## Abstract

The classic Czerny-Turner spectrometer consists of a plane grating and two spherical mirrors. The optical path geometry adopted for incident and grating dispersed light is off-axis reflection, so the spherical collimating and focusing mirrors introduce coma and astigmatism. The conventional configuration is asymmetrical for coma automatic compensation, but suffers from astigmatism. We substitute the off-axis parabolic (OAP) surfaces for spherical surfaces of the collimating mirror and each sub-region of the focusing mirror, to achieve an aberration free configuration. The multiple OAP surfaces are then expanded and mixed, to construct a freeform surface integrating the collimating and focusing mirrors into a single element. Results show that a 0.1 nm spectral resolution is achieved over a bandwidth of 400 nm centered at 800 nm, in the designed spectrometer comprised of a plane grating and one freeform mirror. The construction method is advantageous to integrated optic design, and the resulting freeform mirror spectrometer is compact, and simplifies manufacture and alignment.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Off-axis reflective systems have the advantages of no central obscuration, a wide field of view (FOV), no chromatic aberration, and high transmission [1]. Therefore, they have been widely used in imaging applications such as infrared imaging [2], space cameras [3], and remote sensing [4]. Spherical, conic, aspherical, and freeform surfaces are adopted in off-axis reflective systems [1–4]. Spherical and conic surfaces are free of aberrations for one particular set of conjugate points. A spherical surface forms an aberration free image if the object is at its center of curvature. However, this geometry seldom occurs, and spherical mirrors are often placed in an off-axis reflection geometry with aberrations unavoidable. A parabolic mirror focuses a collimated beam or collimates a divergent source perfectly, and its off-axis design separates the focal point from the rest of the beam path.

The classic Czerny-Turner spectrometer is a typical off-axis reflective system, which consists of a plane grating, a spherical collimating mirror, and a spherical focusing mirror [5]. The oblique incidence on the two spherical mirrors introduces spherical aberration, coma, and astigmatism, but the spectrometer can be configured in a coma-free geometry if the Shafer equation is satisfied [6]. In addition, spherical aberration can be ignored if the numerical aperture of the spectrometer is not too high. Astigmatism in the classic Czerny-Turner imaging spectrometer is the dominant issue, and many efforts have been made to eliminate it by compensating the diffencence between the focal lengths in the tangential and sagittal planes for the two off-axis spherical mirrors [7]. This can be done by introducing divergent illumination on the plane grating through minimizing the distance between the entrance pinhole and the collimating mirror [8]; or using more spherical mirrors to build a concentric off-axis configuration [9,10]; or changing the spherical mirror type, e.g., using a toroidal focusing mirror [11], substituting two cylindrical collimating mirrors for one spherical collimating mirror [12], introducing two freeform mirrors instead of two spherical mirrors [13]; or adding elements to the classic configuration such as inserting a tilted plane-parallel plate [14], a thin piece of glass as a waveguide [15] between the entrance pinhole and the collimating mirror, a cylindrical lens [16], a wedge cylindrical lens [17], a freeform cylindrical lens [18], a customized spherical lens [19], and a toroidal lens together with a special filter [20] between the focusing mirror and the detector. Discussions here do not cover the modifications using a convex grating [21,22], for these change the basic configuration of the classic Czerny-Turner spectrometer using a plane grating.

Using more spherical mirrors [9,10] decreases the compactness of the spectrometer. Moreover, because of the different diffractive angles at the plane grating, the incident angles to the focusing mirror depend on wavelength. For correcting the nonuniform astigmatism introduced by the varying incident angles to the focusing mirror for each wavelength, in the configuration using a toroidal focusing mirror [11], the distance between the grating and the focusing mirror is increased to twice the focal length of the focusing mirror in the tangential mirror, which also decreases the compactness. In the modified configurations, the added elements and the detector are always tilted by a certain angle to solve the nonuniform astigmatism problem [14–20], which complicates the placing of these added elements. In all the aforementioned modifications, the methodology is to compensate astigmatism introduced by the spherical collimating mirror, or by both spherical mirrors in the off-axis geometry, so the collimated beam after reflection from the collimating mirror is aberrated. Therefore, the wavefront aberration cannot be taken as the criteria for placing and aligning the collimating and focusing mirrors, or the added elements. The accurate localization of these separated mirrors and lenses can only be validated from the optical performance of the spectrometer, so the alignment difficultly of the modifications to the classic Czerny-Turner spectrometer should be considered

