Multispectral Earth Polarization Observation Based on the Lagrange L1 Point of the Earth–Moon System

Abstract

We propose a Multispectral Earth Polarization Imager (MEPI), which is located at the Earth–Moon system’s Lagrange point L1. The imager can be used to measure the sunlight reflected by the Earth and the Moon. The measured sunlight has specific polarization information and spectral information, which can provide strong support for a comprehensive understanding of the Earth system and the construction of a perfect Earth–Moon system model. The MEPI provides multispectral images with wavelengths of 400–885 nm, and uses four sub-aperture systems to share a main system. The imager can capture the two-dimensional shape and polarization spectral information of the entire Earth at a spatial resolution of 10 km, and all spectral images can be simultaneously acquired on a single detector. The optical system of the instrument was designed and simulated. The simulation and analysis results showed that the camera can obtain high-quality images of the Earth disc with a 2.5° field of view (FOV). The novel MEPI provides a new way to generate climate-related knowledge from the perspective of global Earth observation. The imager can also be used for lunar observation to obtain spectral polarization information on the lunar surface. In addition, it also shows great potential in other applications of space remote sensing spectral imaging.

1. Introduction

The Multispectral Earth Polarization Observation System is an indispensable instrument for analyzing the polarization information of the Earth’s reflected sunlight. The livability of the Earth is the result of the exquisite synergy of complex systems such as geological structure, atmospheric circulation, marine ecology, and biological evolution. Earth polarization observation, with its keen capture of the microscopic characteristics of each layer of the Earth, breaks the limitations of traditional spectral observation. With its high sensitivity and accuracy, it has become the core pillar technology for understanding global climate change, evaluating ecosystems, and exploring livable planets [1,2,3].

The Multispectral Earth Polarization Observation System is located at the Earth–Moon system’s Lagrange point L1, which is about 327,000 km away from the Earth. At this position, the Earth’s rotation characteristics can be used to realize the stable observation of the whole Earth, and polarization observation is carried out on a daily time scale. The complete Earth image information can be obtained at the same time at the L1 point, which effectively guarantees the time synchronization of data in different regions [4]. The Earth–Moon L1 point is a special position with great scientific research value. At this point, the polarized spectral data generated by the Moon’s reflection of sunlight can be effectively obtained. The analysis of polarization information can accurately identify typical terrain features such as craters and impact debris on the lunar surface. Secondly, due to its high stability of reflectivity and emissivity, the Moon has become an ideal radiation source for on-orbit calibration and attenuation inspection. In the practice of polarimeter calibration, the polarization state of the Moon’s reflected light is used as a reference, and through accurate comparison and calculation, the measurement deviation of the polarimeter can be effectively corrected and the measurement accuracy can be significantly improved. Therefore, the Moon can be regarded as an extended linearly polarized plane source for the relative calibration of polarimeters [5,6,7]. From the system point of view, located at the L1 point of the Earth and the Moon, by integrating the polarization characteristics data of the Earth’s atmosphere and the polarization spectrum information of the Moon, a more perfect and accurate Earth–Moon system model can be constructed, which provides a solid data support and theoretical basis for the interdisciplinary research of astronomy, earth science, space physics and other disciplines.

