How does third-harmonic generation (THG) scale with the numerical aperture (NA) ?

[Nino Karpf]

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Dear list,

I know this is a confocal list, but I noticed that many of you are also working with multiphoton microscopes and since there are many very experienced users in this list, I wanted to ask a question regarding Third-Harmonic Generation (THG) (please let me know in case this is the wrong list or about better alternatives for MPM):

How does the THG signal scale with the NA of the objective?


IMO it should increase with NA. My thoughts are:

1) In TPEF and SHG, the excitation scales with intensity^2. The intensity depends on the NA^2, so overall the excitation efficiency scales with NA^4. Although here there is a dramatic scaling opportunity for the two-photon excitation efficiency, this is only per fluorophore. When we increase the NA, we also reduce the excitation volume with the fourth power (NA reduces rayleigh length quadratically in both dimensions, thus fourth power dependency of the excitation volume on NA).  Thus, effectively we don’t see any increase, as long as we don’t saturate the fluorescence in the focus. This is also what I confirmed experimentally.

2) In 3PEF and THG, the signal scales with I^3, so with NA^6. However, the excitation volume only decreases with NA^4, so overall there should remain a quadratic dependency on NA. Furthermore, in physiological THG, where the effect is generated e.g. by the refractive index change from cytoplasm to nuclear membrane, the „volume“ argument from before does not hold, because a micrometer out of focus there is no structure generating THG. Thus, here the NA should have an even more dramatic influence, in the extreme case of only a 3-4nm thick nuclear membrane the whole NA^6 dependency.

Has anyone of you ever tested the NA dependency of THG? Or know where to read about it?

Of course, the NA (and magnification) can further lead to higher detected signals, but I’m only asking about the excitation. Of course, an overall THG signal (excitation+detection) scaling with NA would also be interesting to know.

Thank you!
Nino


__________________________________
Prof. Dr. Sebastian 'Nino' Karpf
Juniorprofessor
 

University of Lübeck
Institute for Biomedical Optics (BMO)
 
Tel [phone]
E-Mail [hidden email] <mailto:[hidden email]>
www.bmo.uni-luebeck.de <http://www.bmo.uni-luebeck.de/>
 
Peter-Monnik-Weg 4
23562 Lübeck
Germany

[Craig Brideau]

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The thing to keep in mind beyond the math is that multiphoton effects
depend on the probability of multiple photons interacting with a molecule
(or the "lattice" depending on the mechanism) *simultaneously*. The
probability that SHG will happen is dependent on two photons hitting at
once, while for THG it is three photons at once, and so on. You control
photon interaction probability through two mechanisms; shorter pulses,
which crams a higher density of photons into a smaller slice of *time*, or
tighter focus (higher NA) which crams them into a tighter volume of
*space*. Both time and space packing increases the likelihood that multiple
photons will encounter each other and thus interact. From my work with CARS
microscopy I recall that three photon processes go with the third power of
NA squared, or NA^6, as the spatial interaction probability goes to the
power of the number of interacting photons squared as photon density goes
with the square of NA and THG is a third order susceptibility.
Olivier and Beaurepaire went quite a bit beyond this idea and tried
engineering the focal volumes to get even further optimization. In short,
at high NA the polarization state of the beam at the focus becomes a mess,
but by tweaking polarization before focusing you can optimize the result to
satisfy THG conditions at the focus.
https://www.osapublishing.org/oe/fulltext.cfm?uri=oe-16-19-14703&id=171835

