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Modeling and suppressing the wavefront degeneration in a CGH interferometric null test

Mingzhuo Li, Haixiang Hu, Xuejun Zhang, and Sida Lv

Open Access Open Access

Abstract

The computer-generated hologram (CGH) enables the ultra-high accuracy of surface measurement but causes the wavefront degeneration in the optical system. In this article, we give a high-accuracy analytical simulation of the wavefront degeneration in null test by the elliptical Gaussian model. We propose an analytical expression of instrumental transfer function (ITF) for the CGH null test without knowing the phase distribution of CGH, which gives an efficient instruction to suppress the wavefront degeneration. The ITF of the interferometric null test for a ∅3m aspheric mirror can be optimized from 0 to 0.65 at 0.4 Nyquist frequency.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Interferometric null test is one of the most effective methods to measure the surface with ultra-high precision and accuracy [1–3]. Aspheric and freeform surfaces are measured by interferometric null test with custom-made CGH (Fig. 1). For decades, people have studied measurement errors in the CGH null test, such as direct measurement error and geometric retracing error [4–8]. The CGH enables high-accuracy surface test but degenerate the wavefront when the test wavefront travels back from the surface under test (SUT) to interferometer [4,9–11].

 figure: Fig. 1.

Fig. 1. Schematic of a classic setup of null test of mirror under test. SUT: Surface under test; F: Focus of the interferometer; TS: Transmission sphere, transform the collimated beam to spherical wavefront.

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In 2007, C. Zhao and J.H. Burge first reported the wavefront degeneration from the imaging aberration introduced by CGH [9]. In their article, the wavefront propagation is calculated by a two-step method. First, according to infinitesimal ray bundle theory, the propagation of light in a fully coherent system is transformed to a geometric imaging process of SUT through the optical system. Then the imaged SUT is regarded as secondary wavelets and the afterwards light propagation to the detector is calculated by Talbot effect [9,10]. The imaged SUT is not flat anymore but has astigmatic field curve. Such phenomenon is called imaging aberration from null correctors and degenerates the test wavefront. The wavefront degeneration smooths out surface error especially in high-frequency domain and create edge artifacts by Fresnel edge diffraction. The effect of wavefront degeneration to the surface test can be characterized by the instrumental transfer function (ITF) [11–13]. In the null test system of a aspheric surface, J.H. Burge and et al. use geometric optic software to calculate the astigmatic field curve originated from the imaging aberration and propose an ITF equation involving the surface type of optical surfaces to optimize the ITF of optical system [14,15].

However, the geometric process of the two-step method in a freeform null test system is difficult to retrieve an analytical expression and probably has to refer to the professional software. In addition, the ITF equation for the CGH null test needs the second derivative of the phase on CGH, which is corresponding to the surface type of SUT. In the optimization of the CGH null test, we have to modify the phase distribution every time we adjust any parameter of optical element. The optimization procedure becomes iterative and increases the complexity of calculation.

In this paper, first we model the propagation of wavefront in the interferometric CGH null test based on the complex ray tracing method with elliptical Gaussian function. Such a method provides a direct and precise mathematical description of the wavefront degeneration in the optical system. Then, we derive the analytical expression of ITF of CGH null test without knowing the phase of CGH. With such ITF expression, we build the direct relation between ITF and the parameters of optical elements in the system. We quantitatively analyzed the influence of each parameter in the optical system to the ITF and give an effective approach to improve the ITF in the CGH null test. To verify our method, we measure the ITF of the CGH null test of a ∅3m aspheric mirror and compare with the ITF we calculate. After suppressing the wavefront degeneration, the ITF of the CGH null test is significantly improved.

2. Modeling the wavefront degeneration in a CGH null test

2.1 Elliptical Gaussian model in coherent optical system

To model the wavefront propagation in a fully coherent optical system, one can apply the complex ray tracing method [16–18]. Such a method consists of three fundamental steps: decomposition of the incident wave field into a series of equally spaced and mutually coherent Gaussian beamlets, propagation of each Gaussian beamlet via the geometric-diffractive optical approach and coherent recombination of each beamlet to retrieve the total field at any desired position of the optical system [16–18]. We decompose the incident wavefront by a series of Gaussian beamlets [19]

$${u_0}({r,z} )= \frac{{2\pi {A_0}w(z )}}{{|{q(z )} |}}\exp \left( { - \frac{{2i\pi z}}{\lambda } + i{\phi_G}} \right)\exp \left[ {\frac{{i\pi {r^2}}}{{\lambda q(z)}}} \right].$$

Here, w(z) is the beam waist at z, q(z) is the complex propagation parameter, A­0 is the amplitude, ϕG is the Gouy phase and λ is the light wavelength. The beam waist w(z) and the complex propagation parameter q(z) are defined as the following equations

$$w(z )= {w_0}\sqrt {1 + {{\left( {\frac{z}{{{z_R}}}} \right)}^2}} \textrm{ and }q(z )= z - i{z_R}\textrm{ with }{z_R} = \frac{{\pi w_0^2}}{\lambda }.$$

To better simulate the decomposed field, the Gaussian beamlets should be close enough to each other. Otherwise, there will be ripples in the decomposed light field. The distance between the center of each Gaussian beamlet should be beam waist so that ripples could be suppressed to 0.05% of the maximum amplitude [16,20].

After we decompose the light field by Gaussian beamlets, each Gaussian beamlet can be approximated by a cone of the light rays involving the divergent angle of the Gaussian function. The beam waist can be regarded as a small extended object. We can find the imaged beam waist through the optical system via geometric ray tracing. So, the propagated light field at any position is represented by Gaussian beamlet from the imaged beam waist to the desired position (Fig. 2). However, with the application of freeform surfaces, the wavefront in the CGH null test becomes complicated so the geometric ray tracing becomes difficult. The optical surface usually keeps C2 continuity, which is available for the application of differential geometry [21]. If we sample the incident wavefront by a series of tiny areas with Gaussian beamlets, we can apply the differential geometry theory to these tiny areas. Thus, we can find two individual waists along tangential and sagittal directions of the objective beam waist.

 figure: Fig. 2.

