Invisible Fishing Line?
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Transcript Invisible Fishing Line?
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”
– 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.
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
Simple Case - Reflection
from Plane
In region 1:
Ee
ik1 x
k2
k1
ik1 x
re
In region 2:
E te
ik2 x
Match E & B across x=0:
k2
1 r t,1 r t
k1
E
z
k
B=-(k/w)E
y
x
Plane Reflection continued
Solve for reflection coefficient
k2
1
,r
k1
1
R
1 2
R rr*
1
Obviously a strong dependence on ratio of
indexes of refraction
More Complex - Reflection from
a Slab
In region 1:
Ee
ik1 x
k1
re
E ae
k1
ik1 x
In region 2:
ik2 x
k2
be
ik2 x
X=0
X=L
In region 3:
E te
ik1 x
Now we apply BC at x=0 and
X=L
Slab - continued
At x=0:
1 r a b,1 r (a b)
R
At x=L:
ae
ik2 L
be
(ae
ik2 L
ik2 L
ik 2 L
be
te
ik1 L
) te
ik1 L
k1L=10
Solving for r:
( 1) ( 1)e i2k 2 L
r 2
2
2 i 2k2 L 1
( 1) ( 1) e
Also strong
dependence on
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
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
Definition of fields
Incident field
y
z
Ez eik1 x e ik1 r cosq i n Jn k1 rein q
x
n
Br
Bq
r
q
z
1 Ez
1
i n nJ n k1 re inq
iw r q
wr n
1 Ez k1
i n Jn ' k1r ein q
iw r i w n
Ez i nn Hn(1) k1r ein q
Scattered field
n
Bq
Internal field
k1
iw
i H ' k re
n
n
1
n
Ez i n n Jn k2 r ein q
n
n
inq
Bq
k2
iw
i J k re
n
n
n
n
2
inq
Continuity of Tangential
Components
At r=a:
Jn (k1 a) n Hn (k1a) n Jn (k2 a)
Jn ' (k1 a) n Hn ' (k1a) n Jn ' (k 2 a)
Solving for n
n
Jn ' ()Jn () J n ' ( )Jn ()
Jn ' ()Hn () Hn ' ( )Jn ()
Using H(1)=J+iY, we can write this as
n
1
J ' ( )Yn ( ) Yn ' ( )J n ( )
,a n
1 ia
J n ' ( )J n ( ) Jn ' ( )Jn ( )
Scattering Coefficient
Differential intensity scattered
dI dzR dqr •
0
Integrating over q
2p
ExB
R
dz
*
k
n
in q
n
in q
1
dI dzR dq Re
i
H
k
r
e
i
H
'
k
r
e
n
n
1
n
n
1
iw n
n
0
2p
dI 2p k1 R
2
*
Im
H
k
R
H
'
k
R
n
n 1
n
1
dz
w
Im Hn k1 RHn ' k1 R Jn k1 R Yn ' k1 R Yn k1 R J n ' k1 R W(J n ,Yn )
*
dI
4
dz w
2
n
2
pk1 R
Scattering Coefficient - continued
The intensity of the incident plane wave is
dI0 2adz
k1
w
The scattering coefficient is obtained by dividing the total
Scattered light by the incident intensity on the cylinder:
2
2
(
,
k
a)
n 1
k1a
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
1.25
1.5
Calculate as function of
For large
diameter/wavelength
Scattering comes up
Quickly, oscillates around
Mean value
= 10
For visible light, need to integrate
Over wavelength span,
Oscillations will average out
Unless the radius is very small
There will be no observable
Difference between =1.05
and 1.2
= 100
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 -1 is small, the path length is many
wavelengths, so that the wave front is strongly
perturbed
Approximate Refraction Theory
Since 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 (-1)L/l
– Calculate scattered wave using
L
Huygen’s Principle
Result agrees numerically
with Mie theory
Criterion for “invisibility”: (-1)L/l<1
Effect of Salinity on Index of
Refraction
For pure water, n = 1.33
For FC, n=1.4, so =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 goes to 1.04, which is still too large
Conclusion
Fluorocarbon does not live up to its
advertising hype - it is not “invisible”