So far, polarization remote sensing technology has been used to detect and image the Earth, such as DPC of China, HARP2 of NASA, POLDER of France and so on [8,9,10]. However, the current instrument is too small; it cannot fully obtain the desired multi-spectral information, polarization information, and images of the Earth [9]. In the polarization detection technology scheme, the American HARP2 instrument and the French POLDER instrument have adopted two completely different technical routes. The former uses the amplitude-division method combined with a narrow-band filter to measure, and it uses a prism to separate the light of different polarization states in space and guide the three detectors, thereby simultaneously capturing images of the three polarization states. The POLDER instrument in France is mainly based on a wide-field telescope optical system, and the polarization measurement is carried out by using rotating polarizer technology combined with a single detector [10]. These two measurement techniques have inevitable problems. Polarization measurement by prism beam splitting depends on a multi-detector array. Each detector unit has inherent differences in sensitivity, noise and other properties, which makes it difficult to be fully calibrated. This affects the accuracy and consistency of polarization measurements and limits its application in high-precision tasks. The time-sharing measurement method of a rotating polarizer runner faces the bottleneck of time resolution. Because it scans different polarization states with the rotating polarizer, it takes a long time to complete a scanning cycle and cannot be completed instantaneously. Taking the observation of a cloud by DPC as an example, when considering the factors such as cloud movement, Earth rotation and satellite operation, the imaging range change caused by cloud movement (speed 20 m/s) accounts for 14% of a single pixel in the single measurement time of the rotating manipulator structure. Moreover, after the manipulator rotates for one week, the cloud layer is likely to have exceeded the imaging range of the detector. In the face of the rapid dynamic changes of the Earth target, this measurement method can easily miss the key polarization information, resulting in a lack of information in the time dimension. It is difficult to meet the needs of high frequency monitoring of the polarization characteristics of the Earth target, and there are obvious limitations in capturing the rapidly changing polarization characteristics. In addition, airborne and ground-based polarization imagers have obvious shortcomings. The observation range of the two is limited, and they are greatly interfered with by the ground environment and atmospheric boundary layer. There is a certain gap between the observation breadth, accuracy and environmental adaptability and the satellite-borne polarization imager.

Based on the demand for Earth observations and the shortcomings of the existing load, we designed new observation equipment, the Multispectral Earth Polarization Imager (MEPI), which performs multi-spectral and polarization information synchronous with observations of the whole Earth from the Earth–Moon L1 point. We used the advantages of the sub-aperture structure to image multiple polarization states on a single detector at the same time, and adopted a new method of coating the surface of the detector to replace the moving component of the traditional filter to split the light, eliminating vibrations, stray signals, beam drift and other problems. The MEPI has the nature of a snapshot. In addition to its compact structure, the MEPI can also dynamically capture rapidly changing information such as clouds, aerosol and water, effectively improve the real-time performance of satellite remote sensing detection, and provide high-precision and multi-dimensional data support for in-depth study of the characteristics and changes of clouds and aerosols, surface vegetation and soil, so as to help achieve important scientific goals such as a more comprehensive and accurate understanding of the operating mechanism of the Earth’s ecosystem and climate change laws. By accurately obtaining the polarization signal of the whole Earth, it can provide a solid theoretical and data basis for further exploring the key aspects of the atmospheric composition structure and surface physical characteristics of extrasolar terrestrial planets.

In this paper, a new instrument concept combining mosaic coating and simultaneous polarization measurement is proposed, which breaks through the limitations of the asynchronous imaging of polarization information and spectral information in the past. Nine types of spectral information covering three polarization states are simultaneously imaged on a detector so as to realize the simultaneous acquisition of the five-dimensional information of polarization, spectrum and imaging. Figure 1 shows a schematic diagram of the overall system structure. The polarization instrument with a split-aperture structure [11] not only shows the characteristics of being light, small and compact at the structural level, but also significantly improves the real-time performance of satellite remote sensing detection in the functional dimension, which effectively solves the problems of time-sharing imaging of traditional rotating manipulators and imaging calibration and synchronization caused by multiple detectors. At the Earth–Moon system’s Lagrange point L1, the MEPI will measure the Earth’s comprehensive information of nine spectral bands (400–420 nm, 433–453 nm, 460–480 nm, 480–500 nm, 522–542 nm, 545–565 nm, 590–610 nm, 660–680 nm, 845–885 nm), with three polarization angles of 0°, 60°, 120°, and the spatial resolution of Earth observation up to 10 km. The observation of the whole Earth can obtain information about the whole Earth’s disc at the same time, which fundamentally avoids error sources caused by regional splicing, and thus improves the accuracy and reliability of the data.

