## 1.

## Introduction

As aperture increases, the mass of space telescope increases dramatically and the large mirror with high accuracy surface figure is difficult to fabricate. In order to solve these problems, many institutions have begun to study diffractive telescope technology.^{1}^{,}^{2} At present, diffractive telescopes are designed based on similar principles.^{3} A flat thin glass or membrane with diffractive pattern is used as primary lens to focus incident light. For removing the chromatic aberration of the diffractive lens, a relay system with complementary diffractive optical elements (DOE) must be included according to the Schupmann achromatic model. It is known that the DOEs have multiorder diffraction, but only the +1 order is used in the diffractive telescopes and the other orders may propagate to the image plane as stray light. Many publications have reported the stray light effects of DOEs in different optical system. Xiaotong Li et al. studied the ghost images caused by DOEs in the high-power laser system.^{4} Their method is based on the paraxial analysis. Several works studied the effects of the diffraction efficiency on the modulation transfer function in hybrid optical system with weakly powered diffractive lens.^{5}6.^{–}^{7} For diffractive telescope systems, the prototypes in “Fresnel diffractive array imager^{8}^{,}^{9} and “membrane optic imager real-time exploitation” only verified the imaging capability of the diffractive structures.^{10} But, the stray light performance of these prototypes was not considered. Recently, the diffraction efficiency and fabrication errors of the DOEs used in diffractive telescopes were studied,^{11}^{,}^{12} but the relationship between diffraction efficiency and the imaging performance was not studied. In this paper, we thoroughly analyzed and tested the stray light effects in a diffractive telescope. First, a diffractive telescope prototype was developed based on the Schupmann achromatic model. Then, the optical properties and bidirectional scatter distribution function (BSDF) were given to each surface and the stray light analysis model was established. A comprehensive analysis of stray light characteristics was carried out. Nonsequential ray tracing technology and wave optical simulation were used to analyze the ghost images and the point source transmittance (PST). The veiling glare index (VGI) was calculated by integrating the PST over the full hemisphere. Finally, the test of ghost images and the VGI were carried out.

## 2.

## Stray Light Model of Diffractive Telescope

## 2.1.

### Optical Model of Diffractive Telescope Prototype

Based on the Schupmann achromatic model, a diffractive telescope is designed and fabricated. Figure 1(a) shows the optical schematic. There are six optical elements, two DOEs, three doublets, and one filter. The parameters of the optical elements are listed in Table 1.

## Table 1

Main parameters of the optical elements.

Radius of curvature | Length to next surface (mm) | Semidiameter (mm) | |
---|---|---|---|

Front of DOE1 | Infinity | 0.3 | 40 |

Back of DOE1 | Binary 2 | 661.8 | 40 |

Sur1 of L2 | 76.77 | 8 | 13 |

Sur2 of L2 | $-96.83$ | 5 | 13 |

Sur3 of L2 | $-1267.10$ | 215.2 | 13 |

Front of DOE2 | Infinity | 3 | 13.6 |

Back of DOE2 | Binary 2 | 5 | 13.6 |

Sur1 of L4 | 283.1 | 10 | 17 |

Sur2 of L4 | $-33.27$ | 5 | 17 |

Sur3 of L4 | $-50.35$ | 4 | 17 |

Sur1 of L5 | 76.77 | 12 | 17.5 |

Sur2 of L5 | $-40.4$ | 6 | 17.5 |

Sur3 of L5 | $-109.65$ | 70 | 17.5 |

Sur1 of L6 | Infinity | 3 | 9.7 |

Sur2 of L6 | Infinity | 77.73 | 9.7 |

DOE1 is a two-level phase Fresnel diffractive lens for collecting light. Its focal length is 800 mm when the wavelength is 550 nm and the width of the outmost zone is 11 μm. In ZEMAX, the phase function is used to describe the diffractive lens. For DOE1, the coefficients of the phase function are $a1=-7.14$, $a2=2.789\times {10}^{-6}$, $a3=-2.179\times {10}^{-12}$, $a4=2.278\times {10}^{-18}$, $a5=-4.365\times {10}^{-23}$. DOE2 is also a diffractive lens for correcting the chromatic aberration of primary lens, the coefficients of the phase function are: $a1=-58.476$, $a2=2.393\times {10}^{-3}$, $a3=-3.404\times {10}^{-6}$, $a4=-4.340\times {10}^{-9}$, $a5=-1.443\times {10}^{-12}$. L2, L4, and L5 are doublets and L6 is a bandpass filter. The geometry model of the prototype is shown in Fig. 1(b), the outside shade is hidden for the sake of displaying the internal structure.

