## Abstract

The convex reflective diffraction grating is an essential optical component that lends itself to various applications. In this work, we first outline the design principles of convex diffraction gratings from wavefront quality and efficiency perspectives. We then describe a unique fabrication method that allows for the machining of convex diffraction gratings with variable groove structure, which is extendable to rotationally non-symmetric convex diffraction grating substrates. Finally, we demonstrate two quantitative wavefront measurement methods and respective experimental validation.

© 2017 Optical Society of America

## 1. Introduction

The diffraction grating is a fundamental optical component utilized in a wide variety of devices and serves a wide range of scientific fields [1]. Two common forms are gratings fabricated on either flat or spherical concave substrates. Gratings fabricated on spherical convex substrates are less common as historically they have been difficult to test [2]. However, convex reflective gratings are necessary for example in the Offner and Offner-Chrisp optical design geometries that are advantageous for imaging spectrometers [3–5]. These geometries have been utilized in various recent spectrometer designs [6–8]. Most recently, Reimers et al. demonstrated an imaging spectrometer in the Offner-Chrisp geometry where freeform optical surfaces enabled a substantial 5x reduction in volume when compared to all-spherical and aspheric designs [9]. Convex gratings may also be used as Masters in the process of replicating concave gratings for mass production [10]. Thus, while convex gratings are not as prevalent as optical components, they are key enablers in some instrument geometries or manufacturing processes.

To enable the applications of convex diffraction gratings, hereinafter referred to as “convex gratings”, it is important to concurrently investigate the design of these gratings in terms of both wavefront contribution and diffraction efficiency as well as fabrication and testing methods. In this paper, we focus on two types of convex grating designs. The first has equal groove spacing along the arc of the substrate, hereinafter referred to as the “equal-along-arc” grating design. The second has equal groove spacing along the projected plane, hereinafter referred to as the “equal-along-projection” design. These two seemingly similar design forms give rise to drastically different wavefront behaviors, as will be reported in this paper.

In fabrication, existing manufacturing methods for gratings include holographic fabrication [11], electron beam lithography [12], as well as mechanical ruling engines and ultra-precision diamond machine tools [13–16]. Ruling and ultra-precision diamond machining have further been adapted to generate different grating geometries including gratings on curved surfaces [13,14]. In this work, we expand upon traditional processes with an off-axis turning/profiling configuration that we shall describe. Combined with coordinated-axis machining, this technique can be extended to cutting freeform gratings with variable profiles of increased complexity and potentially rotationally non-symmetric substrate designs.

In metrology, we focus on reporting the diffracted wavefront measurements from convex grating components. While a qualitative wavefront test method was reported in [17], to our knowledge no quantitative test method for diffracted wavefronts from convex gratings has been reported or experimentally demonstrated. In this paper, we demonstrate two independent quantitative wavefront metrology methods with their respective experimental validations to certify that the grating was manufactured to geometrical specifications. One method is a natural extension from conventional test methods for flat and concave gratings, while the other is a custom-designed interferometric null test. Ultimately, coatings may be deposited on manufactured gratings to optimize the diffraction efficiency in the specified spectral band. Note that the component testing methods demonstrated here are general in nature and the coated grating can undergo the same procedures to verify wavefront fidelity before assembly.

In this paper, the phase functions of an equal-along-arc and an equal-along-projection convex grating design are explicitly derived in Section 2. For the subsequent sections, simulations and experimental validations pertain to blazed convex gratings that may be specified for Offner-Chrisp geometries. Section 3 discusses the effect of the two grating designs in terms of the optical performance of a spectrometer. Section 4 discusses the design of the gratings from a theoretical diffraction efficiency point of view. Fabrication methods for both gratings are described in Section 5. Two independent quantitative test methods for the diffracted wavefronts of the gratings are described and experimentally validated in Sections 6 and 7.

## 2. Derivation of grating phase functions

The convex gratings discussed in this paper were fabricated with (1) groove spacing that is equal-along-arc or (2) groove spacing that is equal-along-projection. This difference is illustrated in Fig. 1, where the Y-axis is defined to be along the projected plane. Equal-along-arc spacing maps to a nonlinear spacing in the projection plane while equal-along-the-plane spacing maps to a nonlinear spacing along the arc.

