Abstract

The Airy beam (AiB) has attracted a lot of attention recently because of its intriguing features; the most distinctive ones are the propagation along curved trajectories in free space and the weak diffraction. We have previously shown that the AiB is, in fact, a caustic of the rays that radiate from the tail of the Airy function aperture distribution. Here we derive a class of ultra wideband Airy pulsed beams (AiPBs), which are the extension of the AiB into the time domain. We introduce a frequency scaling of the initial aperture field that renders the ray skeleton of the field, including the caustic, frequency independent, thus ensuring that all the frequency components propagate along the same curved trajectory and that the AiPB does not disperse. The resulting AiPB preserves the intriguing features of the time-harmonic AiB discussed above. An exact closed-form solution for the AiPB is derived using the spectral theory of transients. We also derive wavefront approximations for the field in the time window around the pulse arrival, which are valid uniformly in the vicinity of the caustic. These approximations are based on the so-called uniform geometrical optics, which is extended here to the time domain.

© 2011 Optical Society of America

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References

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  1. G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
  2. G. Siviloglou and D. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
    [CrossRef] [PubMed]
  3. M. Bandres and J. Gutiérrez-Vega, “Airy–Gauss beams and their transformation by paraxial optical systems,” Opt. Express 15, 16719–16728 (2007).
    [CrossRef] [PubMed]
  4. G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 (2008).
    [CrossRef] [PubMed]
  5. J. Broky, G. Siviloglou, A. Dogariu, and D. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16, 12880–12891 (2008).
    [CrossRef] [PubMed]
  6. Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18, 8440–8452 (2010). [Note that, in Eq. (16), the factor 2π1/2e−iπ/4ω1/6 is missing due to a typo; see Eq. .]
    [CrossRef] [PubMed]
  7. D. Ludwig, “Wave propagation near a smooth caustic,” Bull. Am. Math. Soc. 71, 776–779 (1965).
    [CrossRef]
  8. Yu. A. Kravtsov, “A modification of the geometrical optics method,” Radiophys. Quantum Electron. 7, 664–673 (1964).
  9. P. Saari, “Laterally accelerating Airy pulses,” Opt. Express 16, 10303–10308 (2008).
    [CrossRef] [PubMed]
  10. E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part I: formulation and interpretation,” IEEE Trans. Antennas Propag. 35, 80–86 (1987).
    [CrossRef]
  11. E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part II: evaluation of the spectral integral,” IEEE Trans. Antennas Propag. 35, 574–580 (1987).
    [CrossRef]
  12. E. Heyman, “Weakly dispersive spectral theory of transients (STT), part III: applications,” IEEE Trans. Antennas Propag. 35, 1258–1266 (1987).
    [CrossRef]
  13. E. Heyman and L. B. Felsen, “Propagating pulsed beam solution by complex source parameter substitution,” IEEE Trans. Antennas Propag. 34, 1062–1065 (1986).
    [CrossRef]
  14. E. Heyman and L. B. Felsen, “Gaussian beam and pulsed beam dynamics: complex source and spectrum formulations within and beyond paraxial asymptotics,” J. Opt. Soc. Am. A 18, 1588–1610 (2001).
    [CrossRef]
  15. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 10th ed. (Dover, 1972).
  16. E. Heyman, and L. B. Felsen, “Real and complex spectra—a generalization of WKBJ seismograms,” Geophys. J. R. Astron. Soc. 91, 1087–1126 (1987).
    [CrossRef]

2010 (1)

2008 (3)

2007 (3)

2001 (1)

1987 (4)

E. Heyman, and L. B. Felsen, “Real and complex spectra—a generalization of WKBJ seismograms,” Geophys. J. R. Astron. Soc. 91, 1087–1126 (1987).
[CrossRef]

E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part I: formulation and interpretation,” IEEE Trans. Antennas Propag. 35, 80–86 (1987).
[CrossRef]

E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part II: evaluation of the spectral integral,” IEEE Trans. Antennas Propag. 35, 574–580 (1987).
[CrossRef]