2. Polarization State and Observation Analysis

2.1. Polarization State

The solar flux and polarization reflected by the Earth can be described by the vector $S$:

$$S=\left[\begin{array}{cccc}I& Q& U& V\end{array}\right]$$

(1)

where $I$ is the total reflection flux and $Q$ and $U$ describe the linear polarization flux. The linear polarization imaging obtains the measurement target from the three components of $I$, $Q$ and $U$. $V$ describes the circular polarization flux. In nature, the circular polarization information is basically 0, so the first three vectors are enough to express the polarization characteristics [12]. That is, through different θ azimuths, different measurements are combined, and $I$, $Q$ and $U$ in the Stokes parameters can be obtained simultaneously. There are two common combinations of three polarization angles: 0°, 60°, 120° and 0°, 45°, 90°. In the case of the same positioning error, different positioning methods have different measurement errors. In comparison, the polarization detection azimuth setting (0°, 60°, 120°) has less measurement error for the polarization detection error, which is an ideal polarization analytical azimuth parameter.

$$I={\displaystyle \frac{2}{3}}\left(I_0+I_{60}+I_{120}\right)$$

(2)

$$Q={\displaystyle \frac{2}{3}}\left(2I_0-I_{60}-I_{120}\right)$$

(3)

$$U={\displaystyle \frac{2\sqrt{3}}{3}}\left(I_{60}-I_{120}\right)$$

(4)

Partially polarized light can be regarded as a mixture of completely polarized light and non-polarized light. The degree of linear polarization (DOLP) is used to represent the ratio of linearly polarized light intensity to total light intensity, which is the ratio of completely linearly polarized light intensity to total light intensity:

$$DOP={\displaystyle \frac{\sqrt{Q^2+U^2}}{I}}$$

(5)

The polarization angle (AOP) describes the angle of rotation of the most obvious direction of the electric vector vibration relative to the x-axis:

$$\epsilon ={\displaystyle \frac{1}{2}}\arctan {\displaystyle \frac{U}{Q}},{\displaystyle \frac{-\pi }{4}}\le \epsilon \le {\displaystyle \frac{\pi }{4}}$$

(6)

When the beam is linearly polarized light, the epsilon angle satisfies $\tan 2\epsilon =0$; when the beam is non-polarized light or circularly polarized light, $\tan 2\epsilon =\pm 1$. When $\tan 2\epsilon$ is another value, it means that the beam is elliptically polarized light or partially polarized light [13].

2.2. Overall Observation

The orbits used by the spaceborne platform to observe the polarization information of the Earth are mostly sun-synchronous orbits. However, there are specific limitations in the observation of this orbit position; that is, only the information of a specific fixed area of the Earth can be observed at the same time within a day, and it is difficult to achieve continuous acquisition of the entire polarization information of the Earth. When implementing the observation of the Earth at the L1 point, the characteristics of the Earth’s rotation are fully utilized, and a single satellite can achieve stable observation of the entire Earth, and polarization observation operations can be carried out on a daily time scale. The Lagrangian gravitational equilibrium point exists in the Earth–Moon system. The Earth’s rotation period is fixed at 23 h, 56 min and 4 s, and its centroid position is constant and the attitude is stable in the Earth–Moon coordinate system, which makes it possible to achieve conditional stability for the satellite’s orbit [4,14,15]. Therefore, the Earth–Moon L1 point becomes an ideal platform for Earth polarization observation. In addition, the position and optical properties of the Moon are relatively stable, and its trajectory in the sky can be accurately predicted, which makes the observation arrangement more convenient to accurately arrange the polarization observation program. The physical properties of the Moon itself change slowly in a long time scale, and the polarization characteristics of its reflected light also have high repeatability, which provides a stable reference standard for long-term and periodic calibration work. Figure 2 shows the Earth–Moon system observed from the Earth–Moon L1 point.

Compared with the regional measurement method, the full Earth measurement can obtain complete Earth image information at the same time, which effectively guarantees the time synchronization of data in different regions. This feature is of great significance for in-depth exploration of natural phenomena with spatio-temporal correlations such as atmospheric circulation and ocean circulation on a global scale. It can accurately analyze its interaction mechanism and dynamic change law, and then significantly improve the data quality and its application value in earth science research. It provides a more reliable and effective observation basis for a comprehensive and in-depth understanding of the complex processes of the Earth’s system.

