Optical Spectrometer with Acousto-Optical Dynamic Grating for Guillermo Haro Astrophysical Observatory

Optical spectrometer of the Guillermo Haro astrophysical observatory (Mexico) realizes investigations in the visible and near-infrared range 350 800 nm and exploits mechanically removable traditional static diffraction gratings as dispersive elements. There is a set of the static gratings with slit-densities 150 600 lines/mm and optical apertures 9 cm × 9 cm that provide the first order spectral resolution from 0.8 to 3.2 Å/pixel, respectively. However, the needed mechanical manipulations, namely, replacing the static diffraction gratings with various resolutions and following recalibration of spectrometer within studying even the same object are practically inconvenient and lead to wasting rather expensive observation time. We suggest exploiting an acousto-optical cell, i.e. the dynamic diffraction grating tunable electronically, as dispersive element in that spectrometer. Involving the acousto-optical technique, which can potentially provide electronic control over the spectral resolution and the range of observations, leads to eliminating the abovementioned demerits and improving the efficiency of analysis.


Introduction: General Characterization of the Spectrometer
The Cassegrain telescope, which is in operation at the Guillermo Haro astrophysical observatory (Mexico), includes classical static grating spectrometer (from Boller & Chivens Corp.).Presently, this spectrometer is available on the observatory at the 2.12-meter telescope with 5 static diffraction gratings, see Table 1.All the static gratings are 9 cm × 9 cm in size, and they are exploited in the first order of light diffraction with the dispersions ranging from 450 to 114 1 Å mm   , allowing a good coverage in both dispersion and wavelength within the CCD matrix photodetector sensitivity ranges.The use of acousto-optics techniques in astronomy starts from the late 1960s when a new kind of spectral devices was developed, electronically tunable acousto-optical filters (AOFs).Later, in the middle of the 1970s [1], the first efforts to use the tunable AOFs for astronomical spectroscopic observations were made at the Harvard Observatory in 1976.Using a collinear filter is based on calcium molybdate crystal.Conceptually and technologically, AOFs at that time were imperfect.The filter had a small optical aperture (4 × 4 mm) and a large interaction length   ~50 mm .Now, the tunable AOFs are technologically mature, compact tunable AOF-based spectrometers and cameras are widely used for research and process control.For example, acousto-optics is widely used currently for spectroscopy in radio-astronomy.In 2002, imaging spectrophotometer on optical range with CCD cameras was tested [2].More examples exist for the use of acoustooptics in astrophysical spectroscopy, such as SPICAM in Mars Express [3].
As usually, the converging light beam from the telescope passes through the spectrometer entrance slit in the telescope focal plane to the collimator, an off-axis parabolic mirror.The reflected parallel beam then falls on to the diffraction grating surface.The diffracted light passes through a Schmidt camera, which images the spectrum on to the CCD matrix.This optical spectrometer can be characterized by the following parameters.

Efficiency of the Static Diffraction Grating
The efficiency as well as the dispersion at the desired working wavelength is an important parameter when choosing a diffraction grating.The efficiencies of the available static gratings have been measured experimentally, so that one of them, exhibiting the maximum diffraction efficiency up to 70%, is presented in Figure 1 as an example.It should be noted that the total system efficiency is the combination of the efficiencies of the telescope, spectrometer, grating, camera, and the detector, but diffraction efficiency of the dispersive element plays the key role.

Spectral Coverage
The static grating dispersion, camera focal length, and detector size determine the joint observable spectral range.For example, a grating with the slit-density 300 lines/mm with a blaze angle , which has a dispersion of