E. Heyman, “Weakly dispersive spectral theory of transients (STT), part III: applications,” IEEE Trans. Antennas Propag. 35, 1258–1266 (1987).
[CrossRef]

1986 (1)

E. Heyman and L. B. Felsen, “Propagating pulsed beam solution by complex source parameter substitution,” IEEE Trans. Antennas Propag. 34, 1062–1065 (1986).
[CrossRef]

1965 (1)

D. Ludwig, “Wave propagation near a smooth caustic,” Bull. Am. Math. Soc. 71, 776–779 (1965).
[CrossRef]

1964 (1)

Yu. A. Kravtsov, “A modification of the geometrical optics method,” Radiophys. Quantum Electron. 7, 664–673 (1964).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 10th ed. (Dover, 1972).

Bandres, M.

Broky, J.

Christodoulides, D.

Dogariu, A.

Felsen, L. B.

E. Heyman and L. B. Felsen, “Gaussian beam and pulsed beam dynamics: complex source and spectrum formulations within and beyond paraxial asymptotics,” J. Opt. Soc. Am. A 18, 1588–1610 (2001).
[CrossRef]

E. Heyman, and L. B. Felsen, “Real and complex spectra—a generalization of WKBJ seismograms,” Geophys. J. R. Astron. Soc. 91, 1087–1126 (1987).
[CrossRef]

E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part I: formulation and interpretation,” IEEE Trans. Antennas Propag. 35, 80–86 (1987).
[CrossRef]

E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part II: evaluation of the spectral integral,” IEEE Trans. Antennas Propag. 35, 574–580 (1987).
[CrossRef]

E. Heyman and L. B. Felsen, “Propagating pulsed beam solution by complex source parameter substitution,” IEEE Trans. Antennas Propag. 34, 1062–1065 (1986).
[CrossRef]

Gutiérrez-Vega, J.

Heyman, E.

Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18, 8440–8452 (2010). [Note that, in Eq. (16), the factor 2π1/2e−iπ/4ω1/6 is missing due to a typo; see Eq. .]
[CrossRef] [PubMed]

E. Heyman and L. B. Felsen, “Gaussian beam and pulsed beam dynamics: complex source and spectrum formulations within and beyond paraxial asymptotics,” J. Opt. Soc. Am. A 18, 1588–1610 (2001).
[CrossRef]

E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part I: formulation and interpretation,” IEEE Trans. Antennas Propag. 35, 80–86 (1987).
[CrossRef]

E. Heyman, and L. B. Felsen, “Real and complex spectra—a generalization of WKBJ seismograms,” Geophys. J. R. Astron. Soc. 91, 1087–1126 (1987).
[CrossRef]

E. Heyman, “Weakly dispersive spectral theory of transients (STT), part III: applications,” IEEE Trans. Antennas Propag. 35, 1258–1266 (1987).
[CrossRef]

E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part II: evaluation of the spectral integral,” IEEE Trans. Antennas Propag. 35, 574–580 (1987).
[CrossRef]

E. Heyman and L. B. Felsen, “Propagating pulsed beam solution by complex source parameter substitution,” IEEE Trans. Antennas Propag. 34, 1062–1065 (1986).
[CrossRef]

Kaganovsky, Y.

Kravtsov, Yu. A.

Yu. A. Kravtsov, “A modification of the geometrical optics method,” Radiophys. Quantum Electron. 7, 664–673 (1964).

Ludwig, D.

D. Ludwig, “Wave propagation near a smooth caustic,” Bull. Am. Math. Soc. 71, 776–779 (1965).
[CrossRef]

Saari, P.

Siviloglou, G.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 10th ed. (Dover, 1972).