2.3. Wavelength Selection

The sunlight reflected by the Earth not only has polarization characteristics but also has wavelength selectivity. The optical properties (such as scattering, absorption, reflection) and microphysical properties (such as particle size, shape, concentration) of the light radiated by different observation targets are all functions of the wavelength. For example, at certain wavelengths, the scattering and absorption of light by aerosol particles will show very different characteristics, and the scattering characteristics of cloud droplets will also change with the wavelength [12,16]. Through the comprehensive utilization of polarization information in different spectral bands, the inversion accuracy of different target characteristic parameters in Earth observation can be effectively improved. This is because the polarization information of different bands can reflect the characteristics of the target from multiple dimensions, and can obtain more comprehensive and accurate information, so as to reduce errors in the inversion of parameters such as aerosol optical thickness and the effective radius of cloud droplets.

Considering the practical application requirements of the Earth’s surface information, atmospheric aerosols and cloud optical and microphysical properties, we refer in depth to the band selection of other orbital polarization detectors [16,17], such as the detailed data shown in

Table 1. Polarization band selection of on-orbit instrument.

Satellite/Sensor	Country	Band Center (nm)
DPC	China	490; 670; 865
APS	U.S.A.	412; 443; 555; 672; 865; 910; 1370; 1610; 2200
POSP	China	380; 410; 443; 490; 670; 865
3MI	Eumetsat	410; 443; 490; 555; 670; 865; 1370; 1650; 2130
MAI	China	560; 670; 865
CAPI	Japan	670; 1640
HARP2	NASA	440; 550; 670; 870
POLDER-1,2	Japan	443; 670; 865
POLDER-3	France	490; 670; 865

. After rigorous screening of a large amount of existing data, a total of seven bands were selected.

On this basis, we innovatively propose two new bands for measurement. Among them, the band near 532 nm has unique optical properties and shows strong penetration to light. In the field of vegetation monitoring, this band has extremely important application value. It can be used to estimate key indicators such as vegetation coverage and leaf area index. Vegetation coverage reflects the degree of vegetation coverage on the Earth’s surface. For the global carbon cycle, the accurate estimation of its coverage is helpful to accurately evaluate carbon absorption and then provide indispensable data support for global carbon cycle and ecological balance research. The 600 nm band shows a significant advantage in the detection of soil characteristics. Through the measurement and analysis of the polarization information of this band, a variety of key characteristic information of the soil can be accurately obtained, which has important guiding significance for agricultural production planning. In summary, the whole instrument is finally determined to have a total of nine bands. The specific selection of each band and its corresponding detailed uses are clearly listed in

Table 2. Wavelength selection targets of the MEPI.

Band Center (nm)	Bandwidth (nm)	Main Purposes
410	20	Absorbent aerosol and ash cloud monitoring
443	20	Aerosol absorption and height, fine mode aerosol size, aerosol refractive index; cloud masking
470	20	Application of joint inversion of liquid water clouds and aerosol characteristics on clouds
490	20	Aerosols, clouds, sea color, surface albedo
532	20	Clouds and aerosols, vegetation
555	20	Surface albedo
600	20	Aerosol fine mode size distribution, cloud screening and characterization, vegetation
670	20	Clouds, aerosols, vegetation
865	40	Clouds, aerosols, vegetation, and surface characteristics

. The comprehensive application of these bands will provide a comprehensive and accurate database for ground polarization measurements, and promote the research and application of related fields.

3. Design

3.1. System Technical Requirements

The polarization imager is deployed at the Earth–Moon system’s Lagrange point L1. The primary task is to accurately determine the spatial coordinates of the L1 point and analyze the field of view requirements of the optical system accordingly. As a special position on the Earth–Moon line, the L1 point is the point where the gravity of the Earth and the Moon reaches a dynamic balance with the centrifugal force generated by the relative motion, and provides a stable platform for the scientific load to be fixed with the relative position of the Earth and the Moon. Under the assumption that the Earth is an ideal sphere, the half field of view (FOV) required for the overall observation of the Earth from the L1 point can be determined by the spatial geometric relationship between the observation point and the celestial body shown in Figure 3. Specifically, the half-field angle refers to the angle between the light reaching the edge of the imaging surface and the main optical axis starting from the main optical axis of the lens. According to Equation (7), we can calculate the minimum half-field angle required to observe the Earth’s edge from point L1 to ensure that the optical system design can meet the specific field coverage requirements.

