Freeform optical surfaces change the optical design landscape by enabling design forms that are truly off-axis. Fabrication technology has evolved to readily produce high quality freeform surfaces. As an example, diamond turning has introduced servos into the axes geometry that allow freeform surfaces to be fabricated to optical quality in the long-wave infrared (LWIR, 8–12 μm). The challenge remains to create a method for measuring the surface quality of the as-fabricated freeform surfaces from these state-of-the-art fabrication processes. One potential method for measuring these surfaces is through the use of a computer-generated hologram (CGH) that acts as a nulling component in an interferometric arrangement [1]. The quality of the measurement obtained with the CGH depends highly on the fabrication of the CGH and the arrangement in which it is placed in the interferometer [2]. Moreover, each CGH is unique to one specific surface, which can be cost prohibitive for multiple surfaces. Another potential method is to arrange optical elements (i.e., lenses or mirrors) in a null configuration. These methods exist for measuring off-axis sections of conics and aspherics [3] but not for freeform surfaces.

In this Letter, we present a more flexible solution compared to a custom CGH interferometric null that utilizes a series of adaptive subsystems that each null a particular aberration type present in the departure of the freeform surface. The concept is applied to measure the surface figure of a freeform mirror that comes from a recent design targeted at an unobscured, F/1.9, 10° full field of view LWIR imager that employs three freeform, $\phi$-polynomial (Zernike) mirrors [4].

The three Zernike polynomial surfaces have been diamond turned by II-VI Inc. in a copper substrate with a gold protective coating. In this Letter, an interferometric null for the secondary mirror surface is presented and the results of the as-fabricated surface figure are reported.

The sag of the secondary mirror surface of the three mirror design presented in [4] is shown in Figs. 1(a)–1(c) where it is evaluated with different Zernike components removed from the base sag. In Fig. 1(a), the sag is evaluated with the piston, power, and tilt Zernike contributions removed so that the dominant astigmatic contribution present in the surface can be seen. When the astigmatism is removed from the surface sag, Fig. 1(b), the remaining sag components are observed. An asymmetry is seen in the sag that results from the comatic departure present in the surface. If the spherical aberration, the next most dominant contribution, is removed from the surface sag, Fig. 1(c), the comatic departure of the surface is more readily visible.

 

Fig. 1. (a) Sag of the secondary mirror surface with the piston, power, and tilt Zernike components removed revealing the astigmatic contribution of the surface, (b) sag with the astigmatic component additionally removed, and (c) sag with the spherical component additionally removed. With the piston, power, tilt, astigmatism, and spherical components removed, the asymmetry induced from the coma being added into the surface can be seen.

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As seen in Figs. 1(a)–1(c), the mirror to be measured has some amount of spherical aberration, astigmatism, coma, and some higher order aberration terms placed into the surface departure. As a result, a conventional interferometer designed for measuring spherical surfaces has insufficient dynamic range to measure the surface because the departure between the spherical reference wavefront and the test wavefront reflected off the surface of the mirror is too great. However, if the test wavefront is manipulated to null or partially null each aberration type present in the mirror, the departure between the test and measurement wavefronts can be minimized and brought within the dynamic range of the interferometer. For this particular design, a planar wavefront, translating to a flat reference surface at the output of the interferometer, is selected because the alignment of the null to the interferometer is less critical since it can lie anywhere within the aperture of the interferometer/transmission flat. From the output of the interferometer, the aberration terms can be nulled in multiple configurations. For this design, the spherical aberration component is first nulled by the use of a refractive Offner null [5], consisting of two refractive elements, one of which is a null lens that introduces the opposite amount of spherical aberration present in the mirror under test and the other is a field lens that conjugates the null lens to the mirror under test. Next, the astigmatic component is removed by operating the mirror off-axis, or tilting the mirror, where the tilt angle is determined by the Coddington equations and the principal radii of curvature of the mirror surface [6]. Last, the residual comatic and higher order terms are nulled by adding their opposite departure on a deformable mirror (DM) that also acts as a reimaging retroreflector to send the light back toward the measurement interferometer [7]. In order to couple the wavefront reflected off the test mirror to the quasi-flat DM, the wavefront must first be collimated with the use of a collimating lens. Together, these three components form a configuration that allows the optical surface to be measured with a conventional interferometer.

The sizing of the various optical components constrains the layout of the interferometric null. The region of interest on the test mirror is 70 mm in diameter and the Zygo interferometer has a 101.6 mm clear aperture. However, only a 45 mm clear aperture is used for the interferometric null to keep the null lenses less than 50.8 mm where commercially available mounting components are readily available. Moreover, the DM to be used is the mirao 52-e, a 15 mm clear aperture, fifty two actuator reflective membrane mirror, from Imagine Eyes. This type of mirror is capable of achieving large deformations of the surface and is well suited for creating the comatic null for the interferometric null system. From these three known parameters, a first-order null solution is created that yields a starting point for further optimization. The end goal for optimization is to produce a double-pass, thick lens solution that provides a null or quasi-null wavefront exiting the interferometric system. Using commercially available lens design software, in this case CODE V, user-defined constraints are written for nulling the fringe Zernike spherical aberration, astigmatism, coma, and any higher order aberration terms while maintaining the imaging conjugates between the null components.

