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. LEC, LCG, LGF and LFD 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. RC and RF 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 αC and αF. 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):

d(sini+sinθ)=mλ
Where m is the diffraction order, usually we take m = −1. λ is the wavelength of the dispersed beam.

 

Fig. 1 Optical layout of the classic Czerny-Turner spectrometer.

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The beam is dispersed spectrally by the grating with the angular spectral spread Δθ, which together with RF determines the spatial length L of the image on the detector. The incidence angle αF on the focusing mirror varies with the wavelength due to diffraction. The wavelength dependence of αF 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.

s=D22R=D24f
Where R represents the radius of curvature of the parent parabolic mirror at the vertex.

 

Fig. 2 Optical layout of the OAP mirror.

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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 f^differs from the parent focal length f, and their relationship is expressed as:

f^=f+s

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

tanβ=Dfs=2RDR2D2
The decenter D can be expressed as a function of R and β:

D=R(1cosβ)sinβ

In this paper, the reflected focal length f^ and the off-axis angle β are defined as the pair (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.

 

Fig. 3 Schemes for the multiple OAP surfaces substitutions.

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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 θmin, θctrl and θmax are calculated according to Eq. (1). βmin, βctrl and βmax 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:

θctrlθmin+βmin=βctrl
θctrlθmax+βmax=βctrl

DB-min, DB-ctrl, and DB-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:

DB-maxDB-min=L
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 βctrl is selected to avoid obstruction in the spectrometer, so, βmin and βmax are derived according to Eqs. (6) and (7). The radius of curvature RB 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 f^B-ctrl of the OAP base equals LGF. The length of the chief ray from each OAP segment to the detector represents its reflective focal length, of which the reflective focal length f^S-ctrl of the OAP segment for central wavelength equals LFD. 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:

f^S-ctrl+f^B-ctrl=f^S-min+f^B-min=f^S-max+f^B-max
Where f^S-minand f^S-maxsignify the reflective focal length of the OAP segment for the minimum and maximum wavelengths, respectively, and f^B-minand f^B-max signify the reflective focal length at the centers located on the OAP base for the minimum and maximum wavelengths respectively. These focal lengths are not labeled in Fig. 4, but it is easy to distinguish them according to the definitions or by reference to Fig. 3.

 

Fig. 4 Specification calculation scheme for the OAP segments.

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With the specifications for the OAP base accomplished, the reflective focal lengths of the OAP base at each center, such as f^B-ctrl,f^B-min, and f^B-max are known. When LFD (f^S-ctrl) is selected, the reflective focal length for each OAP segment, such as f^S-minand f^S-maxcan be derived from Eq. (9). Thus, together with the off-axis angles of βmin, βctrl and βmax, 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 βctrl on the focusing mirror, and distance from the focusing mirror to the detector LFD 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 λ1λ9. Thus, the corresponding diffraction angles were labeled θ1θ9, and off-axis angles of the OAP segments were labeled β1β9. Since λ5 = 800 nm is the central wavelength, θ5 equals θctrl, and β5 equals βctrl. 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 λ1, so that their parameters were the same. Thus, they should satisfy the equation:

βC=β1=θ1i2

To avoid obstruction in the spectrometer, β5 was selected as 20°. Then, according to Eqs. (1), (6) and (10), i was solved for and found to be 5.620°. θ1θ9, and β1β9 were calculated. Note that the criteria for the selection of βC and i are not rigorous.

LFD (f^S5) was then chosen to be 95 mm. Thus, parameters for the OAP base and each OAP segment were derived. The radius of curvature RB 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 β5 = 20° and f^S5=95mm were manually selected. The parameters for the OAP collimating mirror are the same as those for the OAP segment for λ1. 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 (f^S1, β1) – (f^S9, β9) are derived to define each OAP segment, and f^B1f^B9 are used to define their respective locations.

Tables Icon

Table 1. OAP base and OAP segment parameters.

 

Fig. 5 Flowchart showing the parameter calculation for the OAP surfaces.

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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 XF-YF-ZF as illustrated in Fig. 6, for further expansion and mixing. The global coordinate system XF-YF-ZF is defined with its origin at the center of the OAP segment for the central wavelength, and with ZF axis opposite to the surface normal direction. Typically, the OAP segment for the central wavelength is firstly specified in the coordinate system Xctrl-Yctrl-Zctrl as illustrated in Fig. 6, with its origin at the intersection of the chief ray and the plane grating, and with Zctrl axis coincident with the chief ray of the diffracted beam for the central wavelength. The coordinate (xctrl, yctrl, zctrl) of any point on the OAP segment for the central wavelength defined in Xctrl-Yctrl-Zctrl can be expressed by Eq. (11) as

zctrl=f^B-ctrl+sctrl+xctrl2+(yctrl+DS-ctrl)22Rctrl
Where sctrl denotes the sag for the center of the OAP segment for the central wavelength, Dctrl denotes the decenter, and Rctrl 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.

 

Fig. 6 Coordinate transformation for the OAP segments.