$$\sin \omega ={\displaystyle \frac{R_E}{L}}$$

(7)

where $R_E$ is the radius of the Earth, ${\mathrm{R}}_{\mathrm{E}}=6371\mathrm{km}$; L is the distance from the Earth–Moon L1 point to the $L=0.85d$, and d is the average distance of the Earth–Moon, $\mathrm{d}=384403.9\mathrm{km}$. According to the analysis and calculation, under ideal conditions, assuming that the Earth is a perfect sphere, the minimum half-field angle required for the global measurement of the Earth from the L1 point of the Earth is 1.12°. Under the condition of actual orbit operation, the observation system needs an uncertainty of about 0.1°. In the case of meeting the minimum field of view of the Earth’s total observation and reserving a sufficient margin, the half field of view of the system is finally determined to be 1.25° and the full field of view is 2.5°. The accurate analysis of the spatial position of the L1 point and the scientific calculation of the field of view of the optical system can provide a solid theoretical basis and technical support for the deployment and operation of the polarization imager.

3.1.1. Mosaic Coating

As for the acquisition of wavelength, the current spaceborne instruments mainly adopt the forms of filter beam splitting and prism beam splitting. As far as the splitting of the filter is concerned, there are many limitations in the commonly used realization method of the filter wheel. The filter wheel relies on mechanical rotation to switch different filters to filter light of a specific wavelength. This mechanical movement process is relatively slow, resulting in a serious restriction on the imaging speed. Especially in the face of rapidly changing observation scenes, due to the difficulty of improving the imaging frame rate, it is only suitable for shooting tasks for stationary targets. For example, in the observation scene of some surface fixed geomorphological features, the filter wheel may also meet the basic needs. However, once dynamic targets such as fast-moving clouds and dynamic ocean surface phenomena are involved, it is difficult to capture continuous and clear image sequences, resulting in missing key information and unable to meet the demands of high-precision real-time observation. Although the prism splitting technology can decompose the mixed light into beams in different directions according to the wavelength difference to achieve the splitting effect, it requires multiple corresponding detectors to receive these signals separately, which directly leads to the overall structure of the instrument becoming complicated and bloated. Multiple detectors not only occupy a lot of space, but also need to support complex signal transmission and processing lines, which greatly increases the volume of the instrument. This huge instrument architecture, whether in the satellite’s limited carrying space layout, or in the subsequent equipment maintenance, update and upgrade process, is facing a lot of inconvenience and high cost.

In view of the above drawbacks exposed by filter beam splitting and prism beam splitting, in order to effectively reduce the resource and cost costs of multi-spectral images in the acquisition and transmission stages, this paper introduces mosaic coating technology in the system, that is, a pixel-level coating process [18], as shown in Figure 4a. Specifically, we cover the image sensor with a specially designed multilayer dielectric film, which exhibits different spectral transmission characteristics at different locations, achieving selective transmission of light in specific spectral bands. The feasibility of the method has been proven by simulation analysis [19]. This technique allows each pixel of the image sensor to respond to only one specific spectral band, thereby obtaining multiple spectral images in the working band range of 400 nm to 900 nm at one time. The bandwidth of each band is precisely controlled at 20 nm (except for the 865 nm band where the bandwidth is 40 nm). Through this array detection technology, we can obtain high-resolution imaging data and realize the so-called mosaic pixel coated multi-spectral array detector measurement technology. Finally, this technique can produce high-resolution multi-spectral imaging data. In this way, it not only avoids the slow imaging of the filter wheel and the redundancy of the instrument caused by the prism splitting, but also significantly improves the efficiency and convenience of multi-spectral image acquisition and successfully solves the long-term problem of multi-spectral image acquisition and transmission. The problem provides a new solution for the performance optimization and function expansion of spaceborne optical instruments.

