Abstract

In this paper the method of renormalization group (RG) [Phys. Rev. E 54, 376 (1996) ] is related to the well-known approximations of Rytov and Born used in wave propagation in deterministic and random media. Certain problems in linear and nonlinear media are examined from the viewpoint of RG and compared with the literature on Born and Rytov approximations. It is found that the Rytov approximation forms a special case of the asymptotic expansion generated by the RG, and as such it gives a superior approximation to the exact solution compared with its Born counterpart. Analogous conclusions are reached for nonlinear equations with an intensity-dependent index of refraction where the RG recovers the exact solution.

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References

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  1. L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory. Course of Theoretical Physics, Vol. 3 of Addison-Wesley Series in Advanced Physics (Pergamon, 1958).
  2. L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, 1960).
  3. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  4. N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, 1992).
  5. L.-Yuan Chen, N. Goldenfeld, and Y. Oono, “Renormalization group and singular perturbations: multiple scales, boundary layers, and reductive perturbation theory,” Phys. Rev. E 54, 376-394 (1996).
    [CrossRef]
  6. E. Kirkinis, “The renormalization group and the implicit function theorem for amplitude equations,” J. Math. Phys. 49, 073518 (2008).
    [CrossRef]
  7. J. B. Keller, “Accuracy and validity of Born and Rytov approximations,” J. Opt. Soc. Am. 59, 1003-1004 (1969).
  8. D. L. Bosley, “A technique for the numerical verification of asymptotic expansions,” SIAM Rev. 38, 128-135 (1996).
    [CrossRef]
  9. K. Mano, “Interrelationship between terms of Born and Rytov expansions,” Proc. IEEE 58, 1168-1169 (1970).
    [CrossRef]
  10. J. E. Marsden and M. J. Hoffman, Elementary Classical Analysis (Freeman1993).
  11. F. C. Lin and M. A. Fiddy, “The Born-Rytov controversy: I. Comparing analytical and approximate expressions for the one-dimensional deterministic case,” J. Opt. Soc. Am. A 9, 1102-1110 (1992).
    [CrossRef]
  12. F. C. Lin and M. A. Fiddy, “The Born-Rytov controversy: II. Applications to nonlinear and stochastic scattering problems in one-dimensional half-space media,” J. Opt. Soc. Am. A 10, 1971-1983 (1993).
    [CrossRef]
  13. W. James Hadden, Jr., and D. Mintzer, “Test of the Born and Rytov approximations using the Epstein problem,” J. Acoust. Soc. Am. 63, 1279-1286 (1978).
    [CrossRef]
  14. M. L. Oristaglio, “Accuracy of the Born and Rytov approximations for reflection and refraction at a plane interface,” J. Opt. Soc. Am. A 2, 1987-1993 (1985).
    [CrossRef]
  15. A. Ishimaru, “Wave propagation and scattering in random media,” in IEEE/OUP Series on Electromagnetic Wave Theory (IEEE Press, 1997).
  16. E. Kirkinis, “Reduction of amplitude equations by the renormalization group approach,” Phys. Rev. E 77, 011105 (2008).
    [CrossRef]
  17. L. D. Landau and E. M. Lifshitz, “Electrodynamics of continuous media,” in Course of Theoretical Physics (Pergamon1960), Vol. 8.
  18. R. W. Boyd, Nonlinear Optics (Academic, 1992).
  19. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers. I (Springer-Verlag, 1999).

2008 (2)

E. Kirkinis, “The renormalization group and the implicit function theorem for amplitude equations,” J. Math. Phys. 49, 073518 (2008).
[CrossRef]

E. Kirkinis, “Reduction of amplitude equations by the renormalization group approach,” Phys. Rev. E 77, 011105 (2008).
[CrossRef]

1996 (2)

D. L. Bosley, “A technique for the numerical verification of asymptotic expansions,” SIAM Rev. 38, 128-135 (1996).
[CrossRef]

L.-Yuan Chen, N. Goldenfeld, and Y. Oono, “Renormalization group and singular perturbations: multiple scales, boundary layers, and reductive perturbation theory,” Phys. Rev. E 54, 376-394 (1996).
[CrossRef]

1993 (1)

1992 (1)

1985 (1)

1978 (1)

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

1970 (1)

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

1969 (1)

Bender, C. M.

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers. I (Springer-Verlag, 1999).

Bosley, D. L.