Bull. Am. Math. Soc. (1)

D. Ludwig, “Wave propagation near a smooth caustic,” Bull. Am. Math. Soc. 71, 776–779 (1965).
[CrossRef]

Geophys. J. R. Astron. Soc. (1)

E. Heyman, and L. B. Felsen, “Real and complex spectra—a generalization of WKBJ seismograms,” Geophys. J. R. Astron. Soc. 91, 1087–1126 (1987).
[CrossRef]

IEEE Trans. Antennas Propag. (4)

E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part I: formulation and interpretation,” IEEE Trans. Antennas Propag. 35, 80–86 (1987).
[CrossRef]

E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part II: evaluation of the spectral integral,” IEEE Trans. Antennas Propag. 35, 574–580 (1987).
[CrossRef]

E. Heyman, “Weakly dispersive spectral theory of transients (STT), part III: applications,” IEEE Trans. Antennas Propag. 35, 1258–1266 (1987).
[CrossRef]

E. Heyman and L. B. Felsen, “Propagating pulsed beam solution by complex source parameter substitution,” IEEE Trans. Antennas Propag. 34, 1062–1065 (1986).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Radiophys. Quantum Electron. (1)

Yu. A. Kravtsov, “A modification of the geometrical optics method,” Radiophys. Quantum Electron. 7, 664–673 (1964).

Other (1)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 10th ed. (Dover, 1972).

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Figures (9)

Fig. 1
Fig. 1

Ray representation of the AiB. The rays shown in solid blue lines radiate from the aperture at z = 0 and focus onto a caustic (red dashed line), which delineates the beam’s propagation trajectory. The rays shown in dashed green lines originate at distant points where the aperture field is weak. They arrive to the region of the caustic at very late times, beyond the pertinent time window of the AiPB, and do not focus there. In fact, this ray species is not included in the paraxial solution of Eq. (5), but it does exist in the complete ray picture. For the AiPB, the same ray skeleton is used for all frequencies [see Eqs. (3, 4, 5, 6)]. Points A = ( x , z ) = ( 0.023 , 0.3 ) β and B = ( 0.067 , 0.5 ) β are typical points where the field will be calculated in the sequel. All axes are normalized with respect to β. In addition to the rays depicted here, there is another species of rays that emerges in the x direction and therefore does not contribute to the field in the pertinent zone near the caustic ([6], Fig. 1).

Fig. 2
Fig. 2

Rays for a typical observation point r on the lit side of the caustic (dashed red line). Specifically, we consider r near point A in Fig. 1, used as an example to discuss the properties of the field. There are three rays leaving the aperture at x 1 = 0.043 β , x 2 = 0.011 β , and x 3 = 0.9 β , with exit angles θ 1 , 2 , 3 , respectively. Rays r = 1 , 2 (blue solid lines) converge to the caustic and touch it after and before reaching r, respectively. Ray r = 3 (green dashed line) emerges from the aperture at a distant point where the field is very weak and arrives to r at a much later time; hence, its contribution can be neglected. Note that this ray species is not included in the paraxial solution of Eq. (5).

Fig. 3
Fig. 3

Complex ξ plane of the STT integral in Eq. (18) for a given observation point r on the lit side of caustic, shown in Fig. 2. Wavy lines, branch cuts of ζ separating the URS where Re ζ > 0 from the LRS where Re ζ < 0 . C is the integration contour in Eqs. (12, 18). Squares, stationary points ξ 1 = 0.208 , ξ 2 = 0.104 and ξ 3 = 0.951 corresponding to rays r = 1 , 2 , 3 in Fig. 2 arriving at r at t = t 1 , 2 , 3 , respectively. ξ ( p ) ( t ) , p = 1 , 2 , 6 , trajectories of the six poles ξ ( t ) of Eq. (19) as a function of t; poles on the URS and LRS are denoted by solid or dashed curves, respectively. The tags on the trajectories denote values of t there. The poles p = 1 , 3 (blue and red curves) are always in the URS; p = 5 , 6 (cyan and black curves) are always in the LRS; p = 2 , 4 (green and magenta curves) are located first in the LRS and then cross to the URS. In order to improve visibility, we displaced the poles from the real ξ axis by choosing the pulse length parameter T to be rather large, T = 10 2 β / c . In practice, however, we take c T / β = 10 6 , yielding the poles map in Fig. 4 and the subsequent field results. Here and in Figs. 4, 5, 6, 7 we used α = 10 5 .