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The origin of coordinate system Xctrl-Yctrl-Zctrl can also be specified as a point (x0, y0, z0) in the coordinate system XF-YF-ZF, which fulfills the following equations:

y0=f^B-ctrlsinβctrl2
z0=f^B-ctrlcosβctrl2
Then, the point on the OAP segment with coordinate (xctrl, yctrl, zctrl) can be transformed to coordinate (xctrl-F, yctrl-F, zctrl-F) defined in the coordinate system XF-YF-ZF according to the difference in origin position and Z axis direction. The coordinate transformation formulas are:

xctrl-F=xctrl
yctrl-F=yctrlcosβctrl2zctrlsinβctrl2+y0
zctrl-F=yctrlsinβctrl2+zctrlcosβctrl2+z0

Similarly, as illustrated in Fig. 6, the OAP segment for the minimum wavelength is initially specified in the coordinate system Xmin-Ymin-Zmin having an origin coincident with that of Xctrl-Yctrl-Zctrl, and Zmin axis in consistent with the chief ray of the diffracted beam for the minimum wavelength. The coordinate (xmin, ymin, zmin) of any point on the OAP segment for minimum wavelength defined in Xmin-Ymin-Zmin can be expressed by an equation similar to Eq. (11) and it is not presented here. Moreover, the coordinate (xmin-F, ymin-F, zmin-F) defined in the global coordinate system XF-YF-ZF can be derived according to the differences in origin position and Z axis direction:

xmin-F=xmin
ymin-F=ymincos(θminθctrl+βctrl2)zminsin(θminθctrl+βctrl2)+y0
zmin-F=yminsin(θminθctrl+βctrl2)+zmincos(θminθctrl+βctrl2)+z0
The coordinate transformation of any point defined in the OAP segment for a certain wavelength to the global coordinate system XF-YF-ZF 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 Фy and Фx, and they can be derived according the following equations:

Φx=Φcosi
Φy=Φcos(θi)cosi

Therefore, surface sags of the OAP segments for λ1λ9 were uniformly defined in the global coordinate system XF-YF-ZF 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 YF 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.

 

Fig. 7 Surface profiles along YF direction defined in the global coordinate system XF-YF-ZF (a) for each OAP segment and the OAP base (b) for differences between neighboring OAP segments in the overlapping area.

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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 (λ5 here) is the origin of the global coordinate system XF-YF-ZF, it is taken as the reference surface. The OAP segments for the adjacent wavelengths (λ4 and λ6) are adjusted and then derived with updated parameters, to minimize the sag deviations in the overlapping areas. Based on the updated OAP segments for λ4 and λ6 in the previous step, the updated parameters of their neighboring OAP segments for λ3 and λ7 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 YF direction, their profiles along the YF direction instead of the surface sags in the approximately elliptical effective aperture are considered for simplicity. The adjustment of the OAP segment for λ4 is taken as an example as illustrated in Fig. 8, in which only the profiles along the YF direction are plotted. As the OAP segments for both λ4 and λ5 are defined in the global coordinate system XF-YF-ZF, the length of their overlapping area along the YF direction is labeled as δ45. Moreover, the peak to valley value for their sag difference as the red curve plotted in Fig. 7(b) is defined as S45. Thus, to minimize the sag deviations of the neighboring OAP segment in the overlapping area, the OAP segment for λ4 can be tilted by the angle of ε4, which is derived approximately:

 

Fig. 8 The update strategy of the OAP segment for λ4.

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ε4=S45δ45

The update of the OAP segment for λ4 is implemented in two steps. Firstly, it is tilted by an angle of ε4 about its center, and then the off-axis angle β4 for the updated OAP segment for λ4 is derived as:

β4=β4+2ε4
Thus, for the chief ray of λ4, 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 λ4 from the plane grating, with a displacement of Δ4 approximately defined as:

Δ4=d45tanε4
Where, d45 is the distance between the center of the OAP segment for λ4 and the intersection of the two OAP segments along the YF direction. This operation ensures the chief ray will still strike the center of the updated OAP segment.

Thus, the distance LGF4 from the grating to the updated OAP segment for λ4 is calculated, which indicates the new location of this OAP segment.

LGF4=LGF4Δ4=f^B4Δ4

Moreover, the distance LFD4 from the updated OAP segment for λ4 to the detector equals its reflected focal length f^S4, and can be derived approximately from the geometry in Fig. 8.

LFD4=f^S4=LFD4cos2ε4Δ4=f^S4cos2ε4Δ4

As the pair (f^S4,β4) have been derived in Eqs. (23) and (26), the specifications for the updated OAP segment for λ4 are determined. The sag deviations for the neighboring OAP segments for λ4 and λ5 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 (f^S1,β1) to (f^S9,β9) are used to specify the OAP segments, LGF1LGF9 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 XF-YF-ZF supports a base spherical surface upon which the fifth order XY polynomials are added. For any point (xF, yF, zF) on the freeform surface, the surface sag is of the form:

zF=RS+RS2(xF2+yF2)+p20x2+p02y2+p21x2y+p03y3+p40x4+p22x2y2+p04y4+p41x4y+p23x2y3+p05y5
Where, the first two terms on the right-hand side of the equal sign represent the base spherical surface, which is employed to increase the fitting efficiency. The manually selected vertex radius of curvature denoted as Rs = −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 RN, resulting in x = xF/RN and y = yF/RN. Since the freeform surface is symmetric about the tangential plane, only ten XY polynomials terms are used.

 

Fig. 9 Flowchart of OAP surface expansion and mixing.

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Tables Icon

Table 2. Updated OAP segment parameters.

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 λ1 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 (f^S-C,βC) 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. LEC 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.

Tables Icon

Table 3. Constructed and optimized parameters for the freeform mirror.

 

Fig. 10 Optical layout of the freeform spectrometer.

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Tables Icon

Table 4. Constructed and optimized parameters for the freeform spectrometer.

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.

 

Fig. 11 Optical performance of the constructed freeform spectrometer. (a) Wavefront aberration of the collimated beam after the first reflection by the freeform mirror, (b) RMS spot radius on the detector as a function of wavelength across the spectrum range, (c) spot diagrams on the detector for the nine sampling wavelengths.

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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.

 

Fig. 12 Surface sag of the optimized freeform mirror. (a) XY polynomials, (b) with the base spherical surface added.

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Fig. 13 Optical performance of the optimized freeform spectrometer. (a) Wavefront aberration of the collimated beam after the first reflection from the freeform mirror, (b) RMS spot radius on the detector as a function of wavelength across the spectrum range, (c) spot diagrams on the detector for the nine sampling wavelengths.