Fig. 2. Sketch of modeling the propagation of wavefront through the optical system.

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For a single input with Gaussian phase perturbation, the light field in the image space can be represented by an elliptical Gaussian function [19,22]

$${u_i}({x,y,z} )= E(z )\exp \{{i[{{\Phi _{AC}}(z )+ {\Phi _G}(z )+ {\Phi _0} + {\Phi _E}({x,y,z} )} ]} \}.$$

Here, E(z) is the amplitude of the light field, ΦAC is the accumulated phase along propagation, ΦG is the Gouy phase, Φ0 is the initial phase and ΦE is the elliptical wavefront which has the following form

$${\Phi _E}({x,y,z} )\textrm{ = }\frac{{n\pi }}{\lambda }\left[ {\frac{{{x^2}}}{{{q_S}(z )}} + \frac{{{y^2}}}{{{q_T}(z )}}} \right]\textrm{ with }{q_j}(z )= z - {z_j} - \frac{{i\pi w_j^2}}{\lambda }.$$

Here, λ is the wavelength of light, n is the refractive index of the medium, z is the position of the detector and zj is the position of beam waist in the image space along the direction of tangential (T) or sagittal (S). The transversal phase distribution shown in Eq. (4) is similar to the Euler equation

$$\frac{1}{{{R_N}}}\textrm{ = }\frac{{{{\cos }^2}\Phi }}{{{R_S}}} + \frac{{{{\sin }^2}\Phi }}{{{R_T}}}.$$

Here, Φ is the angle between the desired direction and tangential direction, RN is the radius of curvature along direction of Φ and RS, RT are the radii of curvature of tangential and sagittal directions. Comparing with Eq. (4), q(z) can be regarded as complex radius of curvature for the wavefront. We can write down the complex radius of curvature (qN) along arbitrary direction (θ) as Eq. (6). At the intermediate position, there is a certain direction that the complex curvature (1/qN) is zero.

$$\frac{{{x^2}}}{{{q_x}(z )}} + \frac{{{y^2}}}{{{q_y}(z )}}\textrm{ = }\frac{{{\rho ^2}{{\cos }^2}\theta }}{{{q_x}(z )}} + \frac{{{\rho ^2}{{\sin }^2}\theta }}{{{q_y}(z )}} = \frac{{{\rho ^2}}}{{{q_N}(z )}}.$$

For other terms in Eq. (3), the distributions of each propagating phase term are the average of phase along tangential and sagittal directions, which are shown in Eq. (7) and Eq. (8)

$${\Phi _{\textrm{AC}}}(z )\textrm{ = }\frac{1}{2}[{\Phi _{AC}^{(S )}(z )+ \Phi _{AC}^{(T )}(z )} ]\textrm{ = }\frac{{n\pi ({z - {z_S} + z - {z_T}} )}}{\lambda }.$$
$${\Phi _G}(z )\textrm{ = }\frac{1}{2}[{\Phi _G^{(S )}(z )+ \Phi _G^{(T )}(z )} ]\textrm{ = }\frac{1}{2}\left( {\frac{\pi }{2} - \arctan \frac{{z - {z_S}}}{{z_R^{(T )}}} + \frac{\pi }{2} - \arctan \frac{{z - {z_T}}}{{z_R^{(T )}}}} \right).$$

The light field in the image space should follow the law of energy conservation while propagate in the medium regardless of the absorption. At every z-position, the integration of module square of the light field should be a constant E0. So, the amplitude term is

$$E(z )\textrm{ = }\sqrt {\frac{{2\pi n{E_0}\sqrt {z_R^{(S )}z_R^{(T )}} }}{{{\lambda ^2}|{{q_S}(z )} ||{{q_T}(z )} |}}} .$$

The total light field in the image space is sum of every elliptical Gaussian beam. The complex ray tracing is a classic, simple and powerful method to model arbitrary wavefront propagation in the coherent optical system, especially for finite-apertured and astigmatic optical elements [16]. We apply the complex ray tracing method to model light field in the CGH null test for complex optical surfaces.

2.2 Complex ray tracing method in the CGH null test

2.2.1 Parametric model for the CGH in a null test

This section we consider how to apply the geometric imaging process in the CGH null test system. As is shown in Fig. 1, the null test setup consists of a piece of CGH, lens system including the transmission sphere and the imaging system of interferometer. The imaging process of the transmission sphere and the lens system is easily acquired by solving the Gaussian imaging formula of each lens. The imaging process through the CGH should use the generalized Coddington relations with the second derivative of the phase [11,23]. The simulation of the imaging process above could be done by the software. Therefore, if we change any parameter of the optical system, such as the position of CGH, we have to recalculate the phase distribution of the CGH. With the model of beamlet theory and ray tracing, we can derive an imaging formula of CGH without knowing the phase of the CGH.

For light rays in the interferometer, the CGH has two major functions, one is to relay the lights to the focus of the interferometer (Fig. 3(a)), the other is imaging the tiny bundle of rays from P to the position V (Fig. 3(b)). Figure 3 shows the lights propagating through the CGH in tangential plane.

 figure: Fig. 3.

Fig. 3. a) Relay the rays to the focus of interferometer; b) Imaging the tiny bundle of rays from P to V.

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The CGH relays rays from a tiny surface to the focus of interferometer, it is equivalent to imaging the center of local curvature C to the focus of interferometer F. By applying the Coddington equation and refractive equation of phase plate, we can write down equations of light rays passing through CGH

$$\begin{array}{ll} \textrm{Sagittal:}&\frac{{{\partial ^2}\varphi }}{{\partial {x^2}}} = \frac{1}{{R - u}} - \frac{1}{{FG}}\\ \textrm{Tangential:}&\frac{{{\partial ^2}\varphi }}{{\partial {y^2}}} = \frac{{{{\cos }^2}\alpha }}{{R - u}} - \frac{{{{\cos }^2}\beta }}{{FG}}. \end{array}$$

Here, φ is the phase of CGH, R is the radius of curvature of point P (PC), u is the object distance from SUT to CGH (PG), α is defined as the angle between the reflected ray from the SUT (PG) and the normal direction of CGH (parallel to z-axis) and β is defined as the angle between the refractive ray (PG) and the normal direction of CGH (parallel to z-axis). To solve Eq. (10), we have to know the phase distribution φ on CGH.

