<p>Maybe this is a bit too late, but I think I can answer this question, from my experience with developing echelle spectrometers (not for astronomy though).</p>
<p>As pointed out, both gratings and prisms can be used as a cross-disperser.
However, their dispersion property are different.
Prisms are often prefarable to evenly space the spectra with different diffraction orders from echelle grating.</p>
<p>The wavelength separation of each order from echelle grating depends on the wavelength.</p>
<p>We may start from the grating equation,
<span class="math-container">$$
\sin\alpha + \sin\beta = mN\lambda,
$$</span>
where <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> is the incident and diffracted angles (with the quasi litrow configuration <span class="math-container">$\alpha \approx \beta$</span>), <span class="math-container">$m$</span> is the diffraction order, and <span class="math-container">$N$</span> is the groove density of the grating.
The wavelength separation for <span class="math-container">$m$</span>-th and <span class="math-container">$(m+1)$</span>-th order lights is
<span class="math-container">$$
\Delta \lambda = N \lambda.
$$</span>
The separation is proportional to the wavelength. At the longer wavelength, the separation becomes larger.
In order to evenly space each order, the dispersion of the cross-disperser should compensate this dependence, i.e., less dispersion for the longer wavelength would be demanded.</p>
<p>If we use a grating as a cross-disperser, the dispersion angle for <span class="math-container">$m$</span>-th and <span class="math-container">$(m+1)$</span>-th order lights are
<span class="math-container">$$
\Delta \beta' = \frac{\cos\beta'}{N'},
$$</span>
where <span class="math-container">$\beta'$</span> is the diffraction angle by this grating (i.e., the cross-disperser, not the echelle grating), and <span class="math-container">$N'$</span> is the groove density.
<span class="math-container">$\Delta \beta'$</span> is nearly independent on <span class="math-container">$\lambda$</span>, which means that the separation of each order becomes larger in the longer wavelength region.</p>
<p>On the other hand, the dispersion by the prism is determined by the wavelength-dependence of the glass material.
This depends on particular material, but as a very rough approximation, the refractive index <span class="math-container">$n$</span> depends on <span class="math-container">$\lambda$</span>,
<span class="math-container">$$
\frac{d n}{d\lambda} \propto \lambda^{-1},
$$</span>
with a coefficient.
This compensates the wavelength dependence on the order separation.</p>
<p>Thus, prisms are often better as a cross-disperser, particularly when we want to measure wide wavelength range.</p>