In the existing modified configurations, the collimating mirror, focusing mirror, and added elements, are not independent aberration free optics, but they together compose an aberration free spectrometer. We would like to present a different design concept for an aberration free spectrometer in this paper. Replacing the spherical collimating mirror with an off-axis parabolic (OAP) surface, the optical path from the entrance pinhole to the collimating mirror, and then to the plane grating is aberration free. The alignment of the OAP collimating mirror is a routine operation by using the wavefront as the criterion. Similarly, each dispersed beam from the plane grating is incident on a small sub-region of the focusing mirror. Therefore, if each sub-region of the focusing mirror for the corresponding wavelength is an OAP surface, the optical path from the plane grating to the focusing mirror, and then to the detector is also aberration free. The alignment operation for the focusing mirror will benefit from this geometry. However, sub-regions for dispersed beams of neighboring wavelengths overlap on the focusing mirror, and their sags are discontinuous, which necessitates a methodology for mixing these OAP surfaces on the focusing mirror. As we substitute the OAP surfaces for spherical surfaces on the collimating mirror and each sub-region on the focusing mirror, these multiple OAP surfaces are then expanded and mixed, to obtain a freeform mirror integrating the collimating and focusing mirrors. The proposed freeform spectrometer evolved from multiple OAP surfaces consists of a plane grating and a freeform mirror, and exhibits the advantages of increased integration and compactness, as well as reduced manufacturing cost and alignment difficulty.

## 2. Configuration benchmark with multiple OAP surfaces

We substitute OAP surfaces for spherical surfaces for the collimating mirror and each sub-region of the focusing mirror in the classic Czerny-Turner spectrometer. The parameters for each OAP surface consist of the parent focal length, the reflected focal length, the off-axis angle, the decenter, and the surface diameter. The algorithms to solve for these parameters are elaborated. Moreover, all the OAP surfaces are uniformly defined in the global coordinate system through coordinate transformations, to increase the convenience of the freeform spectrometer construction.

#### 2.1 Parameter definitions for the spectrometer

The classic Czerny-Turner spectrometer is illustrated in Fig. 1. A divergent beam originating from the entrance pinhole is reflected by the collimating mirror, and then diffracted in the tangential plane by the plane grating. The focusing mirror converges the dispersed beam onto the detector. *L*_{EC}, *L*_{CG}, *L*_{GF} and *L*_{FD} denote the distances from the entrance pinhole to the collimating mirror, the collimating mirror to the grating, the grating to focusing mirror, and the focusing mirror to the detector respectively. *R*_{C} and *R*_{F} are the radii of curvature of the spherical collimating and spherical focusing mirrors, respectively. The angles of incidence of the off-axis illumination on the collimating mirror and the focusing mirror are labeled as *α*_{C} and *α*_{F}. For the plane grating, *i* denotes the angle of incidence, and *θ* is the diffraction angle. The relationship between *i* and *θ* is determined by the groove interval *d* of the grating, expressed in Eq. (1):

*m*is the diffraction order, usually we take

*m*= −1.

*λ*is the wavelength of the dispersed beam.

The beam is dispersed spectrally by the grating with the angular spectral spread Δ*θ*, which together with *R*_{F} determines the spatial length *L* of the image on the detector. The incidence angle *α*_{F} on the focusing mirror varies with the wavelength due to diffraction. The wavelength dependence of *α*_{F} across the focusing mirror results in the variation of astigmatism. For correcting the nonuniform astigmatism derived from the focusing mirror for each wavelength, the detector is usually tilted at an angle *γ* [14–20]. The inclined incidence on the detector decreases its response, and affects the relative illumination.

The optical path illustrated in Fig. 1 is in the tangential plane (YZ plane). Collimated polychromatic light is diffracted in different directions for each wavelength. The plane grating works as a virtual aperture stop, and the dispersed beams are similar to beams for different FOVs. Furthermore, each dispersed beam is incident on a small sub-region of the focusing mirror. The optical paths before and after the plane grating are considered as asymmetrical. The reverse optical path for the polychromatic light from the plane grating to the spherical collimating mirror and then the entrance pinhole, and the forward optical path for each wavelength from the plane grating to the corresponding sub-region on the spherical focusing mirror and then the detector are all in off-axis refection geometry. Beams after reflection from the spherical collimating mirror and the spherical focusing mirror are aberrated, with coma and astigmatism introduced as the dominant aberrations. Coma aberration is compensated in this Czerny-Turner configuration by satisfying Shafer equation, but their astigmatism aberrations are superimposed.

