Sign conventions
Anamorphic ratio
Dispersion
Free spectral range
Resolution and resolving power
Shift of efficiency
Blazed equation
In a practical way, as shown in Figure 1, the spectrograph is not working in the Littrow configuration and the slit is not a linelike source but it has a width w.
Fig. 1. Layout of a simple grating spectrograph
In this case and assuming a free aberration optics (just “geometrical” collimator and camera) the resolving power is given by
where f is the focal distance of the collimator, w the slit width, ø the angle between the camera and collimator and θ the grating angle (the angle between the grating normal and the bisector of the cameracollimator angle).
Note that the resolving power is higher when ø is positive. This means that the angle of the incident ray must be bigger than the angle of the diffracted ray with respect to the normal to the grating.
Minimum deviation
Dispersion
Resolution and resolving power
Equation + approximation
Prism contribution (Zemax 3D plot)
New post: Spectrograph designs
]]>The resolving power of an spectrograph working in Littrow configuration for a rectangular slit is given by:
where f is the focal distance of the collimator, w the slit width and θ the Littrow angle (the ray incidence angle and diffracted angle with respect to the grating normal are the same). This equation is valid for spectrographs including free aberration optics. However, in a more realistic situation, the image profile of the slit on the CCD is not a perfect rectangular function. Due to aberrations and specially for narrow slits, the profile approaches a gaussian function. In this case, the slit width w is represented by the FWHM of the gaussian function. The same reasoning applies to a circular slit but a different FWHM.
In order to find the resolving power for a circular slit, we need to find the variation of the FWHM between a rectangular and circular slits. In Figure 1 we have superposed the two profiles. For simplicity we consider only the half of a circle. The result is the same for the whole circle.
Figure 1. Rectangular and circular slits. Only the half is shown for simplicity in the calculations
For the rectangular slit, the FWHM is simply the width of the slit. In Figure 1 is 2R. As for the circle, we have to find the profile of the PSF. During the data reduction, the columns of pixels perpendicular to the dispersion are binned to find the PSF. Assuming an evenly illumination of the circle, each pixel will contain the same number of electrons No. If Yi is the number of pixels for a given pixel position Xi, the flux of that column will be YiNo. We can see that the profile of the circle is the circle function y = (R^2 – x^2)^1/2
By doing y = R/2, we find x = R√3/2, therefore the FWHM = R√3
By replacing the FWHM of the circle and slit in equation 1, the resolving power by a circle is
Rcircle = √3 /2 Rslit ~ 1.15 Rslit
Figure 2 shows the spectrum of the sodium doublet obtained with FLECHAS spectrograph using a fibre and a rectangular slit. Note the elliptical shape of the image of the fibre due to the anamorphic magnification of the spectrograph configuration. In order to find the FWHM at different pixel samplings, the fibre and slit were wide opened (400 um) to cover a large number of pixels.
Figure 2. Sodium doublet with a 400 um fibre and with a 400 um slit
The FWHM were
]]>This post discusses how spectrograph features such as the resolving power and the “size” of its optical elements are affected when the spectrograph is linked to a telescope with an optical fibre.
The main aim of the post is to help in the design of a spectrograph taking into account the features of any given telescope together with the specifications of the required spectrograph. The basic parameters of a telescope are its size (aperture diameter), the focal ratio (F/#) and the average seeing at the observing location. For the spectrograph, on the other hand, the specifications are usually the required resolving power, the spectral range and the optical efficiency.
Most of the mentioned parameters above are related in a couple of equations and therefore they will tell you the kind of spectrograph you would like to attach to your telescope. These equations can also help you to know what you can do or what you can expect if you already have available a number of elements, such as an optical fibre, a diffraction grating, a CCD detector, a camera objective or a collimator.
We show first the equation for the case where the spectrograph is directly attached to the telescope. It will tell you the possibilities and limitations of your design. Second, we see how these parameters are affected when an optical fibre is used to link the telescope to the spectrograph.
Note that is not the intention of this post to deduce the equations appearing in the analysis; instead the equations will be used just as a tools to define the most appropriate spectrograph design. In our Spectrographs tab it is described in details the basic concepts on dispersers and spectrograph designs.
Before proceeding let´s clarify a few concepts like the definition of resolution and the resolving power of a spectrograph.