CC BY-SA 4.0
2023-03-22T19:00:24.320Z
I think you have it. I think with a grating cross-disperser, the problem is, especially when you want an echelle that works over a broad wavelength range, that if you set a certain slit height with a dekker, then whilst the orders can be widely separated in the red (and hence wasting silicon that could be filled with data)  they might actually be overlapping in the blue. If you choose your prism correctly then you can keep the order separation at the right size for all wavelengths using a sensible slit height, whilst minimising wasted silicon.
ProfRob
CC BY-SA 4.0
2023-03-22T19:01:08.333Z
You might be able to answer https://astronomy.stackexchange.com/questions/44781/what-are-the-pros-and-cons-of-different-types-of-echelle-spectrograph-cross-disp?noredirect=1&lq=1
ProfRob
CC BY-SA 4.0
2023-03-24T01:02:45.490Z
Very nice: I already upvoted! This helps me understand the order spacing in the echellograms I get with my homemade echelle spectrographs, e.g., https://astronomy.stackexchange.com/a/49116/45954.
Ed V
CC BY-SA 4.0
2023-03-22T18:34:02.953Z
2023-03-22T18:34:02.953Z
Keisuke FUJII
<p>The catchy title <a href="https://arxiv.org/abs/2012.02487" rel="nofollow noreferrer">Down-the-barrel observations of a multiphase quasar outflow at high redshift: VLT/X-shooter spectroscopy of the proximate molecular absorber at z=2.631 towards SDSS J001514+184212</a> mentions <a href="https://it.wikipedia.org/wiki/X-shooter" rel="nofollow noreferrer">X-shooter</a> (Italian only?) which is described in detail in <a href="https://www.aanda.org/articles/aa/full_html/2011/12/aa17752-11/aa17752-11.html" rel="nofollow noreferrer">X-shooter, the new wide band intermediate resolution spectrograph at the ESO Very Large Telescope</a>:</p>
<blockquote>
<p>Abstract</p>
<p>X-shooter is the first 2nd generation instrument of the ESO Very Large Telescope (VLT). It is a very efficient, single-target, intermediate-resolution spectrograph that was installed at the Cassegrain focus of UT2 in 2009. The instrument covers, in a single exposure, the spectral range from 300 to 2500 nm. It is designed to maximize the sensitivity in this spectral range through dichroic splitting in three arms with optimized optics, coatings, dispersive elements and detectors. It operates at intermediate spectral resolution (R ~ 4000−17 000, depending on wavelength and slit width) with fixed échelle spectral format (<strong>prism cross-dispersers</strong>) in the three arms. It includes a 1.8″ × 4″ integral field unit as an alternative to the 11′′ long slits. A dedicated data reduction package delivers fully calibrated two-dimensional and extracted spectra over the full wavelength range. We describe the main characteristics of the instrument and present its performance as measured during commissioning, science verification and the first months of science operations.</p>
</blockquote>
<p>To get large dispersion Echelle gratings are operated in high order so you need some kind of cross-dispersion to separate the orders. The one's I've seen use a first-order grating with lines perpendicular to those of the Echelle for cross-dispersion.</p>
<p>X-shooter uses double passes through prisms in each of the three arms of this 300 to 2500 nm single-slit (though there's a 1.8′′×4′′ to 0.6′′×12′′ slicer), single-target spectrograph.  For UVB and visible there is one prism, for NIR it's a double pass through <em>three prisms!</em></p>
<blockquote>
<p>The collimated beam passes through a <strong>60° silica prism twice to gain enough cross-dispersion.</strong> The main dispersion is achieved through a 180 grooves/mm échelle grating blazed at 41.77°.</p>
</blockquote>
<blockquote>
<p>The optical layout of the VIS spectrograph is very similar to that of the UVB... For cross-dispersion, it uses a <strong>49° Schott SF6 prism in double pass</strong>. The main dispersion is achieved through a 99.4 grooves/mm, 54.0° blaze échelle grating.</p>
</blockquote>
<blockquote>
<p>(For the) NIR spectrograph... In order to get enough cross dispersion, <strong>three prisms are used in double pass</strong>. Prism 1 is a 35° top angle made of Infrasil; prisms 2 and 3 are two 22° top angle ZnSe prisms of the largest available thickness (56 mm). This design provides an almost constant order separation. The main dispersion is provided by a 55 grooves/mm échelle grating with a blaze angle of 46.07°.</p>
</blockquote>
<p><strong>Question:</strong> Why does X-shooter use double passes through prisms for Echelle cross-dispersion instead of gratings? Wouldn't this make it harder to get roughly even order separation?</p>
<p>Dispersion curves (<span class="math-container">$n(\lambda)$</span>) are notoriously nonlinear, they shoot up as you approach short wavelengths were absorption edges happen in the material (<em>ergo</em> <a href="https://en.wikipedia.org/wiki/Refractive_index#Typical_values" rel="nofollow noreferrer">Cauchy's</a> <span class="math-container">$n(\lambda) \approx A + B/\lambda^2 + C/\lambda^4$</span>) so prisms squish longer wavelengths and stretch the shorter ones, whereas gratings disperse roughly linearly with wavelength, as <span class="math-container">$\Delta sin \theta = \lambda m / d$</span>. Why the choice of prisms in this case?</p>
<hr />
<p>Figure 6. shows the NIR spectrograph. &quot;Double pass&quot; means the light passes through the prisms going towards the Echelle and again after being back-diffracted. That's <em>twelve additional optical surfaces!</em></p>
<p><a href="https://i.sstatic.net/MN0zq.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/MN0zq.jpg" alt="Figure 6. The NIR spectrograph optical layout." /></a></p>
<blockquote>
<p>Figure 6. The NIR spectrograph optical layout.</p>
</blockquote>
<hr />
<p><a href="https://i.sstatic.net/oDmm6.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/oDmm6.jpg" alt="Raw images obtained simultaneously by the three detectors of the X-Shooter instrument on the VLT" /></a></p>
<p>Source: <a href="https://www.eso.org/public/images/xshooter_cc/" rel="nofollow noreferrer">Raw images obtained simultaneously by the three detectors of the X-Shooter instrument on the VLT</a></p>
<blockquote>
<p>The light of a celestial object, in this case a quasar, is dispersed according to its wavelength, or colour to form a very long, very high resolution spectrum covering the full wavelength range from ultraviolet to infrared. The optical elements in the spectrograph arrange this long spectrum in a series of consecutive, shorter spectra, called orders. In this illustration, the wavelength increases from bottom to top of each spectrum, and from left to right. The spectrum of the quasar, recording its intensity as a function of the wavelength, is the thin bright lines visible in each of the orders. The short bright nearly horizontal lines correspond to colours where the atmosphere shines brightly.</p>
</blockquote>