Download Full Size | PPT Slide | PDF

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, α2, the incident angle for the chief ray at the central wavelength on the focusing region on the freeform mirror is maintained at 10°, and α1, the incident angle on the collimating region on the freeform mirror changes significantly. Moreover, the change of LFD is small compared with those of LEC, LCG, and LGF. 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 α1, defining the surface information, LEC and LCG, 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. LGF 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.

Tables Icon

Table 5. Inclined incidence angles on the detector.

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).

References

1. D. Korsch, “Anastigmatic three-mirror telescope,” Appl. Opt. 16(8), 2074–2077 (1977). [CrossRef]   [PubMed]  

2. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express 19(22), 21919–21928 (2011). [CrossRef]   [PubMed]  

3. F. Zeng, X. Zhang, J. Zhang, G. Shi, and H. Wu, “Optics ellipticity performance of an unobscured off-axis space telescope,” Opt. Express 22(21), 25277–25285 (2014). [CrossRef]   [PubMed]  

4. M. R. Torr and D. G. Torr, “Compact imaging spectrograph for broadband spectral simultaneity,” Appl. Opt. 34(34), 7888–7898 (1995). [CrossRef]   [PubMed]  

5. M. Czerny and A. F. Turner, “On the Astigmatism of Mirror Spectrometers,” Z. Phys. 61(11–12), 792–797 (1930). [CrossRef]  

6. A. B. Shafer, L. R. Megill, and L. Droppleman, “Optimization of the Czerny–Turner spectrometer,” J. Opt. Soc. Am. 54(7), 879–887 (1964). [CrossRef]  

7. Q. Yuan, D. Zhu, Y. Chen, Z. Guo, C. Zuo, and Z. Gao, “Comparative assessment of astigmatism–corrected Czerny-Turner imaging spectrometer using off-the-shelf optics,” Opt. Commun. 388, 53–61 (2017). [CrossRef]  

8. D. R. Austin, T. Witting, and I. A. Walmsley, “Broadband astigmatism-free Czerny-Turner imaging spectrometer using spherical mirrors,” Appl. Opt. 48(19), 3846–3853 (2009). [CrossRef]   [PubMed]  

9. M. D. Mouriz, E. L. Lago, X. Prieto-Blanco, H. González-Núñez, and R. de la Fuente, “Schwarzschild spectrometer,” Appl. Opt. 50(16), 2418–2424 (2011). [CrossRef]   [PubMed]  

10. T. A. Chen, Y. Tang, L. J. Zhang, Y. E. Chang, and C. Zheng, “Correction of astigmatism and coma using analytic theory of aberrations in imaging spectrometer based on concentric off-axis dual reflector system,” Appl. Opt. 53(4), 565–576 (2014). [CrossRef]   [PubMed]  

11. Q. Xue, S. Wang, and F. Lu, “Aberration-corrected Czerny-Turner imaging spectrometer with a wide spectral region,” Appl. Opt. 48(1), 11–16 (2009). [CrossRef]   [PubMed]  

12. G. Xia, S. Wu, G. Wang, M. Hu, and J. Xing, “Astigmatism-free Czerny-Turner compact spectrometer with cylindrical mirrors,” Appl. Opt. 56(32), 9069–9073 (2017). [CrossRef]   [PubMed]  

13. L. Xu, K. Chen, Q. He, and G. Jin, “Design of freeform mirrors in Czerny-Turner spectrometers to suppress astigmatism,” Appl. Opt. 48(15), 2871–2879 (2009). [CrossRef]   [PubMed]  

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

15. C. Chrystal, K. H. Burrell, and N. A. Pablant, “Straightforward correction for the astigmatism of a Czerny-Turner spectrometer,” Rev. Sci. Instrum. 81(2), 023503 (2010). [CrossRef]   [PubMed]  

16. K. S. Lee, K. P. Thompson, and J. P. Rolland, “Broadband astigmatism-corrected Czerny-Turner spectrometer,” Opt. Express 18(22), 23378–23384 (2010). [CrossRef]   [PubMed]  

17. Q. Xue, “Astigmatism-corrected Czerny-Turner imaging spectrometer for broadband spectral simultaneity,” Appl. Opt. 50(10), 1338–1344 (2011). [CrossRef]   [PubMed]  

18. G. Xia, B. X. Qu, P. Liu, and F. Yu, “Astigmatism–corrected miniature Czerny–Turner spectrometer with freeform cylindrical lens,” Chin. Opt. Lett. 10(8), 23–26 (2012).

19. X. Zhong, Y. Zhang, and G. Jin, “High performance Czerny-Turner imaging spectrometer with aberrations corrected by tilted lenses,” Opt. Commun. 338, 73–76 (2015). [CrossRef]  

20. X. Ge, S. Chen, Y. Zhang, H. Chen, P. Guo, T. Mu, J. Yang, and Z. Bu, “Broadband astigmatism–corrected spectrometer design using a toroidal lens and a special filter,” Opt. Laser Technol. 65, 88–93 (2015). [CrossRef]  

21. S. H. Kim, H. J. Kong, and S. Chang, “Aberration analysis of a concentric imaging spectrometer with a convex grating,” Opt. Commun. 333, 6–10 (2014). [CrossRef]  

22. J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling increased compactness,” Light Sci. Appl. 6(7), e17026 (2017). [CrossRef]   [PubMed]  

23. J. Zhu, X. Wu, T. Yang, and G. Jin, “Generating optical freeform surfaces considering both coordinates and normals of discrete data points,” J. Opt. Soc. Am. A 31(11), 2401–2408 (2014). [CrossRef]   [PubMed]  

24. Q. Meng, H. Wang, K. Wang, Y. Wang, Z. Ji, and D. Wang, “Off-axis three-mirror freeform telescope with a large linear field of view based on an integration mirror,” Appl. Opt. 55(32), 8962–8970 (2016). [CrossRef]   [PubMed]  

25. T. Yang, J. Zhu, and G. Jin, “Compact freeform off-axis three-mirror imaging system based on the integration of primary and tertiary mirrors on one single surface,” Chin. Opt. Lett. 14(6), 26–30 (2016).

