The Correction of Keystone Distortion in Czerny–Turner Spectrometer Using Freeform Surface

Abstract

In the past, conventional Czerny–Turner spectrometers were usually designed to achieve high resolution while often ignoring astigmatism in the sagittal direction. In contrast, by replacing the focusing mirror with a freeform surface in the structure, we can obtain a Czerny–Turner spectrometer with low keystone distortion by controlling the astigmatism. At the same time, the area sensor can receive all of the spectrum from the optical system. In this paper, we briefly describe the formation of keystone distortion and smile in a plane grating. Additionally, the validity of the method is verified through simulation. Finally, we evaluated the smile and keystone distortion of both the initial and final systems. The keystone and smile were reduced to 1.77 μm and 8.3 μm, respectively, over the wavelength range of 535 nm to 630 nm. Concurrently, the resolution achieved was 0.4 nm.

1. Introduction

Spectral detection technology is widely used in various fields, such as gas detection [1], atmospheric remote sensing [2], biochemical detection [3], and more. Aberrations in Czerny–Turner spectrometers affect the detector’s reception of spectral signals and have been a problem that many researchers and scholars seek to resolve [4]. Furthermore, distortion is an important evaluation index in the design of grating spectrometers, which includes keystone and smile. Plane gratings bring about a significant smile and keystone [5]. Smile refers to the deviation of the curved image of the slit from the ideal straight-line image at different operating wavelengths. However, spectral uniformity requires that the image of a slit, having a single monochromatic wavelength, be straight and aligned with the detector array [6]. Keystone distortion is caused by the difference in magnification of the slit image at different working wavelengths. They will produce alignment errors in the spectral and spatial dimensions, respectively [7,8].

Traditional imaging spectrometers have a transmission optical structure, and chromatic aberration affects the image quality. Therefore, reflective optical structures have become a good choice. The Czerny–Turner spectrometer differs from imaging spectrometers like the Offner or Dyson structures, which are known for their low distortion [9]. Due to the different focal lengths in the tangential and sagittal planes, resulting from off-axis incidence on spherical mirrors, traditional Czerny–Turner designs that use two spherical mirrors and a plane grating suffer from astigmatism [10]. In designing a Czerny–Turner imaging spectrometer, researchers have utilized multiple approaches to correct astigmatism, such as integrating a cylindrical lens or a cylindrical mirror [11], conducting theoretical analyses [12], and more. In this paper, we demonstrate the use of freeform surfaces to achieve full coverage of detector surface pixels. Considering the imaging of the slit at different heights within the optical system, the designed Czerny–Turner spectrometer features several fields of view that are linearly aligned along the slit’s height (sagittal direction). Utilizing an area sensor, the Czerny–Turner spectrometer can meet the requirements of spectral detection applications that necessitate a high spectral dynamic range or the detection of weak spectral signals. However, due to the influence of smile and keystone effects in the Czerny–Turner spectrometer, the image of the slit does not align with the detector’s width, leading to spectral crosstalk between adjacent pixels. This can result in the loss of spectral energy and the omission of spectral signals. An overly concentrated spectrum on the detector can cause certain pixels to receive an excessive number of photons, thereby reducing the dynamic range of the spectrometer. Moreover, a significant portion of pixels remain unexposed by the spectrum, leading to an inefficient use of sensor pixels and compromised performance [13].

Chen Wang et al. derived a first-order equation to eliminate the keystone distortion in the crossed Czerny–Turner spectrometer. The convergent illumination on the grating can be achieved solely through adjusting the structural parameters, eliminating the need for additional components and complex optics. The image of the entire spectrum aligns well with the linear array photomultiplier tube [14]. Lifu Wang and colleagues discussed the aberrations of the double-grating spectrometer and derived the first-order condition for keystone distortion correction. The simulation results demonstrated that the keystone distortion in the slit images can be effectively corrected [15]. Sometimes, plane gratings are utilized as the dispersive element in imaging spectrometers. Numerous scholars have also proposed various methods to correct the keystone distortion associated with plane gratings. A method was proposed to correct both smile and keystone distortion using an off-axis lens, and simulations were conducted to validate this approach. The visible spectral imaging systems, both without and with an off-axis lens, were optimized, respectively. The keystone distortion in the latter system was significantly corrected in the image plane [16]. Utilizing vector aberration theory, a method was proposed that involves inserting two tilted lenses, respectively, into the object and image space, which effectively corrects the aberration. Moreover, the Modulation Transfer Function (MTF) performance of the optical system was comparable to that of the convex grating system. Additionally, the keystone distortion was within acceptable limits [17].

