FWHM in z - worse with higher Ri?

[Steffen Dietzel]

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Dear listers,

I am confused about the formulas for the Full Width Half Maximum of the
point spread function along the optical axis. According to this paper:

B. Amos, G. McConnell, T. Wilson: Confocal microscopy. In: E. Egelman
(Hrsg.): Biophysical Techniques for Characterization of Cells (=
Comprehensive Biophysics). Volume2. Elsevier, Academic Press, Amsterdam
2012, ISBN 978-0-12-374920-8
<https://de.wikipedia.org/wiki/Spezial:ISBN-Suche/9780123749208>,
chapter 2, pages3–23, doi
<https://de.wikipedia.org/wiki/Digital_Object_Identifier>:10.1016/B978-0-12-374920-8.00203-4
<https://doi.org/10.1016/B978-0-12-374920-8.00203-4>

(free download here:
http://www2.mrc-lmb.cam.ac.uk/images/groupleaders/Confocal_microscopy_Amos_McConnell_Wilson.pdf)

on page 15 the formula (3)  for a conventional microscope for FWHM(z)
with high NA (> 0.5) is

FWHM(z) = (0.88*lambda) / (n-sqrt(n^2 - NA^2)) where n is refractive
index of the immersion medium.

For NA <0.5 this reduces to formula (4):

FWHM(z) = (1.77*n*lambda)/(NA^2)

Now, let us assume we compare two objectives with the same NA but one
oil immersion (n=1.518), one water immersion (n=1.33). Then, with both
formulas we get bigger (and thus worse resolution) FWHMs with the oil
objective. All other things the same, the bigger the RI of the immersion
medium, the worse is the resolution in z.

That does not sound right. Or is it?

What am I missing?

The same would apply to a confocal microscope, just with a slightly
different formula (see paper, formulas 7 and 8).

Best

Steffen

--
------------------------------------------------------------
Steffen Dietzel, PD Dr. rer. nat
Ludwig-Maximilians-Universität München
Biomedical Center (BMC)
Head of the Core Facility Bioimaging

Großhaderner Straße 9
D-82152 Planegg-Martinsried
Germany

http://www.bioimaging.bmc.med.uni-muenchen.de

[James D. Manton-2]

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Dear Steffen,

The issue here is that you've fixed the NAs of the water and oil
immersion lenses in your comparison, not the semiaperture angles. By
fixing the NA, you've given the oil immersion lens a smaller
semiaperture angle as NA = n sin (semiaperture angle) and n is larger.
As the waves have a smaller range of angles to the optical axis, the
length scale over which interference effects occur is larger. When
considering the lateral resolution, this effect is balanced out by the
reduction is wavelength by a factor of n from the vacuum wavelength,
such that all lenses with the same NA provide the same lateral
resolution, irrespective of n. However, this is not the case axially.

I hope this explanation hasn't made things even more confusing than they
were before...

Best wishes,
James




On 19/02/2021 11:40, Steffen Dietzel wrote:

> *****
> To join, leave or search the confocal microscopy listserv, go to:
> http://lists.umn.edu/cgi-bin/wa?A0=confocalmicroscopy
> Post images on http://www.imgur.com and include the link in your posting.
> *****
>
> Dear listers,
>
> I am confused about the formulas for the Full Width Half Maximum of
> the point spread function along the optical axis. According to this
> paper:
>
> B. Amos, G. McConnell, T. Wilson: Confocal microscopy. In: E. Egelman
> (Hrsg.): Biophysical Techniques for Characterization of Cells (=
> Comprehensive Biophysics). Volume2. Elsevier, Academic Press,
> Amsterdam 2012, ISBN 978-0-12-374920-8
> <https://de.wikipedia.org/wiki/Spezial:ISBN-Suche/9780123749208>,
> chapter 2, pages3–23, doi
> <https://de.wikipedia.org/wiki/Digital_Object_Identifier>:10.1016/B978-0-12-374920-8.00203-4
> <https://doi.org/10.1016/B978-0-12-374920-8.00203-4>
>
> (free download here:
> http://www2.mrc-lmb.cam.ac.uk/images/groupleaders/Confocal_microscopy_Amos_McConnell_Wilson.pdf)
>
> on page 15 the formula (3)  for a conventional microscope for FWHM(z)
> with high NA (> 0.5) is
>
> FWHM(z) = (0.88*lambda) / (n-sqrt(n^2 - NA^2)) where n is refractive
> index of the immersion medium.
>
> For NA <0.5 this reduces to formula (4):
>
> FWHM(z) = (1.77*n*lambda)/(NA^2)
>
> Now, let us assume we compare two objectives with the same NA but one
> oil immersion (n=1.518), one water immersion (n=1.33). Then, with both
> formulas we get bigger (and thus worse resolution) FWHMs with the oil
> objective. All other things the same, the bigger the RI of the
> immersion medium, the worse is the resolution in z.
>
> That does not sound right. Or is it?
>
> What am I missing?
>
> The same would apply to a confocal microscope, just with a slightly
> different formula (see paper, formulas 7 and 8).
>
> Best
>
> Steffen
>