In the case of an ideal free aberration spectrograph where its slit is a pointlike source and this source emits a monochromatic beam, the spectrum is given by the Airy pattern (1). If the slit is a “linelike” source, its image on the detector is a sinc function pattern [sinc(θ)=sin(θ)/θ]. If the slit is not a linelike but has a finite width, the spectrum is the convolution of the sinc function with the rectangular one. In all cases, the monochromatic image of the slit is the socalled point spread function (PSF) of the spectrograph. In a general case, the PSF includes the aberrations of the system.
The resolution of the spectrograph, denoted by Δλ, is the spectral coverage of the image of the slit on the detector. For a spectrograph working in the “visible” domain it is usually measured in nanometres (nm). On the other side, the resolving power R is a unitless value defined by λ/Δλ, where λ is the wavelength of the monochromatic source.
The advantage of using Δλ is that its value is constant along the spectral order, while λ/Δλ is not. The latter, however, is more used as a generic performance of the spectrograph. Astronomers use resolution while engineers characterize the spectrograph by its resolving power.
A second clarification is about the resolving power difference between a diffraction limited spectrograph and a “geometrical” slit spectrograph. Most optical textbooks derive the spectral resolution of a diffraction grating working in Littrow configuration (Figure 1) by
R = mN, (eq. 1)
where m is the diffraction order at which the grating is working and N the total number of grooves in the grating. This equation works for a theoretical spectrograph where the slit is an ideal pointlike source (zero diameter) and the optics is totally aberration free, i.e. diffraction case. Under these conditions and for a circular aperture of the collimator, the PSF is defined by the Airy pattern. The spectral range Δλ is defined by the Rayleigh condition where two spectral lines are resolved when the maximum of one falls on the first minimum of the second (see for example Optics, Hecht, Ch. 10 Diffraction).
Figure 1. Grating working in Littrow configuration. The incident and diffracted beams in red colour are equal with respect to the grating normal (only valid for λ = 500 nm)
Most conventional spectrographs don’t work with pointlike sources but with rectangular slits having a finite width w. In Littrow mode the image of the slit on the detector has the same width w. If this width is much more bigger than the diffraction pattern and the optics of the spectrograph is aberrations free, the profile of the monochromatic slit is practically close to the rectangular function. The resolving power R is then given by
where f is the focal distance of the collimator and θ the Littrow angle (the ray incidence angle and diffracted angle with respect to the grating normal are the same as shown in Figure 1).
Note that this “geometrical” equation is quite different from the ideal case R = mN. It is independent of the number of grooves of the grating or of the working order. In other words, for a given optics configuration ( f and w), the resolving power depends solely of the grating angle with respect to the incident beam!
In the case of real spectrographs where optics are not usually diffraction limited but have small aberrations, the image profile of the slit is not any more rectangular. The edges are rounded and for narrow slits the profile may approach a Lorentzian or Gaussian curve. In any case, the resolution Δλ is now defined by the FWHM (Full Width Half Maximum) of the spectral spread of the image of the slit on the detector. In astronomical spectroscopy Δλ is measured in Angstroms for historical reasons. As for the resolving power, the slit width w is replaced by the FWHM and equation 2 becomes
As a numerical example: for a standard 50 mm grating length, 1200 lines/mm and blaze angle of 17.5 deg, the total number of grooves is 1200 x 50 = 60 000. Working at the first order, the theoretical resolving power (eq. 1) is 60 000.
However for a spectrograph with a collimator having a focal length of 100 mm and a 50 μm slit, the resolving power (eq. 2) is only 1 300 ! (~2 x 100 x tan 17.5 / 50). Using an Échelle grating working at 63 deg, the resolving power rises up to 8 000, still well below the theoretical one.
Actually the optics of the spectrographs are not perfect but introduce some aberrations. The image of the slit on the detector is blurred in some extent, making wider its FWHM and therefore the resolving power is further degraded (eq. 3).
Figure 2 shows a layout of a slit spectrograph directly coupled to a telescope.
Figure 2. A spectrograph directly coupled to a telescope
The spectrograph is composed basically by 4 elements: the collimator, the disperser optics, the objective and the detector. In the astronomical jargon, the objective is called “camera”.
The telescope is represented by an ideal lens with a diameter Φ_{t} and a focal distance f_{t.} The slit subtends an aperture δ on the sky. The focal ratio of the telescope (F/#) matches the one of the collimator.
