a0-6H2PP+5P2+21P2P28H7a0,
B2a0/H3+7PPa0/4H5,
B3a0/4H3,
B4-5a0/8H3.
a1(ξ, z)=a1(ξ, 0)+g[ξ, 0; γ(0), 0]+g[ξ, z; 0, γ(z)].
a1(ξ, z)=g(ξ, 0; , 0)+g[ξ, z; 0, γ(z)].
Δa¯1(ξ, 0, z, γ¯)=zX02 B1+B2 Y¯+Y¯02γ¯X0+B3 Y¯+Y¯02γ¯X0+B4 Y¯2+Y¯Y¯0+Y¯023γ¯2X02,
a1(ξ, z)=-z+iγ(z) a0(ξ)2.
U(x, 0)=U0 exp(-x2/2w2),
X(ξ, 0)=ξ,
P(ξ, 0)=0,
A0(ξ)=a0(ξ)/H(ξ)=U0 exp(-ξ2/2w2),
L(ξ, 0)=0.
X(ξ, z)=ξ,
P(ξ, z)=0,
L(ξ, z)=z.
Uγ(0)(x, z)=U0c exp[ikz-x2/(2w2c2)],
εˆγ(0)(z)=316 zkw2 1+izγ(z).
Uγ(1)(x, z)=U0c 1+1-iγz2kγw2c2 1-x2w2c2×expikz-x22w2c2.
a1(ξ, z)=g[ξ, z; 0, γ(ξ)].
U˜(p, 0)=U˜0 exp-p22q2,
X(ξ, 0)=0,
P(ξ, 0)=ξ,
A0(ξ)=U˜0 exp-ξ22q2,
L(ξ, 0)=0,
X(ξ, z)=ξz/1-ξ2,
P(ξ, z)=ξ,
L(ξ, z)=z/1-ξ2.
limz0 z-1f(γr z, γi z)=10.83γr2+γi2,
limz f(γr z, γi z)=10.75,
n2(x)=n02(1-v2x2),
U(x, z)=m=0McmHm(k¯vx)exp-k¯v2x22×expik¯vz1-(2m-1)k¯1/2+m=M+1cmHm(k¯vx)exp-k¯v2x22×exp-k¯vz(2m-1)k¯-11/2,
cm=v2mm! k¯π U(x, 0)Hm(k¯vx)exp-kv¯2x22dx.
H(ξ)=n0[1-v2X2(ξ, 0)-P2(ξ, 0)/n02]1/2.
X(ξ, z)=X(ξ, 0)cosn0vH(ξ)z+P(ξ, 0)n0v sinn0vH(ξ)z,
P(ξ, z)=P(ξ, 0)cosn0vH(ξ)z-n0vX(ξ, 0)sinn0vH(ξ)z.
L(ξ, z)=L(ξ, 0)+H2(ξ)+n022H(ξ)z+X(ξ, z)P(ξ, z)-X(ξ, 0)P(ξ, 0)2.
Y¯(ξ, z)=Y¯(ξ, 0)exp-iγ¯zH(ξ),
Δa¯1=γ¯2Y¯0H3Y¯0 zρ1 a02-2 HHa0+5H22H2-3H4Ha0-a0-7H4Ha0αα+2αβ1ρ1+Y¯0Y¯0-a04 αα+2α(α2β)1ρ1+Y¯0Y¯0+5a08 α2α2+4αβ1ρ1+Y¯0Y¯0+3α2β21ρ12+Y¯0ρ1Y¯0+Y¯02Y¯02,
limz Δa¯1=-iγ¯Y¯0HHY¯0 a02-HHa0+11 H2H2-23H4H+H4H+7Y¯028Y¯02-9Y¯04Y¯0+15Y¯028Y¯02-Y¯02Y¯0-6 HH-5H4HY¯0Y¯0-Y¯0Y¯0a0.
H(ξ)=n0(1-v2ξ2)1/2.
θγ¯(ξ, z)=-vz(1-v2ξ2)1/2-arctanv3ξ2z(1-v2ξ2)3/2.
a1(ξ, z)=g(ξ, 0; , γ¯)+Δa¯1(ξ, 0, z, γ¯)+g[ξ, z; γ¯, γ(z)],
limz|a1(ξ, z)|2=exp(-ξ2/w2)576ξ4,
U(x, 0)=U0 exp-(x-r)22w2.
ε¯γ(0)0.3vw exp-r22w2,
ε¯γ(0)0.3(q/n0)1/2(1-v2r2)=0.3(k¯vw)1/2(1-v2r2).
a¯˙1=γ¯2H3 a02-2 HHa0+5H22H2-3H4Ha0 1ρ12+a0-7H4Ha0 ρ12-ρ2ρ13+a08 ρ14-4ρ12ρ2+5ρ22-2ρ1ρ3ρ14,
ρm1Y¯ mY¯ξm.
a¯˙1=γ¯2H3 a02-2 HHa0+5H22H2-3H4Ha0 1ρ12-a0-7H4Ha0 ρ1ρ13-a04 ρ1ρ13+5a08 ρ12ρ14,
θγb=π round(θγa/π)+arctan{[Im(γb)X+P]/[Re(γb)X]},
|Uγ(0)(x, z)|2dx=12 kπγ a0(ξ)a0*(τ)Y(ξ, z)Y*(τ, z)H(ξ, z)H(τ, z)1/2×exp-kγ4[X(ξ, z)-X(τ, z)]2-k4γ[P(ξ, z)-P(τ, z)]2×exp-ik2[X(ξ, z)-X(τ, z)][P(ξ, z)+P(τ, z)]+ik[L(ξ, z)-L(τ, z)]dτdξ.

Metrics