Abstract

We report some experimental results concerning the statistical properties of a light beam scattered by a rotating ground glass with average-size inhomogeneities of approximately 1 μm. Photocount statistics measured at different scattering angles and for different angular velocities of the ground glass have confirmed the known result that the scattered-light amplitude is a stochastic gaussian variable. The Bose–Einstein nature of the photocount statistics has been verified with an accuracy of a few parts per thousand. Self-beating measurements on the scattered light of a He–Ne laser in a TEM00 configuration have shown that the power spectrum is a gaussian function of the frequency. The dependence of its half-width on the angular velocity of the ground glass and on the focal length of the lens that focuses the beam on the scattering surface has been measured. The experimental results agree very closely with our theoretical predictions.

© 1971 Optical Society of America

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References

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  1. See, for example, A. Sommerfeld, Optics (Academic, New York, 1954), p. 196.
  2. W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964); F. T. Arecchi, Phys. Rev. Letters 15, 912 (1965).
    [CrossRef]
  3. F. T. Arecchi, M. Giglio, and U. Tartari, Phys. Rev. 163, 186 (1967); N. A. Clark, J. H. Lunacek, and G. B. Benedek, Am. J. Phys. 38, 575 (1970).
    [CrossRef]
  4. F. T. Arecchi, in Quantum Optics, edited by R. J. Glauber (Academic, New York, 1970).
  5. Two authoritative reviews on the problem of light scattering from thermodynamical fluctuations are I. L. Fabelinskii, Molecular Scattering of Light (Plenum, New York, 1968), and G. B. Benedek, in 9th Brandeis Summer Institute in Theoretical Physics (Gordon and Breach, New York, 1968).
    [CrossRef]
  6. N. C. Ford and G. B. Benedek, Phys. Rev. Letters 15, 649 (1965).
    [CrossRef]
  7. V. V. Anisimov, S. M. Kozel, and G. R. Lokshin, Opt. Spektrosk. 27, 258 (1969).
  8. The term 1/4σ2 of Eq. (16) is negligible compared to k2σ2/f2 for the range of focal lengths used in our experiments.
  9. See, for example, G. B. Benedek, Ref. 5.

1969 (1)

V. V. Anisimov, S. M. Kozel, and G. R. Lokshin, Opt. Spektrosk. 27, 258 (1969).

1967 (1)

F. T. Arecchi, M. Giglio, and U. Tartari, Phys. Rev. 163, 186 (1967); N. A. Clark, J. H. Lunacek, and G. B. Benedek, Am. J. Phys. 38, 575 (1970).
[CrossRef]

1965 (1)

N. C. Ford and G. B. Benedek, Phys. Rev. Letters 15, 649 (1965).
[CrossRef]

1964 (1)

W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964); F. T. Arecchi, Phys. Rev. Letters 15, 912 (1965).
[CrossRef]

Anisimov, V. V.

V. V. Anisimov, S. M. Kozel, and G. R. Lokshin, Opt. Spektrosk. 27, 258 (1969).

Arecchi, F. T.

F. T. Arecchi, M. Giglio, and U. Tartari, Phys. Rev. 163, 186 (1967); N. A. Clark, J. H. Lunacek, and G. B. Benedek, Am. J. Phys. 38, 575 (1970).
[CrossRef]

F. T. Arecchi, in Quantum Optics, edited by R. J. Glauber (Academic, New York, 1970).

Benedek, G. B.

N. C. Ford and G. B. Benedek, Phys. Rev. Letters 15, 649 (1965).
[CrossRef]

See, for example, G. B. Benedek, Ref. 5.

Fabelinskii, I. L.

Two authoritative reviews on the problem of light scattering from thermodynamical fluctuations are I. L. Fabelinskii, Molecular Scattering of Light (Plenum, New York, 1968), and G. B. Benedek, in 9th Brandeis Summer Institute in Theoretical Physics (Gordon and Breach, New York, 1968).
[CrossRef]

Ford, N. C.

N. C. Ford and G. B. Benedek, Phys. Rev. Letters 15, 649 (1965).
[CrossRef]

Giglio, M.

F. T. Arecchi, M. Giglio, and U. Tartari, Phys. Rev. 163, 186 (1967); N. A. Clark, J. H. Lunacek, and G. B. Benedek, Am. J. Phys. 38, 575 (1970).
[CrossRef]

Kozel, S. M.

V. V. Anisimov, S. M. Kozel, and G. R. Lokshin, Opt. Spektrosk. 27, 258 (1969).

Lokshin, G. R.