Craig

On Thu, Feb 25, 2021 at 9:54 AM Sebastian 'Nino' Karpf <[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 list,
>
> I know this is a confocal list, but I noticed that many of you are also
> working with multiphoton microscopes and since there are many very
> experienced users in this list, I wanted to ask a question regarding
> Third-Harmonic Generation (THG) (please let me know in case this is the
> wrong list or about better alternatives for MPM):
>
> How does the THG signal scale with the NA of the objective?
>
>
> IMO it should increase with NA. My thoughts are:
>
> 1) In TPEF and SHG, the excitation scales with intensity^2. The intensity
> depends on the NA^2, so overall the excitation efficiency scales with NA^4.
> Although here there is a dramatic scaling opportunity for the two-photon
> excitation efficiency, this is only per fluorophore. When we increase the
> NA, we also reduce the excitation volume with the fourth power (NA reduces
> rayleigh length quadratically in both dimensions, thus fourth power
> dependency of the excitation volume on NA).  Thus, effectively we don’t see
> any increase, as long as we don’t saturate the fluorescence in the focus.
> This is also what I confirmed experimentally.
>
> 2) In 3PEF and THG, the signal scales with I^3, so with NA^6. However, the
> excitation volume only decreases with NA^4, so overall there should remain
> a quadratic dependency on NA. Furthermore, in physiological THG, where the
> effect is generated e.g. by the refractive index change from cytoplasm to
> nuclear membrane, the „volume“ argument from before does not hold, because
> a micrometer out of focus there is no structure generating THG. Thus, here
> the NA should have an even more dramatic influence, in the extreme case of
> only a 3-4nm thick nuclear membrane the whole NA^6 dependency.
>
> Has anyone of you ever tested the NA dependency of THG? Or know where to
> read about it?
>
> Of course, the NA (and magnification) can further lead to higher detected
> signals, but I’m only asking about the excitation. Of course, an overall
> THG signal (excitation+detection) scaling with NA would also be interesting
> to know.
>
> Thank you!
> Nino
>
>
> __________________________________
> Prof. Dr. Sebastian 'Nino' Karpf
> Juniorprofessor
>
>
> University of Lübeck
> Institute for Biomedical Optics (BMO)
>
> Tel [phone]
> E-Mail [hidden email] <mailto:
> [hidden email]>
> www.bmo.uni-luebeck.de <http://www.bmo.uni-luebeck.de/>
>
> Peter-Monnik-Weg 4
> 23562 Lübeck
> Germany

[Steffen Dietzel]

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Hi Nino,

another thing to keep in mind is that objectives do not necessarily
perform the way they theoretically should, once you use wavelengths at
 >1200nm.

That may have improved now, but when we looked at THG resolution many
moons ago (https://doi.org/10.1371/journal.pone.0099615), we found that
FWHM of the PSF at 1275 nm was about twice as large as it should be in
every direction. So that is a big drain on any 3-photon effect. The
better the resolution in xy (smaller PSF) the denser the photon flux,
thus the brighter the signal. I don't think that z-resolution will play
a big role.

The objective we were using at the time (by now over 10 years old
Olympus 20x0.95) was simply not made for that wavelength-range. I assume
that this is now less of a problem with some objectives, but if it
should be important, I'd recommend to check. I doubt that many labs will
have a whole selection of objectives around that is suitable for this
type of microscopy. My guess is that usually it is that one 20-25x with
an NA around 1 with the fat barrel and a WD of ~2 mm for tissue imaging.

Experimental checks of NA-dependency would suffer from the possibility
that different objectives may have a different chirp of the pulse, as
Craig pointed out. And then polarization effects. It is usually said
that a day in the library may save you a week in the lab. On this topic,
my feeling is it may be the other way round. There are so many variables
which are difficult to control that, at least for applications in the
life sciences, it seems faster to just play around and try rather than
to estimate intensity theoretically.

BTW, my favorite test sample for THG is a drop of water on a glass
slide. The water/glass interface is thin and very uniform.

