Abstract

How good are the Born and Rytov approximations for the case of refraction and reflection at a plane interface? Here, we show the following results: For the reflected field, the Born approximation gives a plane wave at the correct reflection angle but approximates the true reflection coefficient by an expression linear in the scattering strength, which in this case involves both the velocity perturbation across the interface and the cosine of the incident angle. The Rytov approximation, on the other hand, can be interpreted as giving an infinite series of reflected plane waves in which the first term is just the Born approximation to the true reflected wave. Both approximations, however, are uniformly valid for the field above the interface. In contrast, the Born approximation to the transmitted field is not a plane wave and is not uniformly valid since it contains a secular term that grows linearly with distance from the interface. The Rytov approximation to the transmitted field is uniformly valid; in fact, the Rytov approximation gives a transmitted plane wave that satisfies a modified form of Snell’s law. Numerical examples indicate that the Rytov approximation to the transmitted field is surprisingly accurate. For velocity contrasts less than 40% and incident angles less than 30°, the Rytov approximation to the transmitted angle and transmission coefficient is never more than 20% in error.

© 1985 Optical Society of America

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References

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  1. J. B. Keller, “Accuracy and validity of the Born and Rytov approximations,” J. Opt. Soc. Am. 59, 1003–1004 (1969).
  2. M. I. Sancer, A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE 58, 140–141 (1970).
    [CrossRef]
  3. K. Mano, “Interrelationship between terms of the Born and Rytov expansions,” Proc. IEEE 58, 1168–1169 (1970).
    [CrossRef]
  4. W. J. Hadden, D. Mintzer, “Test of the Born and Rytov approximations using the Epstein problem,” J. Acoust. Soc. Am. 63, 1279–1286 (1978).
    [CrossRef]
  5. Throughout this paper, we use the expression “uniformly valid” in the following usual sense (see Ref. 6): Given a function g(z. α) and an approximation f(z, α) to g, the statement “g(z, α) = f(z, α) + O(αn) as α→ α0” is called uniformly valid with respect to z in some domain D, if for all z in D there exists a constant k such that, |g−f|⩽kαn as α→ α0, where the constant k can be found independently of the value of z.
  6. J. Kevorkian, J. D. Cole, Perturbation Methods in Applied Mathematics (Springer-Verlag, Berlin, 1981).
    [CrossRef]
  7. A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrasonic Imaging 4, 336–350 (1982).It is actually the complex phase that is used in inverse methods based on the Rytov approximation. Here we consider just the imaginary part of the phase, since in this example the real part of the phase is independent of z and cannot be a source of nonuniformity in the approximation.
    [PubMed]

1982 (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrasonic Imaging 4, 336–350 (1982).It is actually the complex phase that is used in inverse methods based on the Rytov approximation. Here we consider just the imaginary part of the phase, since in this example the real part of the phase is independent of z and cannot be a source of nonuniformity in the approximation.
[PubMed]

1978 (1)

W. J. Hadden, D. Mintzer, “Test of the Born and Rytov approximations using the Epstein problem,” J. Acoust. Soc. Am. 63, 1279–1286 (1978).
[CrossRef]

1970 (2)

M. I. Sancer, A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE 58, 140–141 (1970).
[CrossRef]

K. Mano, “Interrelationship between terms of the Born and Rytov expansions,” Proc. IEEE 58, 1168–1169 (1970).
[CrossRef]

1969 (1)

Cole, J. D.

J. Kevorkian, J. D. Cole, Perturbation Methods in Applied Mathematics (Springer-Verlag, Berlin, 1981).
[CrossRef]

Devaney, A. J.

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrasonic Imaging 4, 336–350 (1982).It is actually the complex phase that is used in inverse methods based on the Rytov approximation. Here we consider just the imaginary part of the phase, since in this example the real part of the phase is independent of z and cannot be a source of nonuniformity in the approximation.
[PubMed]

Hadden, W. J.

W. J. Hadden, D. Mintzer, “Test of the Born and Rytov approximations using the Epstein problem,” J. Acoust. Soc. Am. 63, 1279–1286 (1978).
[CrossRef]

Keller, J. B.

Kevorkian, J.