Furthermore, freeform surfaces not only address typical aberrations and distortions in imaging spectrometers but also offer enhanced performance across various applications [18]. Therefore, we introduced a novel method to address keystone distortion using a freeform surface in a Czerny–Turner spectrometer. Compared to previous methods that eliminate keystone distortion, utilizing a freeform surface only alters the optical path after the collimating mirror. However, previous methods that utilized first-order equations altered the optical path length after the entrance slit and modified the angle of detector tilt. Freeform designs do not require such extensive adjustments, but they present challenges in manufacturing. First, we introduced the theory behind keystone and smile effects, then demonstrated the design of a freeform surface, and finally presented a design example. The difference from previous corrections of keystone distortion was that there was no change to the optical system model, and the spherical focusing reflector in front of the image was replaced by a freeform surface reflector based on the initial structure of the spectrometer. The objective and methodology of optimizing freeform surfaces in Czerny–Turner spectrometers differ from those in imaging spectrometers. Optimizing imaging spectrometers entailed refining multiple surfaces to enhance image quality and incorporating optimization techniques for high-order polynomial surfaces. However, the optimization of the Czerny–Turner spectrometer discussed in this paper focused on eliminating keystone and smile effects, incorporating techniques specifically for low-order polynomial surfaces.

2. Theory

2.1. Smile and Keystone Distortion in Plane Grating Spectrometers

Conventional Czerny–Turner spectrometers typically employ symmetrical structures. Figure 1b illustrates the configuration of a Czerny–Turner spectrometer. Figure 1a displays the image from a Czerny–Turner spectrometer, where smile and keystone distortions are evident. After the center and edge rays of the slit pass through a collimating mirror, two parallel beams of light are obtained. The angles between these parallel beams and the plane grating normal differ. This results in a change in the angle between the differently diffracted rays and the grating normal, ultimately leading to the production of distortion. Based on the definitions of smile and keystone distortion and by referring to the schematic diagram of the plane diffraction grating optical path, we can derive the formulas for smile and keystone distortion.

To facilitate the analysis, a coordinate system was established, as depicted in Figure 1c. The half value of the slit height is denoted as h. Point A represents the center of the incident slit, while points B represent the endpoints of the slit. Point C is the image corresponding to point A, and point D is the image corresponding to point B. The plane of the grating coincides with the xoy plane, and the raster lines are parallel to the incident slit. The incident plane M₁ is defined by the chief ray from point A and the normal $\overrightarrow{n}$ to the grating plane, which coincides with the principal section of the grating. The incident plane M₂ is defined by the chief ray from the slit endpoint B and the normal $\overrightarrow{n}$ to the grating plane. The angle between surfaces M₁ and M₂ is denoted as Φ₂, which is referred to as the incident azimuth angle. In addition, m represents the diffraction order, θ_1,m is the diffractive angle of the central ray, θ_2,m is the diffractive polar angle, Φ_2,m is the diffractive azimuthal angle, and θ_2→1,m is the projection of θ_2,m onto M₁, f₁ is the focal length of the collimating mirror, and f₂ is the focal length of the focusing mirror.

The incident angles of the main light rays at points A and B are θ₁ and θ₂, respectively, and it is obvious that θ₁ ≠ θ₂. The incident beam at endpoint B is not within the main grating cross-section. The diffracted beam is also not in the main grating cross-section, and its diffraction angle is not equal to that of the diffraction angle at the center point A, which results in the generation of a smile. As shown in Figure 1c, the smile caused by the plane grating is the horizontal projection of the line connecting C and D. For different wavelengths, there are differences in the size of the vertical projection of the line connecting C and D, which is called keystone distortion. The related expressions are as follows [5]:

The exact expressions of the smile can be expressed as follows:

$${\Delta L=f}_2\cdot {(tan\theta }_{2\to 1,m}-{tan\theta }_{1,m})$$

(1)

The image height of the incident slit in the focal plane of the focusing mirror can be expressed as shown in Equation (2).

$$h_{m,\lambda }={\displaystyle \frac{h\cdot f_2}{f_1\times {cos\theta }_{2\to 1,m}}}$$

(2)

The expression of the keystone can be expressed as shown in Equation (3).

$${\Delta h}_{m,\lambda }{=hf}_2{\Delta (1/cos\theta }_{2\to 1,m\lambda }{)/f}_1$$

(3)

In Equation (3), θ_2→1,mλ is the projection of θ_2,m onto M₁ at the working wavelength of λ, Δ(1/cosθ_2→1,mλ) and can be expressed as follows:

$${\Delta (1/cos\theta }_{2\to 1,m\lambda }{)=(1/cos\theta }_{2\to 1,m\lambda 1})-{(1/cos\theta }_{2\to 1,m\lambda 2})$$

(4)

where λ1 and λ2 represent different wavelengths.

From Equations (1) and (3), it can be seen that the smile and keystone distortion produced by the grating are related to many physical quantities, such as the grating constant, working wavelength, diffraction order, slit length, incidence angle at the center of the slit, f₁, f₂, etc. In Equation (2), when cosθ_2→1,m varies with the λ and the f₂ varies at the same time with the λ, h_m,λ remain constant. So Δh_m,λ = 0 can be pursued by adjusting the focal length at different positions of the focusing mirror, and a freeform surface can achieve this.

2.2. Design of the Freeform Surface

Freeform surfaces are optical surfaces that lack linear or rotational symmetry. More importantly, the use of freeform surfaces can produce innovative designs that cannot be achieved using spherical or aspherical surfaces [19]. Freeform surfaces are determined by mathematical expressions. Surface parameters such as the radius of curvature at each point are determined by the coefficients of these polynomials, which differ significantly from the properties of a spherical mirror. In this article, the freeform mirror we designed exhibits symmetry around a central axis. While Zernike and other polynomial freeform surfaces are widely accepted for describing rotationally symmetric systems, optimizing or manufacturing these surfaces often becomes challenging when the basic symmetry deviates significantly from rotational symmetry. For non-rotationally symmetric systems, describing the surface in Cartesian coordinates is desirable [20]. The standard Zernike polynomial set is derived using the polar coordinate system; hence, it may not be the optimal choice. In the design process, we quantify the freeform surface using Chebyshev polynomials, which are based on the Cartesian coordinate system. The Chebyshev polynomial equations are as follows:

The first type of Chebyshev polynomial can be derived from the following formula:

$$T_n\left(x\right)=cos(n\cdot arccos\left(x\right)), n=0\cdot \cdot \cdot \infty , x\in \left[-1,1\right]$$

(5)

A two-dimensional Chebyshev polynomial can be obtained from Equation (5) by using the following relation:

$$t_{ij}{(x,y)=T}_i\left(x\right)\cdot T_j\left(y\right), i,j=0\cdot \cdot \cdot \infty , x\in \left[-1,1\right], y\in \left[-1,1\right]$$

(6)

Summing the finite terms of Chebyshev polynomials, the freeform vector height formula is generated as follows:

$$Z={\displaystyle \frac{{c(x}^2{+y}^2)}{1+\sqrt{{1-c}^2{(x}^2{+y}^2)}}}+{\displaystyle \sum _{i=0}^N{\displaystyle \sum _{j=0}^M{a}_{ij}\cdot T_i\left(\overline{x}\right)}}\cdot T_j\left(\overline{y}\right)$$

(7)

In Equation (7), a_ij is the sum of the polynomial coefficient; x and y are the coordinates of the normalized surface; c is the radius of curvature; and N and M are the maximum terms of the polynomial in the x and y directions, respectively. In the software settings, the coefficient of the term C(2,0) is related to the polynomial T₂(x)·T₀(y) (i.e., 2x² − 1), while C(0,2) is affected by the polynomial T₀(x)·T₂(y) (i.e., 2y² − 1), and the other terms are similar. So we could learn about the impact of the optimization parameters on the freeform surface. Such as C(0,1) can adjust the tilt Y and C(0,2) can diminish Y astigmatism in the tangential direction.

