Optical schemes of spectrographs with a diffractive optical element in a converging beam

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$J. ur. Opt. Soc.-api 0, 50 205 www.jeos.org Optical schemes of spectrographs with a iffractive optical element in a converging beam.$ Marx str. Kazan, 420 ussian Feeration Optical schemes of spectrographs base on transmission concave holographic gratings working in converging beams are consiere.

Marx str. Kazan, 420 ussian Feeration Optical schemes of spectrographs base on transmission concave holographic gratings working in converging beams are consiere.

1 J. ur. Opt. Soc.-api 0, Optical schemes of spectrographs with a iffractive optical element in a converging beam.. Muslimov Kazan National esearch Technical University - KAI, Kazan, Tatarstan, ussian Feeration, 0 K. Marx euar.r.muslimov@gmail.com str. Kazan, 420 ussian Feeration N. K. Pavlycheva Kazan National esearch Technical University - KAI, Kazan, Tatarstan, ussian Feeration, 0 K. Marx str. Kazan, 420 ussian Feeration Optical schemes of spectrographs base on transmission concave holographic gratings working in converging beams are consiere. General escription of the esign techniques are provie. ach of them is supporte by a certain example with calculation an moeling results. In particular, it s shown that combination of such element with a spherical wege allows to create a spectrograph with correction of astigmatism an a variable-ispersion spectrograph. [DOI: Keywors: Transmission concave holographic grating, converging beam, astigmatism correction, variable ispersion INTODUCTION Optical schemes of spectrographs with a iffraction grating mounte in a converging beam are known for a long time. They are attractive because of their simplicity an compactness, what explains their use in such fiels as astronomy [, 2]. These remarks remain correct also for combinations of gratings an prisms. owever, the most of these schemes use plane classical gratings i.e. gratings with straight an equiistant grooves, which introuce large aberrations an ecrease spectral resolution. On the other han, it s known that concave non-classical gratings have large potential for aberration correction. There are a few examples of use of non-classical aberration-correcte reflection gratings mounte in converging beam for specific tasks [3, 4]. Nevertheless, use of non-classical gratings in such schemes in t become a self-sufficient optical esign tool an, in our opinion, still remains unerestimate. ecently the authors have extene the theory of concave grating [5] for a case of transmission concave holographic iffraction grating TCDG. It was foun, that such optical element has weak optical power an specific properties. A number of schemes base on TCDG were successfully esigne an implemente [6] [8]. In all of them the grating was mounte in a converging beam. In the present article we iscuss avantages of our approach to esign of the converging-beam schemes. Further we consier options of extension of the esign concept an provie certain examples with necessary calculations an moeling results. 2 DISCUSSION 2. Spectrograph with classical grating Firstly, we consier a spectrograph scheme with a plane classical transmission grating mounte in a converging beam. To illustrate the grating aberrations we use a compact spectrograph scheme for the visible range. The spectral working range spreas from 400 to 800 nm. The grating grooves frequency is 400 mm. It s impose on the secon surface of a plane-parallel plate mae of BK7 glass. We assume that the converging beam is create by an ieal lens f = 00 mm, f/# = 5 set up before the grating an the grating works in normal incience. The spectrum is etecte in the best image plane. The optical scheme rawing is shown on Figure. The ray aberrations for this spectrograph scheme are presente in Table. One can see that the grating has huge aberrations incluing large efocusing an significant astigmatism. This scheme can work only with a wie entrance slit. For instance, if the entrance slit with is 00 µm, the spectral resolution along the spectrum will change between 3.4 an 4.3 nm. ereafter we calculate the spectral resolution as prouct of the spectrograph reciprocal linear ispersion by its instrument function FWM. m M λ = 600 nm λ = 400 nm λ = 800 nm y = 0 mm y = 7.90 mm y = mm δy δz δy δz δy δz TABL Aberrations of the spectrograph with plane classical grating. eceive November 7, 204; revise ms. receive January 3, 205; publishe February 27, 205 ISSN