Spectral Resolution
The theoretical spectral resolution depends on the static grating dispersion, grating position, pixel size, collimator and camera focal lengths, and entrance slit-width.The effective CCD matrix spectral resolution also depends on the detector sampling.As an example, static grating with 300 lines/mm and with the blaze angle will have theoretical resolutions of 2.24 and 4.48 Å for slit-widths of 1" and 2", respectively.Decreasing the entrance slitwidth will improve the spectral resolution.However, this will be possible only when the sampling requirements (Nyquist criterion; one resolution element imaged onto at least two detector elements) are respected and also when the instrumental response is not diffraction limited.Within exploiting a CCD matrix, the effective spectral resolution is determined by convolution between the spectrograph resolution 4 18'    s R of pixels per a resolvable spot and the detector pixel size.With suitable pixel sizes, the spectrum may be sufficiently sampled to avoid spectral information distortion (e.g.instrumental broadening).The most common sampling criterion is 2 pixels s R  (i.e.Nyquist criterion).Due to the current needs of astrophysical observations the resolution of spectrometer has to be changed time to time that can be done only by mechanical substitution of one static diffraction grating with another one.Every time the static grating is substituted, the spectrometer needs to be realigned and recalibrated; however, it leads to potential errors in measurements and losing very important physically and rather expensive time for the observations.
In order to improve this situation, we propose an alternative for the static diffraction gratings, namely, to use specially designed acousto-optical cell as the dynamic diffraction grating, which is electronically tunable and whose capabilities will make it possible in the nearest future to replace the static diffraction gratings from the spectrometer with the new acousto-optical cell.The principal advantages of similar dynamic acousto-optical grating are excluding any mechanical operations within the observation process, avoiding recalibrations (i.e.bringing in additional errors) and any losses of time.Potentially, applying the acousto-optical technique, which can realize tuning of both the spectral resolution and the range of observation electronically, makes possible eliminating these demerits.Additionally, one can expect increasing the efficiency of the spectrometer's operation, because a dynamic diffraction grating in the form of specially designed acousto-optical cell is able potentially to provide close to 100% efficiency of spectrum analysis.the mechanical deformations  or stresses  and the optical refraction index .This effect takes place for all the condensed matters and mathematically can be explained as [5] n   where represents varying the tensor of optical impermeability or, what is the same, the parameters for an ellipsoid of optical refractive indices; while is the tensor of photo-elastic coefficients.Usually, the higher-order terms relative to the deformations Now, let us consider propagation of the traveling single-frequency longitudinal elastic wave along the acoustic axes noted as

 
|| 001 z u -axes.The main property of each acoustic axis in anisotropic acousto-optical material is its ability to support just pure acoustic modes, longitudinal or/and shear ones, within their propagation through.This property is almost the same what can be met in isotropic materials.This is why for simplicity sake one may consider the longitudinal acoustic mode passing through an isotropic medium with scalar refractive index n which can be the average one in a crystal.So that pure longitudinal displacement of particles is described by , where and are the amplitude, cyclic frequency, and wave number of that elastic wave, respectively.The field of linear deforma-, , . The components of the optical impermeability tensor can be written as a) ,   Due to Equation ( 4) does not include any cross-terms, the main axes inherent in a new ellipsoid for the refractive indices will have the same directions as before.Consequently, new main values j N of the refractive indices can explained as These equations mean that in the presence of the traveling acoustic wave, the taken isotropic medium becomes a periodic structure, which is equivalent to a bulk grating with the grating constant equal to the acoustic wavelength 2 K    , because variations in the main refractive in n n p    are proportional to s of d /and deformations in that acoustic wave.An example for a sinusoidal variation of the refractive index is illustrated in Figure 2.This periodic perturbation in a medium is varying in space and in time as well.It represents a traveling wave propagating with the ultrasound velocity the amplitude isplacement or 1 7 10 cm s   .However, the light velocity exceeds th about 5 orders, so that periodic perturbations conditioned by that acous-tic wave can be always considered as quasi-static in be-havior relative to light propagation.Thus, potential reso-lution R of similar diffraction grating measured in the number of slits per unit aperture D (let us say, for 1 mm D is magnitude by


) or, what is the same, the stroke density can ned by the ratio be determi R D   .