References

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  • |

  1. D. Korsch, “Anastigmatic three-mirror telescope,” Appl. Opt. 16(8), 2074–2077 (1977).
    [Crossref] [PubMed]
  2. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express 19(22), 21919–21928 (2011).
    [Crossref] [PubMed]
  3. F. Zeng, X. Zhang, J. Zhang, G. Shi, and H. Wu, “Optics ellipticity performance of an unobscured off-axis space telescope,” Opt. Express 22(21), 25277–25285 (2014).
    [Crossref] [PubMed]
  4. M. R. Torr and D. G. Torr, “Compact imaging spectrograph for broadband spectral simultaneity,” Appl. Opt. 34(34), 7888–7898 (1995).
    [Crossref] [PubMed]
  5. M. Czerny and A. F. Turner, “On the Astigmatism of Mirror Spectrometers,” Z. Phys. 61(11–12), 792–797 (1930).
    [Crossref]
  6. A. B. Shafer, L. R. Megill, and L. Droppleman, “Optimization of the Czerny–Turner spectrometer,” J. Opt. Soc. Am. 54(7), 879–887 (1964).
    [Crossref]
  7. Q. Yuan, D. Zhu, Y. Chen, Z. Guo, C. Zuo, and Z. Gao, “Comparative assessment of astigmatism–corrected Czerny-Turner imaging spectrometer using off-the-shelf optics,” Opt. Commun. 388, 53–61 (2017).
    [Crossref]
  8. D. R. Austin, T. Witting, and I. A. Walmsley, “Broadband astigmatism-free Czerny-Turner imaging spectrometer using spherical mirrors,” Appl. Opt. 48(19), 3846–3853 (2009).
    [Crossref] [PubMed]
  9. M. D. Mouriz, E. L. Lago, X. Prieto-Blanco, H. González-Núñez, and R. de la Fuente, “Schwarzschild spectrometer,” Appl. Opt. 50(16), 2418–2424 (2011).
    [Crossref] [PubMed]
  10. T. A. Chen, Y. Tang, L. J. Zhang, Y. E. Chang, and C. Zheng, “Correction of astigmatism and coma using analytic theory of aberrations in imaging spectrometer based on concentric off-axis dual reflector system,” Appl. Opt. 53(4), 565–576 (2014).
    [Crossref] [PubMed]
  11. Q. Xue, S. Wang, and F. Lu, “Aberration-corrected Czerny-Turner imaging spectrometer with a wide spectral region,” Appl. Opt. 48(1), 11–16 (2009).
    [Crossref] [PubMed]
  12. G. Xia, S. Wu, G. Wang, M. Hu, and J. Xing, “Astigmatism-free Czerny-Turner compact spectrometer with cylindrical mirrors,” Appl. Opt. 56(32), 9069–9073 (2017).
    [Crossref] [PubMed]
  13. L. Xu, K. Chen, Q. He, and G. Jin, “Design of freeform mirrors in Czerny-Turner spectrometers to suppress astigmatism,” Appl. Opt. 48(15), 2871–2879 (2009).
    [Crossref] [PubMed]
  14. E. S. Voropai, I. M. Gulis, and A. G. Kupreev, “Astigmatism correction for a large-aperture dispersive spectrometer,” J. Appl. Spectrosc. 75(1), 150–155 (2008).
    [Crossref]
  15. C. Chrystal, K. H. Burrell, and N. A. Pablant, “Straightforward correction for the astigmatism of a Czerny-Turner spectrometer,” Rev. Sci. Instrum. 81(2), 023503 (2010).
    [Crossref] [PubMed]
  16. K. S. Lee, K. P. Thompson, and J. P. Rolland, “Broadband astigmatism-corrected Czerny-Turner spectrometer,” Opt. Express 18(22), 23378–23384 (2010).
    [Crossref] [PubMed]
  17. Q. Xue, “Astigmatism-corrected Czerny-Turner imaging spectrometer for broadband spectral simultaneity,” Appl. Opt. 50(10), 1338–1344 (2011).
    [Crossref] [PubMed]
  18. G. Xia, B. X. Qu, P. Liu, and F. Yu, “Astigmatism–corrected miniature Czerny–Turner spectrometer with freeform cylindrical lens,” Chin. Opt. Lett. 10(8), 23–26 (2012).
  19. X. Zhong, Y. Zhang, and G. Jin, “High performance Czerny-Turner imaging spectrometer with aberrations corrected by tilted lenses,” Opt. Commun. 338, 73–76 (2015).
    [Crossref]
  20. X. Ge, S. Chen, Y. Zhang, H. Chen, P. Guo, T. Mu, J. Yang, and Z. Bu, “Broadband astigmatism–corrected spectrometer design using a toroidal lens and a special filter,” Opt. Laser Technol. 65, 88–93 (2015).
    [Crossref]
  21. S. H. Kim, H. J. Kong, and S. Chang, “Aberration analysis of a concentric imaging spectrometer with a convex grating,” Opt. Commun. 333, 6–10 (2014).
    [Crossref]
  22. J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling increased compactness,” Light Sci. Appl. 6(7), e17026 (2017).
    [Crossref] [PubMed]
  23. J. Zhu, X. Wu, T. Yang, and G. Jin, “Generating optical freeform surfaces considering both coordinates and normals of discrete data points,” J. Opt. Soc. Am. A 31(11), 2401–2408 (2014).
    [Crossref] [PubMed]
  24. Q. Meng, H. Wang, K. Wang, Y. Wang, Z. Ji, and D. Wang, “Off-axis three-mirror freeform telescope with a large linear field of view based on an integration mirror,” Appl. Opt. 55(32), 8962–8970 (2016).
    [Crossref] [PubMed]
  25. T. Yang, J. Zhu, and G. Jin, “Compact freeform off-axis three-mirror imaging system based on the integration of primary and tertiary mirrors on one single surface,” Chin. Opt. Lett. 14(6), 26–30 (2016).