The prototype is designed to verity the imaging capabilities of diffractive telescope technology and explore possible solutions for the next large-aperture light-weight space telescope. It is also an experiment platform for studying the imaging performance of diffractive telescopes. The main parameters of the prototype are shown in Table 2 and the MTF is shown in Fig. 2. This paper focused on the stray light characteristics and intended to provide prior knowledge and experience for optimizing the imaging performance of larger aperture diffractive telescopes.

## Table 2

Main parameters of the prototype.

Aperture of DOE1 | Focal length | Field of view | Spectral range | Center wavelength |
---|---|---|---|---|

80 mm | 361.5 mm | $\pm 0.2\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$ | 486 to 656 nm | 550 nm |

## 2.2.

### Stray Light Source

Stray light is the light that reaches image plane following an unintended path. These paths include specular reflection and refraction, diffraction, and random scattering.

The reflection and refraction of the optical surface subject to the Snel’s law and the reflectivity and transmittance can be calculated by Fresnel formula if uncoated. Smooth surface of the mechanical structure also has specular reflection, and the reflectivity depends on the processing technology.

For binary DOEs, the diffraction efficiency ${\eta}_{m}$ is given as^{13}

## (1)

$${\eta}_{m}^{L}={\{\frac{\mathrm{sin}[\pi (m-\varphi )]}{\mathrm{sin}(\pi (m-\varphi )/L)}\times \frac{\mathrm{sin}(\pi m/L)}{\pi m}\}}^{2},\varphi =\frac{{\lambda}_{0}}{{\lambda}_{i}}\times \frac{\sqrt{{n}_{i}^{2}-{\mathrm{sin}}^{2}\text{\hspace{0.17em}}\theta}-\mathrm{cos}\text{\hspace{0.17em}}\theta}{{n}_{0}-1},$$## (2)

$${l}^{\prime}=l+\frac{m\lambda}{2\pi}\times \frac{\partial \varphi (r)}{\partial x},\phantom{\rule[-0.0ex]{1em}{0.0ex}}{p}^{\prime}=p+\frac{m\lambda}{2\pi}\times \frac{\partial \varphi (r)}{\partial y},$$Random scatter is often described by BSDF which is defined as the surface scattering radiance normalized by the incident irradiance of the surface

## (3)

$$\mathrm{BSDF}({\theta}_{i},{\varphi}_{i};{\theta}_{s},{\varphi}_{s})=\frac{dL({\theta}_{i},{\varphi}_{i};{\theta}_{s},{\varphi}_{s})}{dE({\theta}_{i},{\varphi}_{i})}.$$For the optically smooth surface, a Harvey BSDF model can simulate the scatter due to RMS surface microroughness $\sigma $. The functional form of Harvey BSDF^{14} is

^{15}where $k$ is the surface transmittance or reflectance, $\mathrm{\Delta}n$ is the refractive index change. Let Eq. (5) equals the integration of Eq. (4), ${b}_{0}$ can be calculated.

The scattering from the sharp transition regions between different zones of DOE is another source to cause stray light. A perfect Lambert scatter model is a reasonable assumption to estimate the scatter distributions and the TIS is the ratio of the transition regions to the total area.^{16}

Particle contaminations may cause random scatter and the numbers of particles on a surface can be specified by the MIL-STD-1246D.^{17} Based on rigorous Mie theory, the scatter characteristics can be addressed as Harvey BSDF model.^{18}

## 2.3.

### Stray Light Analysis Methods

## 2.3.1.

#### Ghost image analysis

Ghost images are often generated by the residual specular reflection of refractive optical elements. The path and the distribution on the image plane of ghost can be obtained directly using the nonsequential ray tracing. But tracing the collimated rays that cover the entrance aperture is a huge amount of computation and very time consuming. To solve the problem, a one-dimensional collimated ray grid is traced first. Then, the critical ghost paths can be selected according to the following conditions:

1. The number of rays in the path is greater than half the number of the imaging rays;

2. The flux of rays in the path is more than 0.01% of the imaging flux.

After that the intensity distribution of ghost images in each critical path can be analyzed one by one. For each analysis, only the reflection or diffraction of the ghost-surface that produces the ghost image in the path needs to be considered.