The ground resolution of MEPI is 10 km, and each pixel unit of its detector is composed of a $3\times 3$ pixel array, so that the total number of pixels required for a single image is $1274.2\times 1274.2$ units. In view of the four-channel architecture of the system, there are four Earth images with the same size but different polarization information, so the total number of pixels required by the whole system is $2548.4\times 2548.4$ units. Under the premise of maintaining the optical aperture spacing, the system uses a CMOS image sensor, which has a pixel size of 3.2 μm and an effective resolution of $16556\left(H\right)\times 9200\left(V\right)$. According to this, the calculated image height d of each image can be deduced to be 12.23 mm.

3.1.2. Scientific Requirements

The focal length f of the optical system is closely related to the maximum image height d on the image plane and the field of view ω of the system. When $f>h$, $\omega \approx h/f$. As shown in Figure 4b, according to the determination of the parameters such as the location L of the MEPI and the size D of the measured target, the focal length and the field of view need to meet the following relationship:

$$f={\displaystyle \frac{d·L}{D}}$$

(8)

According to the main applications of the MEPI, the scientific requirements of the system can be clarified.

• The field of view (FOV) of the instrument should cover the entire Earth at the Earth–Moon system’s Lagrange point L1;
• The instrument measures the polarization of the whole Earth. Instantaneous imaging is applied to avoid losing climate change information during operation [14];
• The resolution is 10 km, which can distinguish between cloudy and clear sky scenes. It can be clearly imaged in nine spectral bands;
• The continental map is derived from the integrated signal of the disc, and the salient features such as vegetation, aerosol, and surface can be identified.

The MEPI will achieve its scientific objectives by recording and analyzing the reflected sunlight of the Earth and the Moon. The target technical requirements of the MEPI are shown in

Table 3. Technical requirements.

Parameter	Specification
Number of channels	4
FOV	2.5°
Effective Focal Length (EFL)	313.67 mm
Entrance Pupil Diameter (EPD)	30 mm
Polarizations	0°; 60°; 120°
Detected wavelength band (nm)	400–420; 433–453; 460–480
	480–500; 522–542; 545–565
	590–610; 660–680; 845–885
Swath (to the Moon)	1127.275 km
Ground Sampling Distance (GSD)	10 km (Earth)
	34.5 km (Moon)

. When the index is set, a certain safety margin is set aside to ensure that the MEPI can function stably and efficiently in the face of a complex environment and potential changes.

3.2. Optical Design of MEPI

As shown in Figure 5, the MEPI consists of a data processing unit with an image sensor, a front main system and a four-aperture subsystem. Each main system and the sub-aperture system constitute a polarization measurement channel. Among them, the front main system plays a key role. It can promote the efficient transmission of light with different polarization states through the same aperture so as to achieve the common field of view effect of the system. In addition, the existence of the pre-system makes the image plane more compact and the image will be more concentrated, which further improves the utilization rate of the detector pixel. The aperture stop is set on the first surface of the subsystem, where the off-axis operation is performed and cooperates with the subsystem. Finally, the entire optical system constructs four independent imaging channels. The aperture is located on the first surface of the subsystem. The MEPI can measure nine spectra, arranged in three rows and three columns. Four sub-aperture systems are used for the measurement of polarized light, three of which are polarized channels and the other is a non-polarized channel. The non-polarized channel provides an additional calibration parameter, which can reduce the error caused by the measurement system, so that the data can more truly reflect the actual physical quantities.

The solar radiation reflected from the Earth enters the system, converges through the main system, and is then divided into four beams of light by the subsystems. The advantage of the system lies in the simultaneous polarization measurement of light in different spectral bands with high resolution and a significant reduction in overall weight and volume. The subsystem efficiency at different wavelengths will be predicted in the laboratory.

3.2.1. Design Approach

In the design process of the sub-aperture polarimeter, it is of great significance to deeply consider the radius of the Earth and the key position of the L1 point of the Earth. With the help of the geometric relationship model constructed by the radius of the Earth and the position of the Earth–Moon L1 point, we accurately derived the key parameters such as the field of view, the height of the image radius, and the focal length of the system. According to these parameters, we have designed the sub-aperture polarimeter so that it can stably and efficiently present the clear image of the whole Earth at the L1 point of the Earth and Moon and achieve the overall detection target. This achievement highlights the importance of the research and application of relevant parameters, and provides strong support for the research and monitoring of earth science.