2 J. ur. Opt. Soc.-api 0, Muslimov, et al. focusing lens plane classical grating 3 53' FIG. Optical scheme of the spectrograph with plane classical grating. spectrum plane 2.2 Flat-fiel spectrograph with transmission concave holographic grating The metho of esign of a flat-fiel spectrograph scheme with TCDG is base on analytical minimization of the grating aberration function terms, which was use, for example, in [9] an [0]. This approach allows to construct a simple esign technique, which provies obtaining of appropriate optical scheme for a wie range of initial parameters. The analytical esign solution can be either use irectly, or serve as a starting point for numerical optimization of the scheme. We must note that the approach has some theoretical limitations, restricting its use for esign of high-aperture an highispersion schemes []. owever in the most cases of practical interest large values of aperture an ispersion aren t achievable because of aberrations of the transmission grating an its substrate. Focusing an aberration properties of the grating can be escribe by its aberration function, which is series expansion of optical path function for a ray emitte from the entrance slit center an iffracte in an arbitrary point on the grating surface. V = yf 0 + y2 2 F + z2 2 F 2 + y3 2 2 F 3 + yz2 2 2 F 4 + y4 8 3 F 5 + y2 z F 6 + z4 8 3 F ere is the grating surface raius, z, y are Cartesian coorinates of the iffraction point an F i are coefficients, efining certain aberrations tangential efocusing, astigmatism, tangential coma, sagittal coma, spherical aberrations etc.. We assume that the coefficients change across the grating surface is negligible. If we consier a st generation holographic grating i.e. grating recore with 2 coherent point sources [2] with efine grooves frequency, there are 3 free parameters for aberration correction. So it s possible to correct efocusing along the spectrum as well as tangential coma an astigmatism in its center. For consierations of simplicity we suppose that the spectrum plane is orthogonal to the chief ray on the meium wavelength. Then the conitions of aberration correction can be written as follows [ 2 [ sin ϕ where a 6 av a 3 av + + cos ϕ cos ϕ + a 2 a 3 N = 0, a 4 + a 5 = 0, N cos ϕ av av + cos2 ϕ + sin ϕ av cos ϕ av + cos2 ϕ av av av ] kλ av 2 = 0, ] kλ av 3 = 0. 2 a = cos 2ϕ av sin ϕ cos 5 ϕ sin 2ϕ av cos 6 ϕ + 5 cos 2ϕ av + 6 sin 2 ϕ av 3 8 ϕ + 4 sin 2ϕ + sin 4ϕ, 32 a 2 = S cos ϕ av sin ϕ 3 sin 3 ϕ S sin ϕ cos 3 ϕ av 3 cos ϕ 3 av 8 ϕ + 4 sin 2ϕ + sin 4ϕ 32 + sin ϕ cos 4 ϕ av, 4 a 3 = cos ϕ cos 4 ϕ av + sin ϕ ϕ sin 4ϕ av sin ϕ cos ϕ av sin ϕ 3 sin 3 ϕ + sin ϕ sin ϕ cos 3 ϕ av av, 3 a 4 = S cos ϕ S sin ϕ sin 2 ϕ 2 + sin ϕ sin ϕ, a 5 = sin 2 ϕ 2ϕ sin2 φ + 2 sin ϕ cos ϕ, S = cos 2 ϕ + cos ϕ. 3 The notations here are:, ϕ are polar coorinates of the entrance slit center,, ϕ are coorinates of its monochromatic image, λ is the working wavelength, is the recoring wavelength, N is the grooves frequency, k is the iffraction orer an i are holographic coefficients similar to those use in [5] 50-2

3 J. ur. Opt. Soc.-api 0, Muslimov, et al. focusing lens transmission concave holographic grating 3 53' ' FIG. 2 Optical scheme of the flat-fiel spectrograph with TCDG. spectrum plane -04,8-52,4 0 λ= 600 nm 52,4 04,8-04,8-52,4 0 52,4 04,8-04,8-52,4 0 λ= 400 nm λ= 800 nm 52,4 04,8 FIG. 3 Instrument functions of the flat-fiel spectrograph. m M λ = 600 nm λ = 400 nm λ = 800 nm y = 0 mm y = 7.90 mm y = mm δy δz δy δz δy δz TABL 2 Aberrations of the flat-fiel spectrograph with TCDG. an [0], while av subscript marks values relating to the average wavelength of the working range. xpressions enote in qs. 2 an 3 are also erive easily from the equations shown in [9]. By solving system 2 one can fin the spectrum position an holographic coefficients, from which coorinates of recoring sources can be efine. Let s consier a flat-fiel scheme similar to that escribe in the previous section. Again we have a grating with grooves frequency of 400 mm, place 0 mm behin the ieal lens f = 00 mm, f/# = 5. The spectrograph working spectral range is nm. In this case the grating substrate is achromatic meniscus mae of BK7 glass. The center of its secon surface coincies with the incient beam focus. Using qs. 2 3 we fin coorinates of the spectrum center mm, The holographic coefficients are = , 2 = , 3 = The corresponing coorinates of recoring point sources for = 44.6 nm e-c laser are mm, 2 5 an mm, 2 2. The spectrograph optical scheme after efinition of the best image plane is shown on Figure 2. To evaluate the spectrograph image quality we consier its geometrical aberrations on the main wavelengths Table 2. It s clear from the table that goo focusing on the plane along all the working range is obtaine, while the tangential coma an astigmatism for the central wavelength are almost correcte. For etaile investigation of the spectral resolution we consier the spectrograph instrument functions Figure 3. The entrance slit with is 50 µm. The instrument function FWMs at 600, 400 an 800 nm are 62.4, 50 an 50 µm, respectively. Accounting for the reciprocal linear ispersion, which is equal to 27.4 nm/mm, we fin that the corresponing values of spectral resolution are.7,.4 an.4 nm. Thus the flat-fiel spectrograph scheme provies quite a high spectral resolution. In aition such features of the scheme as goo astigmatism correction, relatively small eviation angle an weak focusing properties of the grating shoul be emphasize. In the following sections we propose new scheme esigns using these properties of the transmission concave grating. 2.3 Astigmatism-correcte spectrograph As it was shown above, the spectrograph scheme base on TCDG mounte in a converging beam is notable for its low astigmatism. The astigmatism can be completely correcte in the spectrum center, but it slowly increases towars the eges. This phenomenon can be explaine by a tilt between tangential an sagittal s, which intersect each other in the spectrum center. A simple way to reuce this tilt is introucing of a spherical wege into the optical scheme. We assume 50-3

