https://www.youtube.com/watch?v=pFSvM8KD5BY

First, a few caveats. The specification of NA always comes with some assumptions. The specified NA is the capability of the lens, not necessarily the NA you will get with a particular light source. Also, the capability of the lens may be limited by the actual entrance aperture, or by an effective aperture above which the lens will fail to meet other specifications such as spot size. However, most of the time there is an aperture stop which defines the entrance aperture, and the specifications are given for the case of a uniform intensity plane wave flooding the entrance aperture.

So, the case where you flooded the entrance aperture is probably correct for testing the capability of the lens. The grossly under-filling laser beam was never going to show the correct NA. Imagine a single simple lens. The NA is the angle of the rays approaching the focus. You won’t see the largest angle until the incoming beam fills the lens. Anything less makes a smaller angle. A tiny laser beam makes tiny angle.

Now, I think your expectations are incorrect. The NA is not determined by the size of the last lens. Remember the system also has to accommodate some field of view. The closer to the focal plane, the more the lens sizes are determined by the field and less so by the pupil. Also, the 5.7 mm diameter could also be driven by mechanical convenience. Perhaps it would be hard to make it smaller, and there is no need. The beam can and does under-fill every lens in the system except one single aperture that defines the aperture stop. The aperture stop should always be pupil like and not field like, so it will always be at or near the entrance of an objective, not near the focus. The last lens is welcome to be larger than the used aperture.

The NA is the angle of the rays coming into the focus. Or, more precisely, it is the sine of the half angle of the cone of rays coming into the focus. You measured the diameter of the cone at a distance from the focus as 1.4 mm. To calculate the NA you need to know how far away the focus is. You say the working distance is 2.0, but that is the distance in the fluid. For reasons of wanting to be able to compare apples to apples the NA of an objective is always quoted AS IF the lens was being used in air. This leads to some very weird things like NA’s greater than 1 when, of course, the sine CAN’T be greater than 1. Your NA =1 is a perfect example of this. Sine = 1 is 90 degrees. In air the light is at the critical angle and would never leave the lens. Impossible, right? So what they are telling you is that the true NA in the fluid is the quoted NA = 1 divided by the index of the fluid. Assuming the fluid is water, the angle in the fluid is ~asin(.666). At a working distance of 2 mm this gives a beam diameter at the first lens of 2.9 mm.

You measured half that. Why? The most most likely reason is that the entering light was not sufficiently collimating or uniform. Other reasons could be the difficulty of measuring the beam spot in air of an NA =1 lens. The light only exits the lens where it is in contact with something other than air or where there are defects. You might not be measuring the diameter of the beam at the last surface, but rather be measuring the diameter of an area of damage caused by some previous user, or perhaps measuring the contact patch of whatever you are putting against the lens.

In any case, optics works very well and there is essentially no chance the lens is significantly different from what the manufacturer says it is. I feel certain the NA is 1 as advertised.

Dear Mike Albert , thank you for the insightful comment. That's true my non-laser source wasn't collimated because it's just bulb light. But for the laser, would you think I can calculate the actual NA knowing the laser beam diameter when measured without the objective. I also know the focal length of the primary lens (fo) for this objective. So I could calculate NA as n * D/(2fo) but can I assume the aperture diameter D = laser beam diameter?

What you are suggesting is not impossible, but it becomes harder and harder for smaller and smaller beams. I would recommend expanding the laser beam to significantly larger than the entrance aperture (to approach uniform intensity across the aperture) and collimating.

Mike Albert , you mean expand it to make sure I fill the diameter of the back lens, and don't need to recalculate the NA? The problem is that I already did some measurements in this configuration, so now wanted to estimate the NA for those from the beam diameter rather than just assuming the nominal NA. When you say it becomes harder for smaller beams, you mean the beams are altered when entering the objective so I can't think D from the NA equation is the beam diameter, or something else?

If you use a beam that does not fill the aperture, then scaling the measurement to what would happen with a flood beam involves several assumptions and approximations. You must consider not just the nominal “size” of the beam, but also its intensity profile and how that affects the measurement. Further, as the beam gets smaller, the scaling factor gets larger and the resulting precision drops.

However, I think I understand now that you would like to know what NA is achieved with your actual underfilling beam. You would use the measured beam diameter and the working distance (back focal length) with the caveats A) the determined angle is the angle in the fluid and other NAs you may be comparing to are scaled to air (so multiplied by index). B) NA is a simple description. Many phenomena which are described in terms of NA are also making assumptions about the beam‘s transverse intensity profile, usually either the Gaussian assumption, or in the case of lens descriptions, assuming an Airy function. You should be careful that any expressions you use involving NA are under an appropriate assumption OR if using a Gaussian assumption can be appropriately scaled by M^2

Ivan Unksov , if you need to measure your real NA in your working conditions, you should:

• Find position of back focal plane of lens by microscope or your CMOS sensor. It is simple to do by using as object some diffracting grating, like object-micrometer, some optical diffracting grating or even surface of CD disk and determine plane position by sharpest image of diffracted beams;
• by known pitch of your grating (d) and wavelength of illumination (lambda) you can determine scale factor of your focal image plane:
• sin(phi) = k*lambda / d
• where k is diffraction order. So you will know scale factor, how many pixels (millimetres, micrometers) corresponds to given sin(phi) value; you can increase precision of it by averaging values for different diffraction beams;
• measure in this plane the size of light spot of your experimental layout, which you are interesting to know. By known scale factor you can calculate your real NA value.