Fig. 4
Fig. 4

Trajectories of the poles ξ ( p ) ( t ) , p = 1 , 2 , 3 of Eq. (19) in the complex ξ plane for observation points near point A of Fig. 1. The trajectories are marked by the same color code and tags as in Fig. 3. The figure zooms on the spectral zone near the stationary points ξ 1 , 2 of Fig. 3 (squares). Pulse length parameter, c T / β = 10 6 . (a) Observation point on the lit side of the caustic, (b) on the caustic, and (c) on the shadow side. Specifically in this figure, the point in (a) is displaced horizontally from A by Δ x = 2.510 3 β and the point in (c) is displaced by Δ x = 2.310 4 β . On the caustic, the two stationary points coalesce to a second-order stationary point ξ c , and both rays r = 1 , 2 arrive at t = t c .

Fig. 5
Fig. 5

Field near point A as in Fig. 4: points (a), (b) on the lit side of the caustic, (c), (d) on the caustic at point A, and (e), (f) on the shadow side. The waveforms calculated via Eq. (20) (blue) are compared with those calculated via the TD-UGO approximation of Subsection 4C (green) and are seen to be indistinguishable within the scale of the figure. The time axis is centered around t A , time of arrival to point A, and normalized with respect to T. The temporal signal is given by Eq. (17) with m = 2 and γ = 0 in (a), (c), (e) or γ = π / 2 in (b), (d), (f). Note that the waveforms for γ = π / 2 are Hilbert transforms of those for γ = 0 . The field is normalized such that max | u | = 1 on the caustic in (c). Notice also the different vertical scales.

Fig. 6
Fig. 6

Zoom in view of the signals in Figs. 5a, 5b. Rays r = 1 , 2 in Fig. 5a are shown separately in (a) and (c), respectively, while rays r = 1 , 2 in Fig. 5b are shown in (b) and (d). The time axis is centered around t = t 1 , 2 , the time of arrival of the rays, and normalized with respect to T. Notice the differences in the figure scales. The waveforms with γ = π / 2 are Hilbert transformed with respect to those with the same r but with γ = 0 . The waveforms for r = 2 are essentially Hilbert transformed with respect to the r = 1 waveforms with the same γ, but they have larger amplitudes and narrower pulses since they originate at a nearer point | x 2 | < | x 1 | (see Fig. 2) with a narrower pulse length parameter T 2 < T 1 [see Eq. (23)].

Fig. 7
Fig. 7

(a), (b) Snapshots of the field near points A and B on the caustic, which are defined in Fig. 1. The snapshots are taken at t = t A , B = the time of arrival at A and B. The axes are centered around point A and B and are normalized with respect to β. The temporal signal is given by Eq. (17) with m = 2 , γ = 0 , T = 10 6 β / c , and α = 10 5 . The logarithmic color scale retains the sign of the waveform (see color bar). The fields are normalized with respect to the maximal value in (a). One observes that the field consists of wavefronts 1,2, which are generated by rays of species r = 1 , 2 , respectively, depicted in Fig. 2. (c), (d) Cross-sectional cuts of (a) along z = constant lines : the cut in (c) passes exactly through A, while the cut in (d) is along ( z z A ) / β = 0.5 × 10 3 .

Fig. 8
Fig. 8

Same as in Fig. 7 but for α = 10 3 . The fields are normalized with respect to the maximal value in (c). Increasing α with respect to its value in Fig. 7 affects mainly wavefront 1, which arrives first, since it corresponds to ray species r = 1 that originates from aperture points x that are located farther away from the center (see Figs. 1, 2) and are therefore strongly affected by the exponential decay in the aperture. Indeed, the field of wavefront 1 is weaker and has a longer pulse length [see Eq. (23)]. The exponential decay also explains the pulse decay as it propagates away from the aperture from A to B.