As we discussed above, the CGH has another function, imaging the ray bundle from P on the SUT to the position V in Fig. 3(b). With the refractive equation of phase plate, we can write down equations as below

$$\begin{array}{ll} \textrm{Sagittal:}&\frac{{{\partial ^2}\varphi }}{{\partial {x^2}}} ={-} \frac{1}{u} - \frac{1}{v}\\ \textrm{Tangential:}&\frac{{{\partial ^2}\varphi }}{{\partial {y^2}}} ={-} \frac{{{{\cos }^2}\alpha }}{u} - \frac{{{{\cos }^2}\beta }}{v}. \end{array}$$

Here, u is the object distance from SUT to CGH (PG), v is the distance from CGH to focus (GF). By simultaneously solving Eq. (11) and Eq. (10), the phase distribution of CGH φ can be eliminated. The CGH imaging equation is

$$\begin{array}{l} \textrm{Sagittal: }\frac{R}{{u({R - u} )}} + \frac{1}{v} = \frac{1}{{FG}}\\ \textrm{Tangential: }\frac{{{{\cos }^2}\alpha }}{{{{\cos }^2}\beta }}\frac{R}{{u({R - u} )}} + \frac{1}{v} = \frac{1}{{FG}}. \end{array}$$

Equation (12) is similar to the Gaussian imaging formula. Comparing with the Gaussian imaging formula, we can define an equivalent objective length u’ to simplify the formulas.

$$\begin{aligned}&{\rm Sagittal:} \displaystyle{1 \over {u_S^{\prime} }} + \displaystyle{1 \over v} = \displaystyle{1 \over {FG}}{\rm with }u_S^{\prime} = \displaystyle{{u\left( {R_S-u} \right)} \over {R_S}} \\ & {\rm Tangential: } \displaystyle{1 \over {u_T^{\prime} }} + \displaystyle{1 \over v} = \displaystyle{1 \over {FG}}{\rm with }u_T^{\prime} = \displaystyle{{{\cos }^2\beta } \over {{\cos }^2\alpha }}\displaystyle{{u\left( {R_T-u} \right)} \over {R_T}}.\end{aligned}$$

We can characterize the rays passing through the CGH without knowing the phase information. For a certain point P on SUT, the CGH is now equivalent to a tiny piece of lens. The entire optical system for the CGH null test can be regarded as an ideal lens system. So, the geometric optical process in the CGH null test is simplified.

2.2.2 Parametric model for astigmatic field curve in the CGH null test

In this paper, we focus on the influence of wavefront degeneration from CGH. All other influences to the ITF from other sources can be subtracted as the systematic error. So, we suppose the optical system of the interferometer is equivalent to an ideal lens system. Since the CGH is equivalent to the micro lens arrays, we can trace the light from SUT to its image by solving the imaging equation of each lens throughout the optical system (Fig. 4).

 figure: Fig. 4.

Fig. 4. Schematic of modeling the image of SUT through the optical system.

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The astigmatic field curve is defined as the distance between VM and VC along z-axis

$$\delta z ={-} \frac{{D_2^2}}{{D_{CGH}^2}}\left[ {u_C^{\prime} - \frac{{{L_{FC}} \cdot u_M^{\prime}}}{{{{({{L^2}_{FC} + {{D_{CGH}^2} / 4}} )}^{\frac{1}{2}}}}}} \right].$$

Here, u’M and u’C are the effective objective distance, D2 and DCGH are the diameter of the last piece of lens and CGH, LFC is the distance between interferometer focus and the CGH and LCM is the distance between SUT and CGH. In Eq. (14), we can see the δz is independent to transmission sphere and imaging system in the interferometer besides D2. D2 corresponds to beam diameter of the light to the detector. The equivalent objective length u’ and diameter of CGH DCGH are related to the surface profile of the SUT and the LCM. To suppress the astigmatic field curve δz, we have to modify the position of CGH and SUT according to the focus of interferometer F, i.e., the LCM and LFC.

The elliptical Gaussian model gives an accurate simulation of the phase distribution of the CGH null test. However, such a model is not direct and convenient to evaluate the degeneration of the wavefront. The parametric equation Eq. (14) gives us a direct and quantizable understanding of the imaging aberration of CGH. Furthermore, we can suppress the astigmatic field curve δz easily according to Eq. (14).

2.3 Suppressing the astigmatic field curve δz

We apply the model to the CGH null test for an aspheric mirror (conic constant k = −0.99) with central radius of curvature R = 8950mm and diameter of 3m. According to Eq. (14), we can calculate the imaged SUT through the optical system (Fig. 5(a)). We can see an unexpected field curve and astigmatism appear in the image space of the optical system. The test wavefront is not a flat surface anymore. We have to notice that the field curve and astigmatism do not have the same origination as the classic definitions from the ray optics.

 figure: Fig. 5.

Fig. 5. a) The astigmatic field curve δz from imaging aberration of CGH; b) The field curve δz (blue) and diameter of beam on CCD (red) with different LFC; c) The field curve δz (blue) and diameter of CGH (red) with different LCM; d) The size of CGH for different LCM.

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To minimize the unexpected astigmatic field curve according to Eq. (14), we can modify the parameters of optical path. The size of CCD (D2) is limited by the camera in the interferometer. The parameters u’ and DCGH in Eq. (14) are related with LCM. Therefore, the wavefront degeneration is only related to the relative position of SUT and CGH, LFC and LCM. In Fig. 5(b), the astigmatic field curve is suppressed by increasing the LFC while keeping the LCM as constant. Changing LFC changes the F-number of the interferometer as well, which changes the magnification of the optical system. In this case, although the δz is suppressed, but the shrink of beam diameter on the detector needs higher resolution of the optical system.

