## Abstract

This is a proposal and description of a new spectrometer based on the Schwarzschild optical system. The proposed design contains two Schwarzschild optical systems. Light diverging from the spectrometer entrance slit is collimated by the first one; the collimated light beam hits a planar diffraction grating and the light dispersed from the grating is focused by the second system, which is concentric with the first. A very simple procedure obtains designs that are anastigmatic for the center of the slit and for a particular wavelength. A specific example shows the performance of this type of spectrometer.

© 2011 Optical Society of America

## 1. Introduction

The Czerny–Turner mount is one of the most popular optical configurations applied to build monochromators and spectrographs. In a traditional Czerny–Turner design, a plane diffraction grating disperses a collimated beam of light coming from a spherical concave mirror, and the dispersed light is focused with another spherical mirror. This system can be configured to compensate for coma aberration, but it shows astigmatism due to off-axis incidence on the mirrors. This does not affect a monochromator significantly, except for a reduction of throughput, as the exit slit of the instrument is aligned with the astigmatic line image, which is orthogonal to the spectral dimension. However, in applications such as spatially resolved ultrashort pulse characterization, frequency-domain optical coherence tomography, or multistripe spectroscopy, where a broad spectral region is analyzed at one time and spectral information is resolved across one spatial dimension, special care must be taken to reduce astigmatism. Various modifications of the Czerny–Turner spectrometer have been proposed to accomplish this task. Some recent proposals have been the inclusion of a tilted parallel plate before the collimator mirror [1]; the use of free-form mirrors [2]; the use of toroidal mirrors, and placing the grating next to the plane that contains the centers of curvature of the mirrors [3]; illuminating the grating with a divergent beam [4]; adding a small piece of glass, which is used as a one-dimensional waveguide [5]; or placing a cylindrical lens after the focusing mirror [6]. An older proposal concerns the inclusion of additional convex mirrors to compensate for astigmatism in the concave mirrors [7]. Rosendahl presented some general relations that allow for astigmatism compensation, but he did not study specific designs. A specific system was suggested and demonstrated in [8], but the authors used aspheric mirrors. In this paper, astigmatism removal is analyzed by using in the spectrometer one of the simplest two-mirror systems comprising only spherical optics: the Schwarzschild optical system (see Fig. 1). In the configuration chosen, the collimator and the focuser share the same convex mirror. Furthermore, a symmetrical design keeps the radii of curvature of the two concave mirrors equal, allowing them to be replaced with a single mirror, as in an Ebert–Fastie spectrometer. Section 2 contains an analysis of the Schwarzschild optical system in order to derive conditions that remove astigmatism. Section 3 describes the new spectrometer, and Section 4 shows a simple design procedure. Section 5 presents a design example obtained by ray-tracing simulations. Finally, the conclusions are given in Section 6.

## 2. Schwarzschild Optical System

Before discussing the complete Schwarzschild spectrometer, astigmatism in a Schwarzschild imaging system is analyzed in this section. This will establish conditions to allow for optimizing the design of collimators and focusers.

The Schwarzschild optical system [9, 10] is a concentric imaging device comprising two spherical mirrors with curvatures of different signs. It can be used either as a beam collimator (namely, a Schwarzschild objective) or a beam focuser (namely, an inverse Cassegrainian telescope). Figure 2 shows the focusing configuration. A parallel bundle of rays hits first the convex mirror from the left and is reflected to the concave mirror, which brings the rays into focus. The Schwarzschild system is usually illuminated on axis, that is, with the light beam centered along an optical axis (any line that crosses the mirror’s curvature center); in which case, the reflected beam presents a central obscuration. In this section, an off-axis illumination is assumed, with the chief ray running parallel to the optical axis *z*. This applies if an aperture stop is located on the beam path as shown in Fig. 2. From now on, the distance of nearest approach *h* between the principal ray and the center of curvature *C* will be referred to as the ray “impact parameter.” One of the remarkable properties of concentric reflective systems is that the impact parameter remains unaltered after reflection, which can easily be deduced by applying the sine theorem in Fig. 2. Mathematically,

*z*axis, ${\theta}_{p}$, ${\theta}_{s}$, are angles of incidence on the mirrors, and