The wavefront contributions from a curved grating can be obtained by deriving its phase function in the projected plane [18]. We take the equal-along-arc grating as an example to derive its wavefront contributions; the derivations employed here are general across diffractive optics with the assumption that the groove spacing is much smaller than the grating clear aperture. Under this assumption, the spacing between adjacent grooves denoted as${d}_{groove}$can be approximated as the hypotenuse of a triangle when performing the projection, as shown in the cutout in Fig. 1(a). The local angle of projection *θ* is given by the inverse tangent of the slope of the substrate profile. In this case, the substrate is a convex spherical surface with sag *z* given by Eq. (1), where *R* is the substrate radius of curvature. Equations (2)–(3) mathematically define the calculation of ${d}_{proj}$that denotes the groove spacing on the projected plane.

In general, for the *m ^{th}* order of diffraction, each subsequent groove across the clear aperture adds

*m*waves of phase to the wavefront [18]. The slope of this phase function gives the change in phase as a function of position

*y*as expressed in Eq. (4). The phase function is then obtained by integrating Eq. (4) to obtain Eq. (5). A Taylor expansion around the origin as shown in Eq. (6) reveals various orders in

*y*of the phase function. This expansion allows input of this surface into commercial optical design software, as will be discussed in Section 3, and highlights the primary phase contributors.

We see that the dominant first term of the Taylor series expansion is linear and is in fact the exact phase function for the equal-along-projection grating if we substitute${d}_{groove}$ with${d}_{proj}$. The second dominant term is cubic in *y* and is wavelength dependent, giving rise to a wavelength-dependent coma, “coma-chromatism”, as will be shown in Section 3.

## 3. Optical performance modeling and comparison

To investigate the optical aberrations that may arise from different grating phase functions, let us consider an Offner-Chrisp spectrometer design where the aperture stop of the system is located at the grating. This design was developed for a 500 nm – 1100 nm spectral band over a 10 mm full field of view in slit length. The specifications of the convex grating are summarized in Table 1. The groove frequency is defined in the respective equi-spacing zone for the two grating types. As shown in Eq. (6), the first term in the expansion for the equal-along-arc grating is equivalent to the phase function for the equal-along-projection grating. This term characterizes the first order ray trace for the two cases. The higher order terms for the equal-along-arc grating result in optical path differences in the ray trace that produce aberrations in the wavefront. The optical performance simulations were performed in optical design software using the phase polynomial description for a holographic optical element surface. These polynomials are defined to be along the projection plane with dependencies on *x* and *y* but no dependencies on *z*, in a right-handed coordinate system where *z* is along the optical axis [19]. Thus, the relevant coefficients can be calculated directly from the expansion of Eq. (6) and the first three dominant terms are tabulated in Table 2.

With an equal-along-projection grating, the spectrometer achieves $<\lambda /15$ RMS wavefront error (RMSWE) across the entire spectral band and slit length. The spectral full field displays first defined in Reimers et al. (2017) and shown in Fig. 2 plot individual performance metrics against the spectral band and the slit length [9]. The metrics chosen here are the system RMSWE, the Fringe Zernike astigmatism Z5/6, and the Fringe Zernike coma Z7/8 as they are the dominant aberrations. The top row shows the performance with an equal-along-projection grating while the bottom row shows that with an equal-along-arc grating. All six plots are on the same scale and each point on the plots is scaled to the local wavelength reported along the X-axis and is reported in waves. The scale for Fig. 2 was specifically chosen so that the symbols in Figs. 2(a)-(c) are clearly visible, at the expense of having oversized symbols in Fig. 2(d) and Fig. 2(f). After increasing the scale of Fig. 2(f) by 40x and displaying it in Fig. 3, it is shown that the coma scales linearly with wavelength, rendering field constant coma when normalized by the local wavelength in order to display the aberrations in wave units. This “coma-chromatism” behavior was found to be the dominant aberration and cannot be mitigated across the full spectral range in this spectrometer geometry.