The freeform surface design was based on optimizing the surface coefficients using software. With the parameters of the entire optical system held constant, the focusing mirror was designed as a freeform surface to optimize the system. Proper constraints could be chosen and imposed prior to the optimization. Some optimization operands about distortion were added to the merit function, and the coefficients of the freeform surface were set as variables. In the design process, it is essential to take into account that the light spot needs to be symmetrical about the tangential direction of the image plane. Therefore, the optimization process prioritized the inclusion of as many even-order coefficients of x as possible. To prevent the image plane from potentially not receiving the light spot in the tangential direction after the optimization process, it was necessary to adjust the image plane to align with the tangential focusing plane once the optimization was complete. After the optimization process, the initial plane was replaced with this newly designed freeform surface.

The specific design flow chart is shown in Figure 2. Firstly, after designing the Czerny–Turner spectrometer system, we obtained an initial optical system model and evaluated the keystone distortion of the system. Then, we replaced the focusing mirror with a surface described by Chebyshev polynomials. In the design process, several standard operands are employed to maintain the optical system’s resolution. We also add some optimization operands to the merit function, such as DIST (which indicates the degree of distortion, quantified as the wave number induced at a specified wavelength and surface), DISC (which calculates the calibration distortion over the entire field of view at the specified wavelength), etc. In the optimization process, the coefficients of the even terms are designated as variables, and a damped least squares optimization procedure is employed. The damped least squares optimization procedure is an algorithm used to optimize an evaluation function that consists of weighted objective values. This procedure will find and save the smallest evaluation function value. The target value of the system can be obtained by adjusting the weight of the operand.

3. Design Example and Result Analysis

3.1. The Initial Optical System

A conventional symmetrical Czerny–Turner spectrometer was used as the initial structure, and then several symmetrical fields of view were added in the sagittal direction to simulate slits of different heights. The Czerny–Turner spectrometer consists of a slit, collimating mirror, plane diffraction grating, focusing mirror, and area detector. After passing through the 25 μm wide slit, the light passes through a collimating mirror, plane diffraction grating, and focusing mirror and is then captured by the area detector. The optical system parameters are shown in

Table 1. Parameters of the initial Czerny–Turner spectrometer.

Parameters of Initial Spectrometer	Value
Spectral Range	535–630 nm
Object space NA	0.11
LSC (the distance between the light source and collimating mirror)	101.925 mm
RC (radius of curvature of the collimating mirror)	200.00 mm
RF (radius of curvature of the focusing mirror)	200.00 mm
f1 (the distance between the collimating mirror and slit)	101.825 mm
f2 (the distance between the focusing mirror and image plane)	86.365 mm
θc (off-axis angles of the collimating mirror)	10°
θf (off-axis angles of the focusing mirror)	22.3°
G (groove spacing of the grating)	1800 l/mm
m (diffraction order)	1
α (incident angle of the central wavelength on the grating)	18.5°
β (diffraction angle of the central wavelength on the grating)	47°
LCG (distance between the collimating mirror and grating)	80.00 mm
LGF (distance between the grating and focusing mirror)	85.50 mm

.

The magnification in spatial dimension and the spectral resolution can be calculated as follows: For spherical focusing mirror systems, the spatial dimension of the center wavelength is given as follows:

$$H=(1-{\displaystyle \frac{F_T}{F_S}})\cdot 2\cdot Lsc\cdot NA$$

(8)

F_T and F_S can be calculated using the following equations:

$$F_T={\displaystyle \frac{R_C{R}_F{L}_{SC}}{2L_{SC}\left(R_C\sec {\theta }_F+R_F\sec {\theta }_C\frac{{\cos }^2\alpha }{{\cos }^2\beta }\right)-R_C{R}_F\frac{{\cos }^2\alpha }{{\cos }^2\beta }}}$$

(9)

$$F_S={\displaystyle \frac{R_C{R}_F{L}_{SC}}{2L_{SC}\left(R_C\cos {\theta }_F+R_F\cos {\theta }_C\right)-R_C{R}_F}}$$

(10)

F_T represents the distance from the focusing mirror to the tangential direction of the image plane, while F_S represents the distance from the focusing mirror to the sagittal direction of the image plane. NA is the numerical aperture, and Lsc is the distance from the slit to the collimator. Based on the spherical focusing mirror optical system, it was changed to a freeform surface focusing system. Due to the constraints of the freeform surface system design, only an approximate estimation of the spatial dimension can be obtained.

