How to correct z distance in RI mismatch?

[G. Esteban Fernandez]

Hi everyone,

A user needs to measure the volume of fluid inclusions (mostly water)
embedded in a clear mineral of RI ~2.4.  We did this by collecting
z-stacks of reflected light on our LSM 510 confocal and rendering a 3D
model, which turned out pretty reasonable.  We used an oil-immersion
63x Plan-Apo and oil on either side of the coverslip.

I know that the z distances are not accurate because of the grossly
different RIs but I'm not sure how to correct them.  Can I just
multiply the z step size by an n2/n1 factor to get the real Δz (with
different factors for oil/mineral and mineral/water)?  In the
literature I've seen less simplistic correction models that seem more
accurate but no practical ways to implement them that I could decipher
were presented.  Any help would be much appreciated.

Thanks,
Esteban

--
G. Esteban Fernandez, Ph.D.
Associate Director
Molecular Cytology Research Core Facility
University of Missouri
1201 E. Rollins St.
Columbia, MO 65211

573-882-4895
573-884-9676 fax

http://www.biotech.missouri.edu/mcc/

[Kevin Braeckmans]

Search the CONFOCAL archive at
http://listserv.acsu.buffalo.edu/cgi-bin/wa?S1=confocal

Dear Esteban,

A simple ray-tracing diagram in combination with Snell's law will readily
give you the following formula:

Dz = Dz_motor * sqrt((n2^2 - NA^2)/(n1^2 - NA^2))

Where Dz is the actual axial displacement in the sample with refractive
index n2, Dz_motor is the axial displacement as indicated by the microscope
focus motor, n1 is the refractive index of the immersion medium and NA is
the numerical aperture of the objective lens.

Note that, as expected, for a perfectly matched system (n1=n2): Dz =
Dz_motor. Also, for small NA (NA<<n1 & n2) this leads to the much used
approximate correction n2/n1 as you have mentioned in your message. It is
not valid for large NA though.

I have used this formula with good results in my own studies. I seem to
remember there has been a publication during the 90's which compares this
formula with a more rigorous treatment starting from wave optics. I cannot
seem to find the reference, though. Maybe someone else on this list can ...

Hope this helps.

Kevin



Kevin Braeckmans, Ph.D.
Lab. General Biochemistry and Physical Pharmacy
Ghent University
Harelbekestraat 72
9000 Ghent
Belgium
Tel: [phone]
Fax: [phone]

> -----Oorspronkelijk bericht-----
> Van: Confocal Microscopy List [mailto:[hidden email]]
> Namens G. Esteban Fernandez
> Verzonden: vrijdag 15 februari 2008 0:11
> Aan: [hidden email]
> Onderwerp: How to correct z distance in RI mismatch?
>
> Hi everyone,
>
> A user needs to measure the volume of fluid inclusions (mostly water)
> embedded in a clear mineral of RI ~2.4.  We did this by collecting
> z-stacks of reflected light on our LSM 510 confocal and rendering a 3D
> model, which turned out pretty reasonable.  We used an oil-immersion
> 63x Plan-Apo and oil on either side of the coverslip.
>
> I know that the z distances are not accurate because of the grossly
> different RIs but I'm not sure how to correct them.  Can I just
> multiply the z step size by an n2/n1 factor to get the real Δz (with
> different factors for oil/mineral and mineral/water)?  In the
> literature I've seen less simplistic correction models that seem more
> accurate but no practical ways to implement them that I could decipher
> were presented.  Any help would be much appreciated.
>
> Thanks,
> Esteban
>
> --
> G. Esteban Fernandez, Ph.D.
> Associate Director
> Molecular Cytology Research Core Facility
> University of Missouri
> 1201 E. Rollins St.
> Columbia, MO 65211
>
> 573-882-4895
> 573-884-9676 fax
>
> http://www.biotech.missouri.edu/mcc/

[G. Esteban Fernandez]

Thanks for your response Kevin.

How does one deal with the imaginary term for the case of an oil
immersion lens with an aqueous specimen, i.e. when NA > n2, can the
sqrt(-1) just be thrown out?  The numbers seem to work out that way.