CC BY-SA 4.0
2021-07-09T00:42:15.730Z
Related: [How does the ESPRESSO Echelle spectrograph fold the spectrum so nicely?](https://astronomy.stackexchange.com/q/23873/7982) (currently in need of an answer) and in History of Science and Mathematics SE: [Where does “the grating equation” come from? Does it have a another name?](https://hsm.stackexchange.com/q/12591/3869)
uhoh
CC BY-SA 4.0
2021-07-09T07:10:40.270Z
Echelle spectrographs can use either. I think a less specific question might be better.
ProfRob
CC BY-SA 4.0
2021-07-09T08:01:19.287Z
@ProfRob The question already explains that they can use either, and why it seems to me that gratings would be better. I've asked about a specific instrument because I can read about it and include it as my "prior research". One is welcome to answer broadly why *prisms are better or chosen* in some cases and explain why this is likely to be one of those cases. But having a good example of heavy use of prisms in the question may make the answer easier as the example is already here.
uhoh
CC BY-SA 4.0
2021-07-09T08:19:33.923Z
The answer may be specific to Xshooter.
ProfRob
CC BY-SA 4.0
2021-07-09T09:08:40.917Z
@ProfRob then that's good, since that's what I'd like to know!
uhoh
<observational-astronomy><spectroscopy><optics><instruments>
Why does X-shooter use double passes through prisms for Echelle cross-dispersion instead of gratings?
uhoh
# Why does X-shooter use double passes through prisms for Echelle cross-dispersion instead of gratings?

(Asked by: username_0 on 2021-07-09)

The catchy title [Down-the-barrel observations of a multiphase quasar outflow at high redshift: VLT/X-shooter spectroscopy of the proximate molecular absorber at z=2.631 towards SDSS J001514+184212](https://arxiv.org/abs/2012.02487) mentions [X-shooter](https://it.wikipedia.org/wiki/X-shooter) (Italian only?) which is described in detail in [X-shooter, the new wide band intermediate resolution spectrograph at the ESO Very Large Telescope](https://www.aanda.org/articles/aa/full_html/2011/12/aa17752-11/aa17752-11.html):