Another optimization approach is that we set the distance between the SUT to the focus as constant (LCM + LFC = 9000mm) and change the position of CGH (LCM). In Fig. 5(c), the simulation shows the astigmatic field curve δz is reduced with decreasing of the LCM. And the beam diameter does not change too much. In Fig. 5(d), diameter of CGH (DCGH) increases from 170mm to 270mm, requiring a CGH fabrication ability of 300mm-scale with high surface quality. The closer the CGH to the SUT, the larger diameter the CGH will be. So, in the experiment, we set the LCM = 8400mm, corresponding to the CGH diameter of 240mm. With such parameters, we can observe an obvious δz suppression but do not increase too much difficulty to the fabrication of CGH.

3. Optimizing the ITF of the CGH interferometric null test

3.1 Effects of surface test by the wavefront degeneration

In the above section, we discussed how to calculate and suppress the astigmatic field curve. We will discuss how the astigmatic field curve affect the accuracy of surface test. According to Eq. (14), the imaging aberration from the CGH introduces an astigmatic field curve to the test wavefront. The test wavefront will be diffracted if propagates a distance δz. In this section, we discuss the effects of wavefront degeneration to the surface measurement qualitatively. Figure 6(a) shows the surface profiles of the SUT with a Gaussian-shape error under wavefront degeneration according to Eq. (3). The wavefront degeneration introduces artificial features to the surface test and attenuates the maximum height of the real surface feature. Mid- and high-frequency errors will be left on the SUT if we process the surface manufacture under the instruction of wavefront-degenerated surface test (Fig. 6(b)). The mid- and high-frequency errors on the large reflective mirror will affect the imaging quality when it applies in the optical system [24,25]. As we can see in Fig. 6(c), the RMS of iterative surface processing is not going down to 0. Besides, the RMS of the degenerated surface profile is smaller than the real profile. This will give false instruction that the manufacture reaches the desired precision and stop the iterative manufacture. So, to quantify the effects of wavefront degeneration in CGH null test, using RMS or PV is not sufficient.

 figure: Fig. 6.

Fig. 6. The iterative process of SUT by the guidance of wavefront-degenerated surface measurement and the corresponding RMS. a-b) surface profile in the iterative manufacture (blue line) and the wavefront-degenerated surface measurement (red line); c) the RMS of manufactured profile (blue line) and degenerated profile (red line).

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To characterize the surface effect relates to various spatial frequencies, we can use instrumental transfer function (ITF) [10,11]. The ITF characterizes the ratio of the experimental instrument response to different spatial frequency of the features on the SUT [10–12]. So, the ITF can be calculated as the ratio of power spectrum density (PSD) of the ideal surface height h0(x) and the measured surface height hz(x,δz) [12,13]. The corresponding ITF of the CGH null test is defined as the following equation

$$ITF({f,\delta z} )= \sqrt {\frac{{PSD[{{h_z}({x,\delta z} )} ]}}{{PSD[{{h_0}(x )} ]}}} .$$

For simplicity, we consider a wavefront with sinusoidal phase with spatial frequency of f.

$${u_0}(x )= \exp \left[ {i\frac{{2\pi }}{\lambda }{h_0}(x )} \right] = \exp \left( {i\frac{{2\pi \alpha }}{\lambda }\sin 2n\pi fx} \right).$$

Here, λ is the wavelength of the light, f is the spatial frequency of the sinusoidal grating and α is amplitude of sinusoidal grating which is usually a tiny value. So, we can simplify the equation by Taylor expansion at α→0. We keep the linear term and the wavefront becomes

$${u_0}(x )\approx 1 + i\frac{{2\pi \alpha }}{\lambda }\sin 2n\pi fx.$$

As we discussed, the initial beam will be decoupled into a series of Gaussian beamlets. The decoupled initial wavefront can be written as following equation.

$$u_0^{\prime}(x )= {u_0}(x )\ast { {{K_x}({x,\delta z} )} |_{z = 0}} = \sqrt {\frac{{\lambda {z_R}}}{n}} \left[ {1 + i\frac{{2\pi \alpha }}{\lambda }\exp ({ - n\pi \lambda {z_R}{f^2}} )\sin 2n\pi fx} \right].$$

Here, Kx(x,z) is the elliptical Gaussian beamlet, which performs as convolution kernel. The terms, which are only related with z, will not contribute to the wavefront degeneration along x-axis. They usually perform as piston in the surface measurement (Gouy phase term) or the visibility of the interference patterns (the amplitude term). So, the convolution kernel is simplified as

$${K_x}({x,\delta z} )= \exp \left[ {\frac{{in\pi {x^2}}}{{\lambda ({\delta z - i{z_R}} )}}} \right].$$

The phase of the decoupled initial wavefront is given in terms of the Taylor expansion at α→0. We keep the linear term of the phase and calculate the decoupled initial surface height as

$$h_0^{\prime}(x )= \alpha \exp ({ - n\pi \lambda {z_R}{f^2}} )\sin 2n\pi fx.$$

For the degenerated test wavefront is defined as

$$\begin{aligned} {u_z}({x,\delta z} )=& {u_0}({x,\delta z} )\ast {K_x}({x,\delta z} )\\& = \sqrt {\frac{{\lambda ({{z_R} + i\delta z} )}}{n}} \left\{ {1 + i\frac{{2\pi \alpha }}{\lambda }\exp [{ - n\pi \lambda ({\delta z + i{z_R}} ){f^2}} ]\sin 2n\pi fx} \right\}. \end{aligned}$$

By a similar approach, we can calculate the height of the degenerated measured surface

$${h_z}({x,\delta z} )= \frac{1}{2}\arctan \frac{{\delta z}}{{{z_R}}} + \alpha \exp ({ - n\pi \lambda {z_R}{f^2}} )\cos n\pi \lambda \delta z{f^2}\sin 2n\pi fx.$$

In the interferometry, the constant contributes to the piston of the surface. After removing the piston, we plug Eq. (20) and Eq. (22) into Eq. (15) to retrieve the ITF of CGH null test.

$$ITF = \sqrt {\frac{{PSD[{{h_z}({x,\delta z} )} ]}}{{PSD[{h_0^{\prime}(x )} ]}}} = \cos n\pi \lambda \delta z{f^2}.$$

Since the cos-function is a periodic function, we only focus on the spatial frequency regime of the first period of ITF over 0. Otherwise, the surface measurement will have false result or even the reverse measurement if ITF is negative.

