Abstract

The functional relationship between the phase logarithm and the amplitude logarithm of a wave function near its real-plane zero point is found. This result takes the form of the dispersion relation that is deduced analytically and supported by the numerical simulation of the light-wave propagation in an inhomogeneous medium. The sufficient and necessary conditions of existence of this relationship are discussed, and their validity for infinite spectra is shown.

© 1998 Optical Society of America

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References

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  1. D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
    [CrossRef] [PubMed]
  2. N. N. Mayer, V. A. Tartakovski, “Phase dislocations and the minimum phase representation of wave function,” Atmos. Oceanic Opt. 8, 231–234 (1995).
  3. H. M. Nussenzweig, Causality and Dispersion Relations (Academic, New York, 1972).
  4. B. Ja. Levin, Distribution of Zeros of Entire Function (Gostekhizdat, Moscow, 1956), Translation of Vol. 5 of Mathematical Monographs (1964).
  5. J. Peřina, Coherence of Light, (Van Nostrand, London, 1972).
  6. R. E. Burge, M. A. Fiddy, F. H. Greenaway, G. Ross, “The application of dispersion relations (Hilbert transform) to the phase retrieval,” J. Phys. D 7, 165–168 (1974).
    [CrossRef]
  7. V. A. Tartakovski, “On the continuation of interferograms beyond the domain of definition,” Atmos. Oceanic Opt. 6, 898–901 (1993).
  8. N. I. Muschelishvili, Singular Integral Equations: Boundary Problems of the Function Theory and Some of Their Applications to Mathematical Physics (Nauka, Moscow, 1968) [Singulyarnye integralnye uravneniya. Granichnye zadachi teorii funktsyi i nekotorye ikh prilozheniya k matematicheskoi fizike (Nauka, Moskva, 1968), in Russian].
  9. R. C. Singleton, “An algorithm for computing the mixed radix fast Fourier transform,” IEEE Trans. Audio Electroacoust. AU-17, 93–103 (1969).
    [CrossRef]
  10. L. A. Weinstein, D. E. Wakmann, Frequency Separation in the Oscillation and Wave Theory (Nauka, Moscow, 1983) [Razdelenie chastot v teorii kolebanii i voln (Nauka, Moskva, 1983), in Russian].
  11. B. V. Fortes, V. P. Lukin, “Modeling of the image observed through turbulent atmosphere,” in Atmospheric Propagation and Remote Sensing, A. Kohnle, W. B. Miller, eds. Proc. SPIE1668, 477–488 (1992).
    [CrossRef]
  12. N. N. Mayer, V. A. Tartakovski, “Phase dislocations and focal spots,” Atmos. Oceanic Opt. 9, 1457–1461 (1996).

1996

N. N. Mayer, V. A. Tartakovski, “Phase dislocations and focal spots,” Atmos. Oceanic Opt. 9, 1457–1461 (1996).

1995

N. N. Mayer, V. A. Tartakovski, “Phase dislocations and the minimum phase representation of wave function,” Atmos. Oceanic Opt. 8, 231–234 (1995).

1993

V. A. Tartakovski, “On the continuation of interferograms beyond the domain of definition,” Atmos. Oceanic Opt. 6, 898–901 (1993).

1992

1974

R. E. Burge, M. A. Fiddy, F. H. Greenaway, G. Ross, “The application of dispersion relations (Hilbert transform) to the phase retrieval,” J. Phys. D 7, 165–168 (1974).
[CrossRef]

1969

R. C. Singleton, “An algorithm for computing the mixed radix fast Fourier transform,” IEEE Trans. Audio Electroacoust. AU-17, 93–103 (1969).
[CrossRef]

Burge, R. E.

R. E. Burge, M. A. Fiddy, F. H. Greenaway, G. Ross, “The application of dispersion relations (Hilbert transform) to the phase retrieval,” J. Phys. D 7, 165–168 (1974).
[CrossRef]

Fiddy, M. A.

R. E. Burge, M. A. Fiddy, F. H. Greenaway, G. Ross, “The application of dispersion relations (Hilbert transform) to the phase retrieval,” J. Phys. D 7, 165–168 (1974).
[CrossRef]

Fortes, B. V.

B. V. Fortes, V. P. Lukin, “Modeling of the image observed through turbulent atmosphere,” in Atmospheric Propagation and Remote Sensing, A. Kohnle, W. B. Miller, eds. Proc. SPIE1668, 477–488 (1992).
[CrossRef]

Fried, D. L.

