The Optical Invariant, or: Why LEDs make bad light sources for light sheet microscopy

I want to build a light sheet microscope, so recently I’ve been thinking about optical designs for light sheets. This paper [1] describes about the simplest light sheet you can make: collimate a laser beam and use a cylindrical lens to focus it to a sheet in your sample. If you want higher resolution or magnification, you can focus the sheet on the back focal plane of an objective, but if you only need a few micron resolution, the cylindrical lens is alone is enough to achieve that. Since the application I’m interested is the same as that of the paper (imaging whole cleared mouse brains and other large samples), a few micron resolution is enough for many applications, and I’m planning on building a very similar system (partly because we have an unused AZ100 to work with).

Done this way, you only really need two lenses: the cylindrical lens and a second lens to collimate the output of an optical fiber from your laser. With such a simple optical system, the laser is by far the largest cost, and so someone I was talking to recently suggested using an LED instead. I thought this seemed like a good idea – even though the LED doesn’t produce a nice collimated beam, you don’t need a ton of power for a light sheet system – the paper above uses 1.5 mW per mm of beam width, or 15 mW for a 1 cm wide beam, which is probably as big as we’d ever need. It’s no problem to get a 1W (or even multiwatt) LED, so we should be able to throw away a bunch of light to collimate it and still have enough light to illuminate our specimen, right?

One of the fundamental rules of designing optical systems is that of the étendue or optical invariant. These are related quantities that describe the fact that the product of the source size and its emission angle are at best constant in an optical system – no optical system can make the product of the source size and emission angle smaller without throwing away light. In our example above, a perfectly collimated beam would have an optical invariant of 0, since it has no divergence. Obviously this is not possible, but with a laser source, you can have a pretty small optical invariant – an Obis laser has a 0.6 mm beam size with a 1.2 mrad divergence, which corresponds to an optical invariant of 0.6 μm.

Now let’s consider an LED. A high power mounted LED from Thorlabs has a source size of 2 mm and a divergence angle of 40°, which corresponds to an optical invariant of 1.4 mm, about 2000 times worse. So, to get the same performance as a laser, we’d only be left with 1/2000 the output power of the LED. Even with a 1W LED, we’re only left with 0.5 mW of power, not enough to get the job done.  With a smaller LED we can do a little better – a T 1-3/4 LED might have a die size of 300 μm and a divergence angle of 10º, for an optical invariant of 50 μm. But it’s hard to find an LED of that size that emits even 10 mW, so we still wouldn’t get enough power after collimating the beam.  Alternatively, we could use a high power LED and allow a larger beam waist, but to get enough power from the LED, we’re looking at a waist that’s probably at least 10x wider than that produced by a laser beam.

Finally, I want to mention that this is all connected to the M2 parameter used to quantify laser beam quality. Gaussian beam optics is often used to describe laser propagation; this is how the light sheet properties were worked out in the paper above. There is a nice discussion of Gaussian beams in the Melles-Griot chapter on Gaussian beam optics beam optics, which sadly no longer seems to be on the Melles-Griot website, so I’ve posted a copy here. For a Gaussian beam, the optical invariant is given by λ/π, where λ is the wavelength of the beam. Real beams are never pure Gaussians and so only approach this limit. They will always have a larger optical invariant; the amount larger is measured by the beam quality factor M2, which can be 1.2 or less for a good laser (such as the Obis above).  In this case, the optical invariant is given by M2λ/π.

I’m sure optics experts will find some errors in all of the above; if anything I wrote is wrong enough to invalidate my conclusions, please let me know. My optics background in this area isn’t that strong; I confess that this is the first time I’ve actually understood what M2 measures!


  1. H. Dodt, U. Leischner, A. Schierloh, N. Jährling, C.P. Mauch, K. Deininger, J.M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, "Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain", Nature Methods, vol. 4, pp. 331-336, 2007.

2 thoughts on “The Optical Invariant, or: Why LEDs make bad light sources for light sheet microscopy

  1. Thanks for sharing this concept! Etendue has been a tough concept for me to wrap my mind around, so the more I read about it the better. Coherent also wrote about comparing the resolution and power achievable with a laser versus LED. Their approach is a little different, as they pose the question as, “what optics do we need to maintain the same resolution and power at the sample when using an LED vs. a laser?” They reach the same conclusions, but there may be some discrepancy in the factor by which the power decreases. Those calculations may need to be done using etendue, the 3D equivalent of the 2D optical invariant. Here is a link:

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