Best

Steffen

Am 25.02.2021 um 18:57 schrieb Craig Brideau:

> *****
> 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.
> *****
>
> The thing to keep in mind beyond the math is that multiphoton effects
> depend on the probability of multiple photons interacting with a molecule
> (or the "lattice" depending on the mechanism) *simultaneously*. The
> probability that SHG will happen is dependent on two photons hitting at
> once, while for THG it is three photons at once, and so on. You control
> photon interaction probability through two mechanisms; shorter pulses,
> which crams a higher density of photons into a smaller slice of *time*, or
> tighter focus (higher NA) which crams them into a tighter volume of
> *space*. Both time and space packing increases the likelihood that multiple
> photons will encounter each other and thus interact. From my work with CARS
> microscopy I recall that three photon processes go with the third power of
> NA squared, or NA^6, as the spatial interaction probability goes to the
> power of the number of interacting photons squared as photon density goes
> with the square of NA and THG is a third order susceptibility.
> Olivier and Beaurepaire went quite a bit beyond this idea and tried
> engineering the focal volumes to get even further optimization. In short,
> at high NA the polarization state of the beam at the focus becomes a mess,
> but by tweaking polarization before focusing you can optimize the result to
> satisfy THG conditions at the focus.
> https://www.osapublishing.org/oe/fulltext.cfm?uri=oe-16-19-14703&id=171835
>
> Craig
>
> On Thu, Feb 25, 2021 at 9:54 AM Sebastian 'Nino' Karpf <[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 list,
>>
>> I know this is a confocal list, but I noticed that many of you are also
>> working with multiphoton microscopes and since there are many very
>> experienced users in this list, I wanted to ask a question regarding
>> Third-Harmonic Generation (THG) (please let me know in case this is the
>> wrong list or about better alternatives for MPM):
>>
>> How does the THG signal scale with the NA of the objective?
>>
>>
>> IMO it should increase with NA. My thoughts are:
>>
>> 1) In TPEF and SHG, the excitation scales with intensity^2. The intensity
>> depends on the NA^2, so overall the excitation efficiency scales with NA^4.
>> Although here there is a dramatic scaling opportunity for the two-photon
>> excitation efficiency, this is only per fluorophore. When we increase the
>> NA, we also reduce the excitation volume with the fourth power (NA reduces
>> rayleigh length quadratically in both dimensions, thus fourth power
>> dependency of the excitation volume on NA).  Thus, effectively we don’t see
>> any increase, as long as we don’t saturate the fluorescence in the focus.
>> This is also what I confirmed experimentally.
>>
>> 2) In 3PEF and THG, the signal scales with I^3, so with NA^6. However, the
>> excitation volume only decreases with NA^4, so overall there should remain
>> a quadratic dependency on NA. Furthermore, in physiological THG, where the
>> effect is generated e.g. by the refractive index change from cytoplasm to
>> nuclear membrane, the „volume“ argument from before does not hold, because
>> a micrometer out of focus there is no structure generating THG. Thus, here
>> the NA should have an even more dramatic influence, in the extreme case of
>> only a 3-4nm thick nuclear membrane the whole NA^6 dependency.
>>
>> Has anyone of you ever tested the NA dependency of THG? Or know where to
>> read about it?
>>
>> Of course, the NA (and magnification) can further lead to higher detected
>> signals, but I’m only asking about the excitation. Of course, an overall
>> THG signal (excitation+detection) scaling with NA would also be interesting
>> to know.
>>
>> Thank you!
>> Nino
>>
>>
>> __________________________________
>> Prof. Dr. Sebastian 'Nino' Karpf
>> Juniorprofessor
>>
>>
>> University of Lübeck
>> Institute for Biomedical Optics (BMO)
>>
>> Tel [phone]
>> E-Mail [hidden email] <mailto:
>> [hidden email]>
>> www.bmo.uni-luebeck.de <http://www.bmo.uni-luebeck.de/>
>>
>> Peter-Monnik-Weg 4
>> 23562 Lübeck
>> Germany

--
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Ludwig-Maximilians-Universität München
Biomedical Center (BMC)
Head of the Core Facility Bioimaging

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D-82152 Planegg-Martinsried
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