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HARMONI at ELT: line spread functions in a diffraction limited spectrometer

This paper outlines the impact of diffraction and spatial filtering effects on the line spread function of the HARMONI spectrograph for the ELT, identifying key parameters to consider when designing diffraction-limited instruments where slit widths are comparable to the point spread function.

Original authors: Stephen P. Todd (UK Astronomy Technology Centre), Éamonn J. Harvey (UK Astronomy Technology Centre), Anna MacIver (UK Astronomy Technology Centre), William Taylor (UK Astronomy Technology Centre), Edu

Published 2026-07-03

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CC BY 4.0

Original authors: Stephen P. Todd (UK Astronomy Technology Centre), Éamonn J. Harvey (UK Astronomy Technology Centre), Anna MacIver (UK Astronomy Technology Centre), William Taylor (UK Astronomy Technology Centre), Eduard Muslimov (Oxford University), Kjetil Dohlen (Laboratoire d'Astrophysique de Marseille), Matthias Tecza (Oxford University), Magali Loupias (Centre de Recherche Astrophysique de Lyon), Paula Bañares-Palacios (Instituto de Astrofísica de Canarias), Mark Swinbank (Durham University), Ryan Griffiths (Oxford University)

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Super-Camera for the Stars

Imagine the ELT (Extremely Large Telescope) as a giant, super-sharp camera lens. Attached to it is a device called HARMONI, which acts like a prism. Its job is to take the light from a star or galaxy and split it into a rainbow (a spectrum) so scientists can study what it's made of.

The paper discusses a specific problem the team found while designing HARMONI: The "Rainbow" was getting too thin.

The Problem: The "Too-Small" Slit

To understand the problem, imagine you are trying to measure the width of a beam of light using a ruler.

The Old Way (Geometric Optics): In the past, telescopes were a bit blurry (like looking through a foggy window). Scientists assumed the light would act like a solid beam of water coming out of a hose. If you put a narrow slit in front of it, the light coming out would just be the width of that slit.
The New Reality (Diffraction): Because the ELT is so perfect and uses "Adaptive Optics" (which corrects the atmosphere's blur), the light is incredibly sharp. When this super-sharp light passes through a narrow slit, it doesn't just go straight through like a solid beam. It starts to wiggle and spread out, like water rippling through a narrow gap in a dam. This is called diffraction.

The Surprise:
The team realized that because of this "wiggling" (diffraction) and the way the light waves line up perfectly (coherence), the beam of light coming out of the slit is actually narrower than the physical slit itself.

Think of it like this: If you try to squeeze a crowd of people through a narrow door, you expect a crowd to come out the other side. But in this specific physics scenario, the "crowd" (the light) actually organizes itself so tightly that the group coming out is smaller than the door they passed through.

The Consequence: Missing the Details

HARMONI uses a digital camera (a detector) to catch this light. The camera is made of tiny squares called pixels.

The Rule of Thumb: To measure a line of light accurately, you need to cover it with at least two pixels. This is like taking a photo of a thin wire; if the wire is thinner than the pixels, you might miss it or measure its position wrong.
The Issue: Because the light beam is narrower than expected (due to the diffraction effects mentioned above), the team realized that in some settings, the light would only cover less than two pixels.
The Result: This is called undersampling. It's like trying to read a very fine print with thick, blurry glasses. You might miss the exact center of a line or confuse two lines that are close together. This would ruin the scientific measurements.

The Solution: The "Stretchy" Prism

The team needed to make the light beam slightly wider so it would fit onto two pixels, but they couldn't just make the slit wider (that would blur the image too much) or change the main telescope (it's already built).

They found a clever fix using the grisms (prisms with a grating inside) inside the instrument.

The Analogy: Imagine you have a piece of elastic dough. If you pull it sideways, it gets wider but shorter. The team realized they could tweak the angle of the prisms to "stretch" the light beam in one direction (making it wider on the detector) without changing its color or quality.
The Trade-off: They had to be careful. If they stretched the beam too much for the "high zoom" mode (6 mas), it might stretch too much for the "wide angle" mode (25 mas), making the wide-angle view too blurry.

The Outcome

By calculating exactly how much to stretch the light using these prisms, they found a "sweet spot."

For the Wide View (25 mas): The fix works perfectly, ensuring the light covers exactly two pixels.
For the High Zoom (6 mas): The light is still slightly too narrow at the very longest wavelengths (red end of the spectrum), but the team decided this is an acceptable compromise. It's better to have a tiny bit of undersampling in one specific setting than to ruin the resolution for the rest of the instrument.

Summary

The paper is essentially a "physics check-up" for a high-tech telescope camera. The team realized that because the telescope is so sharp, the laws of physics (diffraction) make the light beams thinner than they expected. This threatened to make the camera's measurements inaccurate. They solved it by slightly "stretching" the light using the angles of the prisms inside the machine, ensuring the camera can see the stars clearly without losing detail.

