Concept and optical design of a compact cross-grating spectrometer

Verena Bagusat; Matthias Kraus; Erik Förster; Daniel Thomae; Tobias Hönle; Robert Brüning; Hartmut Hillmer; Robert Brunner

doi:10.1364/JOSAA.36.000345

1. INTRODUCTION

The ongoing trend of implementing spectroscopic analysis for industrial process control and a variety of field applications leads to increasing requirements regarding cost and size, measurement time, as well as optical performance of the spectrometer. Major driving application fields comprise agriculture with a plethora of different tasks, quality assurance in the food industry, or environmental monitoring for the timber industry, to note only a few [1–3]. Such field and high-throughput applications are partly related to contradictory demands, in which, on the one hand, typically high-quality spectrometers are too bulky and costly, and on the other hand, compact spectrometer concepts are not suitable to meet the optical parameters, e.g., allow simultaneously a high-spectral resolution and large spectral range.

Commonly applied spectrometers for such industrial inline and field applications are, for example, monolithic miniature spectrometers [4] or designs based on a Czerny–Turner setup [5], which are optimized regarding system size. Typically, such spectrometers are using the first-diffraction order and small grating periods to achieve a sufficient high spectral resolution, but the detectable spectral range is limited by the length of the linear detector and cross talk from the second-diffraction order. To overcome this limitation, also multiorder configurations are developed, that rely on either changeable spectral filters or adaptable light sources to separate the overlapping spectral intervals [6]. Therefore, such systems are no longer real-time able, and they are not applicable for inline process control.

In contrast, a special high-quality instrument type established in research and high-performance applications is the echelle spectrometer, e.g., commonly used in astronomical spectroscopy [7–9]. The central element is a coarse grating (echelle grating), which is used in several higher diffraction orders simultaneously to provide the required high spectral resolution. Since such a higher-order configuration results in a spatial overlap of various diffraction orders, a second dispersive element is necessary. This cross-dispersing element is oriented perpendicular to the echelle dispersion direction to separate the overlapping higher diffraction orders spatially and allow the simultaneous detection of a large free-spectral range on a two-dimensional detector [10]. The drawback of this instrument type is a complex design due to the required two dispersive elements, resulting in bulky, expensive, and heavy systems, which limits the possibility of size reduction. Hence, typically they are used in laboratory-based applications such as analytical spectrometry [11], for material and element analysis, e.g., inductively coupled plasma [12,13], laser induced breakdown [14,15], or Raman spectroscopy [16]. Apparently, existing solutions are designed and optimized for specific requirements and therefore are limited in their flexibility for the use in different application scenarios. Thus, there is a continuing challenge to implement high-performance optical requirements in a compact and real-time able spectrometer.

In the presented contribution, we propose a new concept of a compact echelle-inspired cross-grating spectrometer to bridge the gap between high-end and compact, portable spectroscopic instruments. For that, the cross grating combining both dispersion functions of the classical echelle spectrometer is integrated into a folded, mirror-based optical system. Furthermore, a cylindrical mirror is added to correct for astigmatism [17]. In total, our new approach enables a compact-sized setup with significantly enhanced performance attributes.

In a first experimental proof-of-concept, the general suitability of a cross grating in a spectrometer has already been demonstrated (CGES, cross-grating echelle spectrometer), but the simplified optical setup was limited in spectral resolution and accessible spectral range [18]. To explore the full capabilities of the new system in detail, we introduce a specific optical concept based on an all-reflective and folded beam path, which is optimized regarding spectral resolution, system size, and detectable spectral range. As a result, the new spectrometer is able to cover a spectral range from the 330 to 1100 nm, with a spectral resolution of about 0.5 to 1 nm and a size of $110 mm \times 110 mm \times 30 mm$ . To discuss our proposed new concept and prove its enhanced optical performance, we present the optimized design in detail and evaluate the image quality based on spot diagrams. Additionally, the diffraction efficiency of the cross grating is analyzed using a simplified model and applying a rigorous simulation method. Finally, we derive the achieved optical resolution from all contributing effects such as residual aberrations, slit width, and grating dispersion.