Novel Schematic Arrangement der proposal, ass through a
A new configuration of the spectrometer un which has already included the dynamic acousto-optical grating as a dispersive element instead of traditional static diffraction grating, is shown in Figure 3.
The light coming from the telescope will p slit: solid lines represent the light beam, being coaxial to the collimator normal, while both dotted and dashed  lines represent light beams slightly tilted from that normal.The ing mirror.The aperture of the slit will limit the angular range of the observation.The collimated beam will arrive to the acousto-optical cell at the Bragg angle, for the previously selected wavelength, and then it will be diffracted.Then, the diffracted light will be imagined in the CCD matrix by a lens or, to have the less possible losses, by a Schmidt camera.
The practical aspects of designing an updated version of the schematic arrangement for spectrometer under consideration lead first of all to creation of a modified optical scheme, which has to include some peculiarities of the AOC. Figure 3 represents the modified configuretion of the spectrometer using the AOC as dynamic diffraction grating instead of the static diffraction gratings; here, B  is the Bragg angle of light incidence for the chosen optical wavelength, see below Equation (9).The light coming from the telescope will pass through the spectrometer entrance slit at the focal plane of the collimator mirror, the reflected beam, a plane wave, will fall on to the AOC at the Bragg angle.Then, the diffracted beams corresponding to the first order will be imaged using a Schmidt-camera and analyzed.An additional modification is connected with the fact that the acoustooptical dynamic diffraction grating operates sufficiently effective in the Bragg transit regime instead of the reflection regime inherent in the above-mentioned classical spectrometer whose static diffraction gratings exhibit about 70% maximum efficiency.

Estimating the Acousto-Optical Materials
bine a large optical aperture with the needed slit-density R, an acceptable level of uniformity for acoustical grooves limited by linear acoustical losses in the chosen material, and possibly high efficiency of operation under an acceptable applied acoustic power.The list of the, currently in use, static diffraction gratings was shown in Table 1.This is why initially we have restricted ourselves by the given slit-density (for example, R = 300 lines/mm), which leads to the inequality , where 2 f    , i.e. to the requirement The o of acoustic attenuation can be expressed as he carrier frequencies f allowing us to realize the AOC, which provide t R he slit-density = 300 lines/mm together with the potential total losses along the AOC aperture.
Then, a given value of   dB

B
for the acoustic attenuation will require the aperture of   V are the slow shear acoustic mode and the longitudinal one, respectively.They both are pureacoustic modes, providing exact coincidence between the wave vectors and the energy flow vectors with the chosen directions (in fact, with the acoustic axes in crystals) of these elastic waves propagation.
The Bragg regime of light diffraction occurs with a large enough length L of acousto-optical interaction.In this case the dynamic acoustic grating is rather thick, so that during the analysis of diffraction one has to take account of the phase relations between light waves in different orders.When the incident light beam is unlimited in a transverse direction, the reflected beam will be placed in the plane of incidence (i.e. in the 2 3 x x -plane) and the angle of reflection should be equal to the angle B  of incidence.The coupled-mode theory predicts that a considerable reflection of the incident light can be expected under condition is the light wavelengths, while m is the w er, which reflects the m th  Four component of the perturbed dielectric permeability.In the case of pure sinusoidal profile peculiar to the acoustic wave, all the Fourier-components with 2 m  will be equal to zero.Thus, the Bragg regime can be realized only when the angle of light incidence  [10].Usually, when an acoustic mode is exited by the applied electric signal, the Bragg regime includes the incident and just one scattered light modes, whose normalized intensities under condition of zero mismatches are described by [11] c where x agatio is the space coordinate almost along the light prop n; is the acoustic power density, P  is the m is the effectiv aterial density, eff p e photo-elastic constants for light scattering, and n is now the averaged effective refractive index of a material.The Bragg regime is preferable for practical applications due to an opportunity to realize a 100% efficiency of light scattering by the acoustic wave.i i th d the e gy flow vectors with the chosen directions (in fact, with the acoustic axes in crystals) of these elastic waves propagation.At this step, it should be explained additionally: applying the needed electronic signals at the electronic input of AOC in such a way that the aboveobtained levels of acoustic power density will be provided makes it possible physically and potentially technically to achieve 100% efficiency of control over the incident light diffraction.By the other words, instead of about 70% maximum efficiency shown in Figure 2 for traditional static diffraction gratings, involving the acousto-optical technique via creating the dynamic acoustooptical diffraction gratings is potentially able to provide close to 100% efficiency of dispersive element over all the range of the above-mentioned spectrum analysis.This fact is caused by an active nature of AOC, which manifests some quantum gain controlled by an external variable electronic signal, in the contrast with traditional static acoustic grating.
Then, in the particular case of 1 m  , one can estimate the resolution of the AOC as a dispersive element.The above-mentioned Bragg condition can be rewritten as 2 sin V f B    and differen with respect to th tiated e acoustic frequency f , so that on obtain e can   Because the frequency resolution peculiar to an AOC in the optical scheme with space integrating is given by f Consequently, in the above-chosen case of R = 300 lines/mm, 500 nm