D. L. Bosley, “A technique for the numerical verification of asymptotic expansions,” SIAM Rev. 38, 128-135 (1996).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 1992).

Chen, L.-Yuan

L.-Yuan Chen, N. Goldenfeld, and Y. Oono, “Renormalization group and singular perturbations: multiple scales, boundary layers, and reductive perturbation theory,” Phys. Rev. E 54, 376-394 (1996).
[CrossRef]

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, 1960).

Fiddy, M. A.

Goldenfeld, N.

L.-Yuan Chen, N. Goldenfeld, and Y. Oono, “Renormalization group and singular perturbations: multiple scales, boundary layers, and reductive perturbation theory,” Phys. Rev. E 54, 376-394 (1996).
[CrossRef]

N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, 1992).

Hoffman, M. J.

J. E. Marsden and M. J. Hoffman, Elementary Classical Analysis (Freeman1993).

Ishimaru, A.

A. Ishimaru, “Wave propagation and scattering in random media,” in IEEE/OUP Series on Electromagnetic Wave Theory (IEEE Press, 1997).

James Hadden, W.

W. James Hadden, Jr., and 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.

Kirkinis, E.

E. Kirkinis, “Reduction of amplitude equations by the renormalization group approach,” Phys. Rev. E 77, 011105 (2008).
[CrossRef]

E. Kirkinis, “The renormalization group and the implicit function theorem for amplitude equations,” J. Math. Phys. 49, 073518 (2008).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory. Course of Theoretical Physics, Vol. 3 of Addison-Wesley Series in Advanced Physics (Pergamon, 1958).

L. D. Landau and E. M. Lifshitz, “Electrodynamics of continuous media,” in Course of Theoretical Physics (Pergamon1960), Vol. 8.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, “Electrodynamics of continuous media,” in Course of Theoretical Physics (Pergamon1960), Vol. 8.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory. Course of Theoretical Physics, Vol. 3 of Addison-Wesley Series in Advanced Physics (Pergamon, 1958).

Lin, F. C.

Mano, K.

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

Marsden, J. E.

J. E. Marsden and M. J. Hoffman, Elementary Classical Analysis (Freeman1993).

Mintzer, D.

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

Oono, Y.

L.-Yuan Chen, N. Goldenfeld, and Y. Oono, “Renormalization group and singular perturbations: multiple scales, boundary layers, and reductive perturbation theory,” Phys. Rev. E 54, 376-394 (1996).
[CrossRef]

Oristaglio, M. L.

Orszag, S. A.

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers. I (Springer-Verlag, 1999).

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

J. Acoust. Soc. Am. (1)

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

J. Math. Phys. (1)

E. Kirkinis, “The renormalization group and the implicit function theorem for amplitude equations,” J. Math. Phys. 49, 073518 (2008).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Phys. Rev. E (2)

E. Kirkinis, “Reduction of amplitude equations by the renormalization group approach,” Phys. Rev. E 77, 011105 (2008).
[CrossRef]

L.-Yuan Chen, N. Goldenfeld, and Y. Oono, “Renormalization group and singular perturbations: multiple scales, boundary layers, and reductive perturbation theory,” Phys. Rev. E 54, 376-394 (1996).
[CrossRef]

Proc. IEEE (1)

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

SIAM Rev. (1)

D. L. Bosley, “A technique for the numerical verification of asymptotic expansions,” SIAM Rev. 38, 128-135 (1996).
[CrossRef]

Other (9)

J. E. Marsden and M. J. Hoffman, Elementary Classical Analysis (Freeman1993).

L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory. Course of Theoretical Physics, Vol. 3 of Addison-Wesley Series in Advanced Physics (Pergamon, 1958).

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, 1960).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, 1992).

L. D. Landau and E. M. Lifshitz, “Electrodynamics of continuous media,” in Course of Theoretical Physics (Pergamon1960), Vol. 8.

R. W. Boyd, Nonlinear Optics (Academic, 1992).

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers. I (Springer-Verlag, 1999).

A. Ishimaru, “Wave propagation and scattering in random media,” in IEEE/OUP Series on Electromagnetic Wave Theory (IEEE Press, 1997).

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Figures (8)

Fig. 1
Fig. 1

Monochromatic wave incident on a dielectric semi-infinite space at z 0 .