Greenaway, F. H.

R. E. Burge, M. A. Fiddy, F. H. Greenaway, G. Ross, “The application of dispersion relations (Hilbert transform) to the phase retrieval,” J. Phys. D 7, 165–168 (1974).
[CrossRef]

Levin, B. Ja.

B. Ja. Levin, Distribution of Zeros of Entire Function (Gostekhizdat, Moscow, 1956), Translation of Vol. 5 of Mathematical Monographs (1964).

Lukin, V. P.

B. V. Fortes, V. P. Lukin, “Modeling of the image observed through turbulent atmosphere,” in Atmospheric Propagation and Remote Sensing, A. Kohnle, W. B. Miller, eds. Proc. SPIE1668, 477–488 (1992).
[CrossRef]

Mayer, N. N.

N. N. Mayer, V. A. Tartakovski, “Phase dislocations and focal spots,” Atmos. Oceanic Opt. 9, 1457–1461 (1996).

N. N. Mayer, V. A. Tartakovski, “Phase dislocations and the minimum phase representation of wave function,” Atmos. Oceanic Opt. 8, 231–234 (1995).

Muschelishvili, N. I.

N. I. Muschelishvili, Singular Integral Equations: Boundary Problems of the Function Theory and Some of Their Applications to Mathematical Physics (Nauka, Moscow, 1968) [Singulyarnye integralnye uravneniya. Granichnye zadachi teorii funktsyi i nekotorye ikh prilozheniya k matematicheskoi fizike (Nauka, Moskva, 1968), in Russian].

Nussenzweig, H. M.

H. M. Nussenzweig, Causality and Dispersion Relations (Academic, New York, 1972).

Perina, J.

J. Peřina, Coherence of Light, (Van Nostrand, London, 1972).

Ross, G.

R. E. Burge, M. A. Fiddy, F. H. Greenaway, G. Ross, “The application of dispersion relations (Hilbert transform) to the phase retrieval,” J. Phys. D 7, 165–168 (1974).
[CrossRef]

Singleton, R. C.

R. C. Singleton, “An algorithm for computing the mixed radix fast Fourier transform,” IEEE Trans. Audio Electroacoust. AU-17, 93–103 (1969).
[CrossRef]

Tartakovski, V. A.

N. N. Mayer, V. A. Tartakovski, “Phase dislocations and focal spots,” Atmos. Oceanic Opt. 9, 1457–1461 (1996).

N. N. Mayer, V. A. Tartakovski, “Phase dislocations and the minimum phase representation of wave function,” Atmos. Oceanic Opt. 8, 231–234 (1995).

V. A. Tartakovski, “On the continuation of interferograms beyond the domain of definition,” Atmos. Oceanic Opt. 6, 898–901 (1993).

Vaughn, J. L.

Wakmann, D. E.

L. A. Weinstein, D. E. Wakmann, Frequency Separation in the Oscillation and Wave Theory (Nauka, Moscow, 1983) [Razdelenie chastot v teorii kolebanii i voln (Nauka, Moskva, 1983), in Russian].

Weinstein, L. A.

L. A. Weinstein, D. E. Wakmann, Frequency Separation in the Oscillation and Wave Theory (Nauka, Moscow, 1983) [Razdelenie chastot v teorii kolebanii i voln (Nauka, Moskva, 1983), in Russian].

Appl. Opt.

Atmos. Oceanic Opt.

N. N. Mayer, V. A. Tartakovski, “Phase dislocations and focal spots,” Atmos. Oceanic Opt. 9, 1457–1461 (1996).

N. N. Mayer, V. A. Tartakovski, “Phase dislocations and the minimum phase representation of wave function,” Atmos. Oceanic Opt. 8, 231–234 (1995).

V. A. Tartakovski, “On the continuation of interferograms beyond the domain of definition,” Atmos. Oceanic Opt. 6, 898–901 (1993).

IEEE Trans. Audio Electroacoust.

R. C. Singleton, “An algorithm for computing the mixed radix fast Fourier transform,” IEEE Trans. Audio Electroacoust. AU-17, 93–103 (1969).
[CrossRef]

J. Phys. D

R. E. Burge, M. A. Fiddy, F. H. Greenaway, G. Ross, “The application of dispersion relations (Hilbert transform) to the phase retrieval,” J. Phys. D 7, 165–168 (1974).
[CrossRef]

Other

L. A. Weinstein, D. E. Wakmann, Frequency Separation in the Oscillation and Wave Theory (Nauka, Moscow, 1983) [Razdelenie chastot v teorii kolebanii i voln (Nauka, Moskva, 1983), in Russian].