#### 2.2 Parameter definitions for the OAP mirror

Unlike the spherical mirror, the OAP mirror focuses a collimated beam or collimates a divergent source perfectly, and its off-axis design facilitate achieving an unobstructed optical path.

As illustrated in Fig. 2, an OAP mirror is simply a side section of a parent parabolic mirror with the diameter Ф and a decenter *D* (Y offset here in the tangential plane), and the light is deviated by an off-axis angle *β*. The off-axis sag *s* is defined as the sag for the center of the OAP mirror with respect to its vertex.

*R*represents the radius of curvature of the parent parabolic mirror at the vertex.

The chief ray plotted as a bold blue line traces along the central ray axis, which departs from the optical axis of the parent parabolic mirror. Moreover, the reflected focal length $\widehat{f}$differs from the parent focal length *f*, and their relationship is expressed as:

For *f*, *D* and *β*, one unknown can be calculated from two known parameters, as they satisfy the equation:

*D*can be expressed as a function of

*R*and

*β*:

In this paper, the reflected focal length $\widehat{f}$ and the off-axis angle *β* are defined as the pair ($\widehat{f}$, *β*) to specify an OAP surface.

#### 2.3 Multiple OAP surfaces substitutions

For monochromatic light in the Czerny-Turner spectrometer, two OAP mirrors instead of the spherical collimating and spherical focusing mirrors will compose an ideal aberration free system. Therefore, for polychromatic light, we substitute one OAP surface for each sub-region on the focusing mirror for each wavelength. Thus, the focusing mirror will be a mixture of multiple OAP surfaces for the dispersed beams.

For exploring the methodology to calculate the parameters specifying these multiple OAP surfaces, we plot only the central ray for each dispersed beam in Fig. 3, in which the OAP surfaces for each wavelength on the focusing mirror are indicated in different colors. These OAP surfaces on the focusing mirror are referred to as OAP segments in this paper. As the plane grating is considered to be a virtual stop, these central rays are the chief rays for each wavelength, which strike the centers of their corresponding OAP segments. The reflected chief rays for each wavelength from the focusing mirror are configured to be parallel. Moreover, the detector is placed perpendicular to the incident chief rays to increase the detector response.

In the reverse optical path from the detector to the focusing mirror in Fig. 3, the chief rays (illustrated as five colored rays) for each of the wavelength comprise a collimated beam. This assumed collimated beam is focused perfectly by the focusing mirror, and the focus is on the plane grating. Only when the centers of the OAP segments for each of the wavelength are located on an OAP surface, can this aberration free optical path be realized. Therefore, the OAP segments are distributed along the OAP surface which is referred to as OAP base in this paper.

The subscripts “min”, “ctrl”, and “max” in Fig. 3 denote the parameters for the minimum, central, and maximum wavelengths, respectively. When a plane grating of known specifications, and the incidence angle *i* are selected, the diffraction angles *θ*_{min}, *θ*_{ctrl} and *θ*_{max} are calculated according to Eq. (1). *β*_{min}, *β*_{ctrl} and *β*_{max} for the minimum, central, and maximum wavelengths, respectively, are not only the off-axis angles of the corresponding OAP segments, but also the off-axis angles at the distributed centers located on the OAP base. Based on the geometrical relationship of the triangles, the diffraction angles and the off-axis angles satisfy the following equations:

*D*_{B-min}, *D*_{B-ctrl}, and *D*_{B-max} are the decenters at the distributed centers located on the OAP base, and their relationships with the spatial length *L* of the image on the detector satisfy the equation:

*L*is determined by the spectral resolution and bandwidth, and the linear detector with the corresponding specifications is then selected.

We take the center of the OAP segment for the central wavelength as the center for the OAP base. The off-axis angle *β*_{ctrl} is selected to avoid obstruction in the spectrometer, so, *β*_{min} and *β*_{max} are derived according to Eqs. (6) and (7). The radius of curvature *R*_{B} of the OAP base is solved by using Eqs. (5) and (8). Thus, the specifications for the OAP base are determined, and the coordinates of the centers for each wavelength can be derived.