Esteban


2008/2/15 Kevin Braeckmans <[hidden email]>:

> Search the CONFOCAL archive at
> http://listserv.acsu.buffalo.edu/cgi-bin/wa?S1=confocal
>
> Dear Esteban,
>
> A simple ray-tracing diagram in combination with Snell's law will readily
> give you the following formula:
>
> Dz = Dz_motor * sqrt((n2^2 - NA^2)/(n1^2 - NA^2))
>
> Where Dz is the actual axial displacement in the sample with refractive
> index n2, Dz_motor is the axial displacement as indicated by the microscope
> focus motor, n1 is the refractive index of the immersion medium and NA is
> the numerical aperture of the objective lens.
>
> Note that, as expected, for a perfectly matched system (n1=n2): Dz =
> Dz_motor. Also, for small NA (NA<<n1 & n2) this leads to the much used
> approximate correction n2/n1 as you have mentioned in your message. It is
> not valid for large NA though.
>
> I have used this formula with good results in my own studies. I seem to
> remember there has been a publication during the 90's which compares this
> formula with a more rigorous treatment starting from wave optics. I cannot
> seem to find the reference, though. Maybe someone else on this list can ...
>
> Hope this helps.
>
> Kevin
>
>
>
> Kevin Braeckmans, Ph.D.
> Lab. General Biochemistry and Physical Pharmacy
> Ghent University
> Harelbekestraat 72
> 9000 Ghent
> Belgium
> Tel: [phone]
> Fax: [phone]
>
> > -----Oorspronkelijk bericht-----
> > Van: Confocal Microscopy List [mailto:[hidden email]]
> > Namens G. Esteban Fernandez
> > Verzonden: vrijdag 15 februari 2008 0:11
> > Aan: [hidden email]
> > Onderwerp: How to correct z distance in RI mismatch?
>
> >
> > Hi everyone,
> >
> > A user needs to measure the volume of fluid inclusions (mostly water)
> > embedded in a clear mineral of RI ~2.4.  We did this by collecting
> > z-stacks of reflected light on our LSM 510 confocal and rendering a 3D
> > model, which turned out pretty reasonable.  We used an oil-immersion
> > 63x Plan-Apo and oil on either side of the coverslip.
> >
> > I know that the z distances are not accurate because of the grossly
> > different RIs but I'm not sure how to correct them.  Can I just
> > multiply the z step size by an n2/n1 factor to get the real Δz (with
> > different factors for oil/mineral and mineral/water)?  In the
> > literature I've seen less simplistic correction models that seem more
> > accurate but no practical ways to implement them that I could decipher
> > were presented.  Any help would be much appreciated.
> >
> > Thanks,
> > Esteban
> >
> > --
> > G. Esteban Fernandez, Ph.D.
> > Associate Director
> > Molecular Cytology Research Core Facility
> > University of Missouri
> > 1201 E. Rollins St.
> > Columbia, MO 65211
> >
> > 573-882-4895
> > 573-884-9676 fax
> >
> > http://www.biotech.missouri.edu/mcc/
>

--
G. Esteban Fernandez, Ph.D.
Associate Director
Molecular Cytology Research Core Facility
University of Missouri
1201 E. Rollins St.
Columbia, MO 65211

573-882-4895
573-884-9676 fax

http://www.biotech.missouri.edu/mcc/

[Kevin Braeckmans]

Search the CONFOCAL archive at
http://listserv.acsu.buffalo.edu/cgi-bin/wa?S1=confocal

Hi Esteban,

No, in that case (n2<n1) it follows from Snell's law that total reflection
will occur for light rays above an angle arcsin(n2/n1), which is about 61
degrees going from oil (n2 = 1.52) to water (n1 = 1.33).

The thing is, the formula mentioned before does not take into account
spherical aberration which means that, if n1 differs from n2, light rays
with different angles are focused to a different point along the optical
axis. In your case, going from water to oil, the most oblique angles will be
focused closer to the cover glass compared to the light rays with smaller
angles. To see this more clearly, replace NA by its definition n1*sin(t1)
and plot the formula as a function of t1. Then you will see that the focal
point decreases rapidly for angles above 40 degrees. The effect becomes more
dramatic when focusing deeper into the sample (i.e. increasing Dz_motor) or
by increasing the difference between n1 and n2.

In short, for a large difference in refractive index and imaging deep into
the sample with a high NA objective lens, there is not a one on one relation
anymore between Dz and Dz_motor.  In this case I think one would need a more
rigorous treatment with wave optics to model Dz vs Dz_motor.

I am quite sure this has been done before. Can someone on this list give a
suitable reference?

Alternatively, as a rough estimate one could maybe use the standard formula
for an angle t1/2 which indicates half of the volume of the illumination
cone. Then half of the light will be focused above the focal point of t1 and
half of the light below that. The volume of a cone of height h and angle t1
is:

1/3 * pi * h^3 * tan(t1)

So, half the volume is found for an angle t1/2 = arctan(0.5*tan(t1)). As
said in the beginning, in your case the effective angle is 61 deg, so t1/2 =
42 deg, which corresponds to an NA of 1.02.

To summarize, for your specific sample a rough estimate could be found by
using the previously mentioned formula for an NA of 1.02.