B. V. Fortes, V. P. Lukin, “Modeling of the image observed through turbulent atmosphere,” in Atmospheric Propagation and Remote Sensing, A. Kohnle, W. B. Miller, eds. Proc. SPIE1668, 477–488 (1992).
[CrossRef]

N. I. Muschelishvili, Singular Integral Equations: Boundary Problems of the Function Theory and Some of Their Applications to Mathematical Physics (Nauka, Moscow, 1968) [Singulyarnye integralnye uravneniya. Granichnye zadachi teorii funktsyi i nekotorye ikh prilozheniya k matematicheskoi fizike (Nauka, Moskva, 1968), in Russian].

H. M. Nussenzweig, Causality and Dispersion Relations (Academic, New York, 1972).

B. Ja. Levin, Distribution of Zeros of Entire Function (Gostekhizdat, Moscow, 1956), Translation of Vol. 5 of Mathematical Monographs (1964).

J. Peřina, Coherence of Light, (Van Nostrand, London, 1972).

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Figures (6)

Fig. 1
Fig. 1

Normalized rms error ∊ϕ of the minimum phase, calculated by the dispersion relation for the zero function W 0(x, y) = x + (a + ib)y + g on a circumference, versus the distance of the zero-function displacement in the xy plane. Gray-scale pictures show squared amplitudes of the zero-function at g = 0. The circumference is signed by inverse contrast; r c is its radius. (a) a = 0.01, b = 1; (b) a = 0.1, b = 10. Displacement of the zero function: along the x direction (circles), along the y direction (dashed curve), along the bisector of the xy plane (solid curve).

Fig. 2
Fig. 2

Components of the zero function W 0(x, y) = x + (a + ib)y at a = 0.01 and at b = 1 on a circumference, centered at (x = 0, y = 0), versus the azimuth angle ϑ. (a) Im is the imaginary part, Re is the real part, and A is the amplitude of the zero function. (b) ϕ is the phase in radians, and ln A is the amplitude logarithm of the zero function. (c) Modulus of the Fourier transform of the zero function. (d) Modulus of the Fourier transform of the logarithm of the zero function.

Fig. 3
Fig. 3

Components of the zero-function W 0(x, y) = x + (a + ib)y at a = 0.1 and b = 10 on a circumference, centered at (x = 0, y = 0), versus the azimuth angle ϑ. (a) Im is the imaginary part, Re is the real part, and A is the amplitude of the zero function. (b) ϕ is the phase in radians, and ln A is the amplitude logarithm of the zero function. (c) Modulus of the Fourier transform of the zero function. (d) Modulus of the Fourier transform of the logarithm of the zero function.

Fig. 4
Fig. 4

Schematic of the numerical experiment. Alignment from left to right: light source, inhomogeneous medium, first lens, focal-plane diaphragm, second lens, and plane of measurements. Two sizes (D = 1, D = 3) of diaphragm are used. Here r 0 is the Fried coherence radius, L is the path length, and r c is the radius of the circumference where the analysis is made.

Fig. 5
Fig. 5

The wave sample with the real-plane zero in the inhomogeneous medium. (a) Squared wave amplitude near the real-plane zero point at the center (pluses) of subaperture; (b) wave phase near the real-plane zero point at the center of subaperture; focal spots (c), (e), (g) and their phases (d), (f), (h), accordingly, formed by the wave (a), (b) when the subaperture size is increased.

Fig. 6
Fig. 6

The samples of the wave with the real-plane zeros (pluses) to check the dispersion relation when the light wave propagates in randomly inhomogeneous medium as shown in Fig. 4. (a) is the squared wave amplitude, and (b) is the wave phase for D = 1. (c) is the squared wave amplitude, and (d) is the wave phase for D = 3.