Figure 4 presents the details of the optical path from the grating to the detector. In addition to the chief rays for each wavelength, the optical path for the OAP segment for the central wavelength is fully illustrated with pink rays. The length of the chief ray from the plane grating to each OAP segment presents the reflected focal length at the corresponding center located on the OAP base, of which the reflective focal length ${\widehat{f}}_{\text{B-ctrl}}$ of the OAP base equals *L*_{GF}. The length of the chief ray from each OAP segment to the detector represents its reflective focal length, of which the reflective focal length ${\widehat{f}}_{\text{S-ctrl}}$ of the OAP segment for central wavelength equals *L*_{FD}. As the assumed collimated beam composed of the chief rays for each wavelength is converged perfectly by the OAP base to the focus on the plane grating, the optical path lengths for each wavelength are the same. Thus, we have:

With the specifications for the OAP base accomplished, the reflective focal lengths of the OAP base at each center, such as ${\widehat{f}}_{\text{B-ctrl}}$,${\widehat{f}}_{\text{B-min}}$, and ${\widehat{f}}_{\text{B-max}}$ are known. When *L*_{FD} (${\widehat{f}}_{\text{S-ctrl}}$) is selected, the reflective focal length for each OAP segment, such as ${\widehat{f}}_{\text{S-min}}$and ${\widehat{f}}_{\text{S-max}}$can be derived from Eq. (9). Thus, together with the off-axis angles of *β*_{min}, *β*_{ctrl} and *β*_{max}, the specifications for each OAP segment are determined.

Based on the spectral resolution and bandwidth specifications of the spectrometer, the incidence angle *i* on the grating, off-axis angle *β*_{ctrl} on the focusing mirror, and distance from the focusing mirror to the detector *L*_{FD} are selected. These three parameters are determined manually in consideration of the desired compactness and obstruction issues. Then, the parameters definitions for the OAP base and OAP segments are derived from the proposed procedure using Eqs. (3-9).

The OAP segments with centers distributed on the OAP base are then mixed to form a freeform focusing mirror. The parameters specifying the OAP collimating mirror should be optimized, to make it easier to integrate both the collimating and focusing mirrors to obtain a single freeform mirror.

## 3. Freeform spectrometer construction

#### 3.1 Design specifications

The following example is intended to elaborate the methodology for freeform spectrometer construction. A wide bandwidth spanning from 600 nm to 1000 nm is considered, and the target for the spectral resolution is 0.1 nm. The numerical aperture (NA) for the divergent beam from the entrance pinhole was set to 0.05, which is consistent with that of a single mode fiber for light delivering. The detector with 4096 pixels was chosen, and a pixel size of 10 μm. A plane grating with a groove interval *d* of 1.2 μm was chosen, and the first negative diffraction order was utilized.

The spectrum from 600 nm to 1000 nm was divided into nine sampling wavelengths with a 50 nm interval, and labeled *λ*_{1}–*λ*_{9}. Thus, the corresponding diffraction angles were labeled *θ*_{1}–*θ*_{9}, and off-axis angles of the OAP segments were labeled *β*_{1}–*β*_{9}. Since *λ*_{5} = 800 nm is the central wavelength, *θ*_{5} equals *θ*_{ctrl}, and *β*_{5} equals *β*_{ctrl}. Here, for the convenience of multiple OAP surfaces mixing, the optical axis for the OAP base shown in Fig. 3 was taken as the axis of symmetry for the OAP collimating mirror and the OAP segment for *λ*_{1}, so that their parameters were the same. Thus, they should satisfy the equation:

To avoid obstruction in the spectrometer, *β*_{5} was selected as 20°. Then, according to Eqs. (1), (6) and (10), *i* was solved for and found to be 5.620°. *θ*_{1}–*θ*_{9}, and *β*_{1}–*β*_{9} were calculated. Note that the criteria for the selection of *β*_{C} and *i* are not rigorous.

*L*_{FD} (${\widehat{f}}_{\text{S5}}$) was then chosen to be 95 mm. Thus, parameters for the OAP base and each OAP segment were derived. The radius of curvature *R*_{B} of the OAP base was 186.852 mm. The parameters for the OAP base and OAP segments are listed in Table 1, in which the emboldened *β*_{5} = 20° and ${\widehat{f}}_{\text{S5}}\text{=95}$mm were manually selected. The parameters for the OAP collimating mirror are the same as those for the OAP segment for *λ*_{1}. The procedure for determination of the parameters of each OAP surface from the design specifications of the spectrometer are presented in the flowchart of Fig. 5. Pairs as (${\widehat{f}}_{\text{S1}}$, *β*_{1}) – (${\widehat{f}}_{\text{S9}}$, *β*_{9}) are derived to define each OAP segment, and ${\widehat{f}}_{\text{B1}}$ – ${\widehat{f}}_{\text{B9}}$ are used to define their respective locations.