Fig. 2
Fig. 2

Upper graph: plot of the real part of the RG solution [Eq. (18)] with the transmission coefficient [Eq. (19)] together with the exact solution [Eq. (20)] and the Born approximation [Eq. (21)] versus the dimensionless parameter k z 0 . Lower graph: plot of the field amplitude ψ of the RG versus exact versus Born for k z 0 . Both graphs are plotted for the value v = 0.1 of the perturbing potential.

Fig. 3
Fig. 3

Monochromatic wave incident on a dielectric slab in the region 0 z d .

Fig. 4
Fig. 4

Log–log plot of the error for the transmission coefficients [Eq. (29)] versus the slab thickness d (i.e., the range of influence of the potential) for v = 10 3 .

Fig. 5
Fig. 5

Log–log plot of the error for the transmission coefficients [Eq. (29)] versus the perturbing parameter v for a finite value of the slab thickness d.

Fig. 6
Fig. 6

Monochromatic wave incident on a dielectric semi-infinite space at z 0 .

Fig. 7
Fig. 7

Upper graph: plot of the real part of the exact-RG solution [Eq. (35)] with the transmission coefficient [Eq. (38)] together with the Born approximation [Eq. (39)] versus the dimensionless parameter k r z 0 . Lower graph: plot of the field amplitude ψ of the exact RG together with the Born for k r z 0 . Both graphs are plotted for a perturbation parameter α = 0.1 .

Fig. 8
Fig. 8

Asymptotic validity of the RG solution [Eq. (18)] for the semi-infinite half-space problem of Subsection 3A. Log–log plot of the error versus the perturbing parameter v. The upper, middle, and lower curves correspond to the zeroth-, first-, and second-order solutions generated by the RG method.

Equations (82)