## 4. Diffraction efficiency modeling and comparison

While groove spacing design contributes to the diffracted wavefront, the groove shape contributes to the diffraction efficiency. From this standpoint, the Offner geometry is advantageous as the concentricity of the surfaces results in a quasi-constant incidence angle on the grating, which allows for a single optimized groove depth based on that incidence angle. Simulations on efficiency were performed using rigorous Fourier modal methods in VirtualLab. Randomly polarized light was assumed and approximated as $(TE+TM)/2$ in all calculations. The model used silver for efficiency calculations as the grating manufactured here, as specified in Table 1, requires silver coating to account for the overall spectral band of 500–1100 nm. Reflectivity and absorption were accounted for in the simulations. Ray trace analysis of the spectrometer system showed that the angles of incidence relative to the local surface normal varied from 37.28° to 38.04° across the aperture. Simulations showed that the + 1 order diffraction efficiency varied $<1\%$ across this incidence angle range. As a result, in subsequent simulations only the angle of incidence at the center of the piece (37.92°) was considered.

For the equal-along-arc grating, the grating period was fixed along the arc of the substrate with the grating sidewall parallel to the local surface normal. Calculations were performed with grating depths of 325 nm, 374 nm, and 425 nm to examine how the + 1 and + 2 orders diffraction efficiency varied across the full spectral band as illustrated in Fig. 4(a). In this design, the 0th order does not make it to the image plane. However, the 2nd order of the 500–550 nm signal overlaps with the 1st order of the 1000–1100 nm signal. To minimize this spectral crosstalk, the 2nd order efficiencies of the shorter wavelengths need to be minimized, and thus the 325 nm groove depth was chosen. The efficiencies for this groove depth over orders −2 to + 10 are shown in Fig. 4(b) to quantify the amount of light in each order. Results show that the light is confined to the zero and + 1 order.

Since the grating geometry for the equal-along-projection grating case varies from the vertex to the edge of the arc, efficiency calculations were also performed for this case with the same basic assumptions as in the equal-along-arc case. For this scenario, the grating period varied along the arc from 3.333 μm at the vertex to 3.371 μm at the edge, which translates to variations in blaze angle from 5.568° to 5.507°. Similarly, we also considered the effects of variations in grating sidewall angles relative to the local surface normal that may occur from the finite tool size and orientation in the single point diamond machining process discussed in Section 5. We examined the two spacing extrema with grating sidewalls parallel to and 10° off the local surface normal and 325 nm groove depth for all cases. Let us note that the case with a 3.333 µm grating period and grating sidewalls parallel to the local surface normal is equivalent to the equal-along-arc case. It was found that these variations yield negligible differences in diffraction efficiencies. In summary, assuming identical groove depths, both the equal-along-arc and equal-along-projection gratings yield equivalent efficiencies shown in Fig. 4(b).

## 5. Fabrication methods

In this work, gratings were produced on a Moore Nanotechnology 350 FG machine with up to 5 axes of positioning capability (3 linear and 2 angular) using single crystal diamond tools. Compared to other fabrication methods, ultra-precision machining and ruling can be time consuming. However, it has distinct advantages in that it is deterministic and generally reproduces directly the programmed geometric tool paths, which makes it suitable for prototyping and low-batch production of gratings. For convex gratings, the grating profile may be required to vary along the curved substrate (as in the case of the equal-along-projection grating) while achieving high fidelity free of adverse material effects such as burr formation and tool wear. Both cutting kinematics (pertaining to geometry) and cutting mechanics (pertaining to fidelity) are investigated and discussed.

#### 5.1 Substrate material and grating fidelity

As shown in Table 3, the substrate material for the convex grating was selected to be naval brass C46400. This selection is based on the following ruling behavior test. Three materials were selected that can be cut with a single crystal diamond tool by ruling or diamond machining, namely (1) super-clean aluminum 6061, (2) naval brass C46400, and (3) electroless nickel with approximately 12% phosphorous content. Their relevant mechanical properties are summarized in Table 3.