2. SPECTROMETER SETUP AND OPTICAL DESIGN

For the design of the CGES, a folded beam path was used to minimize the geometrical dimensions and enable a very compact setup. Therefore, we applied a modified Czerny–Turner configuration with an additional correction mirror to reduce the typical astigmatic aberrations of such optical designs. A second advantage of this mirror-based setup is its applicability on a wide spectral range even in the ultraviolet, by using an enhanced aluminum coating, since normal (inorganic) glasses are fairly or even nontransmissive. For the simultaneous detection of the whole two-dimensional spread spectrum, we use a CCD-array detector with an active area of $15.4 mm \times 15.4 mm$ . In summary, this approach enables the construction of a very compact spectrometer of only $110 mm \times 110 mm \times 30 mm$ size, which is capable of covering a spectral range from 330 to 1100 nm. The CGES setup is shown in Figs. 1 and 2 in the $y - z$ and the $x - y$ cross section, respectively.

Fig. 1. Optical layout in the $y - z$ cross section with ray trace for the fourth order showing the dispersion by the echelle-type grating. The insets show a schematic representation of the crossed grating profile and the two-dimensional spectral image at the detection plan.

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Fig. 2. Optical design layout in the $x - y$ cross section with the ray trace for both the central wavelengths at all orders (a) to illustrate the effect of the cross-dispersion grating and (b) the chief ray for 765 nm in the fifth order highlighting the Littrow configuration and the planar integration of the whole setup.

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The light enters the setup by the entrance pinhole and illuminates a concave mirror, which collimates and directs the light onto the dispersive element. At this point the optical setup of the collimation unit is modified concerning the conventional Czerny–Turner configuration by adding a correction mirror that is used for reducing aberrations by compensating for astigmatism, thus improving spectral resolution. The dispersive element is a blazed crossed grating combining a high dispersive echelle-type grating with a perpendicularly oriented low-dispersion cross disperser. The integration of both functionalities in a single optical element allows a compact design with a high spectral resolution.

The blazed cross-grating profile is shown schematically in the inset of Fig. 1. The vertically oriented echelle grating shows a larger grating period compared to the horizontally aligned cross-dispersion grating. The cross grating splits the collimated light into various diffraction angles depending on the wavelength and diffraction order. The following focusing mirror directs and focuses the light onto the detector. The final recording of the two-dimensional image pattern with separated wavelengths at different diffraction orders is shown schematically as a detector image in Fig. 1. In Fig. 2, the system is depicted in the $x - y$ -plane showing the effect of the cross dispersion with the central wavelengths for the fourth to the 11th diffraction order. Again, the collimating, correction, and focusing mirrors are visible, as are the crossed grating and the detector.

In detail, the concave collimating mirror has a focal length of $+ 101.6 mm$ . The convex, cylindrical correction mirror has a large nominal focal length of 6135.2 mm. Its influence on the total focal length of the collimation optics is rather small, with a change of 0.3% to 101.9 mm in the cylindrical section. The combined demand of a large radius in the cylindrical section and a comparatively small effective diameter of the element is associated with fabrication challenges. As a solution, a mechanical deformation of a standard plane mirror is intended. The final focusing mirror also has a focal length of $+ 101.6 mm$ . The symmetric focal lengths of both collimating and focusing mirror result in an image scale of about one. The values are defined to minimize air distances between the optical elements. The pupil size is set to 10.2 mm, resulting in moderate mirror $f$ -numbers of $f / 10$ for both components. The details on mechanical dimensions and optical parameters for the complete system are summarized in Table 1.

Table 1. Mechanical Dimensions and Optical Parameters for the Optical System Comprising the Mirrors and the Grating for a Pupil Size of 10.2 mm

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All optical elements are tilted to implement the folded ray path. The system is modeled with the commercial optical design software OpticStudio [19] and is defined in a right-handed Cartesian coordinate system with the global coordinate origin on the pinhole aperture. The rotations of the optical elements are performed in the local coordinate systems of each optical element, with the respective origins on the vertices. The local $z$ -axis points in the direction of the incident chief ray at 765 nm in the fifth order. The tilt angles are optimized to achieve a very compact design and to prevent ray vignetting. Because coma is introduced by the oblique illumination of the mirrors, the smallest reasonable tilting angles have been selected for the mirror components. The first two mirrors are tilted by 4 deg and $- 20 \deg$ and the focusing mirror by $- 7 \deg$ about their local $x$ axes, respectively. As can be seen from Fig. 1, the rotation plane for these elements is in the $y - z$ plane. Additionally, the grating is tilted about the local $y$ axis, which is visible in the $x - y$ cross section in Fig. 2. The incident collimated light illuminates the grating at an oblique angle of 22.9 deg. The two-dimensional grating diffracts the incident light into oblique directions in the three-dimensional space described by a normalized vector consisting of the components according to the following equations [20]:

ζ_{m_{x}} = ζ_{0} + \frac{λ m_{x}}{d_{x}}, ξ_{m_{y}} = ξ_{0} + \frac{λ m_{y}}{d_{y}} χ_{m_{x} m_{y}} = \sqrt{1 - ζ_{m_{x}}^{2} - ξ_{m_{y}}^{2}} .