Diffraction of the Light Beam of Finite
Width by a Harmonic Acoustic Wave at Low Acousto-Optic Ef spectrometer, see Figure 4, exhibits potential presenc tical beams whose widths are restricted due to cond n of observations.This is why the diffr beam of finite width by harmonic acoustic wave has to be reviewed and characterized.At first, to illustrate the existing physical tendency simpler let us start from a low acousto-optical efficiency approximation , where now 0   is the angular-frequency mismatch.Due to almost orthogonal geometry of non-collinear acousto-optical interaction the angles of incidence 0  and diffraction 1  do not exceed usually about 10 o t ˚, s hat one can use t plified formulas a) he sim where is average refractive inde are the ces for the incident or diffra ight, respectively.Now, we ass e that the area of propa ic lan n refractive indi the x; 0,1 n cted l um gation for a harmon acoustic wave is bounded by two p es 0 x  and x L  in a crystal.This acoustic wave has the amplitude function  with the am 0 0 0 plitude 0 u , wave number 0 K , and cyclic frequency 0  , and travels along z -axis.Then, let initially monochromatic light beam incidents on the area of interaction under the angle 0  .At the plane x 0  , the light field is described by the compl valued amplitude function ex   e z , reflecting the spatial structure of light field.The spectra of these fields are [12] a) where k is the wave number of the inci 0 dent light.Each individual component of the incident light beam is diffracted by the acoustic harmonic in the interaction area.Using Equations ( 13) and ( 14) within taken low acoustooptical efficiency, the angular spectrum of the diffracted light can be written as [13]  Substituting Equation (17) in Equation (15), one can obtain angular distribution for the diffracted light intensity at lo The functions and represent angular spectra of This relation can be rewritten as  17) and (18a) will be integrated over the corresponding angle ranges: Efficiency of diffraction for the plane inciden light wave has maximum efficiency at q L , describing the acousto-optical effic e incident light wave, is marked out here to highlight the contribution of light beam finiteness.However, Bragg condition cannot be provided now for all the angular co ion mponents described by Equat (17).This is why one can chose the angle of incidence 0  in such a way that the phase synchronism condition will be satisfied for the axis-component of i eam.I case of 2 0 S ncident b n the with 0 B    , one can obtain [15], see Figure 5.
Equation ( 19) should be compared with the above taken relative intensity of diffraction   This modification leads ultimately to another expression for efficiency 0    