The final, optimized system is shown in the $X–Z$ plane in Fig. 2. As can be seen from the figure, an aspect ratio of at least $7:1$ is selected for each lens to aid in manufacturability. The overall package of the interferometric null is roughly $600\mathrm{mm}\times 225\mathrm{mm}$. The theoretical interferogram exiting the interferometric null is shown in Fig. 3(a) before the DM is active and in Fig. 3(b) after the comatic and higher order null has been applied. In Fig. 3(a), the astigmatism and spherical aberration have been nulled from the wavefront, but there is still about $38\lambda$ peak-to-valley (P-V) of departure present in the double-pass wavefront at the testing wavelength of 632.8 nm. After the comatic null has been applied, the residual is on the order of $4\lambda$ P-V or $0.46\lambda$ root mean square (RMS). At the operating wavelength of around 10 μm, the residual in the double-pass null wavefront corresponds to $0.25\lambda$ P-V and $0.03\lambda$ RMS. The residual is nonzero as a result of the tilt angle required to null the astigmatic part of the surface. With a tilted geometry, the pupils cannot be perfectly conjugate to one another since a tilted object must be imaged to a tilted image per the Scheimpflug condition [8]. Moreover, the beam incident on the test mirror is slightly elliptical and will alter the Zernike composition of the wavefront. The residual in the exiting wavefront can be compensated either in hardware or software by using the DM to subtract the residual or by simulating a software null in the lens design software to subtract from the measured data.

 

Fig. 2. Layout of the optimized interferometric null to be coupled to a conventional Fizeau interferometer with a transmission flat. The interferometric null is composed of three nulling subsystems: an Offner null to null spherical aberration, a tilted geometry to null astigmatism, and a retroreflecting DM to null coma and any higher order aberration terms.

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Fig. 3. Simulation of the double-pass interferogram exiting the interferometric null (a) before and (b) after the deformable null has been applied at a testing wavelength of 632.8 nm.

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Before assembling the interferometric null, the shape of the comatic null must be created on the DM. The system for setting this shape is shown in Fig. 4, where a $3:1$ afocal telescope relays collimated light from a 632.8 nm Zygo laser interferometer through a cube beamsplitter and onto the DM. The wavefront then reflects off the surface of the mirror and half the light is directed back to the interferometer and the other half is directed through a $4:1$ afocal telescope that images the DM surface onto a $4.8\mathrm{mm}\times 3.6\mathrm{mm}$ Shack–Hartmann wavefront sensor. Using the wavefront sensor to interrogate the DM surface, it is operated in a closed-loop configuration where the influence functions of the actuators on the DM are known a priori and they are iteratively adjusted in the software to converge to a desired shape. The optimized fringe Zernike coefficients of the comatic null from the lens design are the target for the closed-loop optimization. The laser interferometer is used as an additional aid to measure the shape of the comatic null as the DM is adjusted to its final form. Any aberrations induced from the afocal telescopes can be subtracted from the measured wavefront by first replacing the DM with a flat reference mirror of high quality and using this measurement as a baseline. The final shape of the comatic null measured by the interferometer is shown in Fig. 5(a). The dynamic range of the DM is capable of creating this large departure null with a surface P-V of roughly 12 μm. However, when the theoretical null shape is subtracted from the actual comatic null, there is a large residual as displayed in Fig. 5(b). The residual is on the order of 2 μm P-V and is mostly composed of higher order deformations that result from the local deformation at or near the actuator sites. The voltages of the actuators are near their maximum for this surface shape so some residual is to be expected. Since the comatic null has been measured, it can be applied as a hitmap in software and the residual wavefront can be simulated to create a software null in the lens design software to subtract from the measured data.

 

Fig. 4. Layout of the setup to create the comatic and higher order null on the DM surface. The setup uses a Shack–Hartmann wavefront sensor to run a closed-loop optimization to set the shape of the comatic null. The comatic null is also interrogated with a Fizeau interferometer.

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Fig. 5. (a) Comatic null surface measured by the interferometer and (b) the residual after the theoretical shape has been subtracted. The residual has a P-V error of 2 μm P-V.

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The three lens components, the Offner null lens, Offner field lens, and collimating lens, have been fabricated by Optimax Systems. Each lens is coated with a V-coating that ensures back reflections to the interferometer are minimized. The optics are mounted with commercially available optomechanical components that provide four degrees of freedom movement ($x/y$ decenter and $x/y$ tilt). Moreover, each optical component sits on a kinematic base that provides stable six degrees of freedom positioning. With kinematic couplings, each element can be removed for the alignment of subsequent components and after alignment, the element can be replaced repeatably.