The scalar vector components

ζ_{m_{x}}, ξ_{m_{y}}

and

χ_{m_{x}, m_{y}}

of the diffracted ray related to the

x

,

y

, and

z

axes are dependent on the diffraction orders

m_{x}, m_{y}

of the cross grating, the corresponding grating periods

d_{x}

and

d_{y}

, the wavelength

λ

, and the scalar vector components

ζ_{0}, ξ_{0}

of the incidence ray.

The echelle-type grating is used to improve resolution due to the characteristic of diffracting different wavelengths in large dispersive angles. In contrast, the cross-dispersion grating is used in the first order with lower spectral resolution than the echelle grating but therefore covering the whole spectrum in the first order. The grating constants have been determined considering common values of echelle gratings [10,21] and the maximum feasible image angle of 8.5 deg along the detector height. This angle determines the maximum diffraction angle range and results from the focusing mirrors’ focal length and the detector size. To fulfill this angle, the grating constant of the cross disperser is set to $d_{x} = 5.2 μm$ . The echelle grating is used in the orders four to 11 with a grating period of $d_{y} = 10 μm$ . Using the echelle grating in diffraction orders lower than 10 and at an incidence angle lower than 40 deg is unusual [10,21] and is driven by the main requirement to achieve a compact design and the objective to detect a large spectrum.

Table 2 contains the data for the design wavelengths and spectral ranges with the corresponding diffraction orders for which the system is optimized. The fourth and fifth orders cover the infrared spectrum from 780 to 1100 nm. The visible light ranging from 380 to 780 nm is mainly imaged in the fifth up to the ninth order. The ultraviolet spectrum from 330 to 380 nm is diffracted into the 10th and 11th order. The spectral bandwidth $δ_{m}$ acquired with the detector for each order decreases with an increase in the diffraction order. Values vary from 290 nm at the fourth order to 42 nm at the 11th order. The decrease in spectral range between the fourth and fifth order is 82 nm compared to 9 nm between the 10th and 11th order. The image analysis is performed for five selected wavelengths in the eight diffraction orders defined in a multiconfiguration setting. The two-dimensional diffraction at the crossed grating is simulated as a user-defined surface, which is implemented by a dynamic link library (DLL) file, and the vector components of the diffracted rays are derived from OpticStudio in the $x$ , $y$ , and $z$ directions. The wavelength splitting inside an order is analyzed by calculating the scalar product between the vectors for the maximum and minimum wavelengths. The ray-tracing simulation considers the geometrical orientation of the diffracted rays according to the two-dimensional grating equation, but neglects the diffraction efficiency. The diffraction efficiency of the grating is evaluated by a rigorous simulation technique, which is discussed in Section 3.

Table 2. Maximum and Minimum Detectable Wavelengths and Respective Spectral Bandwidth for the Cross Grating

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The ray trace in the $y - z$ cross section in Fig. 1 shows the wavelength splitting introduced by the echelle-type grating in the main dispersion direction. The separation of the main diffraction orders into the $x$ direction by the cross disperser causes an overlay of the orders in the layout. Hence, for clear distinction, selected wavelengths are drawn for the 290 nm spectral bandwidth in the fourth order. The black rays illustrate the maximum wavelength of 1100 nm and the red the minimum wavelength of 820 nm, denoted as “wave 1” and “wave 5.” The distance between the red and black image points decreases for higher orders with shorter spectral bandwidth and larger dispersion associated with higher spectral resolution. The ray trace in the $x - y$ cross section in Fig. 2(a) shows the cross-dispersion direction resulting in the order splitting induced by the cross grating. Again, to distinguish the images from overlapping wavelengths, rays are only drawn for the central wavelength in all eight orders, denoted as $λ_{central}$ . The gray rays represent the 957 nm in the fourth order and the black rays the 348 nm in the 11th order. The distance between the gray and black image points decreases for shorter wavelengths and increases for longer wavelengths. Figure 3 shows the resulting detector image for the optimized wavelength range at each order. The diffraction angle range for the infrared wavelength from 820 to 1100 nm at the fourth order is 7.4 deg. Light in the ultraviolet at the 11th order is dispersed by 2.7 deg, causing a shorter spectral line of one-third of the detector side length. The detector is slightly decentered and rotated about the $z$ axis. The largest distance of 17.3 mm for the full spectrum from 330 nm to 1100 nm results from an angle of 10 deg and almost fills the detector diagonal.