Aluminum has the advantage that tool wear in diamond machining and ruling is nearly non-existent. However, its mechanical properties lead to a less robust grating as well as undesirable effects such as the formation of large burrs at the tool-material interaction points. Naval brass is more likely to cause tool wear, but has more desirable mechanical properties and is known to produce good cutting mechanics. With this material, burr formation is reduced and thus grating profiles are expected to be of greater fidelity compared to the softer aluminum. Electroless nickel has a much higher tensile strength and hardness and is therefore more mechanically robust than both aluminum and brass. However, it causes diamond tool wear when machined, which can be prohibitive depending on the size of the area to be cut. Representative results from the three materials are shown in Fig. 5. Tools having 60° included angle and a dead-sharp tip (~100 nm nose radius) were used for initial testing. The tools were plunged in the face of the materials at 2000 revolutions per minute (RPM) at a rate of 100 nm/revolution profiling feed rate. Figure 5(a) shows poor quality with no discernable grating in aluminum. The ductility of the material leads to burr formation and deformation of the grating. Figure 5(b) shows some curvature and burr formation in the brass, while a grating is clearly discernable. In electroless nickel [Fig. 5(c)] clean grating lines are observed. For prototyping, naval brass C46400 was chosen as it produces a periodic grating structure without running the risk of catastrophic tool wear. Furthermore, as will be shown in Section 5.3, the continued process and parameter development lead to a significant increase in final grating quality when compared to that of Fig. 5(b).

#### 5.2 Grating geometry generation

Two options were considered for producing the gratings. The first option uses a form-tool having the grating blaze angle incorporated into the tool geometry but requiring continuous rotation along the surface. The second option rules or profiles the groove geometry with multiple passes of the tip of a dead-sharp tool with an included angle less than the excluded angle of the grating profile. The second option was chosen as it has greater flexibility in creating gratings with variable angle pitches and heights, allowing for the generation of both the equal-along-arc and the equal-along-projection grating.

Ruling using the machine axes to produce the needed geometry was considered. It is a possible technique but the total machining time *T* would have been very high. $T$ is given as

^{5}passes. With tool repositioning, each pass required a total of approximately 6 seconds, resulting in a total process time of 166 hours (6.9 days). This is not cost-effective and allows for the possibility of machine errors due to thermal fluctuations. For these reasons, a faster, more cost-effective and controlled technique needed to be developed.

The method developed here builds upon existing diamond turning techniques for spherical and aspherical optical components. Typically, when diamond turning a sphere or asphere the aperture would be positioned concentric to the rotational axis and the tool would trace a path from the edge of the lens to the center of rotation, as shown in Fig. 6(a). If directly adapted to grating fabrication, this configuration produces grooves in a spiral pattern instead of the straight lines required by the grating designs. By moving the workpiece off-axis with the optical axis perpendicular to the axis of rotation as shown in Fig. 6(b), the tool mark pattern becomes linear with the pitch dependent on the feed per spindle revolution. By replacing the typical round-nosed single crystal diamond tool with a dead-sharp single crystal diamond groove tool as shown in Fig. 6(c), the grating lines can then be contour machined. Figure 7 may be considered together with Fig. 6(c) to visualize the orientation of the grating relative to the sphere and the orientation of the workpiece relative to the machine. Given ample reach and access, the configuration in Fig. 6(c) allows for the generation of grating structures with variable angle pitch and height. By allowing the radius of the programmed tool path to differ from the distance of the vertex to the axis of rotation, this configuration also enables the fabrication of gratings without rotational symmetry.

The custom designed fixturing for this configuration is detailed in Fig. 7 with a three-dimensional view as well as views from the top and the front. Prior to cutting the grating, to minimize astigmatism, radii matching was performed between the radius of the programmed toolpath and the radius set by the distance from the axis of rotation to the tip of the tool by adjustment of controller offsets. This matching was quantified by cutting test sphere artifacts with a round-nosed single crystal diamond tool and measuring the sphere in an interferometer (Zygo Verifire) for any resulting astigmatism. Based on these results, the two radii were matched to within 1 μm, meeting the tolerances of this application.