The tilted geometry for the test mirror, collimating lens, and DM is created by using a precision rotation stage with a rail attached to the tabletop of the stage. Since the mirror is to be tilted at $\alpha$, the optical axis or rail must be rotated by $2\alpha$. The mirror, whose vertex lies at the axis of rotation of the rotation stage, also rotates by $2\alpha$ and must be counter-rotated by $\alpha$. The counter-rotation of the test mirror is made possible by the use of a custom kinematic base known as a Kelvin clamp. The two-part base has a bottom plate with three conical cups milled about a radius separated by 120°. In these cups sit three spheres. On the top plate, sets of 120° spaced vee grooves are milled into the plate. Each sphere of the bottom plate sits in one vee groove, constraining two degrees of freedom (DOF). In total, all six DOF are uniquely determined. Each set of grooves defines one index (rotation) of the top plate. For the test mirror measurement there are three vee sets milled into the plate: 0, $\alpha$, and $2\alpha$. Using the specialized base, the mirror under test can first be aligned perpendicular to the optical axis in the zero index position. When the stage is rotated by $2\alpha$, the base can be indexed to the $\alpha$ position, bisecting the rotation angle of the stage. Since the stage is kinematic, the positioning will be repeatable for multiple iterations of positioning.

The assembled and aligned interferometric null configuration is displayed in Fig. 6. The interferometric null is coupled to a phase shifting Zygo DynaFiz interferometer with a reference transmission flat. The optical axis of the null (shown in red) is defined by a line that interests the vertex of the test mirror and is normal to the transmission flat of the interferometer. Because the vertex of the mirror is not readily accessible, a precision external target that couples to the mechanical alignment features of the test mirror is used to locate the vertex. Using this target and the transmission flat, an alignment telescope is aligned to these datums and it defines the optical axis for subsequent alignment of the other optical components. The alignment telescope sits behind the test mirror in the interferometric null. During alignment the test mirror is removed to provide an unobstructed view of the other components. In this fashion, the Offner null can be easily aligned. In order to align the collimating lens and DM, the rotation stage and rail are first aligned to the optical axis defined by the alignment telescope. In this case, the Offner null components are removed. Once the components have been aligned, the rotation stage is set to its angle and the Offner null components are replaced.

 

Fig. 6. Interferometric null configuration realized in the laboratory. A rotation stage with a rail affixed is used to create the tilted geometry. The test mirror is measured using a Zygo Fizeau-type interferometer.

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With the interferometric null aligned, the Zygo interferometer is used to acquire an interferogram of the optical surface. One important consideration for the raw interferogram acquired by the interferometer is the scaling factor, or the relationship between the fringe pattern on the wavefront and surface error on the mirror. A typical Fizeau measurement is double-pass, resulting in one fringe being equivalent to $\lambda /2$ surface error. For this test configuration, the test wavefront reflects off the tilted mirror twice, so, in this case, one fringe on the wavefront is estimated as

 

Fig. 7. (a) Initial surface error map of the test mirror with power and (b) with the power removed. The P-V error of the surface residual before and after the power is removed is 3.821 and 2.025 μm, respectively. (c) Final surface error map of the test mirror after the software null has been subtracted (c) before and (d) after the power has been removed. In this case, the P-V error is 3.230 μm before and 1.140 μm after the power has been removed.

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In order to observe the errors of the mirror surface and not the errors of the comatic null, a software null is created in CODE V. The software null simulates the wavefront at the exit pupil and it includes the effects of the residual aberrations present in the testing setup and incorporates a hitmap of the comatic null. The surface error maps after subtracting the software null from the measured data are depicted in Figs. 7(c) and 7(d). When the power is present in the surface error, Fig. 7(c), the P-V error is now 3.230 and 0.798 μm RMS. After the power is subtracted, Fig. 7(d), the P-V error is reduced to 1.140 and 0.156 μm RMS. At a wavelength of 10 μm, the center operating wavelength of the optical system, the P-V error is $0.114\lambda$ and $0.016\lambda$ RMS. Therefore, for an LWIR application, the surface is almost a tenth wave. In evaluating the features in the surface error, it can be seen that, while small, the error is mostly astigmatism that may be a residual from the mounting process during fabrication.

The interferometric null configuration presented in this work has been successfully implemented and applied to measure the surface figure of a freeform, $\phi$-polynomial (Zernike) mirror. The metrology setup is capable of adapting and measuring a wide variety of surface shapes as the astigmatism can be varied by changing the tilt angle of the mirror surface and the comatic departure can be varied by changing the shape on the DM surface. An interferometric null configuration that combines several adaptable nulling subsystems gives more freedom than a CGH that is designed for measuring only one optical surface. With a methodology now in place to measure $\phi$-polynonmial-type surfaces, designs employing these surfaces are realizable and will push the use of these surfaces to systems operating at shorter wavelengths.