> Abstract
>
> X-shooter is the first 2nd generation instrument of the ESO Very Large Telescope (VLT). It is a very efficient, single-target, intermediate-resolution spectrograph that was installed at the Cassegrain focus of UT2 in 2009. The instrument covers, in a single exposure, the spectral range from 300 to 2500 nm. It is designed to maximize the sensitivity in this spectral range through dichroic splitting in three arms with optimized optics, coatings, dispersive elements and detectors. It operates at intermediate spectral resolution (R ~ 4000−17 000, depending on wavelength and slit width) with fixed échelle spectral format (**prism cross-dispersers**) in the three arms. It includes a 1.8″ × 4″ integral field unit as an alternative to the 11′′ long slits. A dedicated data reduction package delivers fully calibrated two-dimensional and extracted spectra over the full wavelength range. We describe the main characteristics of the instrument and present its performance as measured during commissioning, science verification and the first months of science operations.

To get large dispersion Echelle gratings are operated in high order so you need some kind of cross-dispersion to separate the orders. The one's I've seen use a first-order grating with lines perpendicular to those of the Echelle for cross-dispersion.

X-shooter uses double passes through prisms in each of the three arms of this 300 to 2500 nm single-slit (though there's a 1.8′′×4′′ to 0.6′′×12′′ slicer), single-target spectrograph. For UVB and visible there is one prism, for NIR it's a double pass through *three prisms!*

> The collimated beam passes through a **60° silica prism twice to gain enough cross-dispersion.** The main dispersion is achieved through a 180 grooves/mm échelle grating blazed at 41.77°.

> The optical layout of the VIS spectrograph is very similar to that of the UVB... For cross-dispersion, it uses a **49° Schott SF6 prism in double pass**. The main dispersion is achieved through a 99.4 grooves/mm, 54.0° blaze échelle grating.

> (For the) NIR spectrograph... In order to get enough cross dispersion, **three prisms are used in double pass**. Prism 1 is a 35° top angle made of Infrasil; prisms 2 and 3 are two 22° top angle ZnSe prisms of the largest available thickness (56 mm). This design provides an almost constant order separation. The main dispersion is provided by a 55 grooves/mm échelle grating with a blaze angle of 46.07°.

**Question:** Why does X-shooter use double passes through prisms for Echelle cross-dispersion instead of gratings? Wouldn't this make it harder to get roughly even order separation?

Dispersion curves ($n(\lambda)$) are notoriously nonlinear, they shoot up as you approach short wavelengths were absorption edges happen in the material (*ergo* [Cauchy's](https://en.wikipedia.org/wiki/Refractive_index#Typical_values) $n(\lambda) \approx A + B/\lambda^2 + C/\lambda^4$) so prisms squish longer wavelengths and stretch the shorter ones, whereas gratings disperse roughly linearly with wavelength, as $\Delta sin \theta = \lambda m / d$. Why the choice of prisms in this case?

---

Figure 6. shows the NIR spectrograph. "Double pass" means the light passes through the prisms going towards the Echelle and again after being back-diffracted. That's *twelve additional optical surfaces!*

[![Figure 6. The NIR spectrograph optical layout.](https://i.sstatic.net/MN0zq.jpg)](https://i.sstatic.net/MN0zq.jpg)

> Figure 6. The NIR spectrograph optical layout.

---

[![Raw images obtained simultaneously by the three detectors of the X-Shooter instrument on the VLT](https://i.sstatic.net/oDmm6.jpg)](https://i.sstatic.net/oDmm6.jpg)

Source: [Raw images obtained simultaneously by the three detectors of the X-Shooter instrument on the VLT](https://www.eso.org/public/images/xshooter_cc/)

> The light of a celestial object, in this case a quasar, is dispersed according to its wavelength, or colour to form a very long, very high resolution spectrum covering the full wavelength range from ultraviolet to infrared. The optical elements in the spectrograph arrange this long spectrum in a series of consecutive, shorter spectra, called orders. In this illustration, the wavelength increases from bottom to top of each spectrum, and from left to right. The spectrum of the quasar, recording its intensity as a function of the wavelength, is the thin bright lines visible in each of the orders. The short bright nearly horizontal lines correspond to colours where the atmosphere shines brightly.