Tables (1)

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Table 1 Support of the Dispersion Relation by Numerical Simulationa

Equations (35)

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χ ζ = ln | G ζ | = η π - ln | G s | s - x 2 + η 2 d s + c η ,
c = lim η sup   ln | G 0 + i η | η
- ln | G x | 1 + x 2 d x < .
χ x = ϕ η ,     χ η = - ϕ x ,
d ϕ x ,   η = χ x ,   η η d x - χ x ,   η x d η .
χ x ,   η x = 2 η π - x - s χ s s - x 2 + η 2 - 2 d s , χ x ,   η η = 2 η 2 π -   χ s s - x 2 + η 2 - 2 d s + 1 π -   χ s × s - x 2 + η 2 - 1 d s + c .
ϕ x = 1 2 π   pv   - χ s x - s d s + l x = H x χ x + l x ,
d s x - s 1 2 cot ϑ - θ 2 d θ + i 2 d θ ,
ϕ ϑ = 1 2 π 0 2 π   χ θ cot ϑ - θ 2 d θ + 1 2 π 0 2 π   χ θ d θ + l ϑ .
U x = -   S α exp i α x d α , U s x = - s s   S α exp i α x d α , -   | S α | d α = M .
V s x = - U s y x - y d y = - d y x - y - s s   S α   exp i α y d α = - s s   i   sgn   α S α   exp i α x d α < M .
| U 2 x - U s 2 x | = | U x - U s x | × | U x + U s x | < 2 M ,
2 | χ - χ s | = ln 1 - | G | 2 - | G s | | G | 2 ,
lim s exp χ + i ϕ exp χ s + i ϕ s = exp lim s χ - χ s + i ϕ - ϕ s = 1 .
G x = exp   R x = 1 + k = 1 R k x k ! = 1 + W x
| G x | 2 = 1 + | W x | 2 + W x + W * x > 0 .
ln   G x = ln 1 + W x = k = 1 - 1 k + 1 W k x k !
ln   G x A 0 ,   α 0 > 0 | ln   G x | <   1 + W x = G x A 0 ,   | G x | > 0 W x A 0 ,   | W x | < 1 ,   α 0 > 0 ;
G x = k = n N   c k   exp i 2 π k T   x         c n exp - i 2 π n T   x + k = 1 N - n   c k + n   exp i 2 π k T   x   =   c n   +   W x .
W x ,   y = p R x - x k + q R y - y k + i p I x - x k + q I y - y k .
W x ,   y = exp χ x ,   y + i ϕ x ,   y = x - x k + a + ib y - y k = x - x k + a y - y k 2 + b 2 y - y k 2 1 / 2 × exp   i   arctan b y - y k x - x k + a y - y k .
χ x ,   y = ln x - x k + a y - y k 2 + b 2 y - y k 2 1 / 2 / x x - x k + a y - y k x - x k + a y - y k 2 + b 2 y - y k 2 H x b y - y k x - x k + a y - y k 2 + b 2 y - y k 2 d x arctan b y - y k x - x k + a y - y k + c k ,
ϕ x ,   y =   H x χ x ,   y x d x + c k .
W x ,   y = p R x + q R y + i p I x + q I y + g   y = r   sin   ϑ x = r   cos   ϑ   r [ p   exp i ϑ + q   exp - i ϑ ] + g gr - 1   p q exp i ϑ + exp - i ϑ +   g qr   exp i ϑ   1   +   g qr exp i ϑ +   p q exp i 2 ϑ   =   W o ϑ ,   r .
ϕ ϑ = H ϑ χ ϑ   ±   ϑ   +   c
ϕ = i , j C ϕ 0 i ,   j - ϕ ̑ 0 i ,   j - l c i ,   j 2 ϕ 0 i ,   j - l c i ,   j 2 1 / 2 ,
ϕ ϑ = arctan b   sin   ϑ cos   ϑ + a   sin   ϑ ,
d ϕ d ϑ = b b   sin   ϑ 2 + cos   ϑ + a   sin   ϑ 2 .
d ϕ d ϑ   | W ϑ ,   ρ | 2 = b ρ 2 ,
b = 1 r 2 d ϕ d ϑ   | W ϑ ,   r | 2 ,     a = b   cot   ϕ ϑ - cot   ϑ .
G x ,   y = x + a + ib y exp - x 2 + y 2 2 c 2 .
exp - x 2 2 c 2 F 2 π   c   exp - c 2 α 2 2 , x   exp - x 2 2 c 2 F   i 2 π   c 3 α   exp - c 2 α 2 2 ,
S α ,   β = i 2 π c 4 α + a + ib β exp - α 2 + β 2 c 2 2 ,
F s κ = 0.489   r 0 - 5 / 3 κ 2 + κ 0 2 - 11 / 6 ,     κ 0 2 = 2 π L o , r 0 = 0.423   k 2 L   C n 2 l d l - 3 / 5 ,
= 1 2 π 1 N c i , j C ϕ i ,   j - ϕ ̑ i ,   j - l c i ,   j 2 1 / 2 ,

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