4 J. ur. Opt. Soc.-api 0, Muslimov, et al. that the first surface of the wege coincies with the grating surface. The obtaine optical component, which represents a combination of grating an wege prism is usually calle grism. Such elements are wiely use in imaging an multichannel spectrometers [3] [5]. The esign proceure consists of several stages. During the first stage, the grating parameters are efine from aberration correction equations, similar to qs In contrast with that system, properties of the grism material must be accounte for. So the equations take on the following form: p j= av p j= [n j cos2 φ + cos φ + n 2j 2 = k λ av cos2 φ j j cos φ j [n j cos2 φ + cos φ 3 = k λ av cos 2 φ j + n 2j cos φ j j [ + cos ϕ + n av [ 2 sin ϕ n av 2 sin ϕ av av cos2 ϕ cos 2 ϕ av av ] 2 kλ j = 0, ] 2 kλ j = 0, av + cos ϕ cos ϕ av cos ϕ av ], ]. ere n an n 2 are refraction inexes for meia before an after the grating, respectively. The s for the grating in tangential an sagittal planes are efine by t = cos ϕ 4 n cos 2 ϕ + n cos 5 ϕ cos ϕ sin ϕ n sin ϕ N n s = + n cos 6 ϕ cos ϕ sin ϕ n sin ϕ N 2 On the secon stage we introuce a spherical wege into the system. We suppose that its angle ψ an secon surface raius are variable an axial thickness t is fixe. The astigmatic ifference in the image after wege is x t x s = t cos ϕ n t cos ϕ + n s cos 2 ϕ n s + cos ϕ n ere, ϕ are coorinates of the image point after the wege. The wege parameters can be foun by means of optimization of a simple merit function: f ast, ψ = 7 q [ ] 2 xt, ψ, λ g xs, ψ, λ g 8 g= On the final stage numerical optimization of the scheme in whole is performe. Because the eviation angles for the grating an wege are relatively small an have ifferent signs, after the optimization the scheme can be mae axial. To exemplify the esign algorithm, let s consier a scheme of astigmatism-correcte spectrograph for the visible omain. The working spectral range is nm. The grism is mounte in normal incience after an ieal lens f = 60 mm, f/# = 7.5. Initial istance from the incience beam focus to the grating is 50 mm it s equal to the grating raius, an initial grating grooves frequency is 450 mm. Both of the grating substrate an wege are mae of BK7 glass. Accoring to the escribe proceure we fin the coorinates of spectrum center 5.78 mm, 9 23 an the holographic coefficients = , 2 = , 3 = The recoring point sources coorinates for = 44.6 nm are 5.62 mm, 0 34 an mm, On the secon stage by using of conjugate graient metho we minimize function 8 an obtain the wege parameters: = 42.5 mm, ψ = 3 8. Coorinates of the spectrum center after the wege are 49.2 mm, To illustrate the achieve astigmatism correction we consier the s in tangential an sagittal planes Figure 4. The right plot correspons to the spectrograph scheme with grism an the left plot correspons to an equivalent scheme with single transmission grating. As it is seen on the plots, the s tilt is significantly ecrease. It also can be note that the eviation angle is smaller. After numerical optimization of the scheme we obtain final esign with the following parameters: the grating surface raius mm, the raius of grating substrate first surface 86.2 mm, the grating recoring parameters mm, 9 7 an 96.8 mm, The final view of the optical scheme is presente on Figure 5. Figure 6 shows the spectrograph instrument functions the entrance slit with is 5 µm. The FWM values for the spectrum center an its eges are 5.0, 5.9 an 5.0 µm. As soon as the reciprocal linear ispersion is 5 nm/mm, the spectral resolution is 0.23, 0.24 an 0.23 nm. On Figure 7 the spectrograph spot iagrams are provie to emonstrate the astigmatism correction obtaine in the final scheme. The transverse imensions of the iagrams are µm. 2.4 Variable-ispersion spectrograph In some cases it woul be very attractive to change spectrometer ispersion an spectral resolution uring measurement. There are a few examples of such spectral instruments, but they use complicate zoom optical systems to change the ispersion [6]. Properties of the optical schemes consiere above suggest that it s possible to buil a simple variable-ispersion spectrograph on the basis of TCDG. In such spectrograph the grating is mounte into a converging beam. Weak focusing properties of the grating together with its aberration correction capabilities allow to fin at least two positions of the grating, in which ifferent spectral ranges are focuse on the same plane. In this case the grating, an therefore, its angular ispersion 50-4