## 2.3.2.

#### PST calculation

PST is frequently used to evaluate an optical system’s ability to suppress stray light from outside field of view. It equals to the average detector irradiance ${E}_{\mathrm{det}}$ divided by the incident irradiance ${E}_{\text{inc}}$

## (6)

$$\mathrm{PST}(\theta )=\frac{{E}_{\mathrm{det}}(\theta )}{{E}_{\mathrm{inc}}(\theta )}=\frac{{P}_{\mathrm{det}}(\theta )/{A}_{\mathrm{det}}}{{P}_{\mathrm{inc}}(\theta )/{A}_{\mathrm{inc}}}.$$Before calculating PST, it is necessary to set the reasonable scattering models to each surface. Then tracing a collimated grid rays with different incident angles. The ${E}_{\mathrm{dec}(\theta )}$ is the stray light flux reaching the image plane divided by the area of detector, and the $\mathrm{PST}(\theta )$ is the ratio of ${E}_{\mathrm{dec}(\theta )}$ to ${E}_{\mathrm{inc}(\theta )}$. For the nonrotationally symmetric systems, the PST is the function of zenith angle and azimuth angle between the stray light source and the optical axis.

## 2.3.3.

#### VGI calculation

PST describes the stray light performance of an optical system as the function of incident angle between point source and the optical axis. It cannot give an intuitive elevation of stray light background on the image plane. VGI can be used to quantify the stray light performance, and it is equal to

where ${E}_{s}$ is the irradiance on the center of image plane due to stray light and the ${E}_{0}$ is the irradiance of image. ${E}_{s}$ can be calculated from PST by integrating it over the extended source’s solid angleFor the optical system with small field of view, ${E}_{s}$ can be calculated as

## 3.

## Analysis Results of the Prototype

## 3.1.

### Ghost Images

In the prototype system, the doublets and the filter are coated with antireflection coating. Its reflectivity is 1.0% over the spectral range. The DOEs are composed of substrate and surface microstructure. The substrates are uncoated, so the reflectivity is 4.0%. The diffraction efficiency of microstructure is calculated according to Eq. (1). The ghost images results at the wavelength of 550 nm are shown in Figs. 3(a) and 3(b). The color scale shows the relative irradiance increasing from blue to red. In Fig. 3(a), the map of intensity distribution is in the linear space. Only the ghost image from the -first-order diffraction is obvious, that is to say it is the most serious ghost image. In Fig. 3(b), the map of intensity distribution is in the logarithmic space, where several ghost images appear. The ghost images are not focused at the image plane, but the focal point of these ghost images are near the image plane, so we also be careful to deal with that ghost images. From the results, we can find that the ghost images of the prototype come from two kinds of paths. One is the reflection of filter and the other is the same order diffraction of the DOE1 and the DOE2.

## 3.2.

### PST Calculation Result

The RMS roughness of optical surfaces is tested by white light interferometer; the results are shown in Table 3. The surface TIS is calculated by Eq. (5) and the ${b}_{0}$ is determined by the method described in Sec. 2.2.

## Table 3

RMS roughness and TIS of the optical surface.

Surface | Doublet and filter | Front of DOE | Back of DOE |
---|---|---|---|

RMS roughness (nm) | 0.991 | 1.550 | 5.014 |

TIS, $\lambda =0.55$ (%) | 0.0034% (K9 glass) 0.0054% (ZF1 glass) | 0.0066 | 0.0687 |

${b}_{0}$ | 0.17 for K9 glass and 0.27 for ZF1 glass | 0.033 | 3.46 |

Figure 4 shows the PST curves in log space from 0.5 deg to 80 deg. When the off-axis angle is less than 25 deg, the sample interval is 0.5 deg and 5 deg for off-axis angle greater than 25 deg. There are some interesting features in the plots. The source outside the field of view can directly reach the detector by the nondesign order diffraction of DOE1. For angles less than 22.5 deg, the diffraction of DOE1 contributes the most stray light flux. Beyond 22.5 deg, the light scatter from DOE1 is dominant. Figure 4 shows the PST in different sections and the main contribution path. In the calculation, we only consider the diffraction from $-7$ to $+9$, because the efficiency of higher orders is less than 0.5%.