The role of the sub-aperture imaging system is to divide the optical path into four channels and convert the same object into four images. In the design consideration of the sub-aperture system, the scale parameter selection of the system aperture $d_2$ should be close to the single image plane diameter d, aiming to minimize the insufficient utilization of the image plane. At the same time, the spacing of the aperture deviation needs to be controlled within a reasonable range to ensure that the system energy can be efficiently utilized so as to optimize the overall optical performance and energy conversion efficiency of the sub-aperture system and meet the accuracy and stability requirements of the system in practical applications. As shown in Figure 6, given a single channel aperture $d_2$ and a reserved distance m, the outer tangential aperture $D_2$ of the subchannel can be calculated:

$$D_2=\left(\sqrt{2}+1\right)d_2+\sqrt{2}m$$

(9)

where the total focal length of the system is f, the focal length of the main system is $f_1$, and the focal length of the sub-channel system is $f_2$. The focal length formula between the two systems can be obtained as follows:

$${\displaystyle \frac{1}{f}}={\displaystyle \frac{1}{f_1}}+{\displaystyle \frac{1}{f_2}}-{\displaystyle \frac{d}{f_1{f}_2}}$$

(10)

Then, according to the ideal object–image relationship formula:

$${\displaystyle \frac{1}{l^{\prime }}}-{\displaystyle \frac{1}{l}}={\displaystyle \frac{1}{f}}$$

(11)

$$l_2=l_1^{\prime }-d$$

(12)

We can calculate the total length of the system:

$$L\prime =d+l_2$$

(13)

In the process of system design, it is necessary to comprehensively weigh the geometric scale parameters of the whole system, scientifically plan the focal length distribution scheme of the main system and the sub-channel system, and achieve the expected goals and technical requirements of the system design.

3.2.2. Result of Design

The system consists of a pre-common-aperture objective lens group, an aperture-divided imaging lens group, and a post-polarization component. The polarization information of each channel light is different, and the wavelength is the same (the influence of the polarization element on the optical system can be regarded as a parallel glass plate, so no polarization element is added in the optical design). In the design process, through Table 3, we first determine the focal length of the front and rear systems, which are 1033 mm and 235 mm, respectively. For these two systems, we adopt an independent design strategy and use ray tracing to optimize their optical structures. In the combination process, the combined system is constrained and optimized based on the known parameters to ensure that the system meets the design requirements. Finally, a four-channel optical layout with an F-number of 7.6 is obtained, as shown in Figure 7.

Table 4. Eccentric distance of aperture group.

Channel	X-Axis Eccentric Distance	Y-Axis Eccentric Distance
1st subsystem	15	0
2nd subsystem	0	−15
3rd subsystem	−15	0
4th subsystem	0	15

shows the eccentric distance of each channel in the four systems. The eccentric distance is defined as the distance deviation between the optical axis of the subsystem and the central axis of the main system along the x-axis or y-axis direction, which determines the spatial distribution of the subsystem. Since the MEPI requires imaging in the spectral range of 400–885 nm, the lens material must be able to transmit the corresponding band. We continuously simulated different groups of materials in ZEMAX and finally obtained the optimal solution through hammering and global optimization.

Table 5. MEPI main system lens data.

	Front Surface Curvature	Back Surface Curvature	Thickness	Material
1st lens	353.201	567.661	35.135	H-ZLAF4LB
2nd lens	−525.432	160.852	45.742	H-ZBAF1
3rd lens	153.085	−265.159	29.173	H-FK61
4th lens	−115.491	−123.943	12.487	D-ZF10

and

Table 6. MEPI sub-channel lens data.