3.2 ITF optimization of the CGH null test for a large aspheric mirror

We can optimize the ITF of CGH null test by modify parameters of optical elements according to Eq. (14) and Eq. (23). If we suppress the δz, we can improve the ITF of CGH null test. As we discussed above, to suppress the δz, we should modify the LCM and LFC. So, we base on the CGH null test system of the ∅3m aspheric mirror mentioned above and calculate the ITF with different parameters. We compare different combination of LCM and LFC then draw the corresponding ITF in Fig. 7. We can observe the ITF is almost unchanged with different LFC but varies significantly with different LCM.

 figure: Fig. 7.

Fig. 7. The ITF of CGH null test with different LFC and LCM. Blue line: LFC = 600 mm and LCM = 8600 mm; red dashed line: LFC = 400 mm and LCM = 8600 mm; yellow line: LFC = 600 mm and LCM = 8400 mm.

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As discussed above, the astigmatic field curve δz is suppressed by decreasing the LCM and the ITF of CGH null test is improved. We keep LCM + LFC = 9000mm as constant and change the distance between CGH to SUT (LCM) from 8600mm to 8300mm and calculate the corresponding ITF of the null test in Fig. 8.

 figure: Fig. 8.

Fig. 8. The ITF of CGH null test with different LCM while keeping the LCM + LFC = 9000 mm.

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The ITF has been significantly improved by decreasing the LCM. For LCM = 8600mm, the ITF is cut off at 0.36 Nyquist frequency of the optical system. The ITF drops to 0.5 at 0.29 Nyquist frequency. For LCM = 8400mm, the ITF is cut off at 0.56 Nyquist frequency. The ITF drops to 0.5 at 0.45 Nyquist frequency. In the surface test, the ITF should be over 0.5 for a reliable surface figure. We increased the spatial frequency of surface feature of a reliable measurement 1.5 times by decreasing the LCM from 8600mm to 8400mm.

4. Discussion

Nowadays, the parabolic mirror or the quasi-parabolic mirror, whose conic constant is close to −1 (κ≈−1), are frequently used as the primary mirror in the large and advance telescope designs [26]. In addition, the parabolic surface has a simple surface equation. By inserting the surface equation into Eq. (14), we can simplify the parametric equation for the astigmatic field curve and have more analytical understanding of the imaging aberration in the CGH null test. For a parabolic surface, the sag equation of parabolic surface is

$$z = \frac{{{S^2}}}{{2R}}.$$

Here, R is the radius of the curvature at the center of the optical surface. For an optical surface with large central radius R, the refraction of the CGH is not strong (i.e., α≈β). The sketch of the optical path is shown as Fig. 9.

 figure: Fig. 9.

Fig. 9. The sketch of the optical path imaged through CGH.

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According to Eq. (14), the we calculate the image of C and M on SUT through the rest of the optical system in Fig. 9. The astigmatic field curve (δz) with the Taylor expansion at zM/R≈0 is

$$\delta z\textrm{ = }\frac{{f_2^2}}{{f_1^2}}\frac{{f_t^2}}{{L_{FC}^2}}\left[ { - \frac{{L_{CM}^2}}{R}\frac{{{S^2}}}{{{R^2}}} - {L_{CM}}\frac{{{S^2}}}{{{R^2}}} + \frac{R}{2}\frac{{{S^2}}}{{{R^2}}}} \right].$$

We define a new parameter T, which is the ratio of half-diameter and central radius of the mirror (T = S/R). And P = LCM + LFC is a constant as assumed in the previous sections. Now the Eq. (25) can be written as

$$\delta z ={-} \frac{{f_2^2f_t^2}}{{f_1^2}}{T^2}\frac{{[{{{({{L_{CM}} + {R / 2}} )}^2} - {{3{R^2}} / 4}} ]}}{{{{({P - {L_{CM}}} )}^2}R}}.$$

We set the parameters same as the null test setup for the ∅3m aspheric mirror. We calculated the field curve δz with different S and LCM (Fig. 10). As we can see in the figure, the approximated equation matches the simulated results well especially when the diameter of the mirror is smaller than 1m. The approximated equation is accurate enough to analyze the quasi-parabolic mirror. Equation (26) can be a quick and accurate estimation of the wavefront degeneration in the interferometric null test.

 figure: Fig. 10.

Fig. 10. Comparison of a) simulation results by Eq. (14) and (b) approximated equation of the field curve δz with S and LCM. c) is the difference between the approximated equation and the simulation by Eq. (14).

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According to Eq. (26), the δz can be suppressed to 0 when LCM = (√3−1)R/2. For the aspheric mirror we used, the LCM equals to 3275.9mm. At the minimum field curve position, the semi-diameter of CGH is larger than 1000mm, which exceeds the fabrication ability of the hologram patterns. The best position of wavefront degeneration may not be achieved due to the limitation of fabrication. Nevertheless, for LCM < 8000mm the field curve δz is less than 0.01mm, which will not significantly attenuate the ITF of interferometer.

In addition, Eq. (26) indicates the astigmatic field curve has quadratic relation to the diameter of the parabolic mirror. The wavefront degeneration is not a notable phenomenon when the diameter of SUT is small. If we decrease central radius R of the parabolic mirror, the astigmatic field curve will increase according to Eq. (26). In the CGH null test for a large and strongly bended optical surface, the astigmatic field curve becomes stronger.

5. Simulative and experimental verification

5.1 Optical field in astigmatic system

To verify the elliptical Gaussian model, we simulate the propagation of a single Gaussian beamlet through the interferometric null test system of the astigmatic SUT. We compare the simulation result of light field at the output space via Eq. (3) with the simulation via a powerful simulating software VirtualLab Fusion (LightTrans GmbH, Germany). The simulation is modeled as Fig. 11. We set incident Gaussian function has a beam waist of 20µm. The size of each detector is 1.5mm × 1.5mm square and the distance between each slice is 20mm. The incident beam is a Gaussian beam with waist of 20µm. The cylindrical phase plate will introduce a cylindrical phase along tangential direction (y-direction) with radius of 10mm.

 figure: Fig. 11.

