numerical aperture

[Brian]

Okay this is probably a silly question or two from a newbie just putting things together to get started, but here goes. Do most people just send an unaltered laser beam into the microscope objective? I'm wondering, because it seems that objective magnification and pinhole size pairs I've seen mentioned (for example, 40x and 15 micron) are expecting beam diameters around 1 mm. If this is what is done, then Isn't that a waste of the objective's numerical aperture?

[Din]

Don't forget that the beam is Gaussian. The outer edges get dim pretty fast, (as exp[-k(r^2)], where k takes waist diameter and wavelength into account, as I'm sure you know). But, yes. To an extent, the beam is "wasted", but not completely. The idea is that the objective diverges the beam pretty rapidly, which is what allows holographers to illuminate the entire plate, or to collimate the beam with a large mirror. For example, a 40X has a beam divergence of 80 degrees. This can illuminate an 8in x 10in plate in portrait mode from 6in away. This allows for space in a tight setup. For example , our 12.5in collimater has a focal length of 36in, so a 20X suffices to illuminate the entire collimator.

[Brian]

The 40X (N.A. = 0.65) has beam divergence of 80 degree only if the incident beam fills the objective's entire aperture. The N.A. is roughly proportional to diameter of the aperture/beam size. When you go from the 6 mm diameter aperture of a 40X objective to, say, a 2 mm diameter beam, then the N.A. is reduced to 0.27, which is a 30 degree beam divergence. This is what makes me wonder why people don't telescope the beam to match the diameter of the objective's exit aperture.

[John Klayer]

I remember seeing an article by TJ more than 30 years ago where he suggested putting a concave lens with a z adjustment just before the input of a space filter to adjust the input beam size and output divergence.

[Ed Wesly]

Here is that article! From the Winter 1984 issue of holosphere, the Advocate of Holographic Art, Science, and Technology. I have been doing this for decades. And it also works with converging the beam slightly to zoom down. Great minds thing alike, Brian!

ZoomSF1.jpg (124.7 KiB) Viewed 4583 times

ZoomSF2.jpg (104.26 KiB) Viewed 4583 times

John Klayer, who posted above, might be close to you! You physicists should get in touch. One of John's cave holograms are in the latest edition of the Saxby/Stas book!

[Brian]

I wish I was in John's shoes, having my memory serve me so well.

I love how figure 1 in Dr. Jeong's article shows that a wider collimated input beam produces a larger divergence. But what it cannot show is that the wider input beam produces a narrower spot size (or beam waist) before the beam begins to diverge. So the narrower you want to squeeze light, the wider the beam has to be to start with. And the more you try to squeeze light down, the more it spreads afterward. I think that is so nifty.

Ed, indeed a convex lens can serve the same purpose as the concave lens. Even though Dr. Jeong says the exact choice of focal length doesn't much matter, a particular choice of convex lens can give near optimal performance for a given objective lens. Select a plano convex lens with focal length close to F = 160 (D1/D2) mm, where D1 is incident beam diameter and D2 is objective aperture diameter. Setting the lens on a z translator at a distance S = F + 160 mm behind the aperture and the beam will converge very close to the diffraction limited spot size and subsequently exhibit very close to maximum divergence. Then moving the lens closer to the objective will increase spot size and reduce divergence.

This is the arrangement for DIN standard objectives, which make up about 90% of RMS thread objectives. If you are like me and buy a surplus objective that turns out to be a JIS standard objective (the other 10%) you need to replace 160 in the two equations by 170. This factor optimizes the beam for insertion into the objective, and I recall that the pinhole position only shifts microns as the lens is moved millimeters closer to the objective.

I'll post the reference for the lens choice tomorrow when I'm at a computer instead of an iPad.

And Ed, I don't know if John is close to me, but I can guarantee I'm not close to anything.

[John Klayer]

Brian, I get up to NW Georgia every few weekends. I see you're over in NW Alabama.

[Brian]

reference for the lens
Bechhoefer, J., & Wilson, S. (2002). Faster, cheaper, safer optical tweezers for the undergraduate laboratory. American Journal of Physics, 70(4), 393-400.
lens discussed in section II.B., last paragraph.

John, indeed you located me. If you ever venture toward Huntsville, let me know.

[Din]

Well, one other problem with this wider beam diameter approach is that the beam is more prone to spherical aberration as you veer away from paraxial. Also, if you're not exactly on-axis, you also have the possibility of coma and astigmatism. I've often wondered whether cylindrical lenses in some combination would help with the astigmatism for an off-axis geometry, since there is no tangential/ meridional power, depending on the orientation of the lens.

[Brian]

True, the wide beam puts some stress on the paraxial approximation, but the objective isn't a thin lens either. If microscopists can live with plan achromat objectives, probably my beginner level holography isn't going to suffer much. If I do find spherical aberration causing trouble, then I'll just dial down the D2 value in the lens formula until I achieve acceptable results. Or by then, my holography is so sophisticated that I'll just get apochromat objectives and problem is solved.

Heck, apochromats can come with an iris installed on the aperture so that you can dial down its diameter as you like... that little detail makes it so much easier to align a laser beam through one, it isn't even funny. Too bad they are so expensive.

And true, misalignment adds other problems. But I have a lot of practice aligning optics.

Dinesh, since I only mentioned the possible benefits, it is good to know possible drawbacks. I realize probably most people don't want to add another element to their optical train anyways. But who knows, someone (besides me) may want to give it a go. And this can only help to "inform their decision-making."