**Tags:** observational-astronomy, spectroscopy, optics, instruments
> **Comments:**
> **username_0**: Related: [How does the ESPRESSO Echelle spectrograph fold the spectrum so nicely?](https://astronomy.stackexchange.com/q/23873/7982) (currently in need of an answer) and in History of Science and Mathematics SE: [Where does “the grating equation” come from? Does it have a another name?](https://hsm.stackexchange.com/q/12591/3869)  
> **username_1**: Echelle spectrographs can use either. I think a less specific question might be better.  
> **username_0**: @ProfRob The question already explains that they can use either, and why it seems to me that gratings would be better. I've asked about a specific instrument because I can read about it and include it as my "prior research". One is welcome to answer broadly why *prisms are better or chosen* in some cases and explain why this is likely to be one of those cases. But having a good example of heavy use of prisms in the question may make the answer easier as the example is already here.  
> **username_1**: The answer may be specific to Xshooter.  
> **username_0**: @ProfRob then that's good, since that's what I'd like to know!  

---

## 1 answers

### Answer by username_2 on 2023-03-22. Score: 3

Maybe this is a bit too late, but I think I can answer this question, from my experience with developing echelle spectrometers (not for astronomy though).

As pointed out, both gratings and prisms can be used as a cross-disperser.
However, their dispersion property are different.
Prisms are often prefarable to evenly space the spectra with different diffraction orders from echelle grating.

The wavelength separation of each order from echelle grating depends on the wavelength.

We may start from the grating equation,
$$
\sin\alpha + \sin\beta = mN\lambda,
$$
where $\alpha$ and $\beta$ is the incident and diffracted angles (with the quasi litrow configuration $\alpha \approx \beta$), $m$ is the diffraction order, and $N$ is the groove density of the grating.
The wavelength separation for $m$-th and $(m+1)$-th order lights is
$$
\Delta \lambda = N \lambda.
$$
The separation is proportional to the wavelength. At the longer wavelength, the separation becomes larger.
In order to evenly space each order, the dispersion of the cross-disperser should compensate this dependence, i.e., less dispersion for the longer wavelength would be demanded.

If we use a grating as a cross-disperser, the dispersion angle for $m$-th and $(m+1)$-th order lights are
$$
\Delta \beta' = \frac{\cos\beta'}{N'},
$$
where $\beta'$ is the diffraction angle by this grating (i.e., the cross-disperser, not the echelle grating), and $N'$ is the groove density.
$\Delta \beta'$ is nearly independent on $\lambda$, which means that the separation of each order becomes larger in the longer wavelength region.

On the other hand, the dispersion by the prism is determined by the wavelength-dependence of the glass material.
This depends on particular material, but as a very rough approximation, the refractive index $n$ depends on $\lambda$,
$$
\frac{d n}{d\lambda} \propto \lambda^{-1},
$$
with a coefficient.
This compensates the wavelength dependence on the order separation.

Thus, prisms are often better as a cross-disperser, particularly when we want to measure wide wavelength range.

> **Comments:**
> **username_1**: I think you have it. I think with a grating cross-disperser, the problem is, especially when you want an echelle that works over a broad wavelength range, that if you set a certain slit height with a dekker, then whilst the orders can be widely separated in the red (and hence wasting silicon that could be filled with data)  they might actually be overlapping in the blue. If you choose your prism correctly then you can keep the order separation at the right size for all wavelengths using a sensible slit height, whilst minimising wasted silicon.  
> **username_1**: You might be able to answer https://astronomy.stackexchange.com/questions/44781/what-are-the-pros-and-cons-of-different-types-of-echelle-spectrograph-cross-disp?noredirect=1&lq=1  
> **username_3**: Very nice: I already upvoted! This helps me understand the order spacing in the echellograms I get with my homemade echelle spectrographs, e.g., https://astronomy.stackexchange.com/a/49116/45954.  

---