Fig. 11. Sketch of the model in the VirtualLab simulation. Green line indicates optical path along tangential direction. Red line indicates optical path along tangential direction. The incident Gaussian function has a beam waist of 20µm with wavelength of λ=632.8 nm.

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Figure 12 shows a sketch of the phase distribution directly calculated by Eq. (3). To avoid the influence of ΦAC, we set the wavelength λ equals 1µm. Figure 12(a) shows the phase distribution in x-y plane of 5 planes from z = 0mm to z = 80mm. At the sagittal and tangential focus, the wavefronts have a cylindrical shape. At the intermediate position, the wavefront becomes saddle surface, which satisfies our expectation. Figure 12(b) shows the phase distributions in x-z and y-z planes.

 figure: Fig. 12.

Fig. 12. a) Slices of phase distribution in the image space (λ=1µm); b) Phase distribution of the slice in x-z and y-z plane.

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To numerically verify the simulation accuracy, we compare the real part of the light field calculated by the VirtualLab (VL) and the elliptical Gaussian model (EG). We set the wavelength λ equals to 632.8nm, which is the wavelength of laser source in our laboratory. The simulated results are shown in Fig. 13.

 figure: Fig. 13.

Fig. 13. a-c) are results from the diffractive simulating software, VirtualLab Fusion; d-f) are results from the real part of the light field via Eq. (3).

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The phase simulation by the VirtualLab is related to the intensity of light field. In the low intensity regime, the phase result becomes incorrect. We are interested in the region of intensity over 1/e2 of the maximum intensity and define effective standard deviation as the standard deviation within the region of interests between two approaches. The deviation between the VirtualLab simulation and the elliptical Gaussian function is less than 0.0040V/m. In addition, we as well compare the simulation of amplitude, real and imaginary part of the light field. The results are shown in Table 1. For the simulation results of phase, the effective standard deviations are less than 0.0300rad, which corresponds to ∼λ/200@λ=632.8nm. If we focus on a smaller effective region, which is half of the maximum intensity, the standard deviation is less than 0.0083rad (∼λ/750@λ=632.8nm). Such an accuracy is much higher than the accuracy in the experiments (∼λ/100@λ=632.8nm). The simulation by the software VirtualLab takes about 10s and for our elliptical Gaussian model is about 0.18s.

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Table 1. Effective standard deviations between the simulation of elliptical Gaussian model and VirtualLab

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5.2 Experimental verification of elliptical Gaussian model

To verify the modeling of the light field in the astigmatic optical system, we set up an interferometric null test for a cylindric mirror (Fig. 14). Because a cylindric mirror is a plane mirror along the sagittal direction and is a spherical mirror along the tangential direction. Through the optical system of the interferometer with a CGH, we generate an astigmatic light field in the image space. We can find two separated planes of tangential and sagittal focus in the imaging space of the interferometer (Fig. 15(a)).

 figure: Fig. 14.

Fig. 14. a) The setup of an interferometric CGH test of the cylindric mirror; b) The CGH for measuring the cylindrical mirror; c) The surface measurement of the cylindrical mirror; d) The diffraction effect by the triangular beam block.

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 figure: Fig. 15.

Fig. 15. a) is the sketch of the interferometric null test; b) is the beam block which let the triangular-shape light passes; c-e) are the experimental interferometric surface measurement; f-h) are the simulated surface by the elliptical Gaussian model.

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The interferometer we use is ZYGO MST 1K 4’’. The CGH (Fig. 14(b)) and the concave cylindric mirror have the radius 420mm and diameter of 80mm. The pixel size is 5.5µm. Figure 14(c) shows the surface measurement of the cylindrical mirror. The notable radial ripples are the residual errors from the fabrication. On the cylindrical CGH, there is a bar at its center, which introduces diffraction patterns to the measurement. So, we blocked the central strip to avoid the influence of the diffraction. In the experiment, we put a binary beam block with triangular shape to create defects along different directions. If we put the detector at the position of certain focus, either tangential, sagittal or intermediate focus, we can observe a sharp edge along the corresponding direction. We can simulate the light field after the beam block. Figure 14(d) shows one of the measurements of edge effects, which substrates the surface measurements in Fig. 14(c). The region of interest in Fig. 14(d) is 0.45mm × 0.45mm in the coordinate of camera, corresponding to a 15mm × 15mm region on the SUT.

Since the interferometer retrieve the phase from interference patterns, we extract the phase of the light field in our simulation and compare with the experiment. In Fig. 15, we compare the simulated phase distribution of the interferometric beam with experimental results at different positions by moving the detector. The experimental results are retrieved by the software MetroPro (Zygo Corp, USA). In the simulation, we set the waist of the Gaussian beamlets w to be 1µm. The distance between each Gaussian beamlet is 0.75µm, which will not introduce ripples to the wavefront. The Gaussian beamlets forms a 600 × 600 grid. To avoid the boundary effect, we focus on the central region 300 × 300 grid in the 600 × 600 grid. The simulation takes less than 6s including processing 3 figures (Fig. 15(f)-(h)). At the tangential and sagittal focus, the edges of triangular mask are blurred along the other directions. At the intermediate focus, which is right in the middle of tangential and sagittal directions (Fig. 15(d) and (g)), the edge of triangular mask is sharp along the diagonal direction (Φ=45°). At the vertex of the beam block, it is a quadrangular star shape rather than a blurred corner as we expected.