J. Kevorkian, J. D. Cole, Perturbation Methods in Applied Mathematics (Springer-Verlag, Berlin, 1981).
[CrossRef]

Mano, K.

K. Mano, “Interrelationship between terms of the Born and Rytov expansions,” Proc. IEEE 58, 1168–1169 (1970).
[CrossRef]

Mintzer, D.

W. J. Hadden, D. Mintzer, “Test of the Born and Rytov approximations using the Epstein problem,” J. Acoust. Soc. Am. 63, 1279–1286 (1978).
[CrossRef]

Sancer, M. I.

M. I. Sancer, A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE 58, 140–141 (1970).
[CrossRef]

Varvatsis, A. D.

M. I. Sancer, A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE 58, 140–141 (1970).
[CrossRef]

J. Acoust. Soc. Am. (1)

W. J. Hadden, D. Mintzer, “Test of the Born and Rytov approximations using the Epstein problem,” J. Acoust. Soc. Am. 63, 1279–1286 (1978).
[CrossRef]

J. Opt. Soc. Am. (1)

Proc. IEEE (2)

M. I. Sancer, A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE 58, 140–141 (1970).
[CrossRef]

K. Mano, “Interrelationship between terms of the Born and Rytov expansions,” Proc. IEEE 58, 1168–1169 (1970).
[CrossRef]

Ultrasonic Imaging (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrasonic Imaging 4, 336–350 (1982).It is actually the complex phase that is used in inverse methods based on the Rytov approximation. Here we consider just the imaginary part of the phase, since in this example the real part of the phase is independent of z and cannot be a source of nonuniformity in the approximation.
[PubMed]

Other (2)

Throughout this paper, we use the expression “uniformly valid” in the following usual sense (see Ref. 6): Given a function g(z. α) and an approximation f(z, α) to g, the statement “g(z, α) = f(z, α) + O(αn) as α→ α0” is called uniformly valid with respect to z in some domain D, if for all z in D there exists a constant k such that, |g−f|⩽kαn as α→ α0, where the constant k can be found independently of the value of z.

J. Kevorkian, J. D. Cole, Perturbation Methods in Applied Mathematics (Springer-Verlag, Berlin, 1981).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Geometry of the interface problem.

Fig. 2
Fig. 2

Rytov approximation to the transmitted angle.

Fig. 3
Fig. 3

Exact reflection coefficient (solid lines) and the Born approximation (symbols): (a) velocity contrasts of +10% (squares), +20% (circles) and +40% (triangles); (b) velocity contrasts of −10% (squares), −20% (circles), and −40% (triangles). The filled symbol indicates the first angle at which the approximation is 20% in error; numbers in parentheses give errors for other typical points.

Fig. 4
Fig. 4

Exact transmission coefficient (solid lines) and the Rytov approximation (symbols): (a) velocity contrasts of +10% (squares) +20% (circles) and +40% (triangles); (b) velocity contrasts of −10% (squares), −20% (circles), and −40% (triangles). The filled symbol indicates the lirst angle at which the approximation is 20% in error; numbers in parentheses give errors for other typical points.

Fig. 5
Fig. 5

Exact transmitted angle (solid lines) and the Rytov approximation (symbols): (a) velocity contrasts of+10% (squares), +20% (circles), and +40% (triangles); (b) velocity contrasts of −10% (squares), −20% (circles), and −40% (triangles). The filled symbol indicates the first angle at which the approximation is 20% in error; numbers in parentheses give errors for other typical points.

Fig. 6
Fig. 6

Effective index of refraction for the Rytov approximation (symbols): (a) velocity contrasts of +10% (squares), +20% (circles), and +40% (triangles); (b) velocity contrasts of −10% (squares), −20% (circles), and −40% (triangles). The filled symbol indicates the first angle at which the approximation is 20% in error; numbers in parentheses give errors for other typical points.

