Abstract

The implications of speckle statistics on laser eye-safety considerations are evaluated. The concept of speckled speckle is introduced, and its statistics are shown to correspond to the K0 function. Speckled speckle is defined in terms of the retinal power density when the eye is viewing an optically rough surface that is illuminated by a laser beam diffused through a ground-glass screen—a situation corresponding to subjective speckle modulated by objective speckle. Extensive numerical results are developed relating the ratio of the average power density on the retina over the eye-damage level to the acceptable probability that speckle statistics will cause the damage level to be exceeded. For ordinary speckle and for speckled speckle, for a probability of 10−6 (10−9) of exceeding the damage level, the average power densities must be 0.072 (0.048) and 0.017 (0.0079) of the damage level, respectively.

© 1981 Optical Society of America

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  1. The phrase “adequately monochromatic” implies that the laser wavelength spread Δλ has an associated frequency spread Δf= cΔλ/λ2(where c is the speed of light) whose wavelength Λ = c/Δf= λ2/Δλ is significantly greater than the rms surface roughness of the scattering object.
  2. The term “polarization diversity” carries the implication that the random phase shift associated with the scattering from the rough surface object is different for two orthogonal polarizations.
  3. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Topics in Applied Physics, Vol. 9, Laser Speckle and Related Phemomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975).
    [Crossref]
  4. E. Jakeman and P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP24, 806–814 (1976).
    [Crossref]
  5. We use the term “momentary” here to indicate that the ground-glass screen might move (or something else might change), so that the local objective power density on the object is well defined only for an instant.
  6. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), p. 376, Eq. (9.6.23).
  7. M. Abramowitz and L. A. Stegun, eds., Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), p. 379, Eq. (9.8.6).

1976 (1)

E. Jakeman and P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP24, 806–814 (1976).
[Crossref]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Topics in Applied Physics, Vol. 9, Laser Speckle and Related Phemomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975).
[Crossref]

Jakeman, E.

E. Jakeman and P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP24, 806–814 (1976).
[Crossref]

Pusey, P. N.

E. Jakeman and P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP24, 806–814 (1976).
[Crossref]

IEEE Trans. Antennas Propag. (1)

E. Jakeman and P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propag. AP24, 806–814 (1976).
[Crossref]

Other (6)

We use the term “momentary” here to indicate that the ground-glass screen might move (or something else might change), so that the local objective power density on the object is well defined only for an instant.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), p. 376, Eq. (9.6.23).

M. Abramowitz and L. A. Stegun, eds., Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), p. 379, Eq. (9.8.6).

The phrase “adequately monochromatic” implies that the laser wavelength spread Δλ has an associated frequency spread Δf= cΔλ/λ2(where c is the speed of light) whose wavelength Λ = c/Δf= λ2/Δλ is significantly greater than the rms surface roughness of the scattering object.

The term “polarization diversity” carries the implication that the random phase shift associated with the scattering from the rough surface object is different for two orthogonal polarizations.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Topics in Applied Physics, Vol. 9, Laser Speckle and Related Phemomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975).
[Crossref]

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Figures (1)

Fig. 1
Fig. 1

Speckle and speckled-speckle cumulative probability distributions. The solid line presents the speckled-speckle results, whereas the broken line is for ordinary speckle. The probability PT is the probability that the random power density will exceed some threshold value NT when the average power density is N ¯. For very small values of PT, the probability that the threshold will be exceeded, the speckled-speckle distribution places much more stringent requirements on the allowable average power density N ¯ than does the ordinary speckle distribution.

Equations (19)

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Prob ( N < n < N + d N ) = P Exp ( N ; N ¯ ) d N ,
P Exp ( N ; N ¯ ) = { 0 if N < 0 N ¯ - 1 exp ( - N / N ¯ ) if N 0 .
P T = N T d N N ¯ - 1 exp ( - N / N ¯ ) = exp ( - N T / N ¯ ) .
N ¯ = N T / [ - ln ( P T ) ]
D Ob R S / Ob ( λ / D L ) ,
R E / Ob / D E R S / Ob / D L .
CProb ( N < n < N + d N N ) = P Exp ( N ; N ) d N .
Prob ( N < N < N + d N ) = P Exp ( N ; N ¯ ) d N .
Prob ( N < n < N + d N ) = P K ( N ; N ¯ ) d N ,
P K ( N ; N ¯ ) = 0 d N P Exp ( N ; N ¯ ) P Exp ( N ; N ) .
P K ( N ; N ¯ ) = 0 d N N ¯ - 1 exp ( - N / N ¯ ) N - 1 exp ( - N / N ) .
P K ( N ; N ¯ ) = N ¯ - 1 0 d N N - 1 exp [ - ( N / N ¯ ) - ( N / N ) ] = N ¯ - 1 d x x - 1 exp [ - x - ( N / N ¯ ) x - 1 ] = N ¯ - 1 0 d y y - 1 exp [ - ( N / N ¯ ) 1 / 2 ( y + y - 1 ) ] = N ¯ - 1 - + d t exp [ - ( N / N ¯ ) 1 / 2 ( e t + e - t ) ] = N ¯ - 1 - + d t exp [ - 2 ( N / N ¯ ) 1 / 2 cosh ( t ) ] = 2 N ¯ - 1 0 d t exp [ - 2 ( N / N ¯ ) 1 / 2 cosh ( t ) ] .
K ν ( z ) = [ Γ ( 1 / 2 ) Γ ( ν + 1 / 2 ) ] ( 1 / 2 z ) ν × 0 d t exp [ - z cosh ( t ) ] sinh 2 ν ( t ) .
P K ( N ; N ¯ ) = 2 N ¯ - 1 K 0 [ 2 ( N / N ¯ ) 1 / 2 ] .
P T = N T d N P K ( N ; N ¯ ) = 2 N ¯ - 1 N T d N K 0 [ 2 ( N / N ¯ ) 1 / 2 ] .
K 0 ( 2 x ) ( 2 x ) - 1 / 2 exp ( - 2 x ) P ( x ) ,             for             x 1 ,
P ( x ) = 1.25331414 - 0.07832358 x - 1 + 0.02189568 x - 2 - 0.01062446 x - 3 + 0.00587872 x - 4 - 0.00251540 x - 5 + 0.00053308 x - 6 .
P T = 4 Q x d x K 0 ( 2 x ) = 2 3 / 2 Q d x x 1 / 2 exp ( - 2 x ) P ( x ) ,
Q = ( N T / N ) 1 / 2 .