Dear Valeriy Sterligov , thank you for your answer. However, I understood (please correct me if I'm wrong) from your words and from the additional information I found here: https://www.olympus-lifescience.com/en/microscope-resource/primer/java/imageformation/gratingdiffraction/ that this method works if I send a narrow beam to the objective through, for example, a condenser that is coaxial with the objective. But my system is an inverted microscope for fluorescence detection, which means that I send my beam not from the condenser but through the same objective that I use to collect the signal light from a cone defined by the NA. So, in my case, I have a converging beam coming out of the objective - see the drawing attached.

The method that I have proposed to you is valid when the objective aperture doesn't completely fill by light. This case is drawn in your illustration. In such a way you can determine real, realized in your conditions NA value. In the case of an inverted fluorescence microscope, which you described, the objective's aperture is completely filled by light, so you can trust the NA value which is engraved on the objective.

Valeriy Sterligov , thanks for your comment but I don't quite understand why do you think the situation is different for an inverted microscope, sorry - in my drawing I have the objective exactly like it is in my inverted microscope, below the specimen plane. My point is that my laser beam, which enters the objective, is smaller than the aperture of the objective, hence the 'effective' NA for that beam is smaller than the NA written on the objective.

Ivan Unksov

>My point is that my laser beam, which enters the objective, is smaller than >the aperture of the objective, hence the 'effective' NA for that beam is >smaller than the NA written on the objective.

Yes, you are right, you have two different NA values. As consequence of such layout size of focused laser beam will be bigger relatively case when you fill by laser light total objective aperture. Objective's resolution power is optimized because objective aperture is completely filled by fluorescence light. So, situation is not different for inverted microscope, but different for incident laser and fluorescent light beams.

Valeriy Sterligov, that is true but here I only care about NA for incident light, because my samples are sensitive to NA of that incident light. Then, coming back to the method you suggested, if I place an object that causes diffraction somewhere in the optical path, wouldn't the diffraction pattern be affected by the fact that the beam is converging, and therefore for every angle in the beam the diffraction pattern will be different? If I understood you correctly, you assume that sin(theta) = NA/n can be calculated from the equation sin(theta)/sin(phi) = R/L, where R - radius of the beam in a plane, L - distance from the optical axis to the diffraction pattern of order k, which then has to be measured very precisely?

Ivan Unksov,

>because my samples are sensitive to NA of that incident light.

In air NA is value that measure angular divergence/convergence of the incident beam. It is difficult for me to imagine the sample that is sensitive to such beam property. What samples with such property do you use?

>Then, coming back to the method you suggested, if I place an object that causes diffraction somewhere in the optical path,

From one side, diffraction by object structure present into all spaces before, inside, and outside of a microscope. From another side, I'm proposing you to work into objective's back focal plane. Each point of this wonderful plane corresponds to each directions of light that passed microscope lens independently from divergence/convergence of the incident beam. It is very simple to find it because diffraction orders of diffraction grating (DG) as a sample into this plane are best focused. Light distribution in this plane is linear in terms of NA=n*sin(theta). If you will know how pixels of your camera, micrometers of your screen, etc. contain unity of NA, you will have calibrated NA ruler.

Again, how to calibrate this ruler? By image of diffraction into back focal plane of DG with known pitch d and known wavelengths of illumination lambda. You are measuring in pixels, or any other units distance between diffraction orders: 0..+1; 0..-1; +1..-1; 0..+2; 0..-2; etc. and average measured values, of course taking into account order of diffraction k:

d*n*sin(theta) = k*lambda,

so for each order of diffraction with known sin(theta) you know

gamma = k*lambda/(d*n),

measured in [1/pixels]

Now any measured distance R [pixels] in back focal plane you can calculate in NA units:

R*gamma

>from the equation sin(theta)/sin(phi) = R/L, where R - radius of the beam in a plane, L - distance from the optical axis to the diffraction pattern of order k, which then has to be measured very precisely?

No, you are even don't need to know L, as you see from above, you find answer only by calibrating back focal plane.

If something is not clear, ask questions.

Sometimes back focal plane is placed into body of microscope lens, so direct work, measurements inside it not convenient. But all which I described valid for its conjugated plane, when you image this plane by additional lens somewhere in more convenient place.

Valeriy Sterligov

> 'What samples with such property do you use?'