*ω*the angle made by the emergent ray with the

*z*axis. These three angles are related by Owing to the rotational symmetry around the optical axis, any ray obtained from the chief ray by rotation about this axis crosses it at point

*O*. This is the sagittal focus, and therefore the distance ${z}_{f}$ is the sagittal focal distance. However, rays that are contained in the plane of the paper, i.e., meridional rays, cross in a different position. The meridional focus is located at the point of intersection of those rays that are close to the chief ray. This point can be determined by applying the Coddington equation for meridional rays at each mirror in sequence. However, a simpler and more straightforward graphical approach will be taken here. For this purpose, it is helpful to write the equation of an arbitrary ray after reflection on the concave mirror as where for each ray the angle

*ω*is related to the impact parameter by Eqs. (1, 2). Close rays have impact parameters that differ by an infinitesimal value $dh$. The point of intersection of close rays is obtained by applying the condition that a small change in the impact parameter in Eq. (3) does not change the coordinates of this point. By taking the derivative in Eq. (3) and imposing $\frac{dx}{dh}=\frac{dz}{dh}=0$, the following is obtained: Equations (1, 2) are used to obtain the derivative of

*ω*with respect to the impact parameter as This expression is substituted into Eq. (4); next, Eqs. (3, 4) are solved for $(x,z)$ to determine the coordinates $({x}_{m},{z}_{m})$ of the meridional focus as

*O*has coordinates ${x}_{s}=0$, ${z}_{s}={z}_{f}=h/\mathrm{sin}\omega $. The distance between the two foci gives the longitudinal astigmatism $\mathrm{\Delta}r$. After some algebraic calculations, the following result is obtained:

*β*range between 0 and $\pi /4$ (values of

*β*greater than $\pi /4$, giving real values of the angles of incidence, apply to the configuration in which the concave mirror operates as the primary mirror). Except for a scale factor (for example, the value of ${R}_{p}$), the remaining parameters are determined by Eq. (1). Figure 4 plots the radii ratio and normalized focal distance (${z}_{f}/{R}_{p}$) as a function of the normalized impact parameter ($h/{R}_{p}$) in these anastigmatic configurations. There are only anastigmatic solutions for radii ratios between 2 and $(\sqrt{5}+1)/(\sqrt{5}-1)=2.618$, and a focal distance of less than ${R}_{p}$. The limiting case $h=0$, ${R}_{s}/{R}_{P}=2.618$, ${z}_{f}=0.809{R}_{p}$ corresponds to the classical aplanatic Schwarzschild optical system [9].

It must be noticed that the anastigmatic condition (8) can be also obtained by imposing $\frac{d{z}_{f}}{dh}=0$, that is, imposing meridional rays neighboring the chief ray cutting the *z* axis at the same point. This is similar to the strategy of [10]. However, in that case, the authors performed a Taylor expansion of ${z}_{f}$ (which is not necessary) and did not connect their solutions with astigmatism cancellation. Indeed, we are not aware that the anastigmatic condition (8) has been presented before now. Furthermore, it must be shown that the application of Eq. (8) leads to designs that are analogous to those that can be obtained by the classical procedure of compensating the third- order (Seidel) spherical aberration with spherical aberrations of higher orders. Unlike this procedure, which usually requires the use of numerical ray- tracing software, the method presented here is fully analytical.

So far, a situation has been described where a collimated beam hits the Schwarzschild system and is focused. Nevertheless, an analysis similar to the one presented here can be applied when a point source is placed at point *O* in Fig. 2 and the reference ray emerges from the optical system parallel to the *z* axis. In this case, condition (8) applies for a collimated emerging beam, that is, a beam where emerging rays neighboring the reference ray are parallel.

## 3. Schwarzschild Spectrometer

The proposal of this work is a reflective spectrometer in which both the collimating and the focusing optics are Schwarzschild optical systems optimized according to the procedure detailed in Section 2. At first sight, this means that the spectrometer comprises four mirrors. However, all the mirrors can be arranged so that they have a common center. In this case, the collimator and focuser share the same convex mirror (${R}_{3}={R}_{2}$). Furthermore, a symmetrical configuration can be chosen in which the collimator and focuser are optimized for the same impact parameter. In doing so, the two concave mirrors have the same radius (${R}_{4}={R}_{1}$) and can be built as a single mirror, as in the Ebert–Fastie design, provided there is sufficient spare space for the diffraction grating. A sketch of the proposed system is shown in Fig. 5. The slit center is located at *O*, and the axis *Z* contains the common center of curvature *C* and the grating center ${V}_{g}$. The slit and grating grooves are perpendicular to the plane of the drawing, and it is assumed that the grating acts as the aperture stop. The ray following path $O{V}_{1}{V}_{2}{V}_{g}$ is the chief ray in the object space, and the ray that follows the path ${V}_{g}{V}_{3}{V}_{4}{I}_{s}$ is the chief ray in the image space for an arbitrary wavelength.