V. V. Anisimov, S. M. Kozel, and G. R. Lokshin, Opt. Spektrosk. 27, 258 (1969).

Martienssen, W.

W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964); F. T. Arecchi, Phys. Rev. Letters 15, 912 (1965).
[CrossRef]

Sommerfeld, A.

See, for example, A. Sommerfeld, Optics (Academic, New York, 1954), p. 196.

Spiller, E.

W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964); F. T. Arecchi, Phys. Rev. Letters 15, 912 (1965).
[CrossRef]

Tartari, U.

F. T. Arecchi, M. Giglio, and U. Tartari, Phys. Rev. 163, 186 (1967); N. A. Clark, J. H. Lunacek, and G. B. Benedek, Am. J. Phys. 38, 575 (1970).
[CrossRef]

Am. J. Phys. (1)

W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964); F. T. Arecchi, Phys. Rev. Letters 15, 912 (1965).
[CrossRef]

Opt. Spektrosk. (1)

V. V. Anisimov, S. M. Kozel, and G. R. Lokshin, Opt. Spektrosk. 27, 258 (1969).

Phys. Rev. (1)

F. T. Arecchi, M. Giglio, and U. Tartari, Phys. Rev. 163, 186 (1967); N. A. Clark, J. H. Lunacek, and G. B. Benedek, Am. J. Phys. 38, 575 (1970).
[CrossRef]

Phys. Rev. Letters (1)

N. C. Ford and G. B. Benedek, Phys. Rev. Letters 15, 649 (1965).
[CrossRef]

Other (5)

See, for example, A. Sommerfeld, Optics (Academic, New York, 1954), p. 196.

The term 1/4σ2 of Eq. (16) is negligible compared to k2σ2/f2 for the range of focal lengths used in our experiments.

See, for example, G. B. Benedek, Ref. 5.

F. T. Arecchi, in Quantum Optics, edited by R. J. Glauber (Academic, New York, 1970).

Two authoritative reviews on the problem of light scattering from thermodynamical fluctuations are I. L. Fabelinskii, Molecular Scattering of Light (Plenum, New York, 1968), and G. B. Benedek, in 9th Brandeis Summer Institute in Theoretical Physics (Gordon and Breach, New York, 1968).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Formation of laser spot on ground glass. B, incident laser beam; 1, (u,v) plane of lens; f, focal length of lens; z, distance between lens and ground glass; 2, (x1, y1) plane of ground glass; P, laser-beam propagation axis.

Fig. 2
Fig. 2

Experimental setup. (x1,y1), plane of ground glass; Rj, radius vector to jth microarea of ground glass; r, radius vector from center of laser spot to detector; P, laser-beam propagation axis; ψ, angle between P and r.

Fig. 3
Fig. 3

Photon-counting-system block diagram. L, laser; GG, ground glass; P, photomultiplier; A, wide-band amplifier; S, Lecroy 520A scaler; G1, Advanced Automation 2000 generator; G2, Hewlett-Packard 222A generator; I, computer-system interface; C, PDP-8/L.

Fig. 4
Fig. 4

Block diagram of photocurrent power-density-spectrum measurement apparatus. E(t), fluctuating field amplitude; P, photomultiplier; i(t), photocurrent; SA, spectrum analyzer; G(ω), photocurrent power spectrum; R, strip-chart recorder.

Fig. 5
Fig. 5

Experimental spectrum with theoretical gaussian data points. The peaks at 180 Hz are markers used to center the frequency spectrum. The peak at 0 Hz is due to the modulation carrier.

Fig. 6
Fig. 6

Δ ν 1 2 vs 1/f. v = 1.329 cm/s.

Fig. 7
Fig. 7

Δ ν 1 2 vs v. A, f = 10.15 cm; B, f = 11.30 cm; C, f = 19.39 cm.

Tables (3)

Tables Icon

Table I Independence of statistics with speed. f = 19.39 cm, ψ = 0.

Tables Icon

Table II Independence of statistics with focal length. v = 0.605 cm/s, ψ = 0.

Tables Icon

Table III Independence of statistics with angle. f = 19.39 cm, v = 0.605 cm/s.