5.3 Experimental verification of ITF optimization for large aspheric mirror

We apply the model to an aspheric mirror (conic constant κ=−0.99) with central curvature R = 8950mm and diameter of 3m. To compare with the ITF simulation, we choose LCM = 8400mm and LCM = 8600mm to set up the null test. The ITF of the entire optical system from SUT to CCD in the interferometer can be measured by a phase plate (Fig. 16). After the profile of the step is measured, the ITF of the measurement can be calculated by the ratio of PSD of the measured and ideal step profile as shown in Eq. (23). The interferometer we use is ZYGO HD 2K 4’’. The phase plate was designed with diameter of 80mm and has a transparent step with height of 80nm on it (Fig. 16(c)). The phase plate is mounted on a holder with negative-pressure suckers so that the holder can adhere to the SUT (Fig. 16(b)). We can measure the ITF at different positions on the SUT with different focal status of the interferometer.

 figure: Fig. 16.

Fig. 16. a) Experimental setup of null test; b) Phase plate put on the SUT; c) The transmitted wavefront of phase plate.

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Figure 17 shows the simulated (dashed lines) and experimental (solid lines) ITF with different LCM. The simulated ITF is calculated by the Eq. (23). For the measured ITF, it contains not only the influence of wavefront degeneration, but also the influences of other optics of the system, including imaging system of the interferometer, environmental noises and etc. We put the step plate at the inner and outer edge of the MUT. So, we can subtract the systematic error from the measured ITF by the ratio of the ITF at inner and outer edge of MUT. The step plate has 20 to 50 pixels in the camera of the interferometer, which is sufficient to calculate the ITF from the step. Then, we fit the experimental result by Eq. (23) and the results are shown as solid lines in Fig. 17. The ITF is cut-off at 0.4 Nyquist frequency for LCM = 8600mm and ITF = 0.65 for LCM = 8400mm.

 figure: Fig. 17.

Fig. 17. Comparison of experimental ITF (solid lines) and simulated ITF (dashed lines) with different LCM.

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We can retrieve the δz while fitting the measured ITF by Eq. (23). For LCM = 8600mm case, the measured δz equals to 0.51mm while the simulated δz equals to 0.48mm. The difference between these two δz is 0.03mm. For LCM equals to 8400mm case, the measured δz equals to 0.20mm while the simulated δz equals to 0.18mm. The difference between these two δz are 0.02mm. In Fig. 17, we can barely see the difference between the two ITF curves at LCM = 8600mm. We notice that with the smaller δz, the ITF curves are more sensitive to the error of δz. So, to measure the ITF more precisely for smaller δz, one has to control the environmental noises like air turbulence and etc.

6. Conclusion

Interferometric null test with a custom-made CGH enables the ultra-high accuracy surface test for large and complex optical surfaces. However, when the test wavefront travels back from the SUT, the CGH causes the wavefront degeneration. The coupling of the wavefront degeneration and the surface errors will introduce artifacts to the surface test. It will leave mid- and high-spatial frequency features on SUT if people guide the iterative manufacture of SUT by such surface measurement. We propose an approach to simulate the wavefront degeneration in the CGH interferometric null test by complex ray tracing method with elliptical Gaussian model. Comparing with the professional physical optics software and experimental results, the elliptical Gaussian model gives a high-accuracy result of wavefront simulation in an astigmatic coherent optical system. To quantifiably characterize the influence of the wavefront degeneration, a simple and parametric equation of the ITF containing parameters of optical parameters is presented. We optimize the parameters of optical elements in CGH null test for a ∅3m aspheric mirror to suppress the wavefront degeneration. By placing an ITF probe plate on the SUT, we measure the ITF in the optical system is improved significantly.

Funding

Key Research Program of Frontier Science, Chinese Academy of Sciences (QYZDJ-SSW-JSC038); National Natural Science Foundation of China (61805243, 62005278, 62075218, 62127901); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2019221); Bureau of International Cooperation, Chinese Academy of Sciences (181722KYSB20180015).

Acknowledgments

The authors, especially Haixiang Hu and Mingzhuo Li would like to thank Qiang Cheng, Qingyuan Pang, Jiming Sun, Jinlai Xue for the large aperture optical testing experiments, Site Zhang for physical optics simulation, and Erhui Qi, Zhenyu Liu for the large optical manufacturing.

Disclosures

The authors declare no competing interests.

Data Availability

No data were generated or analyzed in the presented research.

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Data Availability

No data were generated or analyzed in the presented research.

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Figures (17)
Fig. 1.
Fig. 1. Schematic of a classic setup of null test of mirror under test. SUT: Surface under test; F: Focus of the interferometer; TS: Transmission sphere, transform the collimated beam to spherical wavefront.

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Fig. 2.
Fig. 2. Sketch of modeling the propagation of wavefront through the optical system.

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Fig. 3.
Fig. 3. a) Relay the rays to the focus of interferometer; b) Imaging the tiny bundle of rays from P to V.

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Fig. 4.
Fig. 4. Schematic of modeling the image of SUT through the optical system.

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Fig. 5.
Fig. 5. a) The astigmatic field curve δz from imaging aberration of CGH; b) The field curve δz (blue) and diameter of beam on CCD (red) with different LFC; c) The field curve δz (blue) and diameter of CGH (red) with different LCM; d) The size of CGH for different LCM.

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Fig. 6.
Fig. 6. The iterative process of SUT by the guidance of wavefront-degenerated surface measurement and the corresponding RMS. a-b) surface profile in the iterative manufacture (blue line) and the wavefront-degenerated surface measurement (red line); c) the RMS of manufactured profile (blue line) and degenerated profile (red line).

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Fig. 7.
Fig. 7. The ITF of CGH null test with different LFC and LCM. Blue line: LFC = 600 mm and LCM = 8600 mm; red dashed line: LFC = 400 mm and LCM = 8600 mm; yellow line: LFC = 600 mm and LCM = 8400 mm.

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Fig. 8.
Fig. 8. The ITF of CGH null test with different LCM while keeping the LCM + LFC = 9000 mm.

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Fig. 9.
Fig. 9. The sketch of the optical path imaged through CGH.

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Fig. 10.
Fig. 10. Comparison of a) simulation results by Eq. (14) and (b) approximated equation of the field curve δz with S and LCM. c) is the difference between the approximated equation and the simulation by Eq. (14).

