Although M2 and Strehl ratio are both measures of optical quality, they do not measure the same thing.

M2 is a measure of how much a beam deviates from an ideal diffraction-limited Gaussian beam.

Strehl ratio is generally defined as the ratio of peak aberrated image intensity from a point source, with respect to the maximum attainable intensity using an ideal optical system limited only by diffraction over the system aperture https://en.wikipedia.org/wiki/Strehl_ratio#Mathematical_definition

This definition of Strehl ratio implicitly assumes that the aperture of the optical system is uniformly filled. With a circular aperture or entrance pupil, the reference "input beam" has a flat-topped intensity profile, while the diffraction-limited point spread function is an Airy disk https://en.wikipedia.org/wiki/Airy_disk

The behaviour with an approximately Gaussian input beam will depend on the extent to which the Gaussian beam is able to fill the entrance pupil. If it over-fills the aperture, the image spot will not be Gaussian. If it under-fills the aperture, the peak intensity will be greatly reduced, but the profile can be more closely Gaussian, and the output beam more likely to have a low M2.

Measuring Strehl ratio with a Gaussian input beam is problematic. How do you define what the aberration-free maximum intensity should be?

In addition, there is no requirement for M2 to increase when the Strehl ratio of the optical system is less than unity. Consider a Gaussian beam where the input beam quality and M2 are degraded by spherical aberration. Suppose the optical system introduces a similar level of spherical aberration. Depending on the relative signs, the aberrations can either combine and further degrade (increase) M2, or they can cancel, with a decrease in M2.

In general, without knowing the nature and magnitude of the aberrations in the input beam and the optical system, there is no way to predict how the combination will behave, if all you know is the magnitudes of M2 and Strehl Ratio.

Thx

Let me reformulate my question succinctly

Given a M2 is there a maximum strehl ratio

@Samuel: What do you mean by "a maximum strehl ratio

Thx Alan,

I infer from your answers that M2 and Strehl are rather independent metrics.

What I meant was, having an optical system with Strehl = 1 (so perfect system which would perform diffracted limited for a uniform illumination). And now I illuminated this optical system with a Gaussian beam (no under or overfilling of the aperture), whose M2=1 (so a perfect one), and now calculate the Strehl (relative to diffractive limited uniform illumination), this would lead to Strehl = 0,924

(would you agree to that?)

Now the question is, If I illuminate this optical system with a Gaussian beam (no under or overfilling of the aperture), whose M2>1, is there an expression to calculate (or at least rather estimate) the Strehl (relative to diffractive limited uniform illumination)?

And the most general one, If I illuminate an optical system with Strehl 1, is there an expression to calculate (or at least rather estimate) the Strehl (relative to diffractive limited uniform illumination)?

But maybe I am too much focused on my problem, and actually the solution has nothing to do with combining M2 and Strehl into a single metric

Samuel Arba Mosquera wrote: "... an optical system with Strehl = 1 (so perfect system which would perform diffracted limited for a uniform illumination). And now I illuminated this optical system with a Gaussian beam (no under or overfilling of the aperture), whose M2=1 (so a perfect one)"

By "no under or overfilling", do you mean that the diameter of the Gaussian beam is equal to the diameter of the optical system's aperture or entrance pupil? Is the beam diameter measured at 1/e2 intensity, or in some other way?

"and now calculate the Strehl (relative to diffractive limited uniform illumination)". Is your reference uniform illumination with irradiance (W m-2) equal to the Gaussian beam's peak irradiance, or do you normalise so that the input power into the system is equal to the total beam power, or just that part of the beam which enters the system aperture?

Normalising with respect to total power into the entrance pupil is reasonable - but I don't know exactly what you are trying to solve.

"this would lead to Strehl = 0,924". Strictly speaking, the Strehl ratio of the instrument is still 1.0. In the absence of aberrations and waveform distortion, we can work back from your peak intensity ratio to calculate the intensity (irradiance) profile across the aperture, given your choice of reference, and the known Gaussian shape.

"Now the question is, If I illuminate this optical system with a Gaussian beam (no under or overfilling of the aperture), whose M2>1, is there an expression to calculate (or at least rather estimate) the Strehl (relative to diffractive limited uniform illumination)?"

Now you have an aberration-free optical system, whose aperture is illuminated by a truncated nearly Gaussian beam with some unknown combination of phase and amplitude distortion. In general, you need to know the spatial distribution of distortions which result in M2 > 1. If the input wavefront is substantially planar, but with outer diffraction rings which are blocked at the entrance pupil, then you will get a well-imaged spot, with potentially lower M2 than the input beam, and a high peak intensity in the image. If there is significant phase distortion across the parts of the wavefront entering the system aperture, then the peak image intensity will be lower. You need additional information to calculate the peak image intensity ratio.

"And the most general one, If I illuminate an optical system with Strehl 1, is there an expression to calculate (or at least rather estimate) the Strehl (relative to diffractive limited uniform illumination)?"

Again, NO. If the aberrations in the optical system are described by low amplitude phase errors at the outer part of the aperture where the input intensity is low, then the additional distortion may be low. It the phase errors are larger, but concentrated in a smaller region in the centre of the aperture where the Gaussian input beam is strongest, then the distortion will be larger.

In the case of an adaptive optics system, the phase errors can be adjusted to cancel any phase distortion in the Gaussian beam, so that the output is less distorted with lower M2 than the input.

If you have more detailed information about the distortion in the Gaussian beam, and about the nature and distribution of the aberrations in the optical system, you may be able to make statistical predictions about the combination.