Fig. 3. Simulated image of the whole detector plane showing the diffraction orders four to 11 and the spectral sampling points for the optimization.

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Finally, in the layout of Fig. 2(b), solely the chief ray for the central wavelength of 765 nm in the fifth order is depicted to illustrate the Littrow configuration for this central wavelength. The grating rotation about the $y$ axis is adapted for the use in the Littrow configuration, and light is reflected back into the incidence direction for this wavelength [22]. This setting allows for the positioning and mounting of the other mirrors inside the $y - z$ plane, representing a relaxation for the mechanical design, assembly, and integration. In addition, the volume is reduced for implementation of the compact design. The focusing mirror is centered to 765 nm, representing a central wavelength, which is near the middle of the spectrum. The illumination of the focusing mirror by the light cones surrounding this wavelength is well symmetric around and close to the optical axis. This ensures that a minimum of aberrations is introduced to the other wavelengths, and the mirror aperture is reduced as much as possible in terms of volume reduction. Furthermore, the air distances between the focused rays forwarded from the focusing mirror and the correction mirror, as well as the grating, are tight. Thus, this mirror position ensures that the focused rays pass the correction mirror and the grating without vignetting.

3. GRATING PROFILE AND DIFFRACTION EFFICIENCY

The cross grating is the key element of the CGES and is designed as a superposition of two perpendicularly oriented blazed line gratings of different depths and periods. For the implementation of our approach, the diffraction efficiency of the grating plays a decisive role. As far as we know, no rigorous simulation method is available that allows a comprehensive and complete calculation of the diffraction efficiency of a cross grating exhibiting large grating periods and a metal coating with high accuracy in an acceptable computation time. Therefore, we chose a simplified approach and model the cross grating by two independent, perfectly blazed line gratings. After computing the diffraction efficiencies $η_{x}$ and $η_{y}$ of each one-dimensional line grating, the resulting overall efficiency $η$ of the two-dimensional cross grating is obtained by multiplying both line-grating efficiencies. With this approach, the absorption losses from the metal coating are included twice, which has to be compensated by dividing the product of the efficiencies by the reflectance $R$ of the metal coating:

η = \frac{η_{x} η_{y}}{R} .

The line-grating efficiencies are simulated by rigorous coupled-wave analysis (RCWA) using the software UNIGIT (Osires, Ilmenau, Germany). Both blazed profiles are created in a unit cell and are sampled by 200 data values in a lateral direction. The profile heights are approximated by 200 slices. The wavelength is varied between 300 and 1100 nm in steps of 5 nm. The grating periods of

d_{x} = 5.2 μm

and

d_{y} = 10 μm

are derived from the system concept (see Section 2). The cross grating is tilted by 22.5 deg and 4.2 deg about the local

x

and

y

axis, respectively. From this it follows that the one-dimensional line gratings are used in the conical mount, which is taken into account by the RCWA solver. The blaze profile heights

h_{b}

[Eq. (3)] in the

x

and

y

directions are derived from the respective blaze angles

Θ_{b}

, following the blaze condition [Eq. (4)] for the classical mount [23]:

h_{b} = d \tan Θ_{b},

m λ_{b} = d [\sin Θ_{i} + \sin (2 Θ_{b} - Θ_{i})] .

By choosing the grating order

m_{y} = 10

, the blaze wavelength

λ_{b} = 383 nm

, the grating period

d_{y} = 10 μm

, and an angle of incidence

Θ_{i} = 22.5 \deg

for the main grating, a profile height of 1.99 μm can be found. For the cross disperser with a period of

d_{x} = 5.2 μm

, the profile height is 0.28 μm (illumination angle

Θ_{i} = 4.2 \deg

, first diffraction order, and wavelength 550 nm). For the simulation, the gratings are assumed to be surrounded by air and are made of

{SiO}_{2}

with a 100 nm thick reflective aluminum layer. A convergence analysis for selected wavelengths shows that 80 Rayleigh orders are sufficient for the cross-dispersion grating and 200 Rayleigh orders for the main grating, respectively.