2017 (3)

Q. Yuan, D. Zhu, Y. Chen, Z. Guo, C. Zuo, and Z. Gao, “Comparative assessment of astigmatism–corrected Czerny-Turner imaging spectrometer using off-the-shelf optics,” Opt. Commun. 388, 53–61 (2017).
[Crossref]

G. Xia, S. Wu, G. Wang, M. Hu, and J. Xing, “Astigmatism-free Czerny-Turner compact spectrometer with cylindrical mirrors,” Appl. Opt. 56(32), 9069–9073 (2017).
[Crossref] [PubMed]

J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling increased compactness,” Light Sci. Appl. 6(7), e17026 (2017).
[Crossref] [PubMed]

2016 (2)

Q. Meng, H. Wang, K. Wang, Y. Wang, Z. Ji, and D. Wang, “Off-axis three-mirror freeform telescope with a large linear field of view based on an integration mirror,” Appl. Opt. 55(32), 8962–8970 (2016).
[Crossref] [PubMed]

T. Yang, J. Zhu, and G. Jin, “Compact freeform off-axis three-mirror imaging system based on the integration of primary and tertiary mirrors on one single surface,” Chin. Opt. Lett. 14(6), 26–30 (2016).

2015 (2)

X. Zhong, Y. Zhang, and G. Jin, “High performance Czerny-Turner imaging spectrometer with aberrations corrected by tilted lenses,” Opt. Commun. 338, 73–76 (2015).
[Crossref]

X. Ge, S. Chen, Y. Zhang, H. Chen, P. Guo, T. Mu, J. Yang, and Z. Bu, “Broadband astigmatism–corrected spectrometer design using a toroidal lens and a special filter,” Opt. Laser Technol. 65, 88–93 (2015).
[Crossref]

2014 (4)

2012 (1)

G. Xia, B. X. Qu, P. Liu, and F. Yu, “Astigmatism–corrected miniature Czerny–Turner spectrometer with freeform cylindrical lens,” Chin. Opt. Lett. 10(8), 23–26 (2012).

2011 (3)

2010 (2)

C. Chrystal, K. H. Burrell, and N. A. Pablant, “Straightforward correction for the astigmatism of a Czerny-Turner spectrometer,” Rev. Sci. Instrum. 81(2), 023503 (2010).
[Crossref] [PubMed]

K. S. Lee, K. P. Thompson, and J. P. Rolland, “Broadband astigmatism-corrected Czerny-Turner spectrometer,” Opt. Express 18(22), 23378–23384 (2010).
[Crossref] [PubMed]

2009 (3)

2008 (1)

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

1995 (1)

1977 (1)

1964 (1)

1930 (1)

M. Czerny and A. F. Turner, “On the Astigmatism of Mirror Spectrometers,” Z. Phys. 61(11–12), 792–797 (1930).
[Crossref]

Austin, D. R.

Bauer, A.

J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling increased compactness,” Light Sci. Appl. 6(7), e17026 (2017).
[Crossref] [PubMed]

Bu, Z.

X. Ge, S. Chen, Y. Zhang, H. Chen, P. Guo, T. Mu, J. Yang, and Z. Bu, “Broadband astigmatism–corrected spectrometer design using a toroidal lens and a special filter,” Opt. Laser Technol. 65, 88–93 (2015).
[Crossref]

Burrell, K. H.

C. Chrystal, K. H. Burrell, and N. A. Pablant, “Straightforward correction for the astigmatism of a Czerny-Turner spectrometer,” Rev. Sci. Instrum. 81(2), 023503 (2010).
[Crossref] [PubMed]

Chang, S.

S. H. Kim, H. J. Kong, and S. Chang, “Aberration analysis of a concentric imaging spectrometer with a convex grating,” Opt. Commun. 333, 6–10 (2014).
[Crossref]

Chang, Y. E.

Chen, H.

X. Ge, S. Chen, Y. Zhang, H. Chen, P. Guo, T. Mu, J. Yang, and Z. Bu, “Broadband astigmatism–corrected spectrometer design using a toroidal lens and a special filter,” Opt. Laser Technol. 65, 88–93 (2015).
[Crossref]

Chen, K.

Chen, S.

X. Ge, S. Chen, Y. Zhang, H. Chen, P. Guo, T. Mu, J. Yang, and Z. Bu, “Broadband astigmatism–corrected spectrometer design using a toroidal lens and a special filter,” Opt. Laser Technol. 65, 88–93 (2015).
[Crossref]

Chen, T. A.

Chen, Y.

Q. Yuan, D. Zhu, Y. Chen, Z. Guo, C. Zuo, and Z. Gao, “Comparative assessment of astigmatism–corrected Czerny-Turner imaging spectrometer using off-the-shelf optics,” Opt. Commun. 388, 53–61 (2017).
[Crossref]

Chrystal, C.

C. Chrystal, K. H. Burrell, and N. A. Pablant, “Straightforward correction for the astigmatism of a Czerny-Turner spectrometer,” Rev. Sci. Instrum. 81(2), 023503 (2010).
[Crossref] [PubMed]

Czerny, M.