Once the grating is in place, the object position *O* and the radii ratio are chosen in such a way that a perfect collimated beam, as described in Section 2, is incident on the grating in a direction parallel to line $OC$. This direction makes an angle *δ* with the *Z* axis. In the image space, the wavelength diffracted at an angle ${\delta}^{\prime}=\delta $ makes up an anastigmatic image at a point obtained from *O* by reflection across the *Z* axis. The diffraction grating can be rotated about the *Y* axis to change the wavelength that satisfies this condition. For this purpose, the relation between the incidence and diffraction angles $({\theta}_{g},{\theta}_{g}^{\prime})$ and angles *δ*, ${\delta}^{\prime}$, *α* in Fig. 5 must be taken into account ($\delta ={\theta}_{g}-\alpha $, ${\delta}^{\prime}=\alpha -{\theta}_{g}^{\prime}$). This means that the Bragg’s equation for classical diffraction gratings is written as

*m*is the diffraction order and

*g*the groove density. Any other wavelength with ${\delta}^{\prime}\ne \delta $ makes up two anastigmatic line images centered at ${I}_{m}$ (the meridional image) and ${I}_{s}$ (the sagittal image). Specifically, segment $C{I}_{s}={h}^{\prime}/\mathrm{sin}{\omega}^{\prime}=C{V}_{g}\mathrm{sin}{\delta}^{\prime}/\mathrm{sin}{\omega}^{\prime}$ is parallel to segment ${V}_{g}{V}_{3}$, and the distance between meridional and sagittal images is determined by the longitudinal astigmatism [see Eq. (7)].

This spectrometer can be thought of as a modification of four different optical systems:

- A Czerny–Turner spectrometer, in which each concave mirror is replaced by a pair of concentric mirrors, one convex and the other concave.
- An Ebert–Fastie spectrometer, in which the plane diffraction grating is replaced by an optical system composed of a plane grating and a convex mirror.
- An Offner spectrometer in which the convex grating is replaced by a convex mirror plus a plane grating.
- A Schwarzschild system with off-axis illumination from a linear slit and a plane diffraction grating located between the convex and concave mirror.

## 4. Design Strategy

In this section, we present a design procedure of a Schwarzschild type spectrometer based on the configuration depicted in Section 3 and the results of Section 2. This design procedure starts by choosing the parameters that are left free at the beginning of the computation. Even though there are several possible combinations of parameters, for simplicity of calculation, the following ones have been chosen: the convex mirror radius (${R}_{2}={R}_{4}$), the angle of incidence on the grating with respect to the *Z* axis (*δ*), the distance between the vertex of the convex mirror and the vertex of the grating (in Fig. 5 this is $P{V}_{g}\equiv d$), the groove density (*g*), the grating order (typically, $m=\pm 1$), and the design wavelength *λ*. There are some constraints that suggest or restrict the values of these parameters. First, they must be chosen to obtain a spectral image without vignetting in the desired spectral band of the instrument. Second, the convex mirror radius determines the size of the instrument. Third, once the radii of both mirrors have been fixed, the grating density determines the system dispersion. Finally, since the spectral components diffracted further away from the center of curvature *C* present large optical aberration, the design wavelength chosen must be closer to the maximum if $m=-1$, while it is chosen closer to the minimum wavelength if $m=1$.

