Max Ulbrich
Dec28-08, 05:00 AM
Hi,
I am stuck with a problem calculating the image that a single
fluorescent molecule will give on a camera when imaged through a
microscope. Yes, in theory this should be an Airy pattern, but I am also
looking for the pattern when out of focus.
1. I looked into the book from Born and Wolf (Principles of Optics). In
Fig. 8.6 in the 7th Edition, there is the explanation how the angular
intensity seen from a circular aperture in a far distance relates to the
image through a lens. But I don't know how I should transfer this
approach to out of focus images.
2. When a single molecule emits, it emits same intensity in all angles
(let's assume the emission has no angular preference as for a static
dipole but is an average over many orientations due to fast orientation
change of the molecule). When this light is collected by a lens with a
large aperture angle (I have an objective with NA 1.45), would there not
be lower intesity for rays that are far from the center of the lens?
3. However, when the light gets focused to form the image, they should
again have constant intensity per area for any incident ray, similar to
my sketch in http://www.max-ulbrich.com/Graphic1.jpg That means, I would
instead of integrating over the location of the aperture rather
integrate over the angle from the emitting moecule or the forming image,
no?
So what I wrote in Mathematica for the power at a given location is:
amp[x_, z_, ap_] :=
NIntegrate[
Sin[ph]*Cos[(x*Sin[th]*Sin[ph] + z*Cos[ph])],
{th, 0, 2Pi},
{ph, 0, ap*Pi/180}
]^2
where ph and th are the angle to be integrated over, and ap is the
aperture angle. x would be the distance from the focus in the plane of
the image, z the component perpendicular, i.e. towards the lens. To plot
this it takes a while, so I did the same in 2 dimensions:
amp[x_, z_, ap_] :=
NIntegrate[
Cos[(x*Sin[a] + z*Cos[a])],
{a, -ap*Pi/180, ap*Pi/180}
]^2
DensityPlot[amp[x, z, 30], {x, -20, 20}, {z, -20, 20},
Mesh -> False, PlotPoints -> 100];
The Plot shows that the intensity in a certain distance from the focal
plane drops, which I do not understand: Should not the power going
through any plane parallel to the image plane be the same? Also the
pattern does not seem right to me, for small aperture angles I would
expect an Airy pattern. Please help!
Thanks, Max
I am stuck with a problem calculating the image that a single
fluorescent molecule will give on a camera when imaged through a
microscope. Yes, in theory this should be an Airy pattern, but I am also
looking for the pattern when out of focus.
1. I looked into the book from Born and Wolf (Principles of Optics). In
Fig. 8.6 in the 7th Edition, there is the explanation how the angular
intensity seen from a circular aperture in a far distance relates to the
image through a lens. But I don't know how I should transfer this
approach to out of focus images.
2. When a single molecule emits, it emits same intensity in all angles
(let's assume the emission has no angular preference as for a static
dipole but is an average over many orientations due to fast orientation
change of the molecule). When this light is collected by a lens with a
large aperture angle (I have an objective with NA 1.45), would there not
be lower intesity for rays that are far from the center of the lens?
3. However, when the light gets focused to form the image, they should
again have constant intensity per area for any incident ray, similar to
my sketch in http://www.max-ulbrich.com/Graphic1.jpg That means, I would
instead of integrating over the location of the aperture rather
integrate over the angle from the emitting moecule or the forming image,
no?
So what I wrote in Mathematica for the power at a given location is:
amp[x_, z_, ap_] :=
NIntegrate[
Sin[ph]*Cos[(x*Sin[th]*Sin[ph] + z*Cos[ph])],
{th, 0, 2Pi},
{ph, 0, ap*Pi/180}
]^2
where ph and th are the angle to be integrated over, and ap is the
aperture angle. x would be the distance from the focus in the plane of
the image, z the component perpendicular, i.e. towards the lens. To plot
this it takes a while, so I did the same in 2 dimensions:
amp[x_, z_, ap_] :=
NIntegrate[
Cos[(x*Sin[a] + z*Cos[a])],
{a, -ap*Pi/180, ap*Pi/180}
]^2
DensityPlot[amp[x, z, 30], {x, -20, 20}, {z, -20, 20},
Mesh -> False, PlotPoints -> 100];
The Plot shows that the intensity in a certain distance from the focal
plane drops, which I do not understand: Should not the power going
through any plane parallel to the image plane be the same? Also the
pattern does not seem right to me, for small aperture angles I would
expect an Airy pattern. Please help!
Thanks, Max