M. Czerny and A. F. Turner, “On the Astigmatism of Mirror Spectrometers,” Z. Phys. 61(11–12), 792–797 (1930).
[Crossref]

de la Fuente, R.

Droppleman, L.

Fuerschbach, K.

Gao, Z.

Q. Yuan, D. Zhu, Y. Chen, Z. Guo, C. Zuo, and Z. Gao, “Comparative assessment of astigmatism–corrected Czerny-Turner imaging spectrometer using off-the-shelf optics,” Opt. Commun. 388, 53–61 (2017).
[Crossref]

Ge, X.

X. Ge, S. Chen, Y. Zhang, H. Chen, P. Guo, T. Mu, J. Yang, and Z. Bu, “Broadband astigmatism–corrected spectrometer design using a toroidal lens and a special filter,” Opt. Laser Technol. 65, 88–93 (2015).
[Crossref]

González-Núñez, H.

Gulis, I. M.

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

Guo, P.

X. Ge, S. Chen, Y. Zhang, H. Chen, P. Guo, T. Mu, J. Yang, and Z. Bu, “Broadband astigmatism–corrected spectrometer design using a toroidal lens and a special filter,” Opt. Laser Technol. 65, 88–93 (2015).
[Crossref]

Guo, Z.

Q. Yuan, D. Zhu, Y. Chen, Z. Guo, C. Zuo, and Z. Gao, “Comparative assessment of astigmatism–corrected Czerny-Turner imaging spectrometer using off-the-shelf optics,” Opt. Commun. 388, 53–61 (2017).
[Crossref]

He, Q.

Hu, M.

Ji, Z.

Jin, G.

T. Yang, J. Zhu, and G. Jin, “Compact freeform off-axis three-mirror imaging system based on the integration of primary and tertiary mirrors on one single surface,” Chin. Opt. Lett. 14(6), 26–30 (2016).

X. Zhong, Y. Zhang, and G. Jin, “High performance Czerny-Turner imaging spectrometer with aberrations corrected by tilted lenses,” Opt. Commun. 338, 73–76 (2015).
[Crossref]

J. Zhu, X. Wu, T. Yang, and G. Jin, “Generating optical freeform surfaces considering both coordinates and normals of discrete data points,” J. Opt. Soc. Am. A 31(11), 2401–2408 (2014).
[Crossref] [PubMed]

L. Xu, K. Chen, Q. He, and G. Jin, “Design of freeform mirrors in Czerny-Turner spectrometers to suppress astigmatism,” Appl. Opt. 48(15), 2871–2879 (2009).
[Crossref] [PubMed]

Kim, S. H.

S. H. Kim, H. J. Kong, and S. Chang, “Aberration analysis of a concentric imaging spectrometer with a convex grating,” Opt. Commun. 333, 6–10 (2014).
[Crossref]

Kong, H. J.

S. H. Kim, H. J. Kong, and S. Chang, “Aberration analysis of a concentric imaging spectrometer with a convex grating,” Opt. Commun. 333, 6–10 (2014).
[Crossref]

Korsch, D.

Kupreev, A. G.

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

Lago, E. L.

Lee, K. S.

Liu, P.

G. Xia, B. X. Qu, P. Liu, and F. Yu, “Astigmatism–corrected miniature Czerny–Turner spectrometer with freeform cylindrical lens,” Chin. Opt. Lett. 10(8), 23–26 (2012).

Lu, F.

Megill, L. R.

Meng, Q.

Mouriz, M. D.

Mu, T.

X. Ge, S. Chen, Y. Zhang, H. Chen, P. Guo, T. Mu, J. Yang, and Z. Bu, “Broadband astigmatism–corrected spectrometer design using a toroidal lens and a special filter,” Opt. Laser Technol. 65, 88–93 (2015).
[Crossref]

Pablant, N. A.

C. Chrystal, K. H. Burrell, and N. A. Pablant, “Straightforward correction for the astigmatism of a Czerny-Turner spectrometer,” Rev. Sci. Instrum. 81(2), 023503 (2010).
[Crossref] [PubMed]

Prieto-Blanco, X.

Qu, B. X.

G. Xia, B. X. Qu, P. Liu, and F. Yu, “Astigmatism–corrected miniature Czerny–Turner spectrometer with freeform cylindrical lens,” Chin. Opt. Lett. 10(8), 23–26 (2012).

Reimers, J.

J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling increased compactness,” Light Sci. Appl. 6(7), e17026 (2017).
[Crossref] [PubMed]

Rolland, J. P.

Shafer, A. B.

Shi, G.

Tang, Y.

Thompson, K. P.

Torr, D. G.

Torr, M. R.

Turner, A. F.

M. Czerny and A. F. Turner, “On the Astigmatism of Mirror Spectrometers,” Z. Phys. 61(11–12), 792–797 (1930).
[Crossref]

Voropai, E. S.

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

Walmsley, I. A.

Wang, D.

Wang, G.

Wang, H.

Wang, K.

Wang, S.

Wang, Y.

Witting, T.

Wu, H.

Wu, S.

Wu, X.

Xia, G.

G. Xia, S. Wu, G. Wang, M. Hu, and J. Xing, “Astigmatism-free Czerny-Turner compact spectrometer with cylindrical mirrors,” Appl. Opt. 56(32), 9069–9073 (2017).
[Crossref] [PubMed]

G. Xia, B. X. Qu, P. Liu, and F. Yu, “Astigmatism–corrected miniature Czerny–Turner spectrometer with freeform cylindrical lens,” Chin. Opt. Lett. 10(8), 23–26 (2012).

Xing, J.

Xu, L.

Xue, Q.

Yang, J.