The remaining parameters that define the instrument (concave mirror radius, object position, and grating orientation) are calculated by applying the formulas of Section 2 to obtain an anastigmatic collimator and focuser at the design wavelength. The following steps are required:

- The impact parameter is determined by $h={Z}_{g}\mathrm{sin}\delta =(d+{R}_{2})\mathrm{sin}\delta $.
- The angle of incidence ${\theta}_{2}$ on the convex mirror is ${\theta}_{2}={\mathrm{sin}}^{-1}(h/{R}_{2})$.
- The angle of incidence ${\theta}_{1}$ on the concave mirror is ${\theta}_{1}={\theta}_{2}-\beta $.
- The radius of the concave mirror is ${R}_{4}={R}_{1}=h/\mathrm{sin}{\theta}_{1}$.
- The object focal distance of the collimator is ${z}_{f}=h/\mathrm{sin}\omega $.
- The coordinates of the object point are ${X}_{0}=-{z}_{f}\mathrm{sin}(\delta )$, ${Z}_{0}=-{z}_{f}\mathrm{cos}(\delta )$.
- The angle of rotation of the grating (
*α*) is given by $\mathrm{sin}\alpha =mg\lambda /(2\mathrm{cos}\delta )$.

Figure 6 shows typical focal lines obtained with a spectrometer designed in this way (they correspond to the design presented in Section 5). It can be seen that the curves cross only at the design wavelength and at a wavelength that is diffracted on axis (${\delta}^{\prime}$, ${h}^{\prime}=0$), which does not reach the detector. This means that the longitudinal astigmatism does not increase dramatically for wavelengths diffracted to the axis side (longer wavelengths if $m=1$ and shorter wavelengths if $m=-1$). It has been confirmed that it is not possible to make the curves at the design wavelength tangent (this will remove astigmatism to the first order). This results from the fact that, in a Schwarzschild optical system, it is not possible to cancel astigmatism and its derivative with respect to the impact parameter simultaneously.

## 5. Example of Design

To illustrate the performance of a Schwarzschild type spectrometer, an optimized system has been designed using the above formulas. The optical design program Oslo-Edu was used to carry out ray-tracing procedures. The spectrometer was designed to image a spectral band centered on $700\text{\hspace{0.17em}}\mathrm{nm}$ over the long side ($8.8\text{\hspace{0.17em}}\mathrm{mm}$) of a $2/3\text{\hspace{0.17em}}\mathrm{in}\mathrm{.}$ CCD detector. With the parameters of Table 1, this was achieved with a spectral coverage of $270\text{\hspace{0.17em}}\mathrm{nm}$. Other parameters of the spectrometer, such as the object position or the position of its anastigmatic image, can be easily calculated from the formulas in Section 4. A check was made first to ensure that astigmatism was minimized at the design wavelength of $794\text{\hspace{0.17em}}\mathrm{nm}$. Figure 7a shows the spot diagram for an on-axis point in an $F/4$ instrument. The symmetry of this spot along the horizontal dimension is evidence of the absence of Seidel’s coma. In fact, the spot diagram resembles fifth-order cubic astigmatism (or oblique spherical aberration) to a certain extent, but it has not been studied further. In Fig. 7b, sagittal and meridional focal curves for this wavelength and for a $10\text{\hspace{0.17em}}\mathrm{mm}$ long slit are shown. Astigmatism is null for the on-axis object point and remains slight for the full field object point; however, there is a notable field curvature associated with the spherical symmetry of the Schwarzschild optical systems [11], which precludes the use of large slits unless special care is taken to correct the effect. One possibility is to optimize the spectrometer for an off-axis object point, so the spectrometer will be automatically optimized for the off-axis point arranged symmetrically to the center of the slit. It has been confirmed that this stratagem diminishes the overall spot size, as seen in Fig. 8. Unfortunately, it increases spatial and spectral distortions at the same time. Another option is to insert appropriate field lenses, just in front of the slit and the image plane, but this leads to a more complicated instrument.

The values of free parameters in Table 1 were chosen to obtain an optimized instrument over the spectral band specified above, with a slit length of $4\text{\hspace{0.17em}}\mathrm{mm}$ and an *F* number of 4. It must be emphasized that the parameters obtained from the formulas in Section 5 were used directly without further ray- tracing optimization. The only adjustment was to the values of free parameters, in order to obtain small spot sizes over the whole field and spectral range, without vignetting. In this step, the autofocus function of the optical design software was applied at two different wavelengths to obtain the position and orientation of the image plane. RMS spot radii as a function of wavelength for on-axis, 0.7, and full field object height are given in Fig. 9. Similar spot sizes are obtained in different spectral bands of similar width when the grating is rotated to allocate a new band on the detector. This means that the instrument can be tuned over a broad spectral range. Furthermore, the spectrometer can be built with commercial diffraction gratings of different groove density, allowing the instrument’s spectral coverage to vary. Table 2 shows the spectral coverage and RMS spot radii for different gratings.