J. ur. Opt. Soc.-api 0, 50 205.. Muslimov, et al. 50 y, mm tangential sagittal y, mm 50 tangential sagittal 0 x, mm 50 00 50 0 50 00 50 x, mm -50-50 FIG. 4 Focal curves of the spectrographs.

5 J. ur. Opt. Soc.-api 0, Muslimov, et al. 50 y, mm tangential sagittal y, mm 50 tangential sagittal 0 x, mm x, mm FIG. 4 Focal curves of the spectrographs. Left: single grating spectrograph; ight: grism spectrograph. focusing lens transmission concave holographic grating spherical wege spectrum plane FIG. 5 Optical scheme of the astigmatism correcte spectrograph λ= 550 nm λ= 400 nm λ= 700 nm FIG. 6 Instrument functions of the astigmatism-correcte spectrograph. FIG. 7 Spot iagrams of the astigmatism-correcte spectrograph. remain the same as well as the length of the etector, while the linear ispersion an spectral resolution change. The working ranges are centre aroun one wavelength. As we coul see before, introucing of a spherical wege allows to implement an aitional aberration correction an make the scheme completely axial. The latter property coul reuce the ispersion change to simple translation of the iffractive optical component. To show feasibility of such esign an fin an initial point for it, let s examine epenence of the TCDG holographic coefficients on the position of the incient beam focus. We 50-5

6 J. ur. Opt. Soc.-api 0, Muslimov, et al. calculate parameters of a number of flat-fiel spectrograph schemes with the same spectrum length, grating raius an grooves frequency. The positions of incient beam focus an spectrum as well as the working range were ifferent. For each of them we solve qs. 2 3 an foun the holographic coefficients. The obtaine epenencies of normalize values of 3 on the istance to beam focus normalize to the grating raius are shown on Figure 8. It can be seen from the graphs that values of an 3 corresponing to aberration correction vary only slightly for istances more than. It means that the tangential terms of aberrations can be correcte in ifferent positions of the same grating. The astigmatism changes rapily an can t be reuce, so further it can be remove from the correction conitions. The initial point correspons to the longer istance to the focus an wier spectral range. The grating parameters are calculate from qs As soon as the etector length in ifferent position remains the same, an obvious conition shoul be met: tan arcsin kλn ϕ av = const. 9 The wege angle is etermine by axial arrangement of the scheme: n sin ϕ ψ = arctan av n sin ϕ. 0 av The wege secon surface is assume to have the same raius as the grating. After this rough efinition of the starting point, the scheme can be optimize numerically. During the optimization the positions of lens an etector as well as the spectrum length shoul be maintaine. As we mentione before, the astigmatism can be except. As an example of this approach we consier a variable-ispersion spectrograph scheme for the range of nm. In this scheme a movable grism is mounte after a specially esigne projection lens f =9 mm, f/#=6.5. An aspheric fiel lens is use to correct resiual aberrations. In the first position spectrum for the full range of nm is focuse on the etector with reciprocal linear ispersion of 30.7 nm/mm. When the grism moves to the secon position, the central part of the spectrum nm is projecte onto the same etector with ispersion of 5.4 nm/mm. Both positions of the scheme are presente on Figure 9. The corresponing instrument functions are presente on Figure 0. For the first position the spectral resolution is.5.9 nm, an in the secon one it reaches 0.7 nm. i i max 3 2 Thus the esigne spectrograph scheme works in two positions with moerately high spectral resolution. The spectrograph is notable for its compactness an extremely simple mechanical scheme. The preliminary estimates show that the foun solution can be extene for a multi-position esign an also can be applie for other combinations of working spectral ranges CONCLUSIONS - - FIG. 8 Depenencies of holographic coefficients on the focus position. Thus in the presente work we propose a group of new spectrograph optical schemes an approaches to their esign. All of the schemes are built on the basis of a iffractive optical element, mounte in a converging beam. Such setup is very simple, compact an allows to create a irect-vision scheme. It was shown that a transmission concave holographic grating TCDG can be use in a flat-fiel spectrograph scheme. FIG. 9 Optical scheme of the variable ispersion spectrograph. Top: position for working in the nm range; Bottom: position for working in the nm range. 50-6