"But maybe I am too much focused on my problem, and actually the solution has nothing to do with combining M2 and Strehl into a single metric"

What is the problem you are trying to solve?

By "no under or overfilling", I mean that the diameter of the Gaussian beam at 1/e2 intensity is equal to the diameter of the optical system's aperture or entrance pupil.

Actually I have two general questions (also placed hier at RG):

https://www.researchgate.net/post/What_is_the_minimum_focusable_spot_size_in_a_given_medium

https://www.researchgate.net/post/Is_there_an_approximate_equation_to_compute_the_effect_of_Strehl_Ratio_into_spot_size

I want to calculate the spot size at focus of a gaussian laser in a given medium.

I know the following parameters:

Wavelength of the laser source in vaccuumm (air)

M2 of the laser source

Pulse length

NA of the optical system

Strehl of the overall system (computed in Zemax with Gaussian illumination as setting)

Refractive index of the medium (for that particular wavelength)

and I want to calculate the best focusable spot in this condition

Would you have an idea on how to approach it (using a "simple" equation, and not simply letting Zemax do the job for me)

Samuel,

Do you know anything about how the input deviates from an ideal Gaussian beam, and the nature and distribution of the phase distortion in the optical system?

Without such information it is not possible to make a reliable prediction of spot size.

Is the "spot size" you wish to calculate the 1/e2 diameter?

In your question: https://www.researchgate.net/post/What_is_the_minimum_focusable_spot_size_in_a_given_medium you combining the resolution expressed as the diameter of the first zero of the Airy disk diffraction pattern with a Fourier relationship defined in terms of second moments. How do you justify this?

The Airy disk diameter is (1.22 vacuum_wavelength / NA), where NA = n sin θ

What matters is the transverse component of the wave number, and operating in a higher refractive index medium allows this to be larger (smaller wavelength and higher transverse spatial frequency).

Strictly speaking, the relationship between beam waist diameter, far field divergence, wavelength and M2 only applies when the beam diameters are defined in terms of the second moment of the intensity - the D4σ method specified in ISO standard 11146 https://www.rp-photonics.com/beam_radius.html

https://www.spiedigitallibrary.org/conference-proceedings-of-spie/1868/1/Defining-measuring-and-optimizing-laser-beam-quality/10.1117/12.150601.full?SSO=1

Truncation of a near-Gaussian beam by an aperture can have a significant impact on D4σ, which is sensitive to intensity distortion at large distances from the beam centre. The M2 of an Airy disk image spot can be surprisingly large, but is improved considerably when focussed onto a pinhole whose diameter is equal to that of the first dark band in the diffraction pattern.

It follows that the M2 of the image spot could be lower than the M2 of the input beam. It depends on the details of the amplitude and wavefront distortion in the beam, and the magnitude, sign and distribution of phase distortion in the optical system.

A related consideration is that Strehl ratio tells you how much energy is scattered away from the centre of the image spot, but is insensitive to where it is scattered. In contrast, M2 is much more sensitive to intensity changes at large distances from the beam centre than to small changes near the centre.

The two metrics measure different things, and need not be closely correlated. In order to link the two you need to understand the spatial distribution of aberrations and distortions.

Image spots for ideal Airy disk diffraction, Gaussian and truncated Gaussian input beams are shown in the attached figure. The second figure shows that the Airy disk has significant energy in the diffraction rings at larger radii. This significantly increases the second moment and D4σ compared with the FWHM or 1/e2 diameter.

Thank you very much!

excuse me the question, from your expression:

"The Airy disk diameter is (1.22 vacuum_wavelength / NA), where NA = n sin θ"

would you agree that the airy disk is an ideal metric for spot size?

from your expression it reads that Airy disk size is actually independent from the medium in which you are focusing

but I always understood, spots become larger in media with higher refractive index since theta becomes smaller (i.e. effectively less focused)

but this may have been a misperception?

Samuel: "excuse me the question, from your expression:

'The Airy disk diameter is (1.22 vacuum_wavelength / NA), where NA = n sin θ'

would you agree that the airy disk is an ideal metric for spot size?"

Ideal for what? The Airy disk radius is often used as a measure of spatial resolution, in accordance with the Rayleigh criterion. http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/Raylei.html

https://en.wikipedia.org/wiki/Angular_resolution#Explanation

If you wish to maximise the peak intensity, or optimise the power coupled through a circular aperture, then other measures may be more appropriate. If you use the M2 factor to relate near and far field beam diameters, then you should use the D4σ measure of beam width.

"... Airy disk size is actually independent from the medium in which you are focusing, but I always understood, spots become larger in media with higher refractive index since theta becomes smaller (i.e. effectively less focused)"

Think in terms of the spatial frequencies into which the point spread function can be resolved.

The maximum transverse component of spatial frequency is (n/λ0) sinθ = NA/λ0

where λ0 is vacuum wavelength. If the angle θ is reduced when entering a refractive medium (n>1), wavelength is reduced by a similar factor https://en.wikipedia.org/wiki/Snell's_law.

This is why the highest resolution microscopes use oil immersion objectives, so that numerical apertures higher than unity are possible.

https://www.microscopyu.com/microscopy-basics/resolution

https://www.zeiss.com/microscopy/int/solutions/reference/basic-microscopy/numerical-aperture-and-resolution.html

Born & Wolf "Principles of Optics", 6th ed, 1980, section 8.6.3, p 418.