Equations (41)

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u I ( x , z ) = exp ( ik sin θ 1 x + ik cos θ 1 z ) ,
n = c 0 c 1
u R ( x , z ) = R exp ( ik sin θ R x ik cos θ R z ) ,
u T ( x , z ) = T exp ( ikn sin θ T x + ikn cos θ T z ) .
R = cos θ I n cos θ T cos θ I + n cos θ T ,
T = 1 + R = 2 cos θ I cos θ I + n cos θ T ,
n sin θ T = sin θ I ,
n cos θ T = ( 1 + n 2 1 cos 2 θ I ) 1 / 2 cos θ I .
n cos θ T = cos θ I + 1 2 n 2 1 cos θ I + ,
n 2 1 cos 2 θ I < 1 .
u R = 1 ( 1 + α ) 1 / 2 1 + ( 1 + α ) 1 / 2 exp ( ik sin θ I x ik cos θ I z ) ,
u R ( B ) = α 4 exp ( ik sin θ I x ik cos θ I z ) .
u ( R ) = exp [ α 4 exp ( 2 ik cos θ I z ) ] × exp ( ik sin θ I x + ik cos θ I z ) ,
u = [ 1 + 1 ( 1 + α ) 1 / 2 1 + ( 1 + α ) 1 / 2 exp ( 2 ik cos θ I z ) ] × exp ( ik sin θ I x + ik cos θ I z ) .
u ( R ) = [ 1 α 4 exp ( 2 ik cos θ I z ) + α 2 32 × exp ( 4 ik cos θ I z ) + ] u I = u I α 4 exp ( ik sin θ I x ik cos θ I z ) + α 2 32 × exp ( ik sin θ I x 3 ik cos θ I z ) + ,
u T = 2 1 + ( 1 + α ) 1 / 2 exp [ ik sin θ I x + ik ( 1 + α ) 1 / 2 cos θ I z ] .
u T ( B ) = [ 1 α 4 ( 1 2 ik cos θ I z ) ] × exp ( ik sin θ I x + ik cos θ I z ) .
u T ( R ) = e α / 4 exp [ ik sin θ I x + ik ( 1 + α / 2 ) cos θ I z ] ,
k ( n 2 + α 2 cos 2 θ I / 4 ) 1 / 2 .
n ( R ) = ( n 2 + α 2 cos 2 θ I / 4 ) 1 / 2
k ( 1 + α ) 1 / 2 cos θ I z k ( 1 + α 2 ) cos θ I z = k α 2 8 cos θ I z +
| ( 1 + Δ ) 2 1 | < | ( 1 Δ ) 2 1 | .
u ( x , z ) = f ( z ) exp ( ik sin θ I x ) ,
2 u + k 2 n 2 ( x ) = 0 ,
d 2 f d z 2 + k z 2 [ 1 + V ( z ) ] f = 0.
k z = k cos θ I
V ( z ) = { 0 , z < 0 n 2 1 cos 2 θ I α , z > 0 ,
f ( B ) = m = 0 m f m ,
f ( R ) = exp ( m = 0 m ψ m ) .
f 1 ( B ) = f 0 + f 1
f 1 ( R ) = exp ( ψ 0 + ψ 1 ) .
f 1 ( R ) = f 0 exp ( f 1 / f 0 ) = f 0 exp [ f s ( B ) / f 0 ] ,
f 0 = exp ( ik cos θ I z ) = e i k z z
f s ( B ) ( z ) = i k z α 2 0 d z exp ( i k z | z z | ) e i k z z .
f s ( B ) ( z < 0 ) = i k z α 2 e i k z z 0 d z e 2 i k z z . = α 4 e i k z z ,
0 d z e 2 i k z z lim δ 0 + 0 d z exp [ 2 i ( k z + i δ ) z ] = 1 2 i k z .
f s ( B ) ( z > 0 ) = α 4 ( 1 2 i k z z ) e i k z z .
u R ( B ) = α 4 exp ( ik sin θ I x + ik cos θ I z ) .
u T ( B ) = [ 1 α 4 ( 1 2 ik cos θ I z ) ] exp ( ik sin θ I x + ik cos θ I ) .
u ( R ) = exp [ α 4 exp ( 2 ik cos θ I z ) ] × exp ( ik sin θ I x + ik cos θ I z ) ,
u T ( R ) = e α / 4 exp ( ik sin θ I x + ik ( 1 + α / 2 ) cos θ I z ] .

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