Abstract

We suggest a numerical algorithm for complex ray tracing. Such an algorithm is intended for the computation of a wave field in the framework of complex geometrical optics. The main advantage of the complex method is the possibility to take into account diffraction effects by use of only ordinary differential equations of geometrical optics, thus reducing the calculation time. The efficiency of the suggested algorithm is illustrated by several numerical examples that allow comparison with known analytic solutions: the field of a plane wave behind a caustic in a linear layer, uniform field asymptotics on a caustic in a linear layer, and a Gaussian beam field in a homogeneous medium. It is pointed out that the approach under consideration can be readily applied to a great variety of real wave problems that have an analytical solution: nonplane waves, nonplane-stratified media, and the like. In particular, a numerical solution for Gaussian beam propagation through inhomogeneities of Gaussian form is presented.

© 2001 Optical Society of America

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References

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  1. Yu. A. Kravtsov, G. W. Forbes, A. A. Asatryan, “Theory and applications of complex rays,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1999), Vol. XXXIX, pp. 1–62.
  2. Yu. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. Quantum Electron. 10, 719–730 (1967).
    [CrossRef]
  3. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).
  4. Yu. A. Kravtsov, Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990).
  5. Yu. A. Kravtsov, Yu. I. Orlov, Caustics, Catastrophes and Wave Fields. 2nd ed. (Springer-Verlag, Berlin, 1998).
  6. R. A. Egorchenkov, Yu. A. Kravtsov, “Numerical realization of complex geometrical optics method,” Izv. Vyssh. Uchebn. Zaved. Radiodizika 43, 630–637 (2000) [English translation in Radiophys. Quantum Electron. 43, 512–517 (2000)].
  7. E. Poli, G. V. Pereverzev, A. G. Peeters, “Paraxial Gaussian wave beam propagation in an anisotropic inhomogeneous plasma,” Phys. Plasmas 6, 5–11 (1999).
    [CrossRef]

2000 (1)

R. A. Egorchenkov, Yu. A. Kravtsov, “Numerical realization of complex geometrical optics method,” Izv. Vyssh. Uchebn. Zaved. Radiodizika 43, 630–637 (2000) [English translation in Radiophys. Quantum Electron. 43, 512–517 (2000)].

1999 (1)

E. Poli, G. V. Pereverzev, A. G. Peeters, “Paraxial Gaussian wave beam propagation in an anisotropic inhomogeneous plasma,” Phys. Plasmas 6, 5–11 (1999).
[CrossRef]

1967 (1)

Yu. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. Quantum Electron. 10, 719–730 (1967).
[CrossRef]

Asatryan, A. A.

Yu. A. Kravtsov, G. W. Forbes, A. A. Asatryan, “Theory and applications of complex rays,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1999), Vol. XXXIX, pp. 1–62.

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Egorchenkov, R. A.

R. A. Egorchenkov, Yu. A. Kravtsov, “Numerical realization of complex geometrical optics method,” Izv. Vyssh. Uchebn. Zaved. Radiodizika 43, 630–637 (2000) [English translation in Radiophys. Quantum Electron. 43, 512–517 (2000)].

Forbes, G. W.

Yu. A. Kravtsov, G. W. Forbes, A. A. Asatryan, “Theory and applications of complex rays,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1999), Vol. XXXIX, pp. 1–62.

Kravtsov, Yu. A.

R. A. Egorchenkov, Yu. A. Kravtsov, “Numerical realization of complex geometrical optics method,” Izv. Vyssh. Uchebn. Zaved. Radiodizika 43, 630–637 (2000) [English translation in Radiophys. Quantum Electron. 43, 512–517 (2000)].

Yu. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. Quantum Electron. 10, 719–730 (1967).
[CrossRef]

Yu. A. Kravtsov, G. W. Forbes, A. A. Asatryan, “Theory and applications of complex rays,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1999), Vol. XXXIX, pp. 1–62.

Yu. A. Kravtsov, Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990).

Yu. A. Kravtsov, Yu. I. Orlov, Caustics, Catastrophes and Wave Fields. 2nd ed. (Springer-Verlag, Berlin, 1998).

Orlov, Yu. I.

Yu. A. Kravtsov, Yu. I. Orlov, Caustics, Catastrophes and Wave Fields. 2nd ed. (Springer-Verlag, Berlin, 1998).

Yu. A. Kravtsov, Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990).

Peeters, A. G.

E. Poli, G. V. Pereverzev, A. G. Peeters, “Paraxial Gaussian wave beam propagation in an anisotropic inhomogeneous plasma,” Phys. Plasmas 6, 5–11 (1999).
[CrossRef]

Pereverzev, G. V.

E. Poli, G. V. Pereverzev, A. G. Peeters, “Paraxial Gaussian wave beam propagation in an anisotropic inhomogeneous plasma,” Phys. Plasmas 6, 5–11 (1999).
[CrossRef]

Poli, E.

E. Poli, G. V. Pereverzev, A. G. Peeters, “Paraxial Gaussian wave beam propagation in an anisotropic inhomogeneous plasma,” Phys. Plasmas 6, 5–11 (1999).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Izv. Vyssh. Uchebn. Zaved. Radiodizika (1)

R. A. Egorchenkov, Yu. A. Kravtsov, “Numerical realization of complex geometrical optics method,” Izv. Vyssh. Uchebn. Zaved. Radiodizika 43, 630–637 (2000) [English translation in Radiophys. Quantum Electron. 43, 512–517 (2000)].