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A = A R + ϵ Z 1 ( A R , A R * , z ) + ϵ 2 Z 2 ( A R , A R * , z ) + O ( ϵ 3 ) ,
y ̃ ( ϵ ) = y 0 ( A ( ϵ ) , z ) + ϵ y 1 ( A ( ϵ ) , z ) + ϵ 2 y 2 ( A ( ϵ ) , z ) + ,
y ̃ ( ϵ ) = y 0 ( A ( ϵ ) , z ) ϵ = 0 + ϵ [ y 1 ( A ( ϵ ) , z ) + ϵ y 0 ( A ( ϵ ) , z ) ] ϵ = 0 + ϵ 2 [ y 2 ( A ( ϵ ) , z ) + ϵ y 1 ( A ( ϵ ) , z ) + 1 2 2 ϵ 2 y 0 ( A ( ϵ ) , z ) ] ϵ = 0 + O ( ϵ 3 ) .
d A R d z = { ϵ Z 1 z + ϵ 2 ( Z 2 z Z 1 A R Z 1 z Z 1 A R * Z 1 * z ) + O ( ϵ 3 ) } ,
d 2 ψ l ( z ) d z 2 + k 2 ψ l ( z ) = 0 , z < 0 ,
d 2 ψ r ( z ) d z 2 + k 2 ( 1 + v ) ψ r ( z ) = 0 , z 0 , v 0 + .
ψ l ( 0 ) = ψ r ( 0 ) , d ψ l ( 0 ) d z = d ψ r ( 0 ) d z ,
ψ l ( z ) = e i k z + R e i k z ,
ψ r ( z , v ) = ψ r 0 + v ψ r 1 + v 2 ψ r 2 + O ( v 3 ) ,
d 2 ψ r 0 ( z ) d z 2 + k 2 ψ r 0 ( z ) = 0 ,
d 2 ψ r n ( z ) d z 2 + k 2 ψ r n ( z ) = k 2 ψ r ( n 1 ) , n 1 .
ψ r 0 ( z ) = A e i k z ,
ψ r 1 ( z ) = i k 2 z A e i k z , ψ r 2 ( z ) = k 2 8 A z 2 e i k z i k 8 A z e i k z .
ψ r ( z ) = A ( 1 + v i 2 k z v 2 1 8 ( ( k z ) 2 + i k z ) ) e i k z ,
y ̃ ( A R , z , ϵ ) = { A R + v ( i 2 A R k z + Z 1 ) + v 2 ( 1 8 A R ( ( k z ) 2 + i k z ) + i 2 Z 1 k z + Z 2 ) + O ( v 3 ) } e i k z .
Z 1 = i k 2 z A R , Z 2 = k 2 8 A R z 2 + i k 8 A R z .
d A R d z = v i k 2 A R v 2 i k 8 A R + O ( v 3 ) ,
A R ( z ) = A R ( 0 ) e i k ( v 2 v 2 8 + O ( v 3 ) ) z .
ψ r ( z ) = A R ( 0 ) e i k ( 1 + v 2 v 2 8 + O ( v 3 ) ) z .
ψ l ( z ) = e i k z + 1 L 1 + L e i k z , ψ r ( z ) = 2 1 + L e i k L z .
R = k k ( 1 + v ) 1 2 k + k ( 1 + v ) 1 2 1 L 1 + L , T = 2 k k + k ( 1 + v ) 1 2 2 1 + L ,
ψ l ( z ) = e i k z + R e i k z , z < 0
ψ r ( z ) = T e i k ( 1 + v ) 1 2 z , z 0 ,
ψ l ( z ) = e i k z + 1 L 1 + L e i k z ,
ψ r ( z ) = 2 1 + L ( 1 + v i 2 k z v 2 1 8 ( ( k z ) 2 + i k z ) + O ( v 3 ) ) e i k z .
d 2 ψ l ( z ) d z 2 + k 2 ψ l ( z ) = 0 , z < 0 ,
d 2 ψ s ( z ) d z 2 + k 2 ( 1 + v ) ψ s ( z ) = 0 , 0 z d , v 0 + ,
d 2 ψ r ( z ) d z 2 + k 2 ψ r ( z ) = 0 , z > d ,
ψ l ( 0 ) = ψ s ( 0 ) , d ψ l ( 0 ) d z = d ψ s ( 0 ) d z ,
ψ s ( d ) = ψ r ( d ) , d ψ s ( d ) d z = d ψ r ( d ) d z ,
ψ l ( z ) = e i k z + C e i k z , z < 0 ,
ψ s ( z ) = A ( 0 ) e i k L z + B ( 0 ) e i k L z , 0 z d ,
ψ r ( z ) = E e i k z , z > d ,
ψ l ( z ) = e i k z + 2 i L 2 1 F sin ( k L d ) e i k z , z < 0 ,
ψ s ( z ) = 2 1 + L F e i k L ( z d ) 2 1 L F e i k L ( z d ) , 0 z d ,
ψ r ( z ) = 4 L F e i k ( z d ) , z > d ,
ψ l ( z ) = e i k z + R ( 1 exp ( 2 i k ( 1 + v ) 1 2 ) d ) D e i k z , z < 0 ,
ψ s ( z ) = T D e i k ( 1 + v ) 1 2 z R T exp ( 2 i k ( 1 + v ) 1 2 d ) D e i k ( 1 + v ) 1 2 z , 0 z d ,
ψ r ( z ) = T ( 1 R ) exp ( i ( k ( 1 + v ) 1 2 k ) d ) D e i k z , z > d ,