The circular cross section of the collimator beam with a diameter Φo defines the pupil of the spectrograph. This pupil is then deformed by the diffraction grating and takes an elliptical shape where its major axis is represented by Φa. The ratio between Φa and Φo is defined by the anamorphic magnification:
This distortion produces a deformation of the image of the slit on the detector. For example, the image of a circular slit will be an ellipse. Its minor axis will be parallel to the dispersion direction on the detector if the angle of the incidence beam is bigger than the angle of the dispersed beam. This narrowing of the image reduces the FWHM of the PSF and therefore increases the resolving power. In the opposite case, when the angle of the diffracted beam is bigger than the angle of the incident beam, the resolving power is smaller. More about the anamorphic ratio can be found in this post (in preparation).
In the case of needing a spectrograph with a long slit, the instrument should be designed in such a way that the image of the telescope pupil should be placed close to the diffraction grating. This condition helps to reduce the size of the grating and camera. A discussion on pupils can be found in this post (in preparation).
Taking into account the telescope parameters, the resolving power of the spectrograph (Eq. 2) in Littrow configuration takes the form:
For a given telescope and a given seeing at the location of observation, this equation is very useful to have an idea of the pupil diameter in the spectrograph and therefore the “size” of the optical elements in your design. It is the war horse of astronomers and hence we will call it “astronomer’s equation”.
There are a number of interesting points: if you need to observe your stellar object with a given resolving power R and you want to catch as much as possible flux into the spectrograph, you have to wide the slit. But, to preserve the resolution you have to increase the collimator diameter in the same proportion. Note that the grating and camera grow with the collimator aperture.
Note also that the spectrograph size (collimator pupil, grating and camera) increase linearly with the telescope diameter. Unfortunately the price grows exponentially!
Example 1. You have a 30 cm telescope and you wish a spectrograph with a resolving power of R = 20 000. The seeing at your location is 2″ and you open the slit at this value. You want to record the entire spectrum in one shot, so you have in mind to build an échelle spectrograph. Assume you want to buy an échelle grating working at 63 deg blaze angle. With all these parameters in hand the collimator you need will have a diameter of 1.5 cm ( øo = 20000 x 2″ x 30 cm / 2 x tan 63 deg and 2″ = 9.7 x 10^ 6 radians)
This diameter gives you an idea of the “size” of the optical elements (collimator, grating and camera) you need to gather in order to reach this resolving power.
It is important to point out that this equation is valid for a spectrograph working in Littrow configuration. In the post Spectrograph designs we will discuss the equation for a general case.
Figure 3 shows a spectrograph linked to the telescope through an optical fibre. The image of the star is projected on the input fibre end and the output fibre end is also directly coupled to the collimator of the spectrograph.
Figure 3. Spectrograph linked to the telescope through an optical fibre
Unlike the direct telescopetospectrograph coupling, the slit now has a circular shape. This difference affects substantially both, the throughput and resolving power of the spectrograph. A fibre with diameter matching the seeing transmits 50% of the flux, whereas a slit with the same width passes 75%. More on flux calculation for a slit and circular slits can be found in our Flux Calculator here.
Following the data reduction where the illuminated pixels by the image of the fibre are binned perpendicular to the dispersion, it is found that circular slits provide between 15 to 20% more resolving power that the one generated by a square slit (its width equals the diameter of the fibre).
This gain in resolution is compensated by a reduction of the flux entering into the collimator. Indeed, when superposed, the circle reduces the area of a square by around 20% (more precisely by π/4). In our post Resolving power for rectangular and circular slits (to be completed) we propose an explanation of this effect. The resolving power increase by a factor 2/√3 = 1.16 :
Another substantial difference with respect to the direct telescopetospectrograph coupling is the non conservation of the focal ratio by the fibre or focal ratio degradation (FDR), this is the F/# of the output beam is always bigger than the F/# of the input beam. More about FDR can be found in this post. The posts Linking a Telescope to a spectrograph through an optical fibre Part I and Part II describe different ways to inject the telescope beam into the fibre taking into account the focal ratio degradation.
The adaptation of focal ratios between the fibre and the spectrograph collimator has a significant impact on the computation of the resolving power. The astronomer equation (eq. 3) would be valid if:
When designing a fibre linked spectrograph, usually the fibre diameter (“slit width”) and the focal ratio of the fibre beams are fixed. These conditions make sometimes confusion among astronomers who use the astronomer’s equation for a “dynamical” spectrograph where the slit aperture and focal ratio of the collimator follow the parameters of the telescope. This equation however does not take into account the focal ratio degradation (FRD). For a fibre linked spectrograph is better to use the equation 2 and consider the spectrograph as an independent entity.