Figure 4 shows the calculated diffraction efficiencies for the main grating [Fig. 4(a)], the cross -disperser [Fig. 4(b)], and the combined cross grating [Fig. 4(c)]. As expected, the efficiency curves of adjacent main diffraction orders overlap, and so the energy is distributed among these orders. This leads to the typical effect of echelle spectrometers: that most wavelengths are detectable in adjacent spectral bands on the two-dimensional detector. For the analysis of the overall spectral data, the measured intensities of the specific wavelengths can be added by the contributions of the different bands. While the main grating shows a high diffraction efficiency over the whole spectral range, the efficiency of the cross disperser significantly decays for wavelengths with increasing distance to the blaze wavelength. This behavior is known from classical blazed line gratings, which are used over a large spectral range. Hence, the resulting overall diffraction efficiency is mainly limited by the cross disperser. From this it follows that the diffraction efficiency of the cross grating is comparable to classical blazed gratings, which are used in the first order. The calculated diffraction efficiency allows an estimation of the theoretical limit of the cross grating. The main challenge is the manufacturing of the cross-grating element as a superposed blazed grating. A more detailed discussion on manufacturing aspects of cross gratings can be found in Ref. [18].

Fig. 4. Calculated diffraction efficiencies for (a) the third to 11th order of the main grating, (b) the zeroth to second order of the cross-dispersion grating, (c) and the third to 11th order of the cross grating. The resulting overall efficiency is calculated by multiplying the efficiency of the first cross-dispersion order with the third to 11th order of the main grating and dividing the results with the reflectance of an aluminum mirror. A summarized efficiency (resulting efficiency of the third to 11th order) is shown by the black curve with circle markers (c), representing the measurable signal on the detector.

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4. SPECTRAL RESOLUTION

The spectral resolution of the spectrometer is influenced by various effects, such as the grating dispersion, pinhole size, and imaging aberrations. The different contributions on the spectral resolution $Δ λ_{i}$ ( $i$ , indicating the specific effect) are calculated from the respective broadening values $Δ X_{i}$ as well as the geometrical separation of the images, given by the reciprocal linear dispersion $D_{L}^{- 1}$ of the grating [24]:

Δ λ_{i} = Δ X_{i} D_{L}^{- 1} .

Values for the image separation range from 23 nm/mm at the fourth diffraction order up to 9 nm/mm at the 11th diffraction order. The overall system spectral resolution

Δ λ_{system}

comprises individual contributions

Δ λ_{i}

and can be calculated by the following equation, which is based on the approach of spectral line broadening [21]:

Δ λ_{system} = \sqrt{\sum_{i = 1} Δ λ_{i}^{2}} .

The value of

Δ λ_{system}

is dependent on the specific wavelength and varies for the different analyzed spectral images. The six main broadening functions in the proposed system are related to the diffraction from the collimation unit

Δ λ_{1}

, the focusing mirror

Δ λ_{2}

, and the grating

Δ λ_{3}

, as well as the influence of the pixel size

Δ λ_{4}

, the finite pinhole size

Δ λ_{5}

, and imaging aberrations

Δ λ_{6}

. These functions are given in detail in Table 3. There,

λ

is the wavelength and

f_{col}

,

f_{foc}

,

ϕ_{col}

, and

ϕ_{foc}

are the focal lengths and clear apertures of the collimation unit and the focusing mirror, respectively.

l

is the distance between the grating and the detector,

B

is the width of the grating, and

{spot}_{RMS}

is the RMS spot radius with respect to the centroid for the images at a specific wavelength. The detector pixel size and the entrance pinhole size are denoted as pixelsize and pinholesize and specified as 7.4 and 40 μm, respectively. The correction factor

ϒ

is defined as

ϒ = A \frac{f_{foc}}{f_{col}},

with A denoting the anamorphic magnification factor, defined as the relation of the angle of incidence and angle of diffraction for a specific wavelength [21]. Due to the symmetric focal ratio of the collimating and focusing optics, only the anamorphic magnification factor has influence. The angles of incidence and diffraction are derived from OpticStudio.