D = exp ( i k L d ) ( 1 + L ) 2 ( ( 1 + L ) 2 exp ( i k L d ) ( 1 L ) 2 exp ( i k L d ) ) exp ( i k L d ) ( 1 + L ) 2 F .
ψ l ( z ) = e i k z + C e i k z , z < 0 ,
ψ s ( z ) = A 1 ( 1 + v i k z 2 v 2 1 8 ( i k z + ( k z ) 2 ) + O ( v 3 ) ) e i k z + A 2 ( 1 v i k z 2 v 2 1 8 ( i k z + ( k z ) 2 ) + O ( v 3 ) ) e k z , 0 z d ,
ψ r ( z ) = B e i k z , z > d ,
T exact = T ( 1 R ) exp ( i k ( 1 + v ) 1 2 d ) D ,
T RG = 4 L F ,
T Born = A 1 L 1 e i k d A 2 L 2 e i k d .
T exact T RG n = O ( d v n + 1 ) , T exact T Born n = O ( d n + 1 v n + 1 ) .
ψ 0 = ln u 0 , ψ 1 = u ¯ 1 ,
ψ 2 = u ¯ 2 1 2 u ¯ 1 2 , ψ 3 = u ¯ 3 u ¯ 1 u ¯ 2 + 1 3 u ¯ 1 3 , .
d 2 ψ l ( z ) d z 2 + k 2 ψ l ( z ) = 0 , z < 0 ,
d 2 ψ r ( z ) d z 2 + k r 2 ( 1 + α ψ r ( z ) 2 ) 2 ψ r ( z ) = 0 , z 0 , α 0 + ,
ψ r ( z ) = ψ r 0 + α ψ r 1 + α 2 ψ r 2 + O ( α 3 ) ,
ψ r 0 ( z ) + k r 2 ψ r 0 ( z ) = 0 ,
ψ r 1 ( z ) + k r 2 ψ r 1 ( z ) = 2 k r 2 ψ r 0 2 ψ r 0 ,
ψ r 2 ( z ) + k r 2 ψ r 2 ( z ) = 2 k r 2 ( 2 ψ r 0 2 ψ r 1 + ψ r 0 2 ψ r 1 * ) k r 2 ψ r 0 4 ψ r 0 ,
ψ r 3 ( z ) + k r 2 ψ r 3 ( z ) = 2 k r 2 ( 2 ψ r 0 2 ψ r 2 + ψ r 0 * ψ r 1 2 + 2 ψ r 0 ψ r 1 2 + ψ r 0 2 ψ r 2 * ) k r 2 ( 3 ψ r 0 4 ψ r 1 + 2 ψ r 0 2 ψ r 0 2 ψ r 1 * ) ,
ψ r 0 = A e i k r z ,
ψ r 1 = i k r z A 2 A e i k r z ,
ψ r 2 = 1 2 ( k r z ) 2 A 4 A e i k r z ,
ψ r 3 = 1 6 i ( k r z ) 3 A 6 A e i k r z .
Z 1 = i k r z A 2 A ,
Z 2 = 1 2 ( k r z ) 2 A 4 A ,
Z 3 = 1 6 i ( k r z ) 3 A 6 A .
d A R ( z ) d z = α i k A R 2 A R .
ψ r ( z ) = T 0 e i k r ( 1 + α T 0 2 ) z , z 0 ,
ψ l ( z ) = e i k z + R 0 e i k z , z < 0 ,
1 + R 0 = T 0 , 1 R 0 = λ ( 1 + α T 0 2 ) T 0 .
T 0 = 1 3 N 1 3 α λ 1 + λ N 1 3 ,
ψ ( r ) = A ( 1 + α i k r z A 2 α 2 1 2 ( k r z ) 2 A 4 α 3 1 6 i ( k r z ) 3 A 6 + O ( α 4 ) ) e i k r z .
r N ( x ) u exact u N = O ( ϵ N + 1 ) .
ln r N = ( N + 1 ) ln ϵ + ln K ,
d A R d z = A R ϵ { d Z 1 p d z ϵ ( d Z 2 p d z Z 1 p d Z 1 p d z ) ϵ 2 ( d Z 3 p d z Z 1 p d Z 2 p d z Z 2 p d Z 1 p d z + Z 1 p 2 d Z 1 p d z ) + O ( ϵ 3 ) } ,
ln A R = ln A R ( 0 ) + ϵ y 1 p + ϵ 2 ( y 2 p y 1 p 2 2 ) + ϵ 3 ( y 3 p y 1 p y 2 p + y 1 p 3 3 ) + O ( ϵ 4 ) ,
d A R d z = A R ϵ { d y 1 p d z + ϵ ( d y 2 p d z y 1 p d y 1 p d z ) + ϵ 2 ( d y 3 p d z y 1 p d y 2 p d z y 2 p d y 1 p d z + y 1 p 2 d y 1 p d z ) + O ( ϵ 3 ) } ,
A 1 = 2 ( L 2 N 2 ) e i k d ( ( 1 + L ) ( L 2 + N 2 ) e i k d + ( 1 L ) ( L 1 N 1 ) e i k d ) ,
A 2 = 2 ( L 1 N 1 ) e i k d ( ( 1 + L ) ( L 2 + N 2 ) e i k d + ( 1 L ) ( L 1 N 1 ) e i k d ) ,
C = A 1 + A 2 1 ,
B = A 1 L 1 A 2 L 2 e 2 i k d ,
L 1 = 1 + v 1 2 ( i k d + 1 ) + v 2 1 8 ( i k d 1 ( k d ) 2 ) ,
N 1 = 1 + v 1 2 i k d + v 2 1 8 ( i k d ( k d ) 2 ) ,
L 2 = 1 + v 1 2 ( i k d + 1 ) + v 2 1 8 ( i k d 1 ( k d ) 2 ) ,
N 2 = 1 v 1 2 i k d + v 2 1 8 ( i k d ( k d ) 2 ) ,

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