Fig. 9
Fig. 9

Schematic description of the rays (blue lines) near a smooth caustic (red line) characterized by a local radius of curvature R ( σ ) . The region near the caustic is described by the curvilinear coordinate system ( σ , ν ) where σ is the arc length along the caustic and ν is the normal to the caustic, with ν 0 on the lit and shadow sides, respectively. Two rays arrive at any observation point r = ( σ , ν ) on the lit side: ray r = 1 converges to point σ 1 on the caustic, and ray r = 2 has already touched it at σ 2 . The lengths s of these rays are essentially the same (to leading order) while ϕ is the corresponding angular segment on the caustic.

Equations (55)

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U ^ 0 ( x ; ω ) = exp ( α 0 x / x 0 ) Ai ( x / x 0 ) ,
θ 0 ( x ) = sin 1 [ ( x ) 1 / 2 / k x 0 3 / 2 ] ,
x 0 = β 1 / 3 k 2 / 3 , α 0 = α β 1 / 3 k 1 / 3 ,
U ^ 0 ( x ; ω ) = exp ( α k x ) Ai ( β 1 / 3 k 2 / 3 x ) .
U ^ ( r ; ω ) = Ai [ ( k β ) 2 / 3 ( x / β ( z / 2 β ) 2 + i α z / β ) ] exp [ i k ( z + x z / 2 β z 3 / 12 β 2 + α 2 z / 2 ) ] exp [ k α ( x z 2 / β ) ] .
x / β ( z / 2 β ) 2 = 0
u + ( r , t ) = 1 π 0 d ω e i ω t f ^ ( ω ) U ^ ( r ; ω ) , Im t 0 ,
u ( r , t ) = Re u + ( r , t ) , Im t 0.
w + ( t ) = 1 π 0 d ω e i ω t f ^ ( ω ) g ^ ( ω ) = 1 2 d t f + ( t ) g + ( t t ) = 1 2 f + ( t ) g + ( t ) , Im t 0 ,
U ^ 0 ( x ; ω ) = ω 1 / 3 2 π A e i ω τ 0 ( ξ ) e i ω ξ x / c d ξ ,
τ 0 ( ξ ) = β ( ξ + i α ) 3 / 3 c , A = ( β / c ) 1 / 3 ,
U ^ ( r ; ω ) = ω 1 / 3 2 π C A e i ω τ ( ξ ) d ξ ,
τ ( ξ ; r ) = τ 0 ( ξ ) + ξ x / c + ζ z / c ,
u + ( r , t ) = 1 2 π 2 0 d ω e i ω t g ^ ( ω ) C A e i ω τ ( ξ ) d ξ ,
u + ( r , t ) = 1 2 g + ( t ) i 2 π 2 C A t τ ( ξ ) d ξ .
g ^ ( ω ) = e i γ ( i ω ) m exp ( ω T ) ,
g + ( t ) = e i γ δ + ( m ) ( t i T ) = e i γ t m 1 π i 1 t i T ,
u + ( r , t ) = t m i e i γ 2 π 2 C A t τ ( ξ ) i T d ξ .
τ [ ξ ( t ) ] = t i T .
u + ( r , t ) = ± p t m A e i γ π τ ( ξ ) | ξ = ξ ( p ) ( t ) ,
τ ( ξ ) t r + i α | x r | / c + 1 2 τ r ( ξ ξ r ) 2 , r = 1 , 2 ,
ξ ( p ) ( t ) ξ r ± t t r i T r / τ r / 2 ,