Conclusions
We have suggested exploiting an acousto-optical cell (AOC) as a dispersive element in optical spectrometer of th acousto-optical techni n and th large opt -dens materials can be used for designing similar AOC It can be lithium niobate -crystal excited by ustic m e [100]-axis at the e Guillermo Haro astrophysical observatory (Mexico).Potentially, involving que, which can realize tuning both the spectral resolutio e range of observation electronically, makes possible eliminating the above-mentioned practical demerits and excluding filters.The requirements to the cell combine a ical aperture with the needed slit ity and possibly high efficiency of operation under an acceptable acoustic power.This is why initially we have restricted ourselves by the slit-density 300 lines/mm along the 9 cm aperture.The analysis has shown that at least the following .Bi GeO -crystal using the shear acoustic mode along the [110]-axis at 0.53 GHz, so that the slit-density 300 lines/mm appears with the losses ~6.3 dB/aperture.The neighboring figures of acoustooptical merit for these materials promise desirable efficiencies of operation, so that even close to 100% efficiency peculiar to the dynamic acousto-optical dispersive element over all the range of the spectrum analysis can be expected.The figures of acousto-optical merit 2 M for these materials are neighboring and promise efficiencies of operation sufficient for observations.

 
The proposed schematic arrangement for the acoustooptical version of optical spectrometer is able to reduce dimensions and weight of a a view of operating over the spectral range of wavelengths 400 -800 nm.Potential performances of an AOC as the dynamic diffraction grating look rather attractive for practical applications due to their po al resolution as dispersive element is about δλ ≈ 0.184Å, which looks rather attractive in comparison with the data presented in Table 1.Finally, diffracting the light beam of finite width by a harmonic acoustic wave at low acousto-optic efficiency has been briefly discussed.

Figure 1 .
Figure 1.Maximum diffraction (reflection) efficiency of the static diffraction grating with: the slit-density 300 lines/mm, dispersion 224 , and blaze angle of : solid line is for the light polarized parallel to slits and dashed line is for the light polarized perpendicular to slits [4]. symmetry inherent in a medium determines non-zero factors of the tensor .With nonzero external mechanical perturbations, an ellipsoid for the refractive indices can be explained by p behavior, one can use so-called matrix indices, so that here are the components of the photo-elastic tensor with the above-mentioned matrix indices [

Figure 2 .
Figure 2. The instantly frozen ave, which consists of alternating with one anot reas of compressed and acoustic w her a decompressed material density and the corresponding sinusoidal variations of the average refractive index.

Figure 3 .
Figure 3. Schematic arrangement of the acousto-optical version of optical spectrometer: black lines represent the light beam, being coaxial to the collimator normal, while b e MHzVR  ther requirement is connected with the uniformity of acoustical grooves is restricted b te y the acoustic atnuation, whose level B along the total optical aperture D of AOC should not exceed a given value, which can be determined as 6 dB due to usual requirements related to the dynamic range of analysis.The acoustic attenuation is usually almost square-law function of the carrier acoustic frequency f [6].Let us use the conventional factor  of acoustic attenuation [7] expressed in   2 dB cm GHz  .Thus various forms of limitations connected with acoustic attenuation can be written.For examp , the total le level   dB B f  and MHz The requirements to the acousto-optical cell (AOC) com-for the dynamic grating with the slit-density of R = 300 lines/mm.

Figure 4 .
Figure 4. Geometry of interaction between light and acoustic beams.
D T   .These peculiarities of diffrac-ting light beam of finite width are illustrated by Figure 4.The d cted light waves take their origin i points of overlapping light and acoustic beams.Due to their interferencediffracted light width D D can be estimated by Figure 5.The factor   B G versus the Gordon parameter similar to Equation (19).As before, the term   2 sin qL , describing the diffraction of high efficiency for plane incident light wave, is marked out to exhibit the contribution of light beam finiteness, while the function   , B q G reflects the same te B ndency as   B G .Anyway, finally one can conclude that when AOC operates over the light beams of finite width, decreasing the acousto-optical efficiency due to partial asynchronism for the divergent incident light b ot be eliminated.eam cann LiNbO ode along th the longitudinal aco frequency 2 GHz.This selection gives 300 lines/mm with total losses ~5.3 dB/aperture.Then, one can consider bismuth germanate   12 20

Table 2
demonstrates t   at a given optical aper ture D within the chosen acoustooptical material.It is naturally to search for the materials allowing the choice of the carrier acoustic frequency f satisfying the combined inequality low