Phys. Plasmas (1)

E. Poli, G. V. Pereverzev, A. G. Peeters, “Paraxial Gaussian wave beam propagation in an anisotropic inhomogeneous plasma,” Phys. Plasmas 6, 5–11 (1999).
[CrossRef]

Radiophys. Quantum Electron. (1)

Yu. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. Quantum Electron. 10, 719–730 (1967).
[CrossRef]

Other (4)

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Yu. A. Kravtsov, Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990).

Yu. A. Kravtsov, Yu. I. Orlov, Caustics, Catastrophes and Wave Fields. 2nd ed. (Springer-Verlag, Berlin, 1998).

Yu. A. Kravtsov, G. W. Forbes, A. A. Asatryan, “Theory and applications of complex rays,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1999), Vol. XXXIX, pp. 1–62.

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Figures (4)

Fig. 1
Fig. 1

Path of integration on complex plane (τ, τ). The integration necessary every time for calculating value of functions X(ξ, τ) and Y(ξ, τ).

Fig. 2
Fig. 2

Absolute value of wave field for the case of a flat wave incident on a linear layer [Eq. (19)] with b=100λ under angle θ=π/4. Solid curve, results of calculations by the CGO method; dotted curve, the exact analytic solution [Eq. (20)] and simultaneously the asymptotic wave field, calculated by means of the etalon integral method by Kravtsov–Ludwig.

Fig. 3
Fig. 3

(a) Initial Gaussian beam form (curve 1), beam form at distance y=1000λ (curve 2), beam form in a free space (curve 3), and beam form calculated by the TGO method (curve 4). (b) Gaussian beam propagation through the inhomogeneity centered at the beam axis.

Fig. 4
Fig. 4

(a) Curve 1, initial beam; curve 2, beam form at distance y=1000λ; curve 3, at distance y=2000λ. (b) Geometry of Gaussian beam propagation through the inhomogeneity shifted relative to the beam axis.

Equations (43)

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u=U exp(ikψ)=U exp(ikψ-kψ),
(ψ)2=,
2(ψ, U)+UΔψ=0.
drdτ=p,dpdτ=12 (r),
dψ/dτ=(r),
ψ(r)=ψ0(ξ, η)+0τ[r(ξ, η, τ)]dτ,
U=U0(D0/D)1/2,
D(τ)=(x, y, z)/(ξ, η, τ),D0=D(0).
u0(ξ, η)=U0(ξ, η)exp[ikψ0(ξ, η)],
x=x0(ξ),
y=y0(ξ),
ψ=ψ0(ξ),
U=U0(ξ).
dF/dτ=R(F),
R(F)
F3, F4, 12x (F1, F2), 12y (F1, F2), (F1, F2)
F0(ξ)F(τ=0)=[x0(ξ), y0(ξ), p0(ξ), q0(ξ), ψ0(ξ)],
p0x0ξ+q0y0ξ=ψ0ξ,
(p0)2+(q0)2=[x0(ξ),y0(ξ)]0,
q0=y0ξψ0ξ+x0ξ2y0ξ20+x0ξ40-x0ξ2ψ0ξ21/2x0ξ2+y0ξ2,
p0=ψ0ξ-q0y0ξ/x0ξ.
Fn=Fn-1+(K1+2K2+2K3+K4)/6,
K1=R(Fn-1)Δτ,
K2=R(Fn-1+K1/2)Δτ,
K3=R(Fn-1+K2/2)Δτ,
K4=R(Fn-1+K3)Δτ.
X(ξ, τ)=x,Y(ξ, τ)=y
ξa+1τα+1=ξατα-D^-1(ξα, τα)X(ξα, τα)-xY(ξα, τα)-y,
Dˆ(ξ, τ)=X(ξ, τ)ξX(ξ, τ)τY(ξ, τ)ξY(ξ, τ)τ=X(ξ, τ)ξP(ξ, τ)Y(ξ, τ)ξQ(ξ, τ)
X(ξ, τ)-xY(ξ, τ)-y
X(ξα, τα)-xY(ξα, τα)-y<Δρ,
U=U0(ξ)D(ξ, 0)D(ξ, τ)1/2=U0(ξ)×x0(ξ)ξ q0(ξ)-y0(ξ)ξ p0(ξ)X(ξ, τ)ξ Q(ξ, τ)-Y(ξ, τ)ξ P(ξ, τ)1/2,
u=sU(ξs, τs)exp[ikψ(ξs, τs)].
(y)=1-by.
u=2(π cos θ)1/2kb1/6Ai(k2b)1/3y-cos2 θb.
u=(π)1/2[(U1+iU2)(-ζ)1/4Ai(ζ)-i(U1-iU2)×(-ζ)-1/4Ai(ζ)]exp(ikχ-iπ/4),
23(-ζ˜)3/2=12(ψ2-ψ1),
ψ1=ψ+iψ,ψ2=ψ-iψ,ψ>0,
23(-ζ˜)3/2=12(ψ2-ψ1)=-iψ,
u0=exp-ξ22a2,
x=ξ+iξτa2k,y=1+ξ2a4k21/2τ.
u(x, y)=1+iyka2-1/2expiky-x22a21+iyka2-1.
=1+μ exp[-(r-r0)2/2d2],

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