#### 3.2 Coordinate transformation

Since we have obtained the parameters of each OAP segment, the next task is to specify them in the global coordinate system X_{F}-Y_{F}-Z_{F} as illustrated in Fig. 6, for further expansion and mixing. The global coordinate system X_{F}-Y_{F}-Z_{F} is defined with its origin at the center of the OAP segment for the central wavelength, and with Z_{F} axis opposite to the surface normal direction. Typically, the OAP segment for the central wavelength is firstly specified in the coordinate system X_{ctrl}-Y_{ctrl}-Z_{ctrl} as illustrated in Fig. 6, with its origin at the intersection of the chief ray and the plane grating, and with Z_{ctrl} axis coincident with the chief ray of the diffracted beam for the central wavelength. The coordinate (*x*_{ctrl}, *y*_{ctrl}, *z*_{ctrl}) of any point on the OAP segment for the central wavelength defined in X_{ctrl}-Y_{ctrl}-Z_{ctrl} can be expressed by Eq. (11) as

*s*

_{ctrl}denotes the sag for the center of the OAP segment for the central wavelength,

*D*

_{ctrl}denotes the decenter, and

*R*

_{ctrl}denotes the radius of curvature of its parent parabolic mirror at the vertex. All of these parameters can be directly derived once the specifications of each OAP segment are determined.

The origin of coordinate system X_{ctrl}-Y_{ctrl}-Z_{ctrl} can also be specified as a point (*x*_{0}, *y*_{0}, *z*_{0}) in the coordinate system X_{F}-Y_{F}-Z_{F}, which fulfills the following equations:

*x*

_{ctrl},

*y*

_{ctrl},

*z*

_{ctrl}) can be transformed to coordinate (

*x*

_{ctrl-F},

*y*

_{ctrl-F},

*z*

_{ctrl-F}) defined in the coordinate system X

_{F}-Y

_{F}-Z

_{F}according to the difference in origin position and Z axis direction. The coordinate transformation formulas are:

Similarly, as illustrated in Fig. 6, the OAP segment for the minimum wavelength is initially specified in the coordinate system X_{min}-Y_{min}-Z_{min} having an origin coincident with that of X_{ctrl}-Y_{ctrl}-Z_{ctrl}, and Z_{min} axis in consistent with the chief ray of the diffracted beam for the minimum wavelength. The coordinate (*x*_{min}, *y*_{min}, *z*_{min}) of any point on the OAP segment for minimum wavelength defined in X_{min}-Y_{min}-Z_{min} can be expressed by an equation similar to Eq. (11) and it is not presented here. Moreover, the coordinate (*x*_{min-F}, *y*_{min-F}, *z*_{min-F}) defined in the global coordinate system X_{F}-Y_{F}-Z_{F} can be derived according to the differences in origin position and Z axis direction:

_{F}-Y

_{F}-Z

_{F}can be deduced following the procedure for the minimum wavelength.

The divergent beam is collimated by the collimating mirror to obtain a collimated beam with a circular aperture, denoted by Ф. When diffracted from the plane grating, the aperture of the diffracted beam at any wavelength presents an approximately elliptical shape. The major and the minor axes of for this elliptical beam are defined as Ф* _{y}* and Ф

*, and they can be derived according the following equations:*

_{x}Therefore, surface sags of the OAP segments for *λ*_{1}–*λ*_{9} were uniformly defined in the global coordinate system X_{F}-Y_{F}-Z_{F} through coordinate transformation. As the footprint diagram of each diffracted beam on the corresponding OAP segment is approximately elliptical, the nine OAP segments constitute a focusing mirror with a rectangle aperture. The surface profiles along the Y_{F} direction for the nine OAP segments are plotted in Fig. 7(a), in which a spherical base whose radius of curvature is 200 mm was subtracted. The OAP base indicated by the black dotted line was also illustrated in Fig. 7(a). The center of each OAP segment was located on the OAP base, which was in coincidence with the solving schemes for the OAP surfaces proposed in Section 2. Diffracted rays at different wavelengths exhibit crosstalk in the overlapping areas between neighboring OAP segments. However, the sags of the neighboring OAP segments differ in the overlapping areas, and their surface profile differences are illustrated in Fig. 7(b). The differences are on the order of micrometers. If denser wavelength sampling were performed, there would be crosstalk between OAP segments for nonadjacent wavelengths, which are not considered during OAP segments updating.

#### 3.3 OAP surface expansion and mixing

The direction of a ray reflected by the focusing mirror is determined by both the coordinates and the normal vector. Therefore, the discrete data points of the OAP segments are fitted to construct a freeform focusing mirror. However, each OAP segment is designed considering only one wavelength, and the sags of neighboring OAP segments differ substantially. The surface sag deviations in the overlapping areas cause worse mixing results. A freeform mirror with discontinuous surface and discontinuous slope is unphysical.