3.2. The Final Optical System

It is difficult for general spectrometer detectors to ensure the complete reception of all spectral information, and the phenomena of the loss of spectral energy and the missing of spectral signals occur frequently. In the improved Czerny–Turner spectrometer system, the spherical reflector that plays a focusing role was replaced with a freeform surface, while other optical components and parameters were kept unchanged. We used Hamamatsu’s area sensor S12071 as the detector. The main performance indexes are as follows: the number of pixels is 1024 (spatial dimension) × 1024 (spectral dimension), the pixel size is 24 μm × 24 μm, the allowed wavelength is 165–1100 nm, and the whole area of the sensor is 24.576 mm × 24.576 mm. By properly adjusting the height of the slit, the detector’s width in the sagittal direction can fully accommodate all spectra.

The estimated value of the spatial dimension of the optical system after adding the freeform surface is given as follows:

$$D=2\cdot Lsc\cdot NA$$

(11)

The magnification is approximated as follows:

$$K={\displaystyle \frac{D}{H}}={\displaystyle \frac{F_S}{F_S-F_T}}$$

(12)

The designed Czerny–Turner spectrometer system using an area sensor has the advantages of high spectral dynamic range and high sensitivity. However, due to the influence of keystone distortion, the spectral signal cannot be rectangular in spectral dimensions, resulting in the loss of the spectral signal or the detector pixel not being fully utilized. This problem can be addressed by employing a freeform surface, which results in a rectangular spot distribution on the image plane. The layout of the optical system is shown in Figure 3, and the relevant parameters of the Chebyshev polynomial surface are shown in

Table 2. Relevant parameters of Chebyshev polynomial surface.

Parameters of Chebyshev Polynomial Surface	Value
Radius of curvature	Infinity
Highest order of X	2
Highest order of Y	2
Normalized length of X and Y	85
C(1,0)	0
C(2,0)	0
C(0,1)	6.8014 × 10−5
C(1,1)	0
C(2,1)	−1.4764 × 10−6
C(0,2)	−1.2499 × 10−3
C(1,2)	0
C(2,2)	0

. Within the design, the sagittal orientation of the target surface is dictated by the initial parameters. The sagittal direction of the target surface is controlled by several factors: the object height, front focus, back focus, and off-axis angle of the optical system. Based on these factors, the initial value of the sagittal direction at the center wavelength of the detector can be calculated and then optimized by adding a freeform surface. The freeform surface is based on a plane mirror, so the radius of curvature is infinity. The two-dimensional Chebyshev polynomials are composed of two first-type Chebyshev polynomials. The normalized lengths of X and Y for the freeform surface are both set to 85. In the optimized spectrometer system, the distance and angle between the detector and the freeform focusing mirror remain unchanged. To ensure a spectral image centered on the detector, fine-tuning of the Y decenter of the detector aperture is necessary. C(0,1) can adjust the light of the central wavelength to the center of the sensor in the tangential direction, and C(0,2) can enable the light of different wavelengths to focus on the area sensor in the tangential direction. C(2,1) can realize the restriction of astigmatism in the sagittal direction and a small adjustment in the tangential direction.

3.3. Result Analysis

Without altering the optical path, a slit of suitable height was chosen to align the sagittal directional width of the spectral signal on the image plane with the detector. The spot distributions of the image plane before and after optimization are shown in Figure 4, and the two images are both under two heights of slit (0.2 mm and 0.6 mm). The horizontal axis represents the spectral dimension of the detector, and the vertical axis represents the height of the image, matching the spatial dimension of the detector. The gap between the two lines of the same color represents the image height in the spatial dimension for wavelengths ranging from 535 nm to 630 nm. The two lines, each in a different color, encompass areas that correspond to two distinct image heights. The black line represents an image height of 0.2 mm, while the red line represents an image height of 0.6 mm. The initial system’s spot exhibits a trapezoidal shape, whereas the optimized system’s spot pattern is rectangular. The image height of the 0.6 mm slit is approximately 23.662 mm in the sagittal direction, which is smaller than the detector size. Comparing the former and the latter, it is demonstrated that the freeform surface effectively addresses the keystone issue in Czerny–Turner spectrometers. It is difficult for general spectrometer detectors to ensure the complete reception of all spectral information or the full utilization of the detector’s pixels, yet the optimized system can accomplish this.