7 J. ur. Opt. Soc.-api 0, Muslimov, et al λ= 600 nm λ= 400 nm λ= 800 nm λ= 600 nm λ= 500 nm λ= 700 nm FIG. 0 Instrument functions of the variable-ispersion spectrograph. It provies relatively high spectral resolution up to 4 times better in comparison with a classical grating an has such features as low astigmatism, small eviation angle an weak optical power. Two optical schemes of spectrographs using these specific features are propose. In the first one a combination of TCDG an spherical wege - grism is use to correct the astigmatism along the spectrum. The scheme is characterize by axial arrangement an high image quality. In the secon esign the grism is move to obtain ifferent linear ispersion with two spectral working ranges. The mechanical scheme of the variable-ispersion spectrograph is extremely simple, while the spectral resolution in the both of positions is relatively high. We think that the evelope optical schemes an corresponing esign algorithms analytical or semi-analytical open prospects of creation of new spectral evices with extene functionalities an enhance performance. 4 ACKNOWLDGMNTS The authors woul like to thank their colleagues from State Institute of Applie Optics Kazan for help with moelling of the optical schemes. eferences [] K.. Norsieck, A. D. Coe, an C. M. Anerson, xploring ultraviolet astronomical polarimetry: results from the Wisconsin Ultraviolet Photo-Polarimeter xperiment WUPP, Proc. SPI 200, [2] K. M. arrison, Grating spectroscopes an how to use them Springer, Berlin, 202. [3] A. V. Savushkin, Design of stigmatic gratings for grazing incience monochromator spectrographs, Proc. SPI 2805, [4]. Wilkinson, an J. C. Green, First-generation holographic, grazing-incience gratings for use in converging, extremeultraviolet light beams, Appl. Opt. 34, [5] T. Namioka, Theory of the concave grating, J. Opt. Soc. Am. 49, [6] N. K. Pavlycheva, an.. Muslimov, Compact ual-ban spectrograph, Av. Opt. Technol., [7].. Muslimov, Transmission holographic grating with improve iffraction efficiency for a flat-fiel spectrograph, Proc. SPI 8787, 87870B 202. [8]. Muslimov, Optical schemes of spectrographs with transmission concave holographic gratings 203 CIOMP-OSA Summer Session on Optical ngineering, Design an Manufacturing, Changchun, August 4 9, 203. [9]. Noa, T. Namioka, an M. Seya, Geometric theory of the grating, J. Opt. Soc. Am. 64, [0] M. M. Nazmeev, an N. K. Pavlycheva, New generation spectrographs, Opt. ng. 33, [] T. Namioka, M. Koike, an D. Content, Geometric theory of the ellipsoial grating, Appl. Opt. 33, [2] C. Palmer, an. Loewen, Diffraction grating hanbook sixth eition, Newport Corp., ochester, [3] M. Aikio, yperspectral prism-grating-prism imaging spectrograph PhD thesis, VTT Technical esearch Centre of Finlan, 200. [4] O. Pawluczyk, an. Pawluczyk, Applications of multichannel imaging spectrometer, Proc. SPI 5578, [5] T. yvarinen,. errala, an A. Dall Ava, Direct sight imaging spectrograph: a unique a-on component brings spectral imaging to inustrial applications, Proc. SPI 3302, [6] J. Choi, T.. Kim,. J. Kong, an J. U. Lee, Zoom lens esign for a novel imaging spectrometer that controls spatial an spectral resolution iniviually, Appl. Opt. 45,

Exam questions OPTI 517. Only a calculator and a single sheet of paper, 8 X11, with formulas will be allowed during the exam.

Exam questions OPTI 517. Only a calculator and a single sheet of paper, 8 X11, with formulas will be allowed during the exam. Exam questions OPTI 517 Only a calculator an a single sheet of paper, 8 X11, with formulas will be allowe uring the exam. 1) A single optical spherical surface oes not contribute spherical aberration.