Example 2. Let’s take the same telescope and results of example 1: you have a 30 cm telescope, you wish a spectrograph with R = 20 000 and you find that the pupil is 1.5 cm. Given these parameters, you want to know the fibre diameter you need to link the telescope to the spectrograph. Let’s assume that the telescope has a focal ratio of F/10. The plate scale (the projection of the sky on the telescope focal plane in microns per arc seconds) is therefore 29 μm/” (= 1″ x F/10 x 30 cm). As you need R = 20 000 for a sky aperture of 2″, the slit width will have a width of 58 μm. You will need thus a fibre of this size working at F/10. Unfortunately it is not advisable to work with fibres working at F/10 because of the focal ratio degradation. If you look at Figure 9 in this post, the efficiency at F/10 for a 60 or 50 μm fibres is less than 25%. However, If you work at F/5, the efficiency increases up to 50%. In order to reduce the focal ratio at F/5 you have to put a lens in front of the fibre input end, but then the scale plate will be reduced to the half. In other words the 2″ sky aperture becomes 29 μm. The fibre diameter must therefore be 29 μm.
On the market, the fibre standard diameters are 25 and 50 μm. Since the latter is easer to purchase and to work with, you decide to inject at F/10 directly on the fibre end and open your collimator at the fibre output end at F/5. The flux gathering increase significantly but the diameter of your collimator (1.5 cm) must be now 3 cm to cover the F/5 beam. So your optics must be twice bigger if you want to reach 20 000 of resolving power! If you keep your optics to 1.5 cm diameter, you have to reduce the focal distance of your collimator to 7.5 cm to accept the F/5. In this case and following equation 2, you reduce the resolution by half. The alternative (working with a 25 μm fibre and using lenses to convert the F/10 telescope beam in F/5 into the fibre) is certainly not easier to carry out but surely cheaper than buying bigger optics (collimator, grating and camera)!
Notes:
(1) It is valid only for a Littrow configuration, otherwise the Airy pattern is deformed by the anamorphic magnification
Gerardo Avila
European Southern Observatory
Linking a telescope to a spectrograph through an optical fibre by CAOS group is licensed under a Creative Commons AttributionNonCommercialNo Derivative Works 3.0 Germany License.
Based on a work at spectroscopy.wordpress.com.
Injecting a laser beam into an optical fibre is a very common task in optical laboratories. For example it is extremely useful for the alignment and collimation of optical components in instruments like fibrefed spectrographs. Our youtube video Injecting a laser beam into an optical fiber describes this process.
For thin fibres the direct injection of the laser beam does not provide the best coupling efficiency. In addition the pattern of the output projected beam strongly depends on the injection angle to the fibre axis. The video shows:
The numerical aperture (NA) of the fibre has been described in one of our previous posts (look here). The NA is related to the beam aperture into the fibre by
where F/# is the beam aperture of the injected beam. In the video, F/# = d/Φ is the ratio of the distance between the fibre to the screen to the biggest ring on the screen. For small injection angles, the equation can be simplified to
Injecting a laser beam into an optical fibre by CAOS group is an article licensed under a Creative Commons AttributionNonCommercialNo Derivative Works 3.0 Germany License.
Based on a work at spectroscopy.wordpress.com
This article describes a simple laboratory spectroscopy test bench to obtain resolving powers as high as R = 150 000. The optical set up is basically composed of an échelle diffraction grating, a doublet achromatic lens, a beam splitter, an optical fibre and a CCD camera. Among others, this experiment allows to discern and study the longitudinal emission modes of diode lasers. Our 20′ video Building a spectroscopy high resolution experiment explains in details the bench implementation
Figure 1 shows the optical layout of the experiment. The échelle grating has been set up in Littrow configuration where the angles of the incident and diffracted rays with respect to the échelle are the same. A beam splitter has been introduced to get this arrangement. The optical fibre is illuminated by the diode laser, its output end acts as a light source and is placed at the focal plane of the lens. The light beam is reflected by the beam splitter before reaching the lens. The collimated beam after the lens goes to the échelle grating and the diffracted beam is focused on the CCD camera through the beam splitter. The blaze angle of the grating is 63 but it was turned to 62 degrees to centre the diode laser wavelength (532 nm) on the optical axis. The configuration works at the order 42. There is no need of a crossdispersed because we observe tiny wavelength variations of lasers which stay in the same diffraction order.
