# Gaussian Beam

In most cases the diSPIM has been used with Gaussian beams. As the beam waist gets thinner, the distance over which the beam/sheet is thin (called the confocal length) gets smaller according to well-known equations^{1)}. Thus there is a fundamental trade-off between the optical sectioning and the field of view. This tradeoff is controlled by the illumination beam's numerical aperture; in the ASI scanner there is an iris and/or aperture to adjust this. For a sheet created by a swept beam the equations can be directly applied, and for a cylindrical lens the same equations hold along the slice axis.

For an ideal Gaussian beam the waist thickness = k1 * lambda / NA_illum where

- k1 is a constant that depends on how you define the thickness of a Gaussian profile. Commonly k1 is taken to be 0.64, which corresponds to the two-sided beam “diameter” where the intensity has fallen to 1/e^2 of the max intensity (the beam thickness as measured by FWHM is a factor of 1.7 smaller).
- lambda is the wavelength, in this case the excitation laser
- NA_illum is the numerical aperture of the illumination

For Gaussian beams the beam/sheet extent or imaging field of view = k2 * lambda * n / NA_illum^2 where

- k2 is a constant that depends on how you define the acceptable intensity variation across the field of view. Commonly k2 is taken to be 0.64, which corresponds to the two-sided field of view where the beam thickness as measured by the 1/e^2 criteria has increased by a factor of sqrt(2).
- lambda is the wavelength, in this case the excitation laser
- n is the refractive index of the media (1.33 for water)
- NA_illum is the numerical aperture of the illumination

Note that the scaling relationship between the beam waist and length is quadratic; if you tolerate twice as thick of a beam it will be four times longer. In practice the illumination NA is adjusted (via scanner iris) to make the sheet sufficiently uniform across the object and then the sheet thickness simply “is what it is”. The above equations invoke the paraxial approximation so there is ~1% error at NA_illum of 0.3.

NA_illum is determined from the spot size of the beam at the objective back focal plane and the objective's focal length. The spot size depends on the iris diameter and the lenses inside the scanner and the scanner tube lens.

For a typical diSPIM setup, NA_illum = 0.366 * iris_D / EFL where

- 0.366 is the half the total magnification between iris and back focal plane
- iris_D is the iris diameter
- EFL is the effective focal length of the objective, e.g. 5mm for Nikon 40x, 9mm for Olympus 20x, and 12mm for the cleared tissue objective @ RI~1.45

The usual ASI scanner has an iris diameter that varies from ~0.7mm ^{2)} to ~3.5mm ^{3)}. So for 40x objectives the accessible NA range is ~0.05 to ~0.25. Taking k1=k2=0.64 in the equations above, with NA_illum~0.05 and 488nm excitation the beam waist is ~6.25um thick and the confocal length is ~125um, whereas with NA_illum~0.25 the waist is ~1.25um thick and the confocal length is ~5um. However, practical experience suggests that it's hard to make such a thin beam waist (perhaps because the gaussian quality degrades or isn't all captured on the MEMS mirror) and also suggests that the entire camera FOV (>300um) can be filled by a very reduced iris (perhaps because the iris closes more than the spec'ed minimum closure).

**Important note:** rarely is the beam truly Gaussian, which means that it the sheet will be somewhat thicker/shorter than expected. This can be due to a variety of imperfections including that the beam may not be pure Gaussian from the fiber, aberrations in the optical elements, and apertures that cut off the beam before it has decayed far beyond it's 1/e point. The last factor is especially likely because the iris truncates the beam to be somewhat flat-top instead of true Gaussian.

Commonly only a small fraction of the objective's NA is used for illumination (i.e. light sheet generation), which implies that there is usually no resolution benefit from the light sheet, although the out of focus florescence will be reduced (“optical sectioning”) which improves SNR and image quality. There are ways of creating very thin light sheets to increase axial resolution (e.g. Bessel beams) but they generally have other undesirable properties including extra complexity/cost, large amounts of out of focus light, and/or being extremely sensitive to scattering or sample inhomogeneity.

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