Speckle Imaging
Laser speckle: objective and subjective speckle size, contrast from the rms roughness of a Gaussian surface, and contrast reduction by averaging uncorrelated patterns.
Objective speckle λz/D
21.28 µm
first zero 25.95 µm
Subjective speckle λ(1 + M)N
6.38 µm
first zero 7.79 µm
Contrast, one pattern
0.669
σ_φ = 4πσ_h/λ = 1.89 rad, N_c = 100
Contrast, 10 patterns
0.212
C/√N, uncorrelated patterns
Model
Objective speckle forms in free space at distance z from a uniformly lit spot of diameter D. Its intensity correlation is |2J₁(v)/v|² with v = πDr/(λz): λz/D sets the scale and the first zero is at 1.22λz/D. Subjective speckle is the same in an image, with the lens pupil in place of the spot: λ(1 + M)N. Quoted “speckle sizes” differ by these factors (and a diameter is twice the first-zero radius).
A Gaussian surface of rms height σ_h seen in reflection at normal incidence adds a phase of rms σ_φ = 4πσ_h/λ. A resolution cell holding N_c surface correlation cells sums N_c phasors: their mean adds coherently (intensity ∝ N_c² e^(−σ_φ²)) and the rest as a random walk (∝ N_c(1 − e^(−σ_φ²))). With r = N_c e^(−σ_φ²)/(1 − e^(−σ_φ²)) the Rician contrast is C = √(1 + 2r)/(1 + r) (Fujii & Asakura 1974). More cells keep a specular part up to a larger σ_φ: at N_c = 100, C = 0.76 at σ_φ = 2 rad. The form needs σ_φ ≳ 1 rad and overestimates C below. Away from the specular direction the specular part is absent and the speckle is fully developed (C = 1).
Averaging N uncorrelated patterns of equal mean (wavelength, angle or polarization diversity, or a moving diffuser) lowers the contrast to C/√N; partly correlated patterns lower it less.
Speckle contrast vs surface roughness
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Uniformly lit circular spot or pupil (|2J₁(v)/v|² correlation). Rician contrast of a Gaussian surface phase (normal incidence, specular direction) with N_c correlation cells per resolution cell, valid for σ_φ ≳ 1 rad; averaged patterns uncorrelated, equal mean.
- Goodman J. W. (2020). Speckle Phenomena in Optics: Theory and Applications, 2nd ed. SPIE Press. First-order statistics (ch. 3) and spatial structure of speckle (ch. 5). doi:10.1117/3.2548484
- Fujii H., Asakura T. (1974). Effect of surface roughness on the statistical distribution of image speckle intensity. Optics Communications 11, 35–38. doi:10.1016/0030-4018(74)90327-7