T r = T + α | x r | .
u + r ( r , t ) = A r ( r ) p + ( t t r i T r ) e i π M r / 2 ,
A r ( r ) = A / π 2 | τ r | , p + ( t ) = e i γ t m ( t ) 1 / 2 ,
U ^ r ( r ; ω ) = ( i π ) 1 / 2 ω 1 / 6 A r ( r ) e i ω t r ω α | x r | i M r π / 2 ,
U ^ r ( r ; ω ) = A r ( r ) exp ( i ω t r ( r ) i M r π / 2 ) ,
U ^ ( r ; ω ) = 2 π 1 / 2 e i π / 4 ω 1 / 6 e i ω t ¯ [ A ¯ Ai ( ( ω δ ) 2 / 3 ) + i δ A ω 1 / 3 Ai ( ( ω δ ) 2 / 3 ) ] ,
t ¯ = 1 2 ( t 1 + t 2 ) , δ = 3 4 ( t 2 t 1 ) ,
A ¯ = δ 1 / 6 ( A 1 + A 2 ) / 2 , δ A = δ 1 / 6 ( A 2 A 1 ) / 2.
Ai ( ( ω δ ) 2 / 3 ) = 1 2 π ω 1 / 3 e i ω ( η 3 / 3 ( c δ ) 2 / 3 η ) / c d η ,
U ^ ( r ; ω ) = ω 1 / 2 i 2 π B e i ω ( t ¯ + τ ) d η ,
τ ( r , η ) = [ η 3 / 3 ( c δ ( r ) ) 2 / 3 η ] / c ,
B ( r , η ) = 2 π 1 / 2 e i π / 4 [ A ¯ ( r ) η δ A ( r ) ] .
u + ( r , t ) = 1 2 h + ( t ) i 2 π 2 B ( r , η ) t t ¯ ( r ) τ ( r , η ) d η ,
η 3 / 3 ( c δ ) 2 / 3 η c ( t t ¯ ) = 0 ,
u + ( r , t ) = 1 2 h + ( t ) B ( r , η ) π τ ( r , η ) | η = η ( 3 ) ( t ) ,
σ 1 , 2 = σ ± ( 2 ν R ) 1 / 2 .
s = ( 2 ν R ) 1 / 2 + ( 2 ν ) 3 / 2 / 3 R 1 / 2 .
t 1 , 2 = t c ( 2 ν ) 3 / 2 / 3 c R 1 / 2 .
t ¯ = σ / c , δ 2 / 3 = ν ( 2 / c 2 R ) 1 / 3 .
A 1 , 2 ( r ) = a c ( σ 1 , 2 ) / s ,
A 1 , 2 ( r ) [ a c ( σ ) ± a c ( σ ) ( 2 ν R ) 1 / 2 ] ( 2 ν R ) 1 / 4 ,
A ¯ ( r ) a c ( σ ) ( 2 c R 2 ) 1 / 6 , δ A ( r ) a c ( σ ) ( 2 c R 2 ) 1 / 6 .
a c ( r ) = ( 2 ν R ) 1 / 4 ( A 1 + A 2 ) / 2 , a c ( r ) = ( 2 ν R ) 1 / 4 ( A 1 A 2 ) / 2.
h + ( t ) = ( i π ) 1 / 2 e i γ δ + ( m ) ( t i T c ) .
u + ( r , t ) = i e i γ 2 π 2 t m ( i π ) 1 / 2 B ( r , η ) t t ¯ τ ( r , η ) i T c ( r ) d η .
u + ( r , t ) = t m e i γ ( i π ) 1 / 2 B ( r , η ) π τ ( r , η ) | η = η ( 3 ) ( t ) .
sin θ = ( x / β ) 1 / 2 ,
sin θ = ( x x ) / ρ , ρ = ( x x ) 2 + z 2 .
ρ c = [ x θ ( x ) ] 1 cos θ ( x ) = 2 β cos 2 θ ( x ) sin θ ( x ) ,
τ ( ξ ) t r + τ r α α + τ r ξ α α ( ξ ξ r ) + 1 2 τ r ξ ξ ( ξ ξ r ) 2 , r = 1 , 2 ,
τ r α = i β ξ r 2 / c ,
τ r ξ ξ = ( 2 β ξ r ζ r 3 z ) / c = 2 β ξ r ( 1 ρ r / ρ c r ) / c ,
τ r ξ α = 2 i β ξ r / c ,

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