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Fig. 11.
Fig. 11. Sketch of the model in the VirtualLab simulation. Green line indicates optical path along tangential direction. Red line indicates optical path along tangential direction. The incident Gaussian function has a beam waist of 20µm with wavelength of λ=632.8 nm.

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Fig. 12.
Fig. 12. a) Slices of phase distribution in the image space (λ=1µm); b) Phase distribution of the slice in x-z and y-z plane.

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Fig. 13.
Fig. 13. a-c) are results from the diffractive simulating software, VirtualLab Fusion; d-f) are results from the real part of the light field via Eq. (3).

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Fig. 14.
Fig. 14. a) The setup of an interferometric CGH test of the cylindric mirror; b) The CGH for measuring the cylindrical mirror; c) The surface measurement of the cylindrical mirror; d) The diffraction effect by the triangular beam block.

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Fig. 15.
Fig. 15. a) is the sketch of the interferometric null test; b) is the beam block which let the triangular-shape light passes; c-e) are the experimental interferometric surface measurement; f-h) are the simulated surface by the elliptical Gaussian model.

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Fig. 16.
Fig. 16. a) Experimental setup of null test; b) Phase plate put on the SUT; c) The transmitted wavefront of phase plate.

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Fig. 17.
Fig. 17. Comparison of experimental ITF (solid lines) and simulated ITF (dashed lines) with different LCM.

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Tables (1)

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Table 1. Effective standard deviations between the simulation of elliptical Gaussian model and VirtualLab

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Equations (26)

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u 0 ( r , z ) = 2 π A 0 w ( z ) | q ( z ) | exp ⁡ ( − 2 i π z λ + i ϕ G ) exp ⁡ [ i π r 2 λ q ( z ) ] .
w ( z ) = w 0 1 + ( z z R ) 2  and  q ( z ) = z − i z R  with  z R = π w 0 2 λ .
u i ( x , y , z ) = E ( z ) exp ⁡ { i [ Φ A C ( z ) + Φ G ( z ) + Φ 0 + Φ E ( x , y , z ) ] } .
Φ E ( x , y , z )  =  n π λ [ x 2 q S ( z ) + y 2 q T ( z ) ]  with  q j ( z ) = z − z j − i π w j 2 λ .
1 R N  =  cos 2 Φ R S + sin 2 Φ R T .
x 2 q x ( z ) + y 2 q y ( z )  =  ρ 2 cos 2 θ q x ( z ) + ρ 2 sin 2 θ q y ( z ) = ρ 2 q N ( z ) .
Φ AC ( z )  =  1 2 [ Φ A C ( S ) ( z ) + Φ A C ( T ) ( z ) ]  =  n π ( z − z S + z − z T ) λ .
Φ G ( z )  =  1 2 [ Φ G ( S ) ( z ) + Φ G ( T ) ( z ) ]  =  1 2 ( π 2 − arctan ⁡ z − z S z R ( T ) + π 2 − arctan ⁡ z − z T z R ( T ) ) .
E ( z )  =  2 π n E 0 z R ( S ) z R ( T ) λ 2 | q S ( z ) | | q T ( z ) | .
Sagittal: ∂ 2 φ ∂ x 2 = 1 R − u − 1 F G Tangential: ∂ 2 φ ∂ y 2 = cos 2 α R − u − cos 2 β F G .
Sagittal: ∂ 2 φ ∂ x 2 = − 1 u − 1 v Tangential: ∂ 2 φ ∂ y 2 = − cos 2 α u − cos 2 β v .
Sagittal:  R u ( R − u ) + 1 v = 1 F G Tangential:  cos 2 α cos 2 β R u ( R − u ) + 1 v = 1 F G .
S a g i t t a l : 1 u S ′ + 1 v = 1 F G w i t h u S ′ = u ( R S − u ) R S T a n g e n t i a l : 1 u T ′ + 1 v = 1 F G w i t h u T ′ = cos 2 β cos 2 α u ( R T − u ) R T .
δ z = − D 2 2 D C G H 2 [ u C ′ − L F C ⋅ u M ′ ( L 2 F C + D C G H 2 / 4 ) 1 2 ] .
I T F ( f , δ z ) = P S D [ h z ( x , δ z ) ] P S D [ h 0 ( x ) ] .
u 0 ( x ) = exp ⁡ [ i 2 π λ h 0 ( x ) ] = exp ⁡ ( i 2 π α λ sin ⁡ 2 n π f x ) .
u 0 ( x ) ≈ 1 + i 2 π α λ sin ⁡ 2 n π f x .
u 0 ′ ( x ) = u 0 ( x ) ∗ K x ( x , δ z ) | z = 0 = λ z R n [ 1 + i 2 π α λ exp ⁡ ( − n π λ z R f 2 ) sin ⁡ 2 n π f x ] .
K x ( x , δ z ) = exp ⁡ [ i n π x 2 λ ( δ z − i z R ) ] .
h 0 ′ ( x ) = α exp ⁡ ( − n π λ z R f 2 ) sin ⁡ 2 n π f x .
u z ( x , δ z ) = u 0 ( x , δ z ) ∗ K x ( x , δ z ) = λ ( z R + i δ z ) n { 1 + i 2 π α λ exp ⁡ [ − n π λ ( δ z + i z R ) f 2 ] sin ⁡ 2 n π f x } .
h z ( x , δ z ) = 1 2 arctan ⁡ δ z z R + α exp ⁡ ( − n π λ z R f 2 ) cos ⁡ n π λ δ z f 2 sin ⁡ 2 n π f x .
I T F = P S D [ h z ( x , δ z ) ] P S D [ h 0 ′ ( x ) ] = cos ⁡ n π λ δ z f 2 .
z = S 2 2 R .
δ z  =  f 2 2 f 1 2 f t 2 L F C 2 [ − L C M 2 R S 2 R 2 − L C M S 2 R 2 + R 2 S 2 R 2 ] .
δ z = − f 2 2 f t 2 f 1 2 T 2 [ ( L C M + R / 2 ) 2 − 3 R 2 / 4 ] ( P − L C M ) 2 R .
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