	Front Surface Curvature	Back Surface Curvature	Thickness	Material
1st lens	39.171	501.552	8.248	H-FK61B
2nd lens	56.774	31.182	1.458	H-ZLAF92
3rd lens	31.056	49.063	3.381	H-ZF7LAGT
4th lens	−34.272	−24.365	3.937	H-LAK1
5th lens	−22.324	−38.321	5.812	H-ZLAF55D

show the relevant data of the optical lens of the MEPI.

4. Optical Performance Display

The modulation transfer function (MTF) is an important parameter to evaluate the imaging quality of an optical system. Figure 8 is the MTF curve of the MEPI. In Figure 8, the lines in different colors correspond to fields between −1.25° and +1.25°. The black curve in the figure is the diffraction limit MTF curve, and the other curves are the MTF curves under different fields of view and wavelengths.

$$N={\displaystyle \frac{1000}{2D_d}}$$

(14)

At 52 cycles/mm (calculated using Equation (14), $D_{\mathrm{d}}$ refers to the pixel size [20]), the minimum MTF of the system is greater than 0.5, and the image quality and spatial resolution are good.

Figure 9 is the spot diagram on the ray tracing image plane under different fields of view of the system. The RMS radius values of the spot diagram are all less than 9.6 μm, and the image points in the maximum field of view can be relatively concentrated in the range of a single detector pixel.

Both the MTF and spot diagrams have difficulties in fully describing the imaging performance of optical systems. The reason is that neither of them provides an intuitive visual representation of the distortion situation. Therefore, Figure 10 shows the distortion curves of each channel at the maximum FOV, with different colors representing different wavelengths. Statistics show that the slight change of image height is less than 1 pixel at the maximum distortion.

5. Discussion

In this paper, we introduce the MEPI based on the Lagrange L1 point of the Earth–Moon system, which can measure the multi-spectral polarization of the entire Earth and has the ability to produce images. The system provides critical multi-spectral polarization data for various applications, such as Earth cloud and aerosol properties, global water cycle monitoring, and extrasolar planetary exploration.

The MEPI uses a CCD detector, the optical system transmittance is 0.63, the integration time is 1 s, and F is 7.6. We calculated the signal-to-noise ratio of the system when measuring the Earth. The calculation results show that the SNR of the system is better than 400; as shown in Figure 11, it has good observation performance.

As a stable and easy-to-observe natural celestial body, the Moon is an ideal polarization calibration source [5,6,7]. In the field of polarization calibration, the Moon shows significant advantages: (1) Stability: the polarization characteristics of the Moon’s reflected sunlight fluctuate very little in a long time scale, which provides a very stable reference for polarization calibration, ensures the reliability of the reference signal during the calibration process, and effectively reduces the measurement error caused by the instability of the calibration source; (2) Predictability: The parameters such as the trajectory and phase change of the Moon can be accurately predicted, so that researchers can reasonably arrange the time and conditions of calibration observation, which greatly improves the efficiency and success rate of calibration observation. (3) Wide-band response: The spectral coverage of the lunar reflected light is wide, which can be highly adapted to the needs of multi-channel observation systems. The lunar reflected light provides a rich and comprehensive data source, which helps to comprehensively evaluate the polarization response characteristics of the system in different spectral ranges [21].

Using the Moon as the calibration source is the key to verify the reliability and accuracy of the polarization performance of the system. In the on-orbit calibration, the polarization performance of the instrument is evaluated from the dimensions of polarization degree, polarization angle and channel consistency. According to the orbital parameters and phase changes of the Moon, the phase angles (such as 0°, 30°, 60°, 90°) are selected as the observation window to cover the range of polarization characteristics [22], and the random error is reduced by repeated observations at each observation point. The difference between the lunar polarization degree and polarization angle measured by the system and the theoretical value is compared, and the measurement accuracy and polarization direction judgment accuracy are accurately evaluated. By comparing the consistency of the observation results of the three channels in the same area of the Moon, the channel performance balance is judged to help the instrument performance optimization. At the same time, because the reflection characteristics of the lunar surface are stable and the radiation characteristics can be accurately calculated or measured, the satellite observes the sunlight reflected by the Moon and compares it with the known radiation model or reference data to determine the radiation response characteristics of the sensor and realize