That being said, the best solution would be just to calibrate the
measurements experimentally, but then you have to be able to make samples of
known thickness ...

Best regards,

Kevin



Kevin Braeckmans, Ph.D.
Lab. General Biochemistry and Physical Pharmacy
Ghent University
Harelbekestraat 72
9000 Ghent
Belgium
Tel: [phone]
Fax: [phone]

> -----Oorspronkelijk bericht-----
> Van: G. Esteban Fernandez [mailto:[hidden email]]
> Verzonden: vrijdag 15 februari 2008 16:03
> Aan: Confocal Microscopy List; [hidden email]
> Onderwerp: Re: How to correct z distance in RI mismatch?
>
> Thanks for your response Kevin.
>
> How does one deal with the imaginary term for the case of an oil
> immersion lens with an aqueous specimen, i.e. when NA > n2, can the
> sqrt(-1) just be thrown out?  The numbers seem to work out that way.
>
> Esteban
>
>
> 2008/2/15 Kevin Braeckmans <[hidden email]>:
> > Search the CONFOCAL archive at
> > http://listserv.acsu.buffalo.edu/cgi-bin/wa?S1=confocal
> >
> > Dear Esteban,
> >
> > A simple ray-tracing diagram in combination with Snell's law will
> readily
> > give you the following formula:
> >
> > Dz = Dz_motor * sqrt((n2^2 - NA^2)/(n1^2 - NA^2))
> >
> > Where Dz is the actual axial displacement in the sample with
> refractive
> > index n2, Dz_motor is the axial displacement as indicated by the
> microscope
> > focus motor, n1 is the refractive index of the immersion medium and
> NA is
> > the numerical aperture of the objective lens.
> >
> > Note that, as expected, for a perfectly matched system (n1=n2): Dz =
> > Dz_motor. Also, for small NA (NA<<n1 & n2) this leads to the much
> used
> > approximate correction n2/n1 as you have mentioned in your message.
> It is
> > not valid for large NA though.
> >
> > I have used this formula with good results in my own studies. I seem
> to
> > remember there has been a publication during the 90's which compares
> this
> > formula with a more rigorous treatment starting from wave optics. I
> cannot
> > seem to find the reference, though. Maybe someone else on this list
> can ...
> >
> > Hope this helps.
> >
> > Kevin
> >
> >
> >
> > Kevin Braeckmans, Ph.D.
> > Lab. General Biochemistry and Physical Pharmacy
> > Ghent University
> > Harelbekestraat 72
> > 9000 Ghent
> > Belgium
> > Tel: [phone]
> > Fax: [phone]
> >
> > > -----Oorspronkelijk bericht-----
> > > Van: Confocal Microscopy List
> [mailto:[hidden email]]
> > > Namens G. Esteban Fernandez
> > > Verzonden: vrijdag 15 februari 2008 0:11
> > > Aan: [hidden email]
> > > Onderwerp: How to correct z distance in RI mismatch?
> >
> > >
> > > Hi everyone,
> > >
> > > A user needs to measure the volume of fluid inclusions (mostly
> water)
> > > embedded in a clear mineral of RI ~2.4.  We did this by collecting
> > > z-stacks of reflected light on our LSM 510 confocal and rendering a
> 3D
> > > model, which turned out pretty reasonable.  We used an oil-
> immersion
> > > 63x Plan-Apo and oil on either side of the coverslip.
> > >
> > > I know that the z distances are not accurate because of the grossly
> > > different RIs but I'm not sure how to correct them.  Can I just
> > > multiply the z step size by an n2/n1 factor to get the real Δz
> (with
> > > different factors for oil/mineral and mineral/water)?  In the
> > > literature I've seen less simplistic correction models that seem
> more
> > > accurate but no practical ways to implement them that I could
> decipher
> > > were presented.  Any help would be much appreciated.
> > >
> > > Thanks,
> > > Esteban
> > >
> > > --
> > > G. Esteban Fernandez, Ph.D.
> > > Associate Director
> > > Molecular Cytology Research Core Facility
> > > University of Missouri
> > > 1201 E. Rollins St.
> > > Columbia, MO 65211
> > >
> > > 573-882-4895
> > > 573-884-9676 fax
> > >
> > > http://www.biotech.missouri.edu/mcc/
> >
>
>
>
> --
> G. Esteban Fernandez, Ph.D.
> Associate Director
> Molecular Cytology Research Core Facility
> University of Missouri
> 1201 E. Rollins St.
> Columbia, MO 65211
>
> 573-882-4895
> 573-884-9676 fax
>
> http://www.biotech.missouri.edu/mcc/