# Spherical aberration in high NA objectives

#### omg!

hi everybody,

i would appreciate it if someone could clarify the concept of spherical aberrations in the context of high NA objectives in which use lenses are used that are not exclusively of the spherical type.
a common thing that you hear is that some objective is corrected for 0.17mm coverglass of n=1.518, or something like that. i have always interpreted it as in this example:

as the wave front that emerges from a point light source traverses the cover glass and immersion oil towards the objective, the phase relation is changed depending on angle so that the wave front gets deformed. what a "corrected" objective then does is to compensate for this deformed wave front.

is this correct?
thank you very much for your help!

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#### Andy Resnick

hi everybody,

i would appreciate it if someone could clarify the concept of spherical aberrations in the context of high NA objectives in which use lenses are used that are not exclusively of the spherical type.
a common thing that you hear is that some objective is corrected for 0.17mm coverglass of n=1.518, or something like that. i have always interpreted it as in this example:

as the wave front that emerges from a point light source traverses the cover glass and immersion oil towards the objective, the phase relation is changed depending on angle so that the wave front gets deformed. what a "corrected" objective then does is to compensate for this deformed wave front.

is this correct?
thank you very much for your help!
Spherical aberration (SA) is (usually) the largest residual aberration in a well-corrected lens- *any* well corrected lens. Because SA scales as (IIRC) the square of the numerical aperture, well-corrected high-NA (typically immersion) microscope objectives can easily suffer from the presence of SA.

In a well-corrected immersion lens, the coverslip is actually part of the total optical design- the lens has been carefully designed to work with a very precise thickness of coverslip (some specialized lenses are designed to be adjusted for use with a range of coverslip thickness), and if the coverslip is the incorrect thickness, lens aberrations can get really large really fast- and especially SA. I know one (rather extreme) person who goes so far as to measure each coverslip with a micrometer before use, and discards ones that are outside his specification.

The reason the particular index of refraction is specified is also to minimize aberrations- the immersion oil should have the same refractive index as the coverslip, which is also the same refractive index as the mounting medium and glass slide (so-called 'homogeneous conditions').

http://www.microscopyu.com/articles/formulas/formulascoverslipcorrection.html

To be sure, there are lenses designed to be used with no coverslip- generally, reflected light objectives and dipping objectives.

Careful tweaking of the optics can fractionally improve performance: for example, because my laser tweezers use 1064nm light, there is significant (IIRC) positive SA which I can correct by using a higher-than-normal refractive index immersion oil, which introduces a compensating negative SA.

#### omg!

thank you very much for your clarification!

what i'm still asking myself is whether or not the cover glass correction corrects a phase factor of the electromagnetic field that it acquires by traveling through the cover glass. in other words, if the cover glass is thicker or thinner than it is supposed to be, do the waves that propagate towards the objective get an additional phase factor $$\exp ( i f(\theta))$$, where $$\theta$$ is the angle between the k-vector and the optical axis, that leads to spherical aberration? if so, what exactly is the function f ?

thanks again!

#### Andy Resnick

thank you very much for your clarification!

what i'm still asking myself is whether or not the cover glass correction corrects a phase factor of the electromagnetic field that it acquires by traveling through the cover glass. in other words, if the cover glass is thicker or thinner than it is supposed to be, do the waves that propagate towards the objective get an additional phase factor $$\exp ( i f(\theta))$$, where $$\theta$$ is the angle between the k-vector and the optical axis, that leads to spherical aberration? if so, what exactly is the function f ?

thanks again!
Sort of- as light passes through the coverslip, the wavefront picks up a phase factor given by what you wrote. The function f is given by the 'reduced thickness' of the glass (d/n, where d is the actual thickness and n the index of refraction), which will vary with both direction(as d varies) and color (as n varies).

If the phase factor is off-nominal (either d is out of tolerance, or n is out of tolerance), there will be a residual error in the phase that is not corrected by the imaging system. The specific phase function can be complicated, and is usually expanded in Zernike polynomials- orthogonal functions defined on the unit circle.

http://www.telescope-optics.net/zernike_coefficients.htm

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