Abstract

A method based on the measurement of the time-integrated irradiance obtained in an interference experiment is described. This method is of particular utility in cases in which amplitude and phase or both are stochastic fluctuating variables. When only phase fluctuations are present or when amplitude and phase are stochastic independent variables, simple relations can be derived from which the parameters of the distribution can be derived. In these cases, if a gaussian-distribution law for the phase fluctuations is assumed, the mean-square value of the phase fluctuations is directly measured. The method has been applied to the study of phase fluctuations of a laser beam, caused by atmospheric turbulence, with results that agree with the findings of other techniques. The anisotropy of turbulence can also be evaluated.

© 1970 Optical Society of America

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References

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  1. M. Bertolotti, M. Carnevale, L. Muzii, and D. Sette, Appl. Opt. 7, 2246 (1968).
    [CrossRef] [PubMed]
  2. M. Bertolotti, M. Carnevale, L. Muzii, and D. Sette, Appl. Opt. 9, 510 (1970).
    [CrossRef]
  3. M. Bertolotti, M. Carnevale, B. Daino, and D. Sette, Appl. Opt. 9, 962 (1970).
    [CrossRef] [PubMed]
  4. J. D. Gaskill, J. Opt. Soc. Am. 59, 308 (1969).
    [CrossRef]
  5. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965).
  6. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill, New York, 1960).

1970 (2)

1969 (1)

1968 (1)

Bertolotti, M.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965).

Carnevale, M.

Daino, B.

Gaskill, J. D.

Muzii, L.

Sette, D.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill, New York, 1960).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965).

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Figures (7)

Fig. 1
Fig. 1

Schematic arrangement of the reversing-front interferometer. S is a semitransparent mirror, M a totally reflecting mirror, and P a roof prism. The fringes obtained are shown at (b).

Fig. 2
Fig. 2

Three photographs of fringes obtained from the reversing-front interferometer with an exposure of about 30 s. Cases of increasing turbulence are shown from (a) to (c).

Fig. 3
Fig. 3

A densitometric analysis of a photographic recording like that at (c) in Fig. 2. The ordinate shows the photographic density. The abscissa shows the x coordinate from the roof-prism edge.

Fig. 4
Fig. 4

Five recordings of log V=σ2 vs x for different turbulence conditions. The ordinates of curves B and C have been multiplied by a factor 10 to avoid confusion with the other curves. The dotted line represents the x5/3 curve.

Fig. 5
Fig. 5

Schematic arrangement of the reversing-front interferometer used for polar-anisotropy measurements. Notation is the same as Fig. 1(a). The roof prism P of Fig. 1(a) has now been replaced by a corner prism. At (b) are shown the fringes obtained. The straight lines α, β, and γ show the lines along which the density recordings referred to in the text as R, Re1, and Re2, respectively, have been made.

Fig. 6
Fig. 6

Two photographs of fringes obtained with the corner prism. (a) was made with stronger turbulence than (b).

Fig. 7
Fig. 7

Results from a photograph like those of Fig. 6. Circled points are densities recorded along the line α of Fig. 5(a). Solid dots are densities recorded along the line β of Fig. 5, and squares are densities along the line γ of Fig. 5. The points marked 45° along these two last recordings indicate the values of the polar coordinate R at which the angle made by the polar vector with the x axis of Fig. 5 is 45°. Dotted lines c′ and c″ represent the 5/3-slope lines that best fit the recordings Re1 and Re2 (c′), and R (c″), respectively.

Equations (20)

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V 0 = A 0 e i ( ω t ÷ ϕ 0 ) for the reference beam V = A ( x , y , t ) e i [ ω t + ϕ ( x , y , t ) ] for the distorted beam ,
I ( x , y , t ) = V 0 + V 2 = A 0 2 + A 2 ( x , y , t ) + 2 A 0 A ( x , y , t ) cos [ ϕ 0 - ϕ ( x , y , t ) - 2 π sin ϑ · x λ ] ,
I ( x , y ) = A 0 2 + + - { A 2 + 2 A 0 A cos [ Δ ϕ - 2 π sin ϑ · x λ ] × p ( A , Δ ϕ ) d A d ( Δ ϕ ) } .
p ( A , Δ ϕ ) d A d ( Δ ϕ ) = p ( A ) d A p ( Δ ϕ ) d ( Δ ϕ ) .
I ( x , y ) = A 0 2 + A 2 + 2 A 0 A × - + cos [ Δ ϕ - 2 π sin ϑ · x λ ] p ( Δ ϕ ) d ( Δ ϕ ) .
p ( Δ ϕ ) d ( Δ ϕ ) = σ - 1 ( 2 π ) - 1 2 exp { - ( Δ ϕ ) 2 2 σ 2 } d ( Δ ϕ ) ,
I ( x , y ) = A 0 2 + A 2 + 2 A 0 A × cos ( 2 π sin ϑ · x λ ) exp ( - σ 2 2 ) .
V = I max - I min I max + I min = 2 A 0 A e - σ 2 / 2 A 0 2 + A 2 .
V = e - σ 2 / 2
A = A 0 + Δ A ,
A = A 0 ; A 2 = A 0 2 + Δ A 2 .
V = exp ( - σ 2 / 2 ) / [ 1 + ( Δ A 2 / 2 A 0 2 ) ] .
I ( x , t ) = V ( x , t ) + V ( - x , t ) 2 = A 2 ( x , t ) + A 2 ( - x , t ) + 2 A ( x , t ) A ( - x , t ) × cos [ ϕ ( x , t ) - ϕ ( - x , t ) - 2 π sin ϑ · x λ ] .
I ( x ) = - + [ A 2 ( x ) + A 2 ( - x ) + 2 A ( x ) A ( - x ) × cos ( Δ ϕ - 2 π sin ϑ · x λ ) ] × p [ A ( x ) , A ( - x ) , Δ ϕ ] d [ A ( x ) ] d [ A ( - x ) ] d ( Δ ϕ ) ,
P [ A ( x ) , A ( - x ) , Δ ϕ ] d [ A ( x ) ] d [ A ( - x ) ] d ( Δ ϕ ) = P [ A ( x ) , A ( - x ) ] d [ A ( x ) ] d [ A ( - x ) ] × p ( Δ ϕ ) d ( Δ ϕ ) .
I ( x ) = 2 A 2 + 2 A ( x ) A ( - x ) × - + cos [ Δ ϕ - 2 π sin ϑ · x λ ] p ( Δ ϕ ) d ( Δ ϕ ) .
I ( x ) = 2 A 2 ( x ) + 2 A ( x ) A ( - x ) × cos ( 2 π sin ϑ λ x ) exp ( - σ 2 / 2 ) .
V = [ A ( x ) A ( - x ) / A 2 ( x ) ] exp ( - σ 2 ( x , - x ) / 2 ) .
I ( x ) = 2 A 2 [ 1 + cos ( 2 π sin ϑ λ x ) exp ( - σ 2 2 ) ]
V = exp ( - σ 2 ( x , - x ) / 2 ) .