I have three wavelengths. Each of them has a different colour. I want to plot the resulting wave with colour that corresponds to the mixture of these colour intensities at every point. I tried this code:
Plot[I1[y]+I2[y]+I3[y],{y,-0.0000001,0.0000001},ColorFunction ->colf]
colf[y_] := Blend[{{I1[y],Blue},{I2[y],Yellow},{I3[y],Red}}]
But it didn't work out. What's wrong?
This was harder than it should have been. But I learned some stuff along the way.
Let's first set up the functions:
f[l_, y_] = 1/2 + Cos[ Pi y/(l 10^-9)]/2;
Plot[{f[450, y], f[550, y], f[650, y]}, {y, -1 10^-6, 1 10^-6},
PlotStyle -> {Blue, Yellow, Red}]
Now let's make the mixture.
mix[y_] = f[450, y] + f[550, y] + f[650, y];
Plot[mix[y], {y, -1 10^-6, 1 10^-6}]
Now we will plot it using the appropriate color function, turning off ColorFunctionScaling.
Plot[mix[y], {y, -1 10^-6, 1 10^-6}, PlotStyle -> Thickness[0.02],
ColorFunction -> (Blend[{Blue, Yellow, Red}, {f[450, #1], f[550, #1],
f[650, #1]}] &), ColorFunctionScaling -> False]
Or perhaps when all three are low, you might want it to be closer to black (indicating that there isn't as much intensity), in which case:
Plot[mix[y], {y, -1 10^-6, 1 10^-6}, PlotStyle -> Thickness[0.02],
ColorFunction -> (Blend[{Blue, Yellow, Red, Black}, {f[450, #1],
f[550, #1], f[650, #1], 3 - mix[#1]}] &), ColorFunctionScaling -> False]
In debugging some of this, I set up a manipulate (shown below as an animate) to explore the mixing. It is kind of instructive.
Animate[Show[
Plot[{f[450, y 10^-6], f[550, y 10^-6], f[650, y 10^-6]},
{y, -1 , 1}, PlotStyle -> {Blue, Yellow, Red},
PlotRange -> {{-1, 1}, {0, 3.2}}, ImageSize -> Large],
Plot[mix[y 10^-6], {y, -1, 1}, PlotStyle -> Thickness[0.02],
ColorFunction -> (Blend[{Blue, Yellow, Red,
Black}, {f[450, #1 10^-6], f[550, #1 10^-6], f[650, #1 10^-6],
3 - mix[#1 10^-6]}] &), ColorFunctionScaling -> False],
Graphics[{Black, Thick, Circle[{-x , mix[-x 10^-6]}, {rx, ry}],
Circle[{x , mix[x 10^-6]}, {rx, ry}],
Line[{{-x - rx, mix[-x 10^-6]}, {-x + rx, mix[-x 10^-6]}}],
Line[{{-x, 0}, {-x , mix[-x 10^-6] + ry}}],
Line[{{x - rx, mix[x 10^-6]}, {x + rx, mix[x 10^-6]}}],
Line[{{x, 0}, {x , mix[x 10^-6] + ry}}],
Blend[{Blue, Yellow, Red, Black}, {f[450, -x 10^-6],
f[550, -x 10^-6], f[650, -x 10^-6], 3 - mix[x 10^-6]}],
Disk[{-0.5, 2.8}, {2 rx, 2 ry}], ,
Blend[{Blue, Yellow, Red, Black}, {f[450, x 10^-6],
f[550, x 10^-6], f[650, x 10^-6], 3 - mix[x 10^-6]}],
Disk[{0.5, 2.8}, {2 rx, 2 ry}],
Blend[{Black, Blue}, f[450, -x 10^-6]],
Disk[{-0.5, 1}, {rx, ry}],
Blend[{Black, Blue}, f[450, x 10^-6]],
Disk[{0.5, 1}, {rx, ry}],
Blend[{Black, Yellow}, f[550, -x 10^-6]],
Disk[{-0.5, 1.5}, {rx, ry}],
Blend[{Black, Yellow}, f[550, x 10^-6]],
Disk[{0.5, 1.5}, {rx, ry}],
Blend[{Black, Red}, f[650, -x 10^-6]],
Disk[{-0.5, 2}, {rx, ry}],
Blend[{Black, Red}, f[650, x 10^-6]],
Disk[{0.5, 2}, {rx, ry}]}]], {x, 0, 1}]
Or the version without Black (code modifications for this left as an exercise to the reader, unless requested):
This works:
Plot[I1[y] + I2[y] + I3[y], {y, -0.0000001, 0.0000001},
ColorFunctionScaling -> False,
ColorFunction -> (Blend[{Blue, Yellow, Red}, {I1[#], I2[#], I3[#]}] &)]