--
James Manton
MRC Laboratory of Molecular Biology
Cambridge CB2 0QH, UK
[phone]

[Mark Cannell-2]

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Would it help to substitute n.sin(alpha) for NA and think about the behaviour of the cone of rays coming from the sample?

Cheers

Mark B. Cannell. Ph.D. FRSNZ FISHR
Department of Physiology, Pharmacology & Neuroscience
School of Medical Sciences
University Walk
Bristol BS8 1TD
 
[hidden email]
 
 

On 19/02/21, 12:29 PM, "Confocal Microscopy List on behalf of Steffen Dietzel" <[hidden email] on behalf of [hidden email]> wrote:

    *****
    To join, leave or search the confocal microscopy listserv, go to:
    http://lists.umn.edu/cgi-bin/wa?A0=confocalmicroscopy
    Post images on http://www.imgur.com and include the link in your posting.
    *****

    Dear listers,

    I am confused about the formulas for the Full Width Half Maximum of the
    point spread function along the optical axis. According to this paper:

    B. Amos, G. McConnell, T. Wilson: Confocal microscopy. In: E. Egelman
    (Hrsg.): Biophysical Techniques for Characterization of Cells (=
    Comprehensive Biophysics). Volume2. Elsevier, Academic Press, Amsterdam
    2012, ISBN 978-0-12-374920-8
    <https://de.wikipedia.org/wiki/Spezial:ISBN-Suche/9780123749208>,
    chapter 2, pages3–23, doi
    <https://de.wikipedia.org/wiki/Digital_Object_Identifier>:10.1016/B978-0-12-374920-8.00203-4
    <https://doi.org/10.1016/B978-0-12-374920-8.00203-4>

    (free download here:
    http://www2.mrc-lmb.cam.ac.uk/images/groupleaders/Confocal_microscopy_Amos_McConnell_Wilson.pdf)

    on page 15 the formula (3)  for a conventional microscope for FWHM(z)
    with high NA (> 0.5) is

    FWHM(z) = (0.88*lambda) / (n-sqrt(n^2 - NA^2)) where n is refractive
    index of the immersion medium.

    For NA <0.5 this reduces to formula (4):

    FWHM(z) = (1.77*n*lambda)/(NA^2)

    Now, let us assume we compare two objectives with the same NA but one
    oil immersion (n=1.518), one water immersion (n=1.33). Then, with both
    formulas we get bigger (and thus worse resolution) FWHMs with the oil
    objective. All other things the same, the bigger the RI of the immersion
    medium, the worse is the resolution in z.

    That does not sound right. Or is it?

    What am I missing?

    The same would apply to a confocal microscope, just with a slightly
    different formula (see paper, formulas 7 and 8).

    Best

    Steffen

    --
    ------------------------------------------------------------
    Steffen Dietzel, PD Dr. rer. nat
    Ludwig-Maximilians-Universität München
    Biomedical Center (BMC)
    Head of the Core Facility Bioimaging

    Großhaderner Straße 9
    D-82152 Planegg-Martinsried
    Germany

    http://www.bioimaging.bmc.med.uni-muenchen.de

[Zdenek Svindrych-2]

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Perfect explanation, James, thanks!
I made the same observation when I was playing with PSF generators - the
accurate models produced more elongated PSF at higher RI, given the same NA.
zd

On Fri, Feb 19, 2021 at 7:38 AM James D. Manton <[hidden email]>
wrote:

> *****
> To join, leave or search the confocal microscopy listserv, go to:
> http://lists.umn.edu/cgi-bin/wa?A0=confocalmicroscopy
> Post images on http://www.imgur.com and include the link in your posting.
> *****
>
> Dear Steffen,
>
> The issue here is that you've fixed the NAs of the water and oil
> immersion lenses in your comparison, not the semiaperture angles. By
> fixing the NA, you've given the oil immersion lens a smaller
> semiaperture angle as NA = n sin (semiaperture angle) and n is larger.
> As the waves have a smaller range of angles to the optical axis, the
> length scale over which interference effects occur is larger. When
> considering the lateral resolution, this effect is balanced out by the
> reduction is wavelength by a factor of n from the vacuum wavelength,
> such that all lenses with the same NA provide the same lateral
> resolution, irrespective of n. However, this is not the case axially.
>
> I hope this explanation hasn't made things even more confusing than they
> were before...
>
> Best wishes,
> James
>
>
>
>
> On 19/02/2021 11:40, Steffen Dietzel wrote:
> > *****
> > To join, leave or search the confocal microscopy listserv, go to:
> > http://lists.umn.edu/cgi-bin/wa?A0=confocalmicroscopy
> > Post images on http://www.imgur.com and include the link in your
> posting.
> > *****
> >
> > Dear listers,
> >
> > I am confused about the formulas for the Full Width Half Maximum of
> > the point spread function along the optical axis. According to this
> > paper:
> >
> > B. Amos, G. McConnell, T. Wilson: Confocal microscopy. In: E. Egelman
> > (Hrsg.): Biophysical Techniques for Characterization of Cells (=
> > Comprehensive Biophysics). Volume2. Elsevier, Academic Press,
> > Amsterdam 2012, ISBN 978-0-12-374920-8
> > <https://de.wikipedia.org/wiki/Spezial:ISBN-Suche/9780123749208>,
> > chapter 2, pages3–23, doi
> > <https://de.wikipedia.org/wiki/Digital_Object_Identifier>:10.1016/B978-0-12-374920-8.00203-4
>
> > <https://doi.org/10.1016/B978-0-12-374920-8.00203-4>
> >
> > (free download here:
> >
> http://www2.mrc-lmb.cam.ac.uk/images/groupleaders/Confocal_microscopy_Amos_McConnell_Wilson.pdf
> )
> >
> > on page 15 the formula (3)  for a conventional microscope for FWHM(z)
> > with high NA (> 0.5) is
> >
> > FWHM(z) = (0.88*lambda) / (n-sqrt(n^2 - NA^2)) where n is refractive
> > index of the immersion medium.
> >
> > For NA <0.5 this reduces to formula (4):
> >
> > FWHM(z) = (1.77*n*lambda)/(NA^2)
> >
> > Now, let us assume we compare two objectives with the same NA but one
> > oil immersion (n=1.518), one water immersion (n=1.33). Then, with both
> > formulas we get bigger (and thus worse resolution) FWHMs with the oil
> > objective. All other things the same, the bigger the RI of the
> > immersion medium, the worse is the resolution in z.
> >
> > That does not sound right. Or is it?
> >
> > What am I missing?
> >
> > The same would apply to a confocal microscope, just with a slightly
> > different formula (see paper, formulas 7 and 8).
> >
> > Best
> >
> > Steffen
> >
>
> --
> James Manton
> MRC Laboratory of Molecular Biology
> Cambridge CB2 0QH, UK
> [phone]
>

--
--
Zdenek Svindrych, Ph.D.
Research Scientist - Microscopy Imaging Specialist
Department of Biochemistry and Cell Biology
Geisel School of Medicine at Dartmouth

[Steffen Dietzel]

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*****

Thanks guys,

the explanation sounds reasonable even to me. I am now confused on a
much higher level compared to before ;-)

Best

Steffen

Am 19.02.2021 um 14:36 schrieb Zdenek Svindrych:

> *****
> To join, leave or search the confocal microscopy listserv, go to:
> http://lists.umn.edu/cgi-bin/wa?A0=confocalmicroscopy
> Post images on http://www.imgur.com and include the link in your posting.
> *****
>
> Perfect explanation, James, thanks!
> I made the same observation when I was playing with PSF generators - the
> accurate models produced more elongated PSF at higher RI, given the same NA.
> zd
>
> On Fri, Feb 19, 2021 at 7:38 AM James D. Manton <[hidden email]>
> wrote:
>
>> *****
>> To join, leave or search the confocal microscopy listserv, go to:
>> http://lists.umn.edu/cgi-bin/wa?A0=confocalmicroscopy
>> Post images on http://www.imgur.com and include the link in your posting.
>> *****
>>
>> Dear Steffen,
>>
>> The issue here is that you've fixed the NAs of the water and oil
>> immersion lenses in your comparison, not the semiaperture angles. By
>> fixing the NA, you've given the oil immersion lens a smaller
>> semiaperture angle as NA = n sin (semiaperture angle) and n is larger.
>> As the waves have a smaller range of angles to the optical axis, the
>> length scale over which interference effects occur is larger. When
>> considering the lateral resolution, this effect is balanced out by the
>> reduction is wavelength by a factor of n from the vacuum wavelength,
>> such that all lenses with the same NA provide the same lateral
>> resolution, irrespective of n. However, this is not the case axially.
>>
>> I hope this explanation hasn't made things even more confusing than they
>> were before...
>>
>> Best wishes,
>> James
>>
>>
>>
>>
>> On 19/02/2021 11:40, Steffen Dietzel wrote:
>>> *****
>>> To join, leave or search the confocal microscopy listserv, go to:
>>> http://lists.umn.edu/cgi-bin/wa?A0=confocalmicroscopy
>>> Post images on http://www.imgur.com and include the link in your
>> posting.
>>> *****
>>>
>>> Dear listers,
>>>
>>> I am confused about the formulas for the Full Width Half Maximum of
>>> the point spread function along the optical axis. According to this
>>> paper:
>>>
>>> B. Amos, G. McConnell, T. Wilson: Confocal microscopy. In: E. Egelman
>>> (Hrsg.): Biophysical Techniques for Characterization of Cells (=
>>> Comprehensive Biophysics). Volume2. Elsevier, Academic Press,
>>> Amsterdam 2012, ISBN 978-0-12-374920-8
>>> <https://de.wikipedia.org/wiki/Spezial:ISBN-Suche/9780123749208>,
>>> chapter 2, pages3–23, doi
>>> <https://de.wikipedia.org/wiki/Digital_Object_Identifier>:10.1016/B978-0-12-374920-8.00203-4
>>> <https://doi.org/10.1016/B978-0-12-374920-8.00203-4>
>>>
>>> (free download here:
>>>
>> http://www2.mrc-lmb.cam.ac.uk/images/groupleaders/Confocal_microscopy_Amos_McConnell_Wilson.pdf
>> )
>>> on page 15 the formula (3)  for a conventional microscope for FWHM(z)
>>> with high NA (> 0.5) is
>>>
>>> FWHM(z) = (0.88*lambda) / (n-sqrt(n^2 - NA^2)) where n is refractive
>>> index of the immersion medium.
>>>
>>> For NA <0.5 this reduces to formula (4):
>>>
>>> FWHM(z) = (1.77*n*lambda)/(NA^2)
>>>
>>> Now, let us assume we compare two objectives with the same NA but one
>>> oil immersion (n=1.518), one water immersion (n=1.33). Then, with both
>>> formulas we get bigger (and thus worse resolution) FWHMs with the oil
>>> objective. All other things the same, the bigger the RI of the
>>> immersion medium, the worse is the resolution in z.
>>>
>>> That does not sound right. Or is it?
>>>
>>> What am I missing?
>>>
>>> The same would apply to a confocal microscope, just with a slightly
>>> different formula (see paper, formulas 7 and 8).
>>>
>>> Best
>>>
>>> Steffen
>>>
>> --
>> James Manton
>> MRC Laboratory of Molecular Biology
>> Cambridge CB2 0QH, UK
>> [phone]
>>
>

--
------------------------------------------------------------
Steffen Dietzel, PD Dr. rer. nat
Ludwig-Maximilians-Universität München
Biomedical Center (BMC)
Head of the Core Facility Bioimaging

Großhaderner Straße 9
D-82152 Planegg-Martinsried
Germany

http://www.bioimaging.bmc.med.uni-muenchen.de