In the aforementioned procedure, the parameters of each OAP segment were determined simultaneously, and their differences in the overlapping areas were large. Thus, if the OAP segments were developed in a step-by-step method, an attempt to obtain a continuous surface profile can be made. As the center of the OAP segment for the central wavelength (*λ*_{5} here) is the origin of the global coordinate system X_{F}-Y_{F}-Z_{F}, it is taken as the reference surface. The OAP segments for the adjacent wavelengths (*λ*_{4} and *λ*_{6}) are adjusted and then derived with updated parameters, to minimize the sag deviations in the overlapping areas. Based on the updated OAP segments for *λ*_{4} and *λ*_{6} in the previous step, the updated parameters of their neighboring OAP segments for *λ*_{3} and *λ*_{7} are derived following the same strategy. The remaining OAP segments are updated accordingly. Therefore, the OAP segments are gradually expanded and updated from the OAP segment for the central wavelength, to ensure that the surfaces after expansion are continuous, which is of great importance for further surface mixing.

The strategy for OAP segment updating is the crucial process for multiple OAP surface expansion. As the OAP segments are placed along the Y_{F} direction, their profiles along the Y_{F} direction instead of the surface sags in the approximately elliptical effective aperture are considered for simplicity. The adjustment of the OAP segment for *λ*_{4} is taken as an example as illustrated in Fig. 8, in which only the profiles along the Y_{F} direction are plotted. As the OAP segments for both *λ*_{4} and *λ*_{5} are defined in the global coordinate system X_{F}-Y_{F}-Z_{F}, the length of their overlapping area along the Y_{F} direction is labeled as *δ*_{45}. Moreover, the peak to valley value for their sag difference as the red curve plotted in Fig. 7(b) is defined as *S*_{45}. Thus, to minimize the sag deviations of the neighboring OAP segment in the overlapping area, the OAP segment for *λ*_{4} can be tilted by the angle of *ε*_{4}, which is derived approximately:

The update of the OAP segment for *λ*_{4} is implemented in two steps. Firstly, it is tilted by an angle of *ε*_{4} about its center, and then the off-axis angle ${{\beta}^{\prime}}_{4}$ for the updated OAP segment for *λ*_{4} is derived as:

*λ*

_{4}, the condition of normal incident to the detector is no longer met. However, this slightly inclined incidence will not affect the detector response, and the previous rigorous restriction for normal incidence of each chief ray is unnecessary.

Next, the OAP segment is translated along the direction of the incident chief ray of *λ*_{4} from the plane grating, with a displacement of Δ_{4} approximately defined as:

*d*

_{45}is the distance between the center of the OAP segment for

*λ*

_{4}and the intersection of the two OAP segments along the Y

_{F}direction. This operation ensures the chief ray will still strike the center of the updated OAP segment.

Thus, the distance ${{L}^{\prime}}_{\text{GF4}}$ from the grating to the updated OAP segment for *λ*_{4} is calculated, which indicates the new location of this OAP segment.

Moreover, the distance ${{L}^{\prime}}_{\text{FD4}}$ from the updated OAP segment for *λ*_{4} to the detector equals its reflected focal length ${\widehat{{f}^{\prime}}}_{\text{S4}}$, and can be derived approximately from the geometry in Fig. 8.

As the pair (${\widehat{{f}^{\prime}}}_{\text{S4}}$,${{\beta}^{\prime}}_{4}$) have been derived in Eqs. (23) and (26), the specifications for the updated OAP segment for *λ*_{4} are determined. The sag deviations for the neighboring OAP segments for *λ*_{4} and *λ*_{5} are minimized, and the surface mixing would benefit from this OAP segment updating procedure.

With the OAP segment for the central wavelength taken as the initial reference, the OAP segments on the focusing mirror are gradually expanded and updated following the procedure illustrated in Fig. 9. The parameters of the updated OAP segments are listed in Table 2, in which the pairs (${\widehat{{f}^{\prime}}}_{\text{S1}}$,${{\beta}^{\prime}}_{1}$) to (${\widehat{{f}^{\prime}}}_{\text{S9}}$,${{\beta}^{\prime}}_{9}$) are used to specify the OAP segments, ${{L}^{\prime}}_{\text{GF1}}$–${{L}^{\prime}}_{\text{GF9}}$ are used to indicate their locations. The nine updated OAP segments are then mixed to obtain a freeform collimating mirror, by using both the coordinates and the normals of discrete data points on each OAP segment [23]. The freeform surface defined in the coordinate system X_{F}-Y_{F}-Z_{F} supports a base spherical surface upon which the fifth order XY polynomials are added. For any point (*x*_{F}, *y*_{F}, *z*_{F}) on the freeform surface, the surface sag is of the form:

*R*

_{s}= −200 mm is used here to reduce the computational burden. The XY polynomials are power series in the position values

*x*and

*y*which are divided by a normalization radius

*R*

_{N}, resulting in

*x*=

*x*

_{F}/

*R*

_{N}and

*y*=

*y*

_{F}/

*R*

_{N}. Since the freeform surface is symmetric about the tangential plane, only ten XY polynomials terms are used.

The representation of the freeform focusing mirror is initially derived, then the OAP collimating mirror with initial parameters the same as the OAP segment for *λ*_{1} is the next target for mixing. As seen in Fig. 3, there is no overlapping area of the collimating mirror and the focusing mirror. Therefore, the aperture of the freeform focusing mirror is deliberately extended, to cross the collimated beam from the OAP collimating mirror. For the convenience of mixing the OAP collimating mirror and the freeform focusing mirror, the same update strategy is applied on the OAP collimating mirror. The OAP collimating mirror is translated along the chief ray toward the plane grating, until its center locates on the freeform mirror. It can also be tilted, and the off-axis angle and reflected focal length are adjusted, to minimize the sag deviations between the two mirrors. The restrictions on the update procedure of the OAP collimating mirror are less stringent, as only the sag deviations are considered. The pair (${\widehat{{f}^{\prime}}}_{\text{S-C}}$,${{\beta}^{\prime}}_{\text{C}}$) defining the updated OAP collimating mirror is (94.590 mm, 9.472°), and the distance between the updated OAP collimating mirror and the plane grating is 94.125 mm.

The updated OAP collimating mirror and the freeform focusing mirror are then mixed, by using both the coordinates and normals of the discrete data points on each mirror. A freeform mirror integrating both the collimating and focusing mirrors is successfully developed, and the manufacture will benefit from this integration [24,25]. Its parameters are listed in the second column of Table 3. The freeform mirror evolved from multiple OAP surfaces, together with the plane grating and detector comprise a freeform spectrometer, which is illustrated in Fig. 10. The structure parameters of the freeform spectrometer are listed in the second column of Table 4. *L*_{EC} denotes the distance from the entrance pinhole to the collimating region on the freeform mirror, which acts as a collimating mirror. In other words, for any wavelength of the beam originating from the entrance pinhole, when it initially strikes the freeform mirror, and as the effective region on the freeform mirror closely approximates an OAP surface, the beam is well collimated. After diffraction from the plane grating, it strikes the freeform mirror again, and the effective region on the freeform mirror still closely approximates an OAP surface, thus the beam is well focused.

## 4. Further optimization and performance analysis

The parameters of the constructed spectrometer were then imported to OpticStudio 16.5 SP2 optical design software (Zemax LLC, Kirkland, WA, USA) for optical performance evaluation, and the results are presented in Fig. 11. The realization of our design concepts as substituting multiple OAP surfaces for spherical surfaces were firstly evaluated. An ideal divergent beam originating from the entrance pinhole is reflected by the freeform mirror to obtain an aberration constrained collimated beam. The resulting wavefront aberration is shown in Fig. 11(a), with a peak-to-valley (PV) value of 0.633 λ and a root-mean-square (rms) value of 0.123 λ. The imaging quality in terms of the spots on the detector was considered next. The RMS spot radius across the 400 nm bandwidth as well as the spot diagrams distribution are presented, as shown in Figs. 11(b) and 11(c). The spots are close to being diffraction limited, and those for the shorter wavelengths are more aberrated.