Table 3. Calculation of the Various Broadening Functions Defining the Overall Spectral Resolution $λ_{system}$

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Due to the identical aperture ratios of the collimation and focusing unit, the diffraction effects on spectral resolution are in the same range. Values for the image point broadening $Δ X_{1}$ and $Δ X_{2}$ range from 14 μm in the fourth order to 4 μm in the 11th order, which corresponds to a decrease in spectral resolution of $Δ λ_{1} = 0.3 nm$ for $λ = 1100 nm$ and $Δ λ_{2} = 0.04 nm$ for $λ = 330 nm$ in the 11th order. Also, the broadening originating from the grating and the finite pixel size are of a similar magnitude. The image point broadening from the grating is the size of $Δ X_{3} = 19 μm$ in the fourth order to 5 μm in the 11th order, causing a decrease in spectral resolution of $Δ λ_{3} = 0.4 nm$ for $λ = 1100 nm$ in the fourth order and 0.05 nm for $λ = 330 nm$ in the 11th order. The reduction of the spectral resolution from the pixel size is in a range of $Δ λ_{4} = 0.2 nm$ in the fourth order and 0.07 nm in the 11th order. However, the main broadening effects are introduced from the pinhole size and the system aberrations; thus, they dominate the achievable spectral resolution $Δ λ_{system}$ . The values $Δ λ_{5}$ for the pinhole range from 0.85 nm at $λ = 1100 nm$ and 0.33 nm at $λ = 330 nm$ in the fourth and 11th order, respectively. The broadening effects from aberrations on the spectral images $Δ X_{6}$ are analyzed in OpticStudio by the RMS spot size shown in Fig. 5 for two exemplary wavelengths and diffraction orders. Figure 5(a) shows the spot diagram at 765 nm of the fifth diffraction order with an RMS of 37.5 μm and represents exemplarily the aberrations in the nominal system without the correction mirror. In this case, the image points are mainly affected by astigmatism, coma, and field curvature. Spot size RMS values for the other wavelengths and diffraction orders are up to 61.6 μm, as shown in Fig. 5(b) for the 1100 nm spot diagram in the fourth diffraction order. The influence on the overall spectral resolution is between $Δ λ_{6} = 0.31 - 1.36 nm$ , depending on the diffraction order. By using the cylindrical correction mirror in the system, the astigmatism is corrected, and the resulting spot sizes are reduced by factors of 1.5 up to 2.5. The RMS is significantly improved to 14.2 μm for 765 nm in the fifth order and to a maximum value of 42.5 μm for 1100 nm in the fourth order, as depicted in Figs. 5(c) and 5(d), respectively.

Fig. 5. Spot diagrams of the central wavelength (a) 765 nm in the fifth order and (b) one at an edge position at 1100 nm in the fourth order before correction by the cylindrical mirror. The effect of correction is shown in (c) and (d) for the central and edge wavelength, respectively. The corresponding RMS spot size radii are (a) 37.5 μm, (b) 61.6 μm, (c) 14.2 μm, and (d) 42.5 μm.

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The best-case value is associated with the image at 765 nm because the focusing and the collimation mirrors are centered in the $x$ and $y$ planes, and thus rays pass the mirrors centric, and the system is focused on that. The spot diagrams, corresponding to the peripheral wavelength area, are somewhat larger because the ray bundles hit the focusing mirror under more off-axis conditions due to larger diffraction angles. The small amount of nonsymmetrical aberrations in the central image is a result of the system mirror tilts. The optical axis is tilted relating to the chief ray with the effect of a moderate off-axis condition for the centered image at 765 nm in the fifth order. Both effects cause the residual coma and field curvature on the wavelength images after the astigmatism optimization. The influence on the overall spectral resolution is between $Δ λ_{6} = 0.95 nm$ at $λ = 1100 nm$ and 0.21 nm at $λ = 330 nm$ , respectively, in the fourth and 11th order. In the improved system, the aberration-dependent spectral resolution $Δ λ_{6}$ is reduced by a factor of up to 1.6 (40%), compared to the system without the correction mirror.

Finally, according to Eq. (6), the spectrometer quality for the optimized system is characterized by its spectral resolution combining the separate influences discussed above. Figure 6 shows the spectral resolution of the overall system for the eight relevant diffraction orders, where three spot series are displayed. The best resolution $Δ λ_{system} (central)$ is achieved for the central wavelength position of each spectral interval (green spots). The upper limit of the spectral resolution $Δ λ_{system} (wavel)$ is related to the maximum wavelength within a spectral interval (red spots). Additionally, the sole contribution of the pinhole size $Δ λ_{5}$ is also plotted (blue spots), which represents the ideal case of maximum resolution. The depicted values for the different cases of the central (green) and the critical wavelengths (red) show a similar dependency.