Technical Summary: HARMONI at ELT: Line Spread Functions in a Diffraction Limited Spectrometer

Problem Statement
The paper addresses a critical design challenge for the HARMONI spectrograph, the first-light near-infrared integral field spectrograph for the European Extremely Large Telescope (ELT). HARMONI operates in a diffraction-limited regime, assisted by adaptive optics (SCAO and MCAO), with spatial sampling of 6 mas and 25 mas. In such systems, the slit (or slice) width is comparable to the diffraction-limited Point Spread Function (PSF).

The core problem is that standard geometric optics assumptions, which predict the Line Spread Function (LSF) width based solely on the geometric image of the slit, fail in this regime. Due to diffraction and spatial coherence effects caused by the finite size of the pupil aperture relative to the slit, the resulting LSF can be significantly narrower than the geometric slit width. The authors identified that, under the initial design parameters, HARMONI would suffer from spectral undersampling (LSF FWHM < 2 detector pixels) across all operating modes, particularly at longer wavelengths in the 6 mas scale. Undersampling risks systematic errors in wavelength calibration, centroid determination, and sky subtraction, degrading scientific performance.

Methodology
The authors developed a theoretical framework to model the LSF of a diffraction-limited spectrograph, moving beyond simple geometric optics to include spatial coherence.

Analytic Modeling: The study utilizes analytic expressions derived by Casini and Mielenz for ideal spectrographs. The model considers the entrance pupil, slit width ( $w_S$ ), grating aperture width ( $w_G$ ), and focal lengths of the collimator ( $f_C$ ) and camera ( $f_L$ ).
Coherence Regimes: The authors analyze three regimes:
- Fully Coherent: Occurs when the pupil is small and distant relative to the slit.
- Fully Incoherent: Occurs when the pupil image is large and close to the slit.
- Partial Coherence: The realistic case for astronomical instruments, calculated by integrating the complex amplitude transfer function over the entrance pupil, accounting for the circular nature of the telescope pupil.
Dimensionless Parameters: The analysis is normalized using two key parameters:
- $\phi = \frac{w_S w_G}{\lambda f_C}$ : Relates slit and grating dimensions to wavelength and focal length.
- $\Omega_G$ : The oversizing factor of the grating aperture relative to the geometric pupil image.
- $\alpha$ : The ratio of slit width to the diffraction-limited spot size at the slit.
Application to HARMONI: These models were applied to HARMONI's specific optical parameters (including the ELT, MORFEO, and IFU pre-optics). The authors calculated the expected FWHM across the 0.8–2.45 $\mu$ m wavelength range for both the 6 mas and 25 mas spatial scales.
Mitigation Strategy: A trade study was conducted to identify mitigation strategies that did not require redesigning the Integral Field Unit (IFU) or adding cryogenic mechanisms. The proposed solution involves adjusting the geometry of the Volume Phase Holographic (VPH) grisms to introduce a small anamorphic magnification factor, effectively widening the spectral line on the detector.

Key Results

Undersampling Prediction: The modeling confirmed that without mitigation, the HARMONI LSF would have an FWHM of less than 2 pixels at all wavelengths in both spatial scales. The undersampling is most severe in the 6 mas scale at long wavelengths (approaching the fully coherent regime where $\phi \approx 2$ ).
Coherence Impact: The study quantified that even when the slit width is matched to 10 times the diffraction limit ( $10\lambda/D$ ), a significant degree of spatial coherence ( $\mu_{RMS} > 0.2$ ) remains, causing the LSF to be narrower than geometric predictions.
Mitigation Efficacy: By utilizing the anamorphic magnification provided by the grism geometry, the authors demonstrated that it is possible to achieve the target 2-pixel sampling in the 25 mas scale across the full wavelength range.
Trade-offs: In the 6 mas scale, the mitigation strategy results in a small degree of residual undersampling at the longest wavelengths. The authors judge this to be an acceptable compromise to avoid excessive oversampling (and consequent loss of spectral resolution) in the 25 mas scale and at shorter wavelengths.

Significance and Claims
The paper claims to provide a practical methodology for incorporating diffraction and spatial coherence effects into the top-level design of astronomical spectrographs. The authors emphasize that while these effects have been known since Wadsworth (1897), they are frequently overlooked in initial design phases.

The significance of this work lies in:

Design Correction: Identifying a potential performance failure (undersampling) in a major instrument (HARMONI) during its design phase, allowing for a corrective strategy before finalization.
Generalizable Framework: Offering a set of dimensionless parameters ( $\phi$ , $\Omega_G$ ) that allow designers to quickly estimate the significance of diffraction effects and adjust optical parameters (slit width, grating aperture) to ensure adequate sampling before detailed optical design begins.
Realistic Constraints: Demonstrating how to mitigate diffraction-induced undersampling within the strict constraints of an advanced instrument where major optical components (like the IFU) are already in manufacture, avoiding costly redesigns.

The authors conclude that further work is required to develop robust tools for modeling these diffraction effects in combination with geometric optical aberrations in real, complex systems like HARMONI, which currently relies on computationally intensive Fourier optics simulations or spreadsheet-based error budgeting.