#### 5.3 Cylindrical cutting test and final spherical grating

The final grating on the spherical component was placed on the front side of the workpiece as shown in Fig. 7. However, prior to cutting the grating on the surface of the sphere, the profiling method was first tested on a simpler configuration to ensure grating fidelity. In this configuration, the grating was cut on the outer radial surface of a cylinder at a constant surface speed and measured. Once the grating fidelity has been verified, the final spherical grating was then cut. The blazed grating tested on the outer radial surface of the cylinder has similar parameters to the spherical convex grating, with a pitch of 300 lines/mm and groove depth of 400 nm. This 25 mm diameter cylindrical workpiece was turned with a dead-sharp diamond tool with 60° included angle, at a spindle speed of 800 RPM with a profiling feed rate of 100 nm/revolution and a 5 μm depth of cut. The grooves were profiled in this case rather than plunge cuts as in the initial tests presented in Fig. 6. A representative measurement was generated by a scanning white light interferometer (Zygo NewView), as shown in Fig. 8, confirming the groove spacing. Note that this measurement method underestimates the groove depth. In this turning configuration the qualitative character of groove lines appears significantly improved when compared with that of the ruling configuration in Fig. 5. This is likely caused by the continuous profiling of the grooves rather than the individual plunge cutting of each groove sequentially.

Based on these results, similar parameters were used to cut the final spherical grating. The parameters for cutting using the configuration shown in Fig. 7 were 800 RPM with a profiling feed rate of 100 nm/revolution and a 3 μm depth of cut. To allow for better chip evacuation, the included angle of the dead-sharp tool used was decreased to 30°. The spherical grating was examined in SEM with representative results shown in Fig. 9. In addition to the grating structure at 300 lines/mm, a linear structure can be seen on the surfaces of the grating facets that results from the profiling feed of the tool at 100 nm/revolution. Surface roughness was measured on the top facets of the grating and was approximately 10-15 nm RMS and 30-45 nm Rz. This structure produced on the grating facets by the profiling have a subwavelength spacing and a depth that is approximately an order of magnitude less than the grating groove depth and is therefore assumed to have minimal effect on overall grating performance.

## 6. Direct wavefront metrology method and results

#### 6.1 Method

Direct measurement of the diffracted wavefront at various orders was performed with a phase shifting Fizeau interferometer (Zygo Verifire). The methodology outlined can be extended to other interferometer types. Positioning the grating so that it is concentric with a sufficiently fast transmission sphere allows testing of the 0th order. Naturally, the beam is expected to fill the part. To obtain autoreflection at other orders, the grating surface must be tilted about its vertex at a specific angle that is unique to each order. This angle, known as the Littrow angle *θ _{L}* is derived from the grating equation by setting the incidence angle equal to the outgoing angle and is given by Eq. (8), where

*λ*is the test wavelength,

*d*is the grating groove spacing, and

*m*is the diffraction order that can be either positive or negative. When a spherical grating is tilted at a Littrow angle, the diffracted wavefront consists of contributions from both the grating itself and the mirror substrate, as shown in Eq. (9). We refer to the former as the diffractive term

*D*and the latter as the mirror term

*M*. Note that each term contains contributions from any errors present on the part as well as from alignment.

In the case of a spherical mirror substrate, the wavefront contribution ${M}_{m}$ is in general symmetric with tilt angle. However, as the tilt introduces additional terms into the wavefront, for example most noticeably astigmatism from using a spherical surface at a non-zero angle of incidence, the mirror term will generally be different across orders and require additional computation to quantify the exact contribution terms and amounts. This is expressed in Eq. (10), where *j* and *k* are different diffraction orders and are defined to be non-negative. From these expressions, one can see that if data is available for an order on both the positive and negative side, the mirror term and the diffractive term can be isolated via Eqs. (11) and (12).

For small tilt angles or equivalently low orders of diffraction, the main contribution from the spherical mirror substrate is astigmatism and can be estimated via the Coddington trace equations [20], given as

#### 6.2 Results

As both the equal-along-projection and equal-along-arc gratings were designed to have maximum diffraction efficiency in the + 1 order, quantitative measurement results were only obtainable at the + 1 order under our laboratory conditions. Optical design software was used to simulate the testing conditions and the expected interferograms with the coefficients in Table 2. The simulation results for both gratings are shown in Fig. 10, after removing piston, tilt, and power. Using Eq. (17), the Coddington trace equations give 9.1 waves of PV astigmatism at the 632.8 nm He-Ne wavelength. This is in excellent agreement with the model interferogram and can be clearly seen in Fig. 10(b) for the equal-along-projection grating. For the equal-along-arc grating, one can see the comatic cubic term balancing with the astigmatism in Fig. 10(c).