## 6. Conclusions

This is a proposal for a new spectrometer, which can be considered as a modification of a Czerny–Turner spectrometer in which Schwarzschild optics are used both to collimate light before the grating, and to focus the dispersed light. First, research was done into astigmatism in a Schwarzschild focuser/collimator, and conditions for an anastigmatic configuration were established. Based on this analysis, a specific concentric and symmetrical spectrometer design (except for grating orientation) that minimizes the number of mirrors was proposed. The concentric design will facilitate the optical alignment in a practical instrument. Furthermore, avoiding the use of refractive elements or curved diffraction gratings allows the system to be tuned in a broad spectral range, as with the classic Czerny–Turner mount. Other concentric designs, such as the Offner spectrometer, which resembles the proposed design, certainly present high imaging performance, since they can handle larger slits and lower *f* numbers; but they lack the capability of tuning.

Using a specific design procedure, an illustrative example was presented showing the effectiveness of astigmatic correction. The main drawback of this new instrument is the field curvature generated by the Schwarzschild optical systems. In applications which require long slits, some means must be devised to reduce this optical aberration.

The authors hereby express their acknowledgment to the Xunta de Galicia for providing financial support under the contract 07MDS035166PR. Héctor González-Núñez acknowledges support to the Xunta de Galicia by the María Barbeito Program.

**1. **E. S. Voropai, I. M. Gulis, and A. G. Kupreev, “Astigmatism correction for a large-aperture dispersive spectrometer,” J. Appl. Spectrosc. **75**, 150–155 (2008). [CrossRef]

**2. **L. Xu, K. Chen, Q. He, and G. Jin, “Design of freeform mirrors in Czerny–Turner spectrometers to suppress astigmatism,” Appl. Opt. **48**, 2871–2879 (2009). [CrossRef] [PubMed]

**3. **Q. Xue, S. Wang, and F. Lu, “Aberration-corrected Czerny–Turner imaging spectrometer with a wide spectral region,” Appl. Opt. **48**, 11–16 (2009). [CrossRef]

**4. **D. R. Austin, T. Witting, and I. A. Walmsley, “Broadband astigmatism-free Czerny–Turner imaging spectrometer using spherical mirrors,” Appl. Opt. **48**, 3846–3853 (2009). [CrossRef] [PubMed]

**5. **C. Chrystal, K. H. Burrell, and N. A. Pablant, “Straightforward correction for the astigmatism of a Czerny–Turner spectrometer,” Rev. Sci. Instrum. **81**, 023503 (2010). [CrossRef] [PubMed]

**6. **K.-S. Lee, K. P. Thompson, and J. P. Rolland, “Broadband astigmatism-corrected Czerny–Turner spectrometer,” Opt. Express **18**, 23378–23384 (2010). [CrossRef] [PubMed]

**7. **G. R. Rosendahl, “Contributions to the optics of mirror systems and gratings with oblique incidence. III. Some applications,” J. Opt. Soc. Am. **52**, 412–415 (1962). [CrossRef]

**8. **T. H. Kim, H. J. Kong, T. H. Kim, and J. S. Shin, “Design and fabrication of a $900\u20131700\text{\hspace{0.17em}}\mathrm{nm}$ hyper-spectral imaging spectrometer,” Opt. Commun. **283**, 355–361 (2010). [CrossRef]

**9. **I. A. Artioukov and K. M. Krymski, “Schwarzschild objective for soft x-rays,” Opt. Eng. **39**, 2163–2170 (2000). [CrossRef]

**10. **A. Budano, F. Flora, and L. Mezi, “Analytical design method for a modified Schwarzschild optics,” Appl. Opt. **45**, 4254–4262 (2006). [CrossRef] [PubMed]

**11. **W. B. Wetherell and M. P. Rimmer, “General analysis of aplanatic Cassegrain, Gregorian, and Schwarzschild telescopes,” Appl. Opt. **11**, 2817–2832 (1972). [CrossRef] [PubMed]