At the same time, as an important index to evaluate optical systems, spectral resolution still needs to be considered. The spectral resolution can be calculated using the following formula:

$$\Delta \lambda ={\displaystyle \frac{ad}{m{f}_1}}$$

(13)

where a is the width of the slit and d is the constant of the grating. The calculated spectral resolution is 0.138 nm. In the optical system, the spectral resolution is determined by the capability to distinguish between two adjacent wavelength spectral spots. To maintain this spectral resolution, we incorporated operands for optimizing the Y spot in the optimization design.

4. Discussion

The evaluation methods of keystones in a conventional Czerny–Turner spectrometer and imaging spectrometers are different. The keystone distortion is due to the fact that the imaging size of the slit varies with light of different wavelengths. The keystone distortion in an imaging spectrometer can be ascertained by the positional difference in the centroids of spots at various wavelengths within the same field of view. However, the image of the Czerny–Turner spectrometer is not a point spot but a spot with a certain width of sagittal astigmatism. Therefore, the spot centroid of a conventional Czerny–Turner optical system cannot determine the actual keystone distortion. In this paper, the difference in the position of edge rays of different wavelengths in the space dimension was used as the keystone distortion size. The positional differences of edge rays at various fields of view for the same wavelength can be assessed in terms of smile. In this evaluation method, the initial optical system exhibits maximum smile and keystone values of approximately 10.186 μm and 0.79 mm, respectively. Notably, the keystone distortion significantly exceeds the acceptable range of the detector’s pixel size. The smile and keystone of the initial optical system are shown in Figure 5a,c. The maximum values of the smile and keystone of the final optical system are approximately 8.3 μm and 1.77 μm, respectively. The smile and keystone of the final optical system are shown in Figure 5b,d. The scale of Figure 5c,d is approximately 100:1, and the subfigure in Figure 5d shows the data from Figure 5d at the scale of Figure 5c.

Comparing the former and the latter and analyzing the light spot data, it could be seen that the keystone distortion in the system has been significantly eliminated. The keystone distortion and smile are smaller than the size of the pixel, which can meet the requirements of the detector. The resolution of the optical system before and after optimization has a certain change. Before optimization, the spectral resolution of the center wavelength and edge wavelength are 0.15 nm and 0.3 nm, respectively, and after optimization, the spectral resolution of the center wavelength and edge wavelength are 0.4 nm. The spectral dimensions of the initial and final spectrometers in the tangential plane are shown in Figure 6. The image heights of the initial and modified optical systems are 24.564 mm and 23.4973 mm, respectively, and the area sensor width in the tangential direction is 24.576 mm. So, the ratios of the RMS Y spot to the detector spectral dimension of the initial and final optical systems are 0.9995 and 0.9561, respectively. It is evident that the optical system’s keystone is substantially corrected through the use of freeform surfaces, and this method has a minimal impact on spectral resolution.

5. Conclusions

In this paper, the keystone distortion issue of conventional Czerny–Turner spectrometers was examined and analyzed, and a method to eliminate the keystone distortion in Czerny–Turner spectrometers was proposed. The simulation results indicated that the keystone distortion problem can be addressed by replacing the spherical focusing mirror with a freeform surface and optimizing the relevant parameters of the freeform surface. From Figure 4 and Figure 5, we can see that the keystone distortion has been effectively eliminated. The keystone distortion and smile are smaller than the size of a pixel, which meets the detector’s requirements. Concurrently, the area sensor can fully receive the spectrometer’s optical signal and ensure the complete utilization of spectral signal energy. Not only did the freeform surface solve the problem of keystone distortion, but it also maintained the system’s compactness.