This test setup was then implemented in a Fizeau phase shifting interferometer to obtain the results shown in Fig. 11, after removing piston, tilt, and power. The obtained interferograms are in excellent agreement with predictions in simulations. After removing the dominant astigmatism term, the underlying form is then representative of the diffracted term in the wavefront. For the equal-along-projection grating, the wavefront shows 14 nm (~λ/50) RMS flatness, which is within typical spherical grating specifications. The equal-along-arc grating exhibit a cubic behavior as expected from the Eq. (6) expansion. Note the line artifacts present in the interferograms that are indicative of the groove ruling.

## 7. Nulling wavefront metrology method and results

A custom Offner type null test was designed to independently test the equal-along-projection grating under similar conditions as in use in the spectrometer of Section 3. This null test utilizes a concave spherical mirror placed concentric with the convex grating substrate with the grating acting as the aperture stop. An object numerical aperture of F/3.8 was chosen to match that of the spectrometer. A schematic diagram of the setup layout is shown in Fig. 12(a) and specifications of the setup are summarized in Table 4. The selected field bias of the Offner geometry operates at the crossing of the tangential and sagittal field curves allowing for minimal astigmatism [9]. Testing at the + 1 order disrupts the symmetry of the design about the stop and thus there exists residual coma as well as trefoil balancing the coma. Simulations show that the system is nevertheless diffraction limited with a RMSWE of 0.04 waves and the expected null interferogram is shown in Fig. 12(b).

To obtain the null interferogram in experiment, one can couple this design with an interferometric configuration. A qualitative interferogram may be obtained for example by placing a point diffraction interferometer plate at the outgoing focus of the system. To obtain quantitative measurements, we operated in double pass with an autoreflector and a phase shifting Fizeau interferometer (Zygo Verifire). The concave spherical mirror was custom made (Kreischer Optics, Ltd.). In house metrology on this mirror shows a RMS figure error of better than λ/20 at the 632.8 nm test wavelength.

To minimize alignment-induced errors in the final wavefront, a detailed alignment procedure was designed and alignment aberrations were analyzed in simulation through perturbations in design. The following steps minimized the setup misalignment up to the concave mirror, leaving the main misalignment degrees of freedom (DOF) to be the 6 DOFs on the convex grating (*x*, *y*, *z*, *α* tilt, *β* tilt, *γ* tilt). Before coupling to the commercial Fizeau instrument, the setup was first aligned on a transportable breadboard with a standalone He-Ne laser source. This breadboard set-up is shown on the right-hand side of Fig. 13. The laser was aligned to the optical table, spatially filtered with a 0.65 NA microscope objective and a 10 μm pinhole, and collimated with collimation checked with a shear plate interferometer (ThorLabs, Inc.) (not shown here). A 0.13 NA microscope objective was used to form the incoming beam in the null test setup and the collimated He-Ne beam was made to overfill the objective. The concave mirror was first positioned concentric with the focus of the 0.13 NA objective. By checking the collimation of the beam autoreflected from the concave surface and back through the objective, the tip/tilt as well as the X position of the concave mirror were fixed. The mirror was then translated 23.7 mm along Y on a linear micrometer stage as required by the field bias of the design. As squareness errors may exist, the object conjugate position (i.e. the distance from the object plane to the concave mirror) as well as field bias were also kept as DOFs and analyzed in simulation, as shown next.