Fig. 6. Achievable spectral resolution of the complete setup for the main diffraction orders four to 11 in comparison to the theoretical maximum limited by the pinhole size ( $Δ λ_{5}$ ). The resolution of each order varies between the best case at the respective central wavelength positions [ $Δ λ_{system}$ (central)] and the worst case at the edge positions, denoted as [ $Δ λ_{system}$ (wave1)].

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Due to increasing dispersion for higher-diffraction orders of the echelle grating, the best overall resolution of 0.4 nm is found in the 11th order and decreases towards the lower orders. Comparing worst- and best-case wavelengths, the spectral resolution is similar for high orders, while for lower diffraction orders, a difference of up to 0.3 nm results. For the entire spectral range, a spectral resolution better than 1.5 nm is achieved. In the lower orders, the image resolution of the longest wavelengths is affected by the discussed residual aberrations and the lower dispersion. The upper-limit spectral resolution $Δ λ_{system} (wavel)$ is slightly decreased in the orders four to six compared to the central wavelength resolution. In particular, for the upper resolution limit at $λ = 1100 nm$ , a maximum value of 1.42 nm is found in the fourth order. However, a larger spectrum is available compared to the higher resolution in the 11th order.

Despite the upper limit defined by the maximum wavelengths, the values labeled $Δ λ_{system} (central)$ show the system resolution for the central wavelengths in all eight orders and are well representative of the spectral range. This means that the resolution varies between 1.1 nm in the fourth up to 0.4 nm in the highest 11th order for most of the analyzed wavelengths, and the system is well corrected for a large spectrum. With the spectral resolving power defined as the relation of the wavelength and the respective spectral resolution, our cross-grating spectrometer has a resolving power of about 900. In summary, the implementation of the high spectral separation from the echelle-type grating combined with a narrow pinhole size and aberration control leads to a high system spectral resolution. In particular, the introduction of the correction mirror offers an improved overall spectral resolution of up to 25%, which gets close to the ideal case and allows the exploration of the full design potential.

Certainly, classical echelle spectrometers achieve considerably much higher spectral resolutions. For example the astronomical spectrograph “HIRDES” has a spectral resolving power of $\sim 55,000$ and is based on a complex design with large dimensions and a weight of 155 kg [8]. Also, echelle-based setups used in analytical spectrometry are of larger sizes and weight (e.g., an instrument with a comparable spectral range of 250–900 nm has a size of $438 mm \times 200 mm \times 232 mm$ and weighs 12 kg [16]). Therefore, on the one hand, our proposed spectrometer is of remarkably reduced size and, consequently, low weight compared to the established echelle spectrometer setups. On the other hand, in comparison to an existing simplified cross-grating optical setup [18], the proposed spectrometer has a 3 times enlarged spectral resolving power and additionally covers the ultraviolet and near-infrared spectrum.

5. CONCLUSION

We proposed a new optical design for a compact CGES with significantly enhanced optical performance. To achieve this, we combined a folded mirror design based on the Czerny–Turner configuration, a cylindrical correction mirror, and a two-dimensional echelle-type cross grating. By the superposition of an echelle-type grating with an order separating cross-dispersion grating in one two-dimensional dispersion element, the CGES is able to simultaneously detect a wide spectral range with a high resolution. We described the complete optimized optical setup including element apertures, positions, rotations, and incidence, reflection, and diffraction angles in detail. By using the cross-dispersing grating in Littrow configuration, a planar integration of the whole spectrometer is possible, which minimizes, together with the folded design, the required space to $110 mm \times 110 mm \times 30 mm$ . This strong size reduction compared to classical echelle spectrometer is possible due to the cross-grating, which is the key element of the system. We determined the diffraction efficiency of the grating and demonstrated that it is comparable to classical blazed line gratings. From the evaluation of the optical performance, such as collimation errors and image aberrations, we derived the spectroscopic capabilities of the whole system in dependence of the wavelength and the used diffraction order. As a result, our proposed CGES covers a wide spectral range from 330 to 1100 nm with a spectral resolution between 0.4 nm and 1.1 nm in dependence of the considered main diffraction order, which varies between four and 11. Considering the respective imaged wavelength interval of each diffraction order, this results in a spectral resolving power of about 900. In summary, our new echelle-inspired cross-grating spectrometer offers great compactness, a high spectral resolution, and a wide accessible spectral range, representing a valuable design approach for industrial and field applications.