X. Ge, S. Chen, Y. Zhang, H. Chen, P. Guo, T. Mu, J. Yang, and Z. Bu, “Broadband astigmatism–corrected spectrometer design using a toroidal lens and a special filter,” Opt. Laser Technol. 65, 88–93 (2015).
[Crossref]

Yang, T.

T. Yang, J. Zhu, and G. Jin, “Compact freeform off-axis three-mirror imaging system based on the integration of primary and tertiary mirrors on one single surface,” Chin. Opt. Lett. 14(6), 26–30 (2016).

J. Zhu, X. Wu, T. Yang, and G. Jin, “Generating optical freeform surfaces considering both coordinates and normals of discrete data points,” J. Opt. Soc. Am. A 31(11), 2401–2408 (2014).
[Crossref] [PubMed]

Yu, F.

G. Xia, B. X. Qu, P. Liu, and F. Yu, “Astigmatism–corrected miniature Czerny–Turner spectrometer with freeform cylindrical lens,” Chin. Opt. Lett. 10(8), 23–26 (2012).

Yuan, Q.

Q. Yuan, D. Zhu, Y. Chen, Z. Guo, C. Zuo, and Z. Gao, “Comparative assessment of astigmatism–corrected Czerny-Turner imaging spectrometer using off-the-shelf optics,” Opt. Commun. 388, 53–61 (2017).
[Crossref]

Zeng, F.

Zhang, J.

Zhang, L. J.

Zhang, X.

Zhang, Y.

X. Zhong, Y. Zhang, and G. Jin, “High performance Czerny-Turner imaging spectrometer with aberrations corrected by tilted lenses,” Opt. Commun. 338, 73–76 (2015).
[Crossref]

X. Ge, S. Chen, Y. Zhang, H. Chen, P. Guo, T. Mu, J. Yang, and Z. Bu, “Broadband astigmatism–corrected spectrometer design using a toroidal lens and a special filter,” Opt. Laser Technol. 65, 88–93 (2015).
[Crossref]

Zheng, C.

Zhong, X.

X. Zhong, Y. Zhang, and G. Jin, “High performance Czerny-Turner imaging spectrometer with aberrations corrected by tilted lenses,” Opt. Commun. 338, 73–76 (2015).
[Crossref]

Zhu, D.

Q. Yuan, D. Zhu, Y. Chen, Z. Guo, C. Zuo, and Z. Gao, “Comparative assessment of astigmatism–corrected Czerny-Turner imaging spectrometer using off-the-shelf optics,” Opt. Commun. 388, 53–61 (2017).
[Crossref]

Zhu, J.

T. Yang, J. Zhu, and G. Jin, “Compact freeform off-axis three-mirror imaging system based on the integration of primary and tertiary mirrors on one single surface,” Chin. Opt. Lett. 14(6), 26–30 (2016).

J. Zhu, X. Wu, T. Yang, and G. Jin, “Generating optical freeform surfaces considering both coordinates and normals of discrete data points,” J. Opt. Soc. Am. A 31(11), 2401–2408 (2014).
[Crossref] [PubMed]

Zuo, C.

Q. Yuan, D. Zhu, Y. Chen, Z. Guo, C. Zuo, and Z. Gao, “Comparative assessment of astigmatism–corrected Czerny-Turner imaging spectrometer using off-the-shelf optics,” Opt. Commun. 388, 53–61 (2017).
[Crossref]

Appl. Opt. (10)

D. R. Austin, T. Witting, and I. A. Walmsley, “Broadband astigmatism-free Czerny-Turner imaging spectrometer using spherical mirrors,” Appl. Opt. 48(19), 3846–3853 (2009).
[Crossref] [PubMed]

M. D. Mouriz, E. L. Lago, X. Prieto-Blanco, H. González-Núñez, and R. de la Fuente, “Schwarzschild spectrometer,” Appl. Opt. 50(16), 2418–2424 (2011).
[Crossref] [PubMed]

T. A. Chen, Y. Tang, L. J. Zhang, Y. E. Chang, and C. Zheng, “Correction of astigmatism and coma using analytic theory of aberrations in imaging spectrometer based on concentric off-axis dual reflector system,” Appl. Opt. 53(4), 565–576 (2014).
[Crossref] [PubMed]

Q. Xue, S. Wang, and F. Lu, “Aberration-corrected Czerny-Turner imaging spectrometer with a wide spectral region,” Appl. Opt. 48(1), 11–16 (2009).
[Crossref] [PubMed]

G. Xia, S. Wu, G. Wang, M. Hu, and J. Xing, “Astigmatism-free Czerny-Turner compact spectrometer with cylindrical mirrors,” Appl. Opt. 56(32), 9069–9073 (2017).
[Crossref] [PubMed]

L. Xu, K. Chen, Q. He, and G. Jin, “Design of freeform mirrors in Czerny-Turner spectrometers to suppress astigmatism,” Appl. Opt. 48(15), 2871–2879 (2009).
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Q. Xue, “Astigmatism-corrected Czerny-Turner imaging spectrometer for broadband spectral simultaneity,” Appl. Opt. 50(10), 1338–1344 (2011).
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Q. Meng, H. Wang, K. Wang, Y. Wang, Z. Ji, and D. Wang, “Off-axis three-mirror freeform telescope with a large linear field of view based on an integration mirror,” Appl. Opt. 55(32), 8962–8970 (2016).
[Crossref] [PubMed]

Chin. Opt. Lett. (2)

T. Yang, J. Zhu, and G. Jin, “Compact freeform off-axis three-mirror imaging system based on the integration of primary and tertiary mirrors on one single surface,” Chin. Opt. Lett. 14(6), 26–30 (2016).

G. Xia, B. X. Qu, P. Liu, and F. Yu, “Astigmatism–corrected miniature Czerny–Turner spectrometer with freeform cylindrical lens,” Chin. Opt. Lett. 10(8), 23–26 (2012).

