Abstract

Multiple scattering of light by particles suspended in liquid can be a cause of degraded optical resolution. This is especially significant when the particles are relatively large and numerous, but most of the scattered power is deflected through very small angles, owing to a near match between the refractive indices of the particles and the medium. Formulas which convert the volume scattering function to an image modulation transfer function are derived for this case. This conversion allows calculation of image degradation, given the scattering properties of the medium. The inverse transform is also derived so that the scattering properties can be calculated, given the loss of resolution. The latter is best measured by imaging a set of paralled bar patterns having a wide range of spatial frequencies. The conversion from scattering function to optical transfer function is interpreted in general and evaluated for one published scattering function measured in sea water. The results are compared to published image measurements in fresh water.

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  1. In this paper, s instead of the usual σ denotes the VSF to avoid a notation difficulty in Appendix A.
  2. Small-angle-scattering meter built by Visibility Lab., Scripps Institute of Oceanography under Contract N62269-3097, NADC.
  3. J. W. Coltman, J. Opt. Soc. Am. 44, 468 (1954).
  4. This can be seen by an intuitive study of Eqs. (2) and (3). Since s(θ) is everywhere positive, it follows that u(θ) is everywhere positive and monotonically decreasing. The Bessel function has plus and minus oscillations that tend to cancel out during integration. As ψ increases, the first sign reversal occurs at lower values of θ, where it cancels larger values of u(θ).
  5. Proof: J0 → 1, Q → 2π∫0θ u (θ)dθ. Integrate by parts, differentiating u(θ) and integrating dθ. Q → 2π∫θs(θ)dθ ≍ 2π ∫ sinθs(θ)dθ = ∫s(θ)dω = st.
  6. S. Q. Duntley, W. H. Culver, F. Richey, and R. W. Preisen-dorfer, J. Opt. Soc. Am. 53, 351 (1963).
  7. C. B. Rogers, J. Opt. Soc. Am. 55, 1151 (1965).
  8. Tables of Integral Transforms (The Bateman Manuscript Project), A. Erdelyi, Ed. (McGraw—Hill Book Co., New York, 1954), Vol. II.
  9. Robert E. Morrison, doctoral thesis, Department of Meteorology and Oceanography, New York University (1967).
  10. That is, they are singular to the best of our ability to extrapolate. Either u(θ) or both s(θ) and u(θ) may turn at extremely small angles to approach a finite value. Singularities create no special problem so long as they are integrable to give finite total cross sections, as this one does.
  11. We are unaware of any special significance of θ½, but we did note that this power gives a significantly smoother plot of V near the origin than does the 0.47 or 0.53 power.
  12. We chose θ = 0.5 rad for this integration.
  13. F. S. Replogle and I. B. Steiner, J. Opt. Soc. Am. 55, 1149 (1965).
  14. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press Inc., New York, 1965), 4th ed.; also M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions. Natl. Bur. Std. (U. S.) Appl. Math. Ser. 55 (U. S. Gov't. Printing Office, Washington, D. C., 1964; Dover Publications, Inc., New York, 1965).

Coltman, J. W.

J. W. Coltman, J. Opt. Soc. Am. 44, 468 (1954).

Culver, W. H.

S. Q. Duntley, W. H. Culver, F. Richey, and R. W. Preisen-dorfer, J. Opt. Soc. Am. 53, 351 (1963).

Duntley, S. Q.

S. Q. Duntley, W. H. Culver, F. Richey, and R. W. Preisen-dorfer, J. Opt. Soc. Am. 53, 351 (1963).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press Inc., New York, 1965), 4th ed.; also M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions. Natl. Bur. Std. (U. S.) Appl. Math. Ser. 55 (U. S. Gov't. Printing Office, Washington, D. C., 1964; Dover Publications, Inc., New York, 1965).

Morrison, Robert E.

Robert E. Morrison, doctoral thesis, Department of Meteorology and Oceanography, New York University (1967).

Preisen-dorfer, R. W.

S. Q. Duntley, W. H. Culver, F. Richey, and R. W. Preisen-dorfer, J. Opt. Soc. Am. 53, 351 (1963).

Replogle, F. S.

F. S. Replogle and I. B. Steiner, J. Opt. Soc. Am. 55, 1149 (1965).

Richey, F.

S. Q. Duntley, W. H. Culver, F. Richey, and R. W. Preisen-dorfer, J. Opt. Soc. Am. 53, 351 (1963).

Rogers, C. B.

C. B. Rogers, J. Opt. Soc. Am. 55, 1151 (1965).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press Inc., New York, 1965), 4th ed.; also M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions. Natl. Bur. Std. (U. S.) Appl. Math. Ser. 55 (U. S. Gov't. Printing Office, Washington, D. C., 1964; Dover Publications, Inc., New York, 1965).

Steiner, I. B.

F. S. Replogle and I. B. Steiner, J. Opt. Soc. Am. 55, 1149 (1965).

Other (14)

In this paper, s instead of the usual σ denotes the VSF to avoid a notation difficulty in Appendix A.

Small-angle-scattering meter built by Visibility Lab., Scripps Institute of Oceanography under Contract N62269-3097, NADC.

J. W. Coltman, J. Opt. Soc. Am. 44, 468 (1954).

This can be seen by an intuitive study of Eqs. (2) and (3). Since s(θ) is everywhere positive, it follows that u(θ) is everywhere positive and monotonically decreasing. The Bessel function has plus and minus oscillations that tend to cancel out during integration. As ψ increases, the first sign reversal occurs at lower values of θ, where it cancels larger values of u(θ).

Proof: J0 → 1, Q → 2π∫0θ u (θ)dθ. Integrate by parts, differentiating u(θ) and integrating dθ. Q → 2π∫θs(θ)dθ ≍ 2π ∫ sinθs(θ)dθ = ∫s(θ)dω = st.

S. Q. Duntley, W. H. Culver, F. Richey, and R. W. Preisen-dorfer, J. Opt. Soc. Am. 53, 351 (1963).

C. B. Rogers, J. Opt. Soc. Am. 55, 1151 (1965).

Tables of Integral Transforms (The Bateman Manuscript Project), A. Erdelyi, Ed. (McGraw—Hill Book Co., New York, 1954), Vol. II.

Robert E. Morrison, doctoral thesis, Department of Meteorology and Oceanography, New York University (1967).

That is, they are singular to the best of our ability to extrapolate. Either u(θ) or both s(θ) and u(θ) may turn at extremely small angles to approach a finite value. Singularities create no special problem so long as they are integrable to give finite total cross sections, as this one does.

We are unaware of any special significance of θ½, but we did note that this power gives a significantly smoother plot of V near the origin than does the 0.47 or 0.53 power.

We chose θ = 0.5 rad for this integration.

F. S. Replogle and I. B. Steiner, J. Opt. Soc. Am. 55, 1149 (1965).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press Inc., New York, 1965), 4th ed.; also M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions. Natl. Bur. Std. (U. S.) Appl. Math. Ser. 55 (U. S. Gov't. Printing Office, Washington, D. C., 1964; Dover Publications, Inc., New York, 1965).

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