The samples are high aspect ratio nanostructures, the dependence on the incident light angle for them is discussed for example in Article Applied electromagnetic optics simulations for nanophotonics

> 'It is very simple to find it because diffraction orders of diffraction grating (DG) as a sample into this plane are best focused.'

So if I observe the diffraction patten in the back focal plane, where the pattern is best focused, does it matter at what distance from the objective do I place the diffraction grating itself? I mean it should be in a plane other than the back focal plane, right?

> 'Now any measured distance R [pixels] in back focal plane you can calculate in NA units: R*gamma' - yes, so you mean there will be proportionality between sin(phi) corresponding to a distance between for example 0th and 1st diffraction orders, and sin(theta) that corresponds to NA.

Ivan Unksov ,

>The samples are high aspect ratio nanostructures, the dependence on the incident light angle for them is discussed for example in ...

So, would it be more correctly tell that samples are sensitive to angular distribution of incident beam?

>So if I observe the diffraction patten in the back focal plane, where the pattern is best focused, does it matter at what distance from the objective do I place the diffraction grating itself?

I suppose that you place DG into front focal plane, which is usual layout for microscopy observation. You don't need to move it somewhere else.

>I mean it should be in a plane other than the back focal plane, right?

Again, DG have to be if front focal plane. In such case you observe by microscope image of its groves. With such layout back focal plane contains distribution of light, diffracted by the sample, set of diffracted beams for the case of DG.

>so you mean there will be proportionality between sin(phi) corresponding to a distance between for example 0th and 1st diffraction orders, and sin(theta) that corresponds to NA.

Yes, sure! Even more, proportionality will be to n*sin(theta), which is NA. I don't know, what is phi, it is absent on your plot?

Ivan Unksov , if you are satisfied by answers you can recommend them.

Valeriy Sterligov , yes, thank you for the answers and discussion!

>So, would it be more correctly tell that samples are sensitive to angular distribution of incident beam?

Yes, that's correct.

>I don't know, what is phi, it is absent on your plot?

I meant the angles of diffraction: sin(phi)=m*lambda/d

Ivan, first of all, let me slightly refine the terminology here, NA is defined as:

NA=n*sin(AOI), namely it does contain the refractive index of the medium the light propagates into after the lens. Strictly speaking, NA is the fraction of wavevector projected on the plane perpendicular to optical axis of the system

There are several ways to estimate the marginal ray of your incident beam - here are several:

1. You can use a knife-edge test at the back focal plane of the objective (pupil plain or conjugated plain) and estimate, based on energy distribution the angular content of you beam.

To do that, i have prepared my own "knife edge" - simply a microscope slide with several nm of silver deposited on it. During deposition I had another slide attached to the first one, so when I removed it, I got a sharp edge of silver.

2. You can use plasmonic effect of attenuated reflection - again a thin layer of Ag on your slide and change the AOI, while measuring the reflected light. From there, you can estimate your angular profile.

3. High aspect ratio well defined diffraction grating. It can be demonstrated that for really high aspect ratio (grating depth >> wavelength), the high orders are generated exclusively for Littrow condition (the diffraction order is diffracted back into the incident beam direction) - it is not difficult to prove, for instance from Born approximation. Practically, you can scan your AOI and collect light within 1st DO - again at the back focal plane of the objective, this way you can monitor the exact angular distribution of the beam.

Please note that the last two methods assume monochromatic illumination - they both have to do with wavevector components project either on surface (plasmonics/light phase matching) or on axes (Littrow phase matching) in both cases, spectral and angular effects are very similar, so if you use significant bandwidth, it will alert your measurement.

But, in any case, using significant bandwidth may be challenging in your case, and you need to give a concern on chromatics of your microscope.

I could propose several other approaches, but I guess these should cover the main principles.

BTW, if I understand correctly your description of the sample, the last method could be used simply as calibration for your measurement. Just get several calibration samples with various grating periods, and normalize your measurement to the reflected signal from these known diffraction gratings.

Best

Yuri Paskover , thanks for the comment. Yes, above in this thread we were discussing the way to measure the NA by seeing the diffraction pattern from a diffraction grating on the back focal plane of the objective.

>'High aspect ratio well defined diffraction grating'

What do you mean by high aspect ratio? Is it gamma in this picture I found: https://www.thorlabs.com/images/TabImages/Littrow_Configuration_D1-780.gif ? Also, do you know any literature where this method of NA measurements and the conditions you mentioned would be described in more detail?

Ivan, I used the high aspect ratio term incorrectly. What I was talking about is grating height, I would propose to have a set of gratings with groove height of, say, 3-5 micrometers, and varying grating period (a in your illustration). This way, you could observe very narrow diffraction that would be diffracted only at the the AOI NAill=wvl*m/2a:

which is the angle of incidence satisfying Littrow conditions for m-th diffraction order at wavelength=wvl for grating with period a.

I am not sure there is a particular literature for this topic. I just was running into somewhat similar problems working on my research, and those were some of the solutions I came up with, or saw others to come up with over time.