J. Appl. Spectrosc. (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Light Sci. Appl. (1)

J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling increased compactness,” Light Sci. Appl. 6(7), e17026 (2017).
[Crossref] [PubMed]

Opt. Commun. (3)

Q. Yuan, D. Zhu, Y. Chen, Z. Guo, C. Zuo, and Z. Gao, “Comparative assessment of astigmatism–corrected Czerny-Turner imaging spectrometer using off-the-shelf optics,” Opt. Commun. 388, 53–61 (2017).
[Crossref]

S. H. Kim, H. J. Kong, and S. Chang, “Aberration analysis of a concentric imaging spectrometer with a convex grating,” Opt. Commun. 333, 6–10 (2014).
[Crossref]

X. Zhong, Y. Zhang, and G. Jin, “High performance Czerny-Turner imaging spectrometer with aberrations corrected by tilted lenses,” Opt. Commun. 338, 73–76 (2015).
[Crossref]

Opt. Express (3)

Opt. Laser Technol. (1)

X. Ge, S. Chen, Y. Zhang, H. Chen, P. Guo, T. Mu, J. Yang, and Z. Bu, “Broadband astigmatism–corrected spectrometer design using a toroidal lens and a special filter,” Opt. Laser Technol. 65, 88–93 (2015).
[Crossref]

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C. Chrystal, K. H. Burrell, and N. A. Pablant, “Straightforward correction for the astigmatism of a Czerny-Turner spectrometer,” Rev. Sci. Instrum. 81(2), 023503 (2010).
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[Crossref]

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Figures (13)

Fig. 1
Fig. 1 Optical layout of the classic Czerny-Turner spectrometer.
Fig. 2
Fig. 2 Optical layout of the OAP mirror.
Fig. 3
Fig. 3 Schemes for the multiple OAP surfaces substitutions.
Fig. 4
Fig. 4 Specification calculation scheme for the OAP segments.
Fig. 5
Fig. 5 Flowchart showing the parameter calculation for the OAP surfaces.
Fig. 6
Fig. 6 Coordinate transformation for the OAP segments.
Fig. 7
Fig. 7 Surface profiles along YF direction defined in the global coordinate system XF-YF-ZF (a) for each OAP segment and the OAP base (b) for differences between neighboring OAP segments in the overlapping area.
Fig. 8
Fig. 8 The update strategy of the OAP segment for λ4.
Fig. 9
Fig. 9 Flowchart of OAP surface expansion and mixing.
Fig. 10
Fig. 10 Optical layout of the freeform spectrometer.
Fig. 11
Fig. 11 Optical performance of the constructed freeform spectrometer. (a) Wavefront aberration of the collimated beam after the first reflection by the freeform mirror, (b) RMS spot radius on the detector as a function of wavelength across the spectrum range, (c) spot diagrams on the detector for the nine sampling wavelengths.
Fig. 12
Fig. 12 Surface sag of the optimized freeform mirror. (a) XY polynomials, (b) with the base spherical surface added.
Fig. 13
Fig. 13 Optical performance of the optimized freeform spectrometer. (a) Wavefront aberration of the collimated beam after the first reflection from the freeform mirror, (b) RMS spot radius on the detector as a function of wavelength across the spectrum range, (c) spot diagrams on the detector for the nine sampling wavelengths.

Tables (5)

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Table 1 OAP base and OAP segment parameters.

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Table 2 Updated OAP segment parameters.

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Table 3 Constructed and optimized parameters for the freeform mirror.

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Table 4 Constructed and optimized parameters for the freeform spectrometer.

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Table 5 Inclined incidence angles on the detector.

Equations (27)

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d(sini+sinθ)=mλ
s= D 2 2R = D 2 4f
f ^ =f+s
tanβ= D fs = 2RD R 2 D 2
D= R(1cosβ) sinβ
θ ctrl θ min + β min = β ctrl
θ ctrl θ max + β max = β ctrl
D B-max D B-min =L
f ^ S-ctrl + f ^ B-ctrl = f ^ S-min + f ^ B-min = f ^ S-max + f ^ B-max
β C = β 1 = θ 1 i 2
z ctrl = f ^ B-ctrl + s ctrl + x ctrl 2 + ( y ctrl + D S-ctrl ) 2 2 R ctrl
y 0 = f ^ B-ctrl sin β ctrl 2
z 0 = f ^ B-ctrl cos β ctrl 2
x ctrl-F = x ctrl
y ctrl-F = y ctrl cos β ctrl 2 z ctrl sin β ctrl 2 + y 0
z ctrl-F = y ctrl sin β ctrl 2 + z ctrl cos β ctrl 2 + z 0
x min-F = x min
y min-F = y min cos( θ min θ ctrl + β ctrl 2 ) z min sin( θ min θ ctrl + β ctrl 2 )+ y 0
z min-F = y min sin( θ min θ ctrl + β ctrl 2 )+ z min cos( θ min θ ctrl + β ctrl 2 )+ z 0
Φ x = Φ cosi
Φy= Φcos(θi) cosi
ε 4 = S 45 δ 45
β 4 = β 4 +2 ε 4
Δ 4 = d 45 tan ε 4
L GF4 = L GF4 Δ 4 = f ^ B4 Δ 4
L FD4 = f ^ S4 = L FD4 cos2 ε 4 Δ 4 = f ^ S4 cos2 ε 4 Δ 4
z F = R S + R S 2 ( x F 2 + y F 2 ) + p 20 x 2 + p 02 y 2 + p 21 x 2 y+ p 03 y 3 + p 40 x 4 + p 22 x 2 y 2 + p 04 y 4 + p 41 x 4 y+ p 23 x 2 y 3 + p 05 y 5

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