To guide the fine alignment, the analysis of wavefront aberrations induced by the convex grating’s six DOFs as well as the two DOFs of the object conjugate and the field bias were performed in optical design software. As the underlying mirror substrate for the grating and the concave mirror formed a coaxial, rotationally symmetric system, an *α* tilt on the grating about its vertex produced similar aberrations as a *y* decenter, and the same occurred for the *β* tilt and the *x* decenter. A *γ* tilt on the grating was found to be negligible at small angles and only shows significant aberration contribution for angles larger than ~30°. Therefore, only three of the convex grating’s six DOFs needed to be considered. For this setup, it was found that the dominant decenter-induced aberrations were astigmatism (Fringe Zernike Z5/6), coma (Fringe Zernike Z7/8), and trefoil (Fringe Zernike Z10/11). The coefficients of these dominant aberrations with respect to perturbations in the DOFs of relevance are plotted in Fig. 14. It was found that object conjugate error only significantly affected Z5, Z8, and Z11, which is expected as this error is confined within the *y-z* plane. Moreover, as field bias error maintains system bilateral symmetry, the only significant error this misalignment introduces is Z5, as shown in Fig. 14(h). A Zernike normalization radius of 60 mm was used for these analyses. These relationships were then utilized to guide the fine alignment of the setup to the null location.

The setup on the breadboard was then transported and coupled with a commercial Fizeau phase shifting interferometer, shown also in Fig. 13. A fold mirror was placed at the focus of the + 1 order beam to send the light to an autoreflecting concave spherical lens. The autoreflecting lens was made in house with its front surface better than λ/10 in RMS figure error. Wavefront measurements were taken and fit to Fringe Zernike coefficients. Using knowledge of the coefficient dependencies shown in Fig. 14, the alignment was walked toward the null. At the final null location, 100 phase measurements were made consecutively with the interferometer with no phase averaging. To mitigate for vibration noise and air turbulence noise, these measurements were then grouped quasi-randomly into 5 sets and averaged within each set. Over these 5 data sets, the RMS of the obtained null wavefront was calculated to be 14 ± 0.2 nm. One instance of these measurements is shown in Fig. 15(b) and exhibits excellent agreement with the expected null wavefront shown in Fig. 15(a). The measured wavefront was consistent with design expectations. With this null obtained, the grating was verified to meet design specifications.

## 8. Conclusion and future work

In this paper, the design, fabrication, and wavefront testing of convex blazed reflective diffraction gratings were addressed. In design, the grating phase function derivation process for wavefront contribution was explicitly shown and can be extended to gratings with other spacing profiles. It was found that an equal spacing along the arc of the spherical substrate introduces a dominant cubic term that manifests itself as wavelength dependent coma in the diffracted wavefront. For diffraction efficiencies, it was found in this case that they were mainly determined by the groove depth and substrate material, and that the spacing profile and variations in groove sidewall angle have negligible effects. As such, for gratings similar to the ones presented here, the design in wavefront and in efficiency can largely be treated as orthogonal processes.

The gratings were manufactured with a unique diamond turning configuration. This configuration expands upon traditional processes, enables the fabrication of gratings with variable structures, and relieves the limitation of rotational symmetry, allowing for a new degree of freedom in optical grating design.

Two quantitative wavefront tests were designed and carried out to independently verify that the grating was manufactured to specifications. While the direct method is simpler in hardware implementation, a key point to note is that the wavefront contribution from the substrate is in general larger for substrates of higher power and will scale with the tilt needed to obtain autoreflection. The wavefront contribution from the grating will also scale with the order used; this may not be an issue as most gratings are used in the first order, but should be kept in mind in case a design calls for the use of higher orders. This method is also susceptible to an issue faced by convex spherical optical components in general, namely that the distance from the transmission sphere to the part needs to be sufficient, which is harder to achieve the faster the component is. For the null test method, the increased complexity in hardware means that it is necessary to have a quantitative understanding regarding misalignment DOFs and their contributions. Provided good alignment, the null test will always be operating with low fringe density.

For the Offner-Chrisp spectrometer geometry considered in this paper, a convex grating with linear spacing in projection was shown to be imperative. However, the authors believe that the equal-along-arc grating as well as other gratings with varying spacing in projection may hold potential for other optical designs to be investigated in the future.

## Funding

National Science Foundation (NSF) I/UCRC Center for Freeform Optics (IIP-1338877 and IIP-1338898).

## Acknowledgments

We thank Synopsys, Inc. for the education license for CODE V® used in the simulations. Gratitude is also expressed to Jinxin Huang for stimulating discussions, Michael Pomerantz for making the in-house autoreflective lens as well as for his laboratory support together with Brandon Dube.