## 3.3.

### Estimate of the VGI

The flux reaching the detector is determined by

## (10)

$$\mathrm{\varphi}={\int}_{{\lambda}_{1}}^{{\lambda}_{2}}S(\lambda )\tau (\lambda )V(\lambda )\mathrm{d}\lambda ,$$## 4.

## Test Results of the Prototype

## 4.1.

### Ghost Images

In Sec. 3.1, the ghost irradiance was analyzed at the wavelength of 550 nm. But in application, the light source is polychromatic. At different wavelength, the diffraction efficiency and the light deflection ability of DOEs will change according to Eqs. (1) and (2). So the distribution of the ghost irradiance will change correspondingly. Considering that factor, the spectral range was divided into 17 small segments. The ghost irradiance was analyzed at the middle wavelength of each segment. The 17 results were superposed and shown in Fig. 5(a). The center of the plot has been truncated to show the detail of ghost irradiance around the image point. To obtain the ghost distribution, the exposure time of charge-coupled device was improved when getting the star image of the prototype. When the intensity of star image is saturation, the ghost images appear around the image as shown in Fig. 5(b). The test results are in accordance with the simulation results; while the actual size of the ghost distribution is 0.58 mm, larger than the calculated value 0.53 mm. This difference may be caused by the fabrication error and assembly error of the prototype.

## 4.2.

### VGI of the Prototype

According to the ISO9358:1994,^{19} a VGI test platform was built as shown in Fig. 6. The integral sphere with diameter of 1.5 m is used to simulate the background. The luminance uniformity is better than 3%. The detector is low light level illumination meter and the indication error is less than 4%.

The calculated value and the test result of VGI are shown in Table 4. The calculation value is 57.88% and the test result is 67.11%. The agreement in magnitude shows the correctness of the analytical methods. The difference between calculated value and test value may be caused by two factors. First, only the diffractive order between $-7$ and $+9$ was considered. The total diffractive efficiency is about 95%, the other orders will obviously increase the stray flux. Second, the processing errors of the DOE were not taken into account. These errors will reduce the efficiency of the +1 order and increase the efficiency of some nondesign orders.

## Table 4

The VGI of the prototype.

Test result | Calculated value | |
---|---|---|

VGI | 67.11% | 57.88% |

## 5.

## Summary and Conclusions

In this paper, the stray light characteristics of the diffractive telescope were analyzed comprehensively. For the diffractive telescope system, except for the residual reflections of the optical elements, the nondesign order of DOEs is an important cause of the ghost. The ghost path is mainly the same order diffraction of the diffractive primary lens and the achromatic DOE. The source outside the field of the view can directly illuminate the image plane through the nondesign order of diffractive primary lens according to the analysis of the PST. This stray light path contributes to more than 90% of the stray light flux on the image plane. Therefore, improving the diffraction efficiency of the diffractive primary lens is the key to suppression of stray light in diffractive telescopes. In the future, further studies on effective measures to suppress stray light and improve imaging performance will be in progress.

## Acknowledgments

This work was supported by the National Key R&D Program of the Ministry of Science and Technology of the People’s Republic of China, grant No. 2016YF0500200 and the Chinese Academy of Sciences, grant No. YA16K010.

## References

## Biography

**Dun Liu** was a PhD student at the University of Chinese Academy of Sciences (UCAS). In 2013, he received his bachelor’s degree at Wuhan University. He has been conducting the research of diffractive telescope technology at the Institute of Optics and Electronics (IOE), Chinese Academy of Sciences, from 2013 to present. His current research interests include optical design and stray light analysis and testing.

**Lihua Wang** has been an associate researcher at IOE since 2017. Her research topics include optical design and subaperture stitching testing of large aperture optical systems.

**Wei Yang** has been an associate researcher at IOE since 2010. Her research topics include membrane optical imaging technology, integrated simulation, and automatic testing technology of optical systems.

**Shibin Wu** has been a research director since 2006 at IOE. His research topics include optical testing and precision instruments. He is a member of the Chinese National Optical Metrology Committee.