Equations (22)

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E ( u , v ) = E 0 exp { - [ 1 / 4 σ 2 + i ( k / 2 f ) ] ( u 2 + v 2 ) } ,
E ( x 1 , y 1 ) = - i k 2 π z d u d v E ( u , v ) × exp { i k [ ( x 1 - u ) 2 + ( y 1 - v ) 2 + z 2 ] 1 2 } .
[ ( x 1 - u ) 2 + ( y 1 - v ) 2 + z 2 ] 1 2 z { 1 + 1 2 z 2 [ ( x 1 - u ) 2 + ( y 1 - v ) 2 ] }
E ( x 1 , y 1 ) = E 0 exp ( - ( k 2 / f - i k / 2 σ 2 ) ( x 1 2 + y 1 2 ) 4 z [ 1 / 4 σ 2 + ( i k / 2 ) ( 1 / f - 1 ) / z ] ) ,
E 0 = - i k 2 z E 0 e i k z 1 / 4 σ 2 + ( i k / 2 ) ( 1 / f - 1 / z ) .
E j ( r , t ) = E [ R j ( t ) ] s j exp { i [ φ j + k r - ω 0 t - k n ˆ R j ( t ) ] } ,
E ( r , t ) = j E j ( r , t ) = exp [ i ( k r - ω 0 t ) ] j s j e i φ j d 2 r 1 E ( r 1 ) e - i k n ˆ r 1 δ ( 2 ) [ r 1 - R j ( t ) ] ,
E ( r , t ) E * ( r , t + τ ) φ j , s j , R j = e i ω 0 τ j , l s j s l e i ( φ j - φ l ) d 2 r 1 d 2 r 1 E ( r 1 ) E * ( r 1 ) × e i k n ˆ ( r 1 - r 1 ) δ ( 2 ) [ r 1 - R j ( t ) ] δ ( 2 ) [ r 1 - R l ( t + τ ) ] .
E ( r , t ) E * ( r , t + τ ) = e i ω 0 τ s 2 d 2 r 1 d 2 r 1 E ( r 1 ) E * ( r 1 ) e - i k n ˆ ( r 1 - r 1 ) G ( r 1 , r 1 ) ,
G ( r 1 , r 1 ) = j δ ( 2 ) [ r 1 - R j ( t ) ] δ ( 2 ) [ r 1 - R j ( t + τ ) ] R j .
G ( r 1 , r 1 ) = j d 2 q 1 d 2 q 1 ( 2 π ) 4 × exp { i q 1 · ( r 1 - v t ) + i q 1 · ( r 1 - v ( t + τ ) } · exp [ - i ( q 1 + q 1 ) · R j ( 0 ) ] ,
exp [ - i ( q 1 + q 1 ) · R j ( 0 ) ] ( 2 π ) 2 A δ ( 2 ) ( q 1 + q 1 )
G ( r 1 , r 1 ) = ( N / A ) δ ( 2 ) ( r 1 - r 1 + v τ ) .
E ( r , t ) E * ( r , t + τ ) = N s 2 A exp [ i ( ω 0 + k · v ) τ ] · d 2 r 1 E ( r 1 ) E * ( r 1 + v τ ) ,
E ( r , t ) E * ( r , t + τ ) = 2 π σ 2 A N s 2 · E 0 2 e i ω 0 τ exp [ - v 2 τ 2 2 ( k 2 σ 2 f 2 + 1 4 σ 2 ) ] .
I ( ω ) = ( 2 π ) - 1 2 - d τ e - i ω τ E ( r , t ) E * ( r , t + τ ) = 2 π σ 2 A N s 2 E 0 2 [ v 2 ( k 2 α 2 / f 2 + 1 / 4 σ 2 ) ] 1 2 · exp ( - ( ω - ω 0 ) 2 2 v 2 ( k 2 σ 2 / f 2 + 1 / 4 σ 2 ) )
Δ ω 1 2 = ( 2 ln 2 ) 1 2 v ( k 2 σ 2 / f 2 + 1 / 4 σ 2 ) 1 2 .
Γ ( τ ) = i 0 2 + i 0 2 | γ ( τ ) γ ( 0 ) | 2 + G e i 0 δ ( τ ) ,
Γ ( τ ) = i ( t ) i ( t + τ ) , γ ( τ ) = E ( t ) E * ( t + τ ) ;
G ( ω ) = ( 2 π ) - 1 2 - d ω Γ ( τ ) e - i ω τ = ( 2 π ) 1 2 i 0 2 δ ( ω ) + G e i 0 ( 2 π ) - 1 2 + i 0 2 [ 2 v 2 ( k 2 σ 2 f 2 + 1 4 σ 2 ) ] - 1 2 × exp ( - ω 2 4 v 2 ( k 2 v 2 / f 2 + 1 / 4 σ 2 ) )
H 2 = ( n 2 - n - n 2 ) / n 2 H 3 = ( n 3 - 3 n 2 + 2 n - n 3 ) / n 3 .
Δ ν 1 2 = ( 2 v / λ f ) σ ( 2 ln 2 ) 1 2 .