Does “n” belong? (or, getting light sheet equations correct)

A few months ago I stumbled across a discrepancy between formulas used to describe the imaging and illumination paths of light sheet microscopes.  I believe that one of the oft-cited equations in the world of light sheet microscopy is missing a term.  I could be wrong, but hear me out.  Comments or corrections welcome.

The TLDR; version: I claim the confocal parameter for light sheet microscopy (the length of sheet thinness) depends on the refractive index of the medium, but this term is not present in any formulation I have seen. For most light sheet microscopy the error amounts to 33%, which is small enough that most people wouldn’t notice. But if you are going to write an equation down then you may as well get it right!

The symbol n represents refractive index, which is a measure of the relative speed of light in the material.  n is why light bends when it enters a lens [fn1].  Microscopists go to great lengths to match n from the the medium to the sample in order to minimize blurring from spherical aberrations.  But because n is dimensionless, it is easy to forget in equations and you still get an answer which makes sense.  Also it is a modest scale factor, usually in microscopy between 1.00 (air) and 1.51 (oil, coverslip glass).

To tell the story I will use the correspondence between expressions for the light sheet dimensions and resolution of imaging optics, which have the same underlying physics. The imaging objective lens has a focal plane which we idealize as being infinitely thin, but it has finite thickness due to the wavelength of light and the inability of the lens to capture all the rays from the focal plane.  This thickness is called the depth of field [fn2], and it can be expressed mathematically as:

DoF = k1 *  λ * n / NA2

where the symbol k1 is a numeric constant whose value depends on the threshold chosen to define the “edge” of the focal plane even though the defocus roll-off is continuous, λ is the wavelength of the light, n is the refractive index of the medium, and NA is the numerical aperture of the objective lens (which represents the angular range of the rays captured by the objective).  The depth of field is equivalent to the intrinsic axial resolution of the objective, potentially with a different choice of prefactor k1. There is a separate equation which represents the lateral resolution of the objective lens which is proportional to (λ0/NA) without a factor of n.

In light sheet microscopy you shine light through a lens to create the light sheet.  The light sheet has a thickness and a “confocal parameter” or length, which is essentially the distance over which the sheet is thin before it diverges.  To the consternation of light sheet aficionados, the relationship between thickness and the confocal parameter is quadratic, e.g. having a sheet half as thick means it is only a quarter as long. The oft-cited but (I believe) incorrect equation for the confocal parameter for the typical case of a Gaussian light sheet is

CP = k2 *  λ  / NAi2   ( incorrect?!)

where the symbol k2 is a numeric constant whose value depends on the threshold chosen to define the “edge” of the sheet even though the intensity roll-off is continuous, λ is the wavelength of the light, and NAi is the numerical aperture of the illumination (which represents the angular range of rays exiting the illumination objective).  There is a separate equation which represents the thickness of the light sheet which is proportional to (λ/NA) without a factor of n.

The depth of field and confocal parameter represent the same thing, “the range of thinness” from the point of view of the imaging and illuminating optics respectively. If you stare at the equations for DoF and CP you will notice the latter is missing the factor of n.  Why?  The physics of light don’t change if you run it backwards through a lens.  The numeric constants k1 and k2 might be different due to different threshold definitions, but the equations disagree on whether the refractive index matters or not. The other set of corresponding equations — imaging lateral resolution and illumination sheet thickness — line up nicely with each other, why not the equations for imaging depth of field and light sheet confocal parameter?

I believe the answer is sloppiness [fn3] and that the factor of n belongs in the expression for light sheet confocal parameter.  The Gaussian beam equations are usually derived in vacuum/air which has refractive index of 1.00, so in that particular instance it is safe to omit the n to simplify both the derivation and the result.  I suspect everyone simply has been applying the in-air equation (without the n) to light sheet microscopy (where n is most often 1.33) without noticing the simplification made in the earlier derivation.  I am among the guilty ones parroting an equation that I now believe is not quite accurate.

Next I will derive what I claim to be the correct result up to a constant factor starting with two key equations from the Wikipedia entry for Gaussian beam:

(1) zR = π * ω02 * n / λ0
(2) θ = λ0 / ( π * n * ω0 )

where zR is the one-sided Rayleigh range (equivalent to the half the confocal parameter), ω0 is the one-sided thickness of the beam waist, and θ is the divergence or long-range half-angle of the light. Note we have made it explicit than the wavelength of light in a particular medium depends on the refractive index of the medium: λ(n) = λ0/n where λ0 is the vacuum wavelength. This fact is obvious for physicists but not widely known among microscopists because the source and detector are usually in air/vacuum. From here on we will be careful to make it explicit that we are using the vacuum wavelength λ0, which is the wavelength that everyone quotes in microscopy as well as physics (e.g. 488 nm laser for GFP excitation).

To be able to express things in term of NA we use a third equation where we invoke the small-angle (“paraxial”) approximation:

(3) NA = n * sin(θ) ≈ n * θ

By rearranging equation (2) and using equation (3) to replace θ by NA we see:

ω0 = λ0 / ( π * n * NA / n) = λ0 / ( π * NA )

Notice that both the numerical aperture and the wavelength in the medium both have factors of n but the terms cancel out, so the equation for the beam waist thickness depends only on the vacuum wavelength and numerical aperture [fn4], regardless of the medium refractive index.

Now let’s plug in that result into equation (1) to express the Rayleigh length as a function of NA:

zR = ( π * n * λ02) / ( λ0 * π2 * NA2) = ( λ0 * n ) / ( π * NA2)

Notice that the Rayleigh range has a factor of n in the numerator, as I am claiming that it should.

Finally, I will propose an experiment for testing my assertion. It is this: measure the light sheet thickness t and confocal parameter CP for different values of NAi and in two different media with different value of n, e.g. using ASI’s multi-immersion objective lens. The criteria used to determine the thickness and confocal parameter doesn’t matter as long as it is consistent. Compute the ratio of these values (t/CP) and plot the ratio versus NAi. If my reasoning is correct, then the points will form two lines with different slopes depending on the value of n. If the traditional equation is correct, all the points will lie on a single line.

I’m happy to receive rebuttals, corrections, or learn of experimental results.  If you agree with me and are writing and/or presenting about light sheet microscopy, let’s be more accurate and write the confocal parameter as

CP = k2 *  λ0  * n / NAi2

OK, enough nitpicking nerdiness for one day!  Oh, and special thanks to the people who have so far assured me via Twitter and email that I am not crazy about this.

[fn1] To be more precise, the ratio between refractive indexes across the interface determines the bend (~1.5x the angle for air-glass), and the fractional difference determines how much light is reflected (~4% for air-glass).

[fn2] Depth of field and depth of focus are often used interchangeably, but to be precise the depth of focus is on the side of the lens where the camera goes and the depth of field is on the side of the lens of the object.

[fn3] Take one look at my office and you’d see that I have to be careful about calling anyone sloppy!

[fn4] The numerical aperture value is based on air (i.e. it explicitly contains a term to reflect the refractive index).