Funding

Carl Zeiss Spectroscopy GmbH; Fraunhofer-Gesellschaft; Thüringer Ministerium für Wirtschaft, Wissenschaft und Digitale Gesellschaft (2015-0024); Bundesministerium für Bildung und Forschung (BMBF) (13FH027IX5).

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Optical Element	Mechanical Diameter in mm	Radius of Curvature in mm	Focal Length in mm	$f$ -Number	Coating
Optical Element	$D$	$r$	$F$	$f / N$	$c$
Collimation mirror	25.4	−203.2 (concave)	$+ 101.6$		Enhanced aluminum
Correction mirror	20	$\infty / + 12270.5$ (convex)	$\infty / - 6135.2$	$f / 10$	Enhanced aluminum
Focusing mirror	25.4	−203.2 (concave)	$+ 101.6$	$f / 10$	Enhanced aluminum
Grating	15	–	–	–	Enhanced aluminum

$m_{y}$	$λ_{\min} - λ_{\max}$ (μm)	$λ_{central}$ (μm)	$δ_{m}$ (μm)
4	0.820–1.110	0.957	0.290
5	0.675–0.883	0.765	0.208
6	0.574–0.717	0.638	0.143
7	0.499–0.604	0.547	0.105
8	0.441–0.521	0.478	0.080
9	0.396–0.459	0.425	0.063
10	0.359–0.410	0.383	0.051
11	0.328–0.370	0.348	0.042

$Δ λ_{1}$	Spectral resolution from natural line of the collimation unit	$Δ λ_{1} = (ϒ \frac{1.22 λ f_{col}}{ϕ_{col}}) D_{L}^{- 1}$
$Δ λ_{2}$	Spectral resolution from natural line of the focusing unit	$Δ λ_{2} = (\frac{1.22 λ f_{foc}}{ϕ_{foc}}) D_{L}^{- 1}$
$Δ λ_{3}$	Spectral resolution from grating	$Δ λ_{3} = (λ l / B) D_{L}^{- 1}$
$Δ λ_{4}$	Spectral resolution from pixel size	$Δ λ_{4} = pixelsize \times D_{L}^{- 1}$
$Δ λ_{5}$	Spectral resolution from pinhole size	$Δ λ_{5} = (ϒ \times pinholesize) D_{L}^{- 1}$
$Δ λ_{6}$	Spectral resolution from instrumental line broadening (system aberrations)	$Δ λ_{6} = {spot}_{RMS} \times D_{L}^{- 1}$

Optical Element	Mechanical Diameter in mm	Radius of Curvature in mm	Focal Length in mm	$f$ -Number	Coating
Optical Element	$D$	$r$	$F$	$f / N$	$c$
Collimation mirror	25.4	−203.2 (concave)	$+ 101.6$		Enhanced aluminum
Correction mirror	20	$\infty / + 12270.5$ (convex)	$\infty / - 6135.2$	$f / 10$	Enhanced aluminum
Focusing mirror	25.4	−203.2 (concave)	$+ 101.6$	$f / 10$	Enhanced aluminum
Grating	15	–	–	–	Enhanced aluminum

$m_{y}$	$λ_{\min} - λ_{\max}$ (μm)	$λ_{central}$ (μm)	$δ_{m}$ (μm)
4	0.820–1.110	0.957	0.290
5	0.675–0.883	0.765	0.208
6	0.574–0.717	0.638	0.143
7	0.499–0.604	0.547	0.105
8	0.441–0.521	0.478	0.080
9	0.396–0.459	0.425	0.063
10	0.359–0.410	0.383	0.051
11	0.328–0.370	0.348	0.042

Concept and optical design of a compact cross-grating spectrometer

Abstract

1. INTRODUCTION

2. SPECTROMETER SETUP AND OPTICAL DESIGN

3. GRATING PROFILE AND DIFFRACTION EFFICIENCY

4. SPECTRAL RESOLUTION

5. CONCLUSION

Funding

REFERENCES

Cited By

Figures (6)

Tables (3)

Equations (7)

Journal of the Optical Society of America A