## References and links

**1. **E. G. Loewen and E. Popov, *Diffraction Gratings and Applications* (CRC, 1997).

**2. **E. G. Loewen, “Diffraction gratings for spectroscopy,” J. Phys. Educ. **3**(12), 953 (1970).

**3. **A. Offner, “New concepts in projection mask aligners,” Opt. Eng. **14**(2), 142130 (1975). [CrossRef]

**4. **M. P. Chrisp, “Convex diffraction grating imaging spectrometer,” U.S. patent US5880834 A (March 9, 1999).

**5. **D. R. Lobb, “Theory of concentric designs for grating spectrometers,” Appl. Opt. **33**(13), 2648–2658 (1994). [CrossRef] [PubMed]

**6. **P. Mouroulis, R. O. Green, and T. G. Chrien, “Design of pushbroom imaging spectrometers for optimum recovery of spectroscopic and spatial information,” Appl. Opt. **39**(13), 2210–2220 (2000). [CrossRef] [PubMed]

**7. **R. B. Lockwood, T. W. Cooley, R. M. Nadile, J. A. Gardner, P. S. Armstrong, A. M. Payton, T. M. Davis, and S. D. Straight, “Advanced responsive tactically effective military imaging spectrometer (ARTEMIS): system overview and objectives,” in *Proc. SPIE 6661*, S. S. Shen and P. E. Lewis, eds. (2007), p. 666102.

**8. **X. Prieto-Blanco, C. Montero-Orille, H. González-Nuñez, M. D. Mouriz, E. L. Lago, and R. de la Fuente, “The Offner imaging spectrometer in quadrature,” Opt. Express **18**(12), 12756–12769 (2010). [CrossRef] [PubMed]

**9. **J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling increased compactness,” Light Sci. Appl. **6**, e17026 (2017).

**10. **Q. Zhou and L.-F. Li, “[Design method of convex master gratings for replicating flat-field concave gratings],” Guangpuxue Yu Guangpu Fenxi **29**(8), 2281–2285 (2009). [PubMed]

**11. **E. G. Loewen, E. K. Popov, J. Hoose, and L. V. Tsonev, “Experimental study of local and integral efficiency behavior of a concave holographic diffraction grating,” J. Opt. Soc. Am. A **7**(9), 1764–1769 (1990). [CrossRef]

**12. **P. Mouroulis, D. W. Wilson, P. D. Maker, and R. E. Muller, “Convex grating types for concentric imaging spectrometers,” Appl. Opt. **37**(31), 7200–7208 (1998). [CrossRef] [PubMed]

**13. **M. Neviere and W. R. Hunter, “Analysis of the changes in efficiency across the ruled area of a concave diffraction grating,” Appl. Opt. **19**(12), 2059–2065 (1980). [CrossRef] [PubMed]

**14. **M. C. Hettrick, “Varied line-space gratings: past, present and future,” in 29th Annual Technical Symposium (International Society for Optics and Photonics, 1986), pp. 96–108.

**15. **Y. Takeuchi, Y. Yoneyama, T. Ishida, and T. Kawai, “6-Axis control ultraprecision microgrooving on sculptured surfaces with non-rotational cutting tool,” CIRP Ann. - Manuf. Technol. **58**(1), 53–56 (2009). [CrossRef]

**16. **M. A. Davies, B. S. Dutterer, T. J. Suleski, J. F. Silny, and E. D. Kim, “Diamond machining of diffraction gratings for imaging spectrometers,” Precis. Eng. **36**(2), 334–338 (2012). [CrossRef]

**17. **P. Mouroulis and M. M. McKerns, “Pushbroom imaging spectrometer with high spectroscopic data fidelity: experimental demonstration,” Opt. Eng. **39**(3), 808–816 (2000). [CrossRef]

**18. **W. T. Welford, “Aberration Theory of Gratings and Grating Mountings,” Prog. Opt. **4**, 241–280 (1965). [CrossRef]

**19. **Page 317–321, CODE V® Version 10.7 Reference Manual (Synopsys, Inc.) (2016).

**20. **R. Kingslake, “Who discovered Coddington’s equations?” Opt. Photonics News **5**(8), 20–23 (1994). [CrossRef]