Further optimization was performed on the constructed spectrometer leveraging the optical design software. The coefficients of the XY polynomials representing the freeform surface, as well as the distances to specify the relative locations of the entrance pinhole, freeform mirror, plane grating, and detector, are taken as variables. A freeform mirror with a rectangular effective aperture is finally achieved with it surface sag illustrated in Fig. 12, the minus sign indicates that it is concave surface. The surface of the XY polynomials up to fifth order is smooth, and its PV value is 190.9 μm as shown in Fig. 12 (a). According to its surface sag as shown in Fig. 12 (b), a best-fit sphere for interferometric testing of the freeform mirror can then be calculated, and the freeform surface sag departure from the best-fit sphere is tens of micrometers, which is within the current testing ability of a stitching interferometer. The wavefront aberration of the collimated beam reflected by the collimating region on the freeform mirror is decreased to a PV value of 0.395 λ and rms value of 0.069 λ, with trefoil as the dominant aberration as shown in Fig. 13(a). A near diffraction limited optical performance was accomplished through further optimization, as illustrated in Figs. 13(b) and (c). Thus, the optical path from the entrance pinhole to the collimating region on the freeform mirror, and then the plane grating, as well as the optical path from the plane grating to the focusing region on the freeform mirror, and then the detector, are both aberrations constrained. Then, the aberration constrained wavefront can be taken as the criterion for positioning and aligning the collimating region and the focusing region. Furthermore, the collimating region and the focusing region are integrated into the freeform mirror. Therefore, it is convenient to locate the entrance pinhole, freeform mirror, plane grating and detector sequentially.

The optimized parameters for the freeform mirror are listed in the third column of Table 3, and the optimized parameters for the freeform spectrometer are listed in the third column of Table 4. As deduced from Tables 3 and 4, other than the changes to the coefficients of the XY polynomials, *α*_{2}, the incident angle for the chief ray at the central wavelength on the focusing region on the freeform mirror is maintained at 10°, and *α*_{1}, the incident angle on the collimating region on the freeform mirror changes significantly. Moreover, the change of *L*_{FD} is small compared with those of *L*_{EC}, *L*_{CG}, and *L*_{GF}. Several OAP segments are mixed to develop the focusing region, and only one OAP segment is used to retrieve the collimating region, so the restrictions on the collimating region are less stringent compared with those on the focusing region. Thus, it is understandable that *α*_{1}, defining the surface information, *L*_{EC} and *L*_{CG}, defining the location information of the collimating region, exhibit a relatively large change during optimization. The optimization is leveraged to obtain a better collimated beam with constrained aberrations as the comparison of Fig. 11(a) and 13(a) shows. *L*_{GF} is also increased, to minimize the spot diagrams on the detector especially for the short wavelengths. As have been referred in the strategy for OAP segments update, the condition of normal incidence on the detector is not met, thus the incidence angles for the nine sampling wavelengths are listed in Table. 5. The maximum incidence angle tilt is only 1.266° for the maximum wavelength, which is acceptable considering the detector response, and is quite small compared with that in other configurations proposed for astigmatism compensation.

Note that optimization by optical design software remains necessary, but the configuration constructed through multiple OAP surface expansion and mixing is a good starting point. Creating the initial constructed configuration is critical to achieving the ultimate optical requirements because it constrains the design form of the spectrometer. Moreover, the design concepts employed to obtain independent aberration free or aberration constrained optics are maintained during the optimization process. A spectrometer with a spectral resolution of 0.1 nm is established for wavelengths ranging from 600 nm to 1000 nm, which is composed of a freeform mirror, plane grating, and detector. The integration of the collimating and the focusing mirrors into a single freeform surface results in a freeform spectrometer that is more compact, and simplifies the manufacture and alignment process.

## 5. Conclusion

The OAP mirror is an aberration free optical element, which focuses a collimated beam or collimates a divergent source perfectly. This characteristic is leveraged for aberration correction in the classic Czerny-Turner spectrometer, by replacing the spherical collimating mirror and the spherical focusing mirror with multiple OAP surfaces. The design concepts in this paper describe how to develop an optical system with independent aberration free or aberration constrained optics. Thus, the wavefront aberration can be utilized as the criteria for alignment of each optical element sequentially. The specifications of the OAP surfaces are gradually updated and expanded from the OAP surface for the central wavelength on the focusing mirror, to minimize the surface sag deviations of the neighboring OAP surfaces. Therefore, the multiple OAP surfaces are mixed to construct a freeform surface integrating the collimating and focusing mirrors. By adopting the proposed construction method, a freeform spectrometer composed of an entrance pinhole, a freeform mirror, plane grating, and detector is developed. The spectrometer with a 0.1 nm spectral resolution over the bandwidth of 400 nm centered at 800 nm demonstrates advantages of compactness, integrated optical manufacture, and simplified alignment. The construction method employing multiple OAP surface expansion and mixing to obtain a freeform surface provides a feasible route to design an optical system with large FOVs by gradually developing it from an initial system with a limited FOV, and this idea is useful for freeform optics generation and optimization.

## Funding

Fundamental Research Funds for the Central Universities (30917011107, 30918014115-004), National Natural Science Foundation of China (61505080, 61377015).

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