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Mathematical Theory of Fishing Line Visibility

Mathematical Theory of Fishing Line Visibility. Is there such a thing as an “invisible” fishing line? Jeff Thomson. Invisible Fishing Line?. 2 types of line - monofilament & fluorocarbon (FC) FC has index of refraction near water’s Touted by manufacturers as “invisible”

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Mathematical Theory of Fishing Line Visibility

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  1. Mathematical Theory of Fishing Line Visibility Is there such a thing as an “invisible” fishing line? Jeff Thomson

  2. Invisible Fishing Line? • 2 types of line - monofilament & fluorocarbon (FC) • FC has index of refraction near water’s • Touted by manufacturers as “invisible” • Much more expensive than mono • How much less visible is FC than mono? • We will analyze scattering using Mie theory • Limit in which diameter of line is much larger than wavelength of light.

  3. Outline of Presentation • Simple cases suggest a strong dependence on difference of refraction index: • Reflection from plane • Reflection from slab • Mie theory analysis does not show this • Tabletop comparison of mono and FC line in water shows no observable differences • Approximate refraction theory agrees with Mie results gives insight into reason

  4. Simple Case - Reflection from Plane In region 1: k2 k1 In region 2: E z y k x Match E & B across x=0: B=-(k/w)E

  5. Plane Reflection continued • Solve for reflection coefficient R g Obviously a strong dependence on ratio of indexes of refraction

  6. More Complex - Reflection from a Slab In region 1: k2 k1 k1 In region 2: X=0 X=L In region 3: Now we apply BC at x=0 and X=L

  7. Slab - continued At x=0: R At x=L: k1L=10 Solving for r: g Also strong dependence on g

  8. Reflection from Cylinder • We will use Mie scattering theory and obtain an exact answer • Incident plane wave is expressed in cylindrical wave functions • Scattered and internal fields are also expressed in cylindrical wave functions • Boundary conditions are applied at edge of cylinder • I.e. tangential fields (Ez, Bq) are continuous

  9. Definition of Some Terms • J and Y are Bessel functions of the first kind • H(1) is the Hankel function of the first kind • The Wronskian is W(J,Y) = J(z)Y(z)/ - Y(z)J(z)/ = 2/pz • Fields - E electric, B magnetic

  10. Definition of fields r z q Incident field y z x Scattered field Internal field

  11. Continuity of Tangential Components At r=a: Solving for bn Using H(1)=J+iY, we can write this as

  12. Scattering Coefficient Differential intensity scattered R dz Integrating over q

  13. Scattering Coefficient - continued The intensity of the incident plane wave is The scattering coefficient is obtained by dividing the total Scattered light by the incident intensity on the cylinder:

  14. Why you can believe this Formal solution is the same as given in van de Hulst: Light Scattering by Small Particles Numerical solutions (using Mathematica) are identical with Van de Hulst’s s s r r

  15. Calculate s as function of g For large diameter/wavelength Scattering comes up Quickly, oscillates around Mean value s r = 10 g For visible light, need to integrate Over wavelength span, Oscillations will average out s Unless the radius is very small There will be no observable Difference between g=1.05 and 1.2 r = 100 g

  16. Why is This? • Cylinders do not collapse, in any limit, to slabs • All normally incident rays on slab will propagate with some reduction in intensity • Only one ray incident on a cylinder will do so, all others are bent, thus scattered • Even if g-1 is small, the path length is many wavelengths, so that the wave front is strongly perturbed

  17. Approximate Refraction Theory • Since g is near unity and cylinder is very large, can develop approximate theory • Ray is not deflected much entering and leaving • Phase change during passage is (g-1)L/l • Calculate scattered wave using Huygen’s Principle • Result agrees numerically with Mie theory • Criterion for “invisibility”: (g-1)L/l<1

  18. Effect of Salinity on Index of Refraction • For pure water, n = 1.33 • For FC, n=1.4, so g=1.05, i.e. 5% difference • Water n changes with wavelength, temperature, and salinity • Temperature effect very small • Over visible range, n changes by about .7% • Over salinity change of 0 - 40 ppt (fresh water to very salty) n changes by about .5% • At most g goes to 1.04, which is still too large

  19. Conclusion • Fluorocarbon does not live up to its advertising hype - it is not “invisible”

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