The beam splitter was introduced to work in Littrow mode but at the cost of some drawbacks: first, only 50% of the incoming light goes to the collimatoréchelle group. The échelle efficiency is about 50%, to that we have to consider that the échelle didn’t cover all the collimator aperture. We estimated a vignetting of 60 %. So 25% x 0.4 = 10% is directed to the detector. But another 50% is taken by the beam splitter again, so the resulting beam reaching the detector is only 5%. To that we have to subtract another 4% losses by the doublet and around 8% more by the reflection losses in the beam splitter. At the end, around 4.5% of the original beam reaches the CCD. Another problem is the introduction of aberrations by the beam splitter. These aberrations may significantly reduce the resolving power of the spectrograph.
Figure 2 shows the spot diagram for 4 wavelengths: 532, 530, 528 and 526 nm. The first is the laser wavelength; the fourth is the extreme wavelength of the free spectral range of order 42. Note the rapid degradation of the image quality at the extreme wavelengths. However, in our case we have worked close to the centre of the field where the aberrations are close to the diffraction limit (Figure 2, right).
Figure 3. Spot diagrams. Left: all wavelengths. Right: central wavelengths
In order to accurately measure the resolution, the pixel matching (projection of the monochromatic image of the fibre, or point spread function) should cover between 2.5 and 3 pixels of the CCD. Since one of the fibres is very small (10 um core diameter) we tried to use a detector with the smallest pixel size. The SBIG 1603ME has pixels of 9 um. With this pixel size the image of the 25 um fibre covers 2.8 pixels. However, for the 10 um fibre, the pixel matching is only 1.1, well below the required 2.5. The measurements with this undersampling are therefore quite inaccurate. For this case we have placed a divergent lens between the beam splitter and the detector to increase the size of the image of the fibre on the CCD. The drawback was a small degradation of the image quality and therefore a degradation of the resolution.
The complete implementation of this experiment is better shown in this 20′ video under the title “High resolution spectroscopy experiment”.
It is important to mention that the the green laser used here plays important different roles:
The spectral resolution in Littrow configuration takes a simple form:
where f is the focal distance of the doublet lens, w the slit width and θ the incident (= diffracted) angle to the grating.
Table below shows 4 examples using 2 doublets, 2 fibres and 2 gratings. Note that the resolving power does NOT depends of the number of grooves but by the Littrow angle!
Configuration 
Theoretical R 
Measured R 
f = 500 mm, w = 25 µm, θ = 62^{o}  75 000  85 000 
f = 500 mm, w = 10 µm, θ = 62^{o}  188 000  116 000, 140 000 with divergent lens 
f = 400 mm, w = 25 µm, θ = 62^{o}  60 000  68 000 
f = 400 mm, w = 25 µm, θ = 75.2^{o}  121 000  106 000 
We have used the standard method to measure the resolving power of the arrangement: we measure the spectral coverage of the point spread function (image of the fibre) on the detector. The resolution is given by the ratio of this spread to the wavelength of the light beam, in other words R = /. Now to measure the spectral coverage of the image of the fibre we have to find and observe with the spectrograph an adequate doublet or triplet from a known spectral lamp. We have used the doublet emission lines of the Hg lamp (576.96 and 579.07 nm). Once we have recorded these lines on the CCD, we can deduce the spectral dispersion per pixel of the experimental set up. With this value, we measure the FWHM (full width half maximum) of any of the Hg lines and apply the equation above. The last column of the Table above shows the resolutions we measured with our experimental set up. Note that the resolving power for the configurations using the 25 um fibre and the 62 deg échelle is slightly higher than the theoretical one. This effect is due to the circular shape of the fibre: the resolving power equations are derived by assuming a rectangular slit uniform illuminated. A circular fibre may be composed by a number of slides [parallel to the dispersion direction on the CCD. Each slide has its own "width". Only the central slide has the width of the equivalent slit. The other slides decrease their width. Therefore there is an associated resolving power for each slide. The resolution of the slides on the poles has a higher resolution than the one in the equator. In a rectangular slit all the slices contribute with the same resolution. Therefore the the average resolution for all slices in a circular fibre is higher than the slit counterpart. This increase in resolution is around 20 % with respect to the slit counterpart. The price to pay for this increase is a reduction in flux efficiency: the fibre shape picks up less than photons than the slit shape. The amount of vignetting by the circular shape is precisely 20%!
When using a 10 um fibre, the sampling factor by the detector is well below the required one (2.5 to 3 pixels per FWHM of the PSF). The image of the spectral line can felt in one pixel or covering 2 pixels. This undersampling creates a big error in the measurement of the FWHM of the spectral line (PSF).
In the configuration using a 25 um fibre but with another échelle where the blaze angle is bigger (75.2 deg), we obtained resolving powers below the theoretical one. Since we did just one test, we suspect that the measurement of the dispersion per pixel was not properly done.
Most of green diode laser pointers are not monochromatic but they emit radiation in several longitudinal modes having slightly different wavelengths. These modes are also polarized. The number of modes, the intensity and the separation between them depends largely in the type of laser cavity. The mode separation is about 0.05 nm only.
To separate these modes you need a spectrograph with a minimum resolving power of R=10 000. However, the lateral lines separation is so small that a powerful microscope is needed if you want to observe them with your eye instead of the CCD. With our spectrograph working at resolutions beyond 60 000 their visual observation is simpler and more spectacular.
The image below shows a sequence of 7 exposures starting (top) with a current of 100 mA feeding the green laser and with increases of 10 mA up to 160 mA (bottom). In this current range there are 4 modes in total but they do not lase simultaneously, they appear as a function of the current applied to the diode. The first mode (left) only appears with current 100 mA or less, while the forth mode (right) appears first with 110 mA very weak but increases in intensity when increasing the current.
Layout used for this observation: Fibre 10um, collimator: 500 mm focal length, échelle 79gr/mm working at 62 degrees to centre the 532 nm and a CCD camera SBIG 1603ME with 9 um square pixel. Under this configuration we got a dispersion of 2.547×103 nm/pixel. Theoretical R: 188000, Measured R: 120000
Figure 4 shows the wavelength shifts for each mode as a function of the diode current intensity. we measured roughly a factor of 2.5×104nm/mA
In astronomical observations, the sodium doublet appears in lots of spectra. When quickly analysing the spectrum on the CCD or its respective graphical profile, the separation of the D1 and D2 lines (588.9950 and 589.5924 nm) tell us a rough idea of the resolving power of the spectrograph. Usually with resolutions of few hundreds the doublet is barely resolved. At 20 000 you can widely resolve the doublet and detect the Nickel line in the middle. Figure 5 shows the Na doublet as seeing by our test bench
We have built a test bench spectrograph with an échelle grating but without crossdisperser to study the behaviour of the longitudinal modes of pointer green diode lasers. With a proper pixel sampling of the PSF (25 um fibre), we measured resolving powers a bit higher than the theory. The discrepancy is due to the fact that the fibre has a circular shape instead of a rectangular slit.
In one setup using a f 500 mm doublet achromat, a 10 um fibre and an R2 (63 deg blaze) échelle we could reach 140 000 of resolving power. The theoretical value is 188 000. The big discrepancy was caused by the strong undersamplig of the 10 um fibre, only 1.1 pixels per resolution element (FWHM of the PSF).
Using a low power microscope (60x), we can observe with the naked eye the longitudinal modes of a 532 nm diode laser and study their behaviour as appearance, polarization and intensity as a function of the current feeding the diode.
Gerardo Avila and Carlos Guirao
European Southern Observatory
Experimenting with Very High Resolution by CAOS group is licensed under a Creative Commons AttributionNonCommercialNo Derivative Works 3.0 Germany License.
Based on a work at spectroscopy.wordpress.com.
The purpose of this post is to show the alignment procedure of an onaxis parabolic mirror. The methodology has been applied to align the collimator of two of our fibre linked spectrographs:
An offaxis parabolic (OAP) mirror consists of a small section cut out from a larger, socalled “parent” parabolic mirror
The alignment was highly simplified by using a high quality (better than λ) mirror with a central aperture:
The steps we followed were:
Detailed procedure:
Preparation of the paper mask (optical target). Concentric circles were printed on a piece of paper where the outer circle matched exactly the diameter of the parabolic mirror (Ø 108 mm). Each concentric ring corresponded to specific F/#. In a parabola of 444 mm of focal length, a circle with Ø 44.4 mm represents an F10, another circle of Ø 55.5 mm for an F8 , Ø 74 mm for an F6 and so on.
Cutting the paper mask. Centre the mirror with respect to the mask and with its face down to avoid touching the aluminium surface. With care and a very sharp knife cut half a diameter of the mirror. In the very same centre make a hole of Ø 2 mm.
Mount the parabolic mirror on its support. Fix the parabolic mirror on its final support perpendicular to the optical table and hang the paper mask over it. The small hole Ø 2mm in the mask defines the centre of the mirror. We assume that the vertex of the parabola is on the centre of the mirror!
Definition of the optical axis. The laser was mounted on a support containing X, Y displacements (green) and tiptilt (yellow). The laser beam should be parallel to the optical table and its heigh must be the one defined by the centre of the parabola.
Fix a ruler to the optical table and parallel to one side. Another ruler in “L” shape will be used to mark the height of the axis marked by the laser beam. This ruler will slide along the other ruler fixed to the table. Adjust the position in X, Y (red) of the laser so that the laser hits the edge of the “L” ruler along the fixed ruler. Adjust tilt (yellow) and height (green) until the laser beam hits the marked height in the “L” ruler along the fixed ruler. Once the axis is set, clamp the laser mount to the table.
Alignment of the parabola with the laser. Remove the rulers and set the mirror assembly at the other end of the optical table. Set the paper mask on the mirror and adjust the position of the mount until the laser beam hits the hole in the centre of the paper mask. The laser beam is now reflected. A fine adjustment of the height of the laser beam is permitted until the laser beam hits with precision the very same centre of the mirror.
While keeping the laser beam in the centre, tilt and tip the mirror assembly until the reflected laser beam returns in the same axis (autocollimation). Once the autocollimation is achieved clamp the mirror assembly to the table.
Alignment of the hollow mirror. Set the flat mirror facing the parabolic mirror and at a distance beyond the focal length (444 mm) of the parabola, so that it will not interfere later with the fibre output and its support.
Adjust the height of the mirror to pass the laser through the central hole. Place a small flat mirror in the centre of the hole. Tilt and tip the mirror to adjust the perpendicularity with the laser beam (autocollimation with respect to the laser beam.
Alignment of the optical fibre. Inject the laser beam on one end of the fibre. To increase the flux of the laser into the fibre we used a microscope objective: first, you place the fibre input end in front of the laser beam, place the microscope objective (< 10X) between the two with the entrance end of the objective pointing to the fibre. Adjust its high and lateral position until the output beam is centred with respect to the fibre. Approach the objective to the fibre to focus the beam into the fibre. For that look through the objective and look for the image of the fibre end. DO NOT look at the laser beam!! On the other fibre end we glued a small white screen on the SMA connector and made a small hole just in front of the fibre.
The output fibre end was mounted on a mechanical support allowing tilttip and height adjustments. The fibre assembly is set in front of the parabolic mirror at a distance close to its focal length (444 mm). Use the laser beam projected on the paper mask to coarse adjust the fibre output by tiptilt.
Remove the paper mask from the parabola, the laser beam is now reflected from the parabolic mirror toward the flat mirror behind the fibre. And again back from the flat mirror toward the parabolic mirror and reflected back again toward the fibre. On the small white screen you should observe the image of the fibre. Adjust focus and lateral movements of the fibre output until the reflection concurs with the fibre position.
To reach the sharpest focus position along the axis the best method is to visualize the image of the fibre output with an objective set at infinite and acting as a collimator. We have a 500mm focal distance telephoto objective where we put a reticle on the focal plane and an eyepiece t see the reticle.The focal length of the objective should be larger than the one of the parabola (444 mm) in order to increase the accuracy of the focusing. The calibrated objective was put just behind the fibre assembly and aligned in height with the axis. With a ruler along the optical axis (laser beam) we displaced the fibre until its image observed through the objective is in focus. When achieved we clamp the fibre assembly to the the optical table.
Another method to achieve focus position can be obtained with a paper mask with two holes converting our onaxis parabolic mirror in two smaller offaxis parabolic mirrors. Each of these parabolic mirrors will produce individual images of the fibre.
By displacing the fibre assembly along the axis, the focus position will be reached when both images coincide together with the same position of the fibre. Once this is achieved clamp the fibre assembly to the optical table. At this point the alignment is complete.
Below a gallery of pictures taken during the alignment of our collimator for our LECHES spectrograph
Carlos Guirao and Gerardo Avila
Alignment of an onaxis parabolic mirror by CAOS group is an article licensed under a Creative Commons AttributionNonCommercialNo Derivative Works 3.0 Germany License.
Based on a work at spectroscopy.wordpress.com
We present here our based prism low resolution spectrograph baptized BESOS or BEst Simple Optical Spectrograph (kisses in spanish). Designed in 2003, the spectrograph was proposed to overcome the low throughput of our previous instrument LOROS (coming soon to this blog) which was an instrument based on an onaxis dispersion prism obtained from a commercial spectroscope. The total efficiency of LOROS was only 25% in the visible spectral range. BESOS was built with only two doublets and a prism. This configuration reached almost 87 % at 620 nm. With such efficiency and low resolution, we expected to measure the red shift of the most bright galaxies and quasars.
In this post we provide a description of the instrument, features, performances and the set of mechanical drawings.
The Figure below shows the optical design of BESOS. The description of the elements is given in the Table below. The prism works at the minimum deviation and for this configuration the incidence and refracted angle to the input and output prism surface is 59.6960º respectively.
Two LINOS (now Qioptiq) doublets were chosen as collimator and objective respectively. An equilateral 60º prism from Edmund Optics was used as disperser. The image quality at 400, 550 and 700 nm is illustrated in the following Figure. The square is 20 μm side. The images for these 3 wavelengths were optimized to have their minimum spot size perpendicular to the dispersion.
Instrument  Prism spectrograph 

Name  BESOS (BEst Simple Optical Spectrograph)  
Description  Low resolution spectrograph based on an equilateral prism  
Slit  Ni plate Ø 10 mm, 25 µm thick, centered slit:25 µm width, 2 mm long. Labelcomat (B)  
Collimator  Doublet f = 30 mm, Ø 12.5 mm, external mount. Ø 25 mm. Linos ref. G052008000. Glass: SF18. Coated: ARB2 VIS  
Prism  Dispersion prism. Material SF18, 15x15mm. AR coating. Edmund Scientific Ref U43494  
Objective  Doublet f = 40 mm, Ø 12.5 mm, external mount. Ø 25 mm. Linos ref. G052010000. Glass: SF18. Coated: ARB2 VIS  
Telescope interface  Standard 1.25” tube.  
Camera interfaces  T2 thread  
Eye piece.  f = 20 mm, standard barrel Ø 1.25”, with extender15 mm, Ø 1.25” (included). For direct spectrum or slit observation  
Slit illumination  Red LED, 10 mA.  
Power supply  Battery 12 V Model MN21/A23/K23A/LRV08  
Slit viewer lens  Doublet f = 30 mm, Ø 12.5 mm, external mount Ø 25 mm. Linos ref. 063130  
Slit viewer interface  Internal tube standard Ø 1.25”  
Total transmission (without CCD) 

CCD test camera  SBIG ST7 XMEI, KAF0401E 756×510, 9×9µm  
Dispersion  0.72 nm/pixel @ 650nm  
Spectral range  362 nm – 812 nm  
Resolution 

All BESOS technical drawing ara available for download in this PDF document: BesosAllDrawings
This image describe the BESOS assembly with both optical and mechanical components. The only remark worth being mentioned is that the mechanical part 3 – Folding Mirror is made enterily of black anodized aluminium except the inclined surface that has been polished till it achieved the quality of a mirror surface (probably would habe been simpler solution to glue a small mirror).
The following photos show BESOS in different configurations
“BESOS: a prism spectrograph” by CAOS group is licensed under a Creative Commons AttributionNonCommercialNo Derivative Works 3.0 Germany License.
Based on a work at spectroscopy.wordpress.com.
This is the principle explaining the mirages, gradient index lenses and fibres. Amaze your friends by telling them that you put a black hole behind the recipient in order to create a huge gravitational field which bends light beams (General theory of relativity). A friend was convinced that we put a strong magnet below !!!
When properly poured, the sweet water will create a mixture with a gradual refraction index. The bottom will have a higher refraction index than the top. When a light beam travels inside, its direction will bend continuously (principle of Fermat)
In order to success the mixture:
Bending a laser beam by CAOS group is licensed under a Creative Commons AttributionNonCommercialNo Derivative Works 3.0 Germany License.
Based on a work at spectroscopy.wordpress.com.
Till present we have used several different Canon objectives as cameras on our spectrographs:
The article presents our experimental results with the calculation of the transmission for these objectives.
Carlos Guirao and Gerardo Avila
Optical efficiency of the 200 ln/mm Newport transmission grating by CAOS group is licensed under a Creative Commons AttributionNonCommercialNo Derivative Works 3.0 Germany License.
Based on a work at spectroscopy.wordpress.com.
We present here the the optical efficiency (transmission vs. wavelength) for its 1st order for two of Newport (former Richardson gratings) transmission gratings:
and a description of the laboratory setup to take these measurements.
and compared the results with the manufacturer (ones):
Measurements made with nonpolarized light and perpendicular to the grating (not Littrow). Date: 17 Dec 2008
Measurements made with nonpolarized light and perpendicular to the grating (not Littrow). Date: 11 April 2013
Carlos Guirao and Gerardo Avila
Optical efficiency of the 200 ln/mm Newport transmission grating by CAOS group is licensed under a Creative Commons AttributionNonCommercialNo Derivative Works 3.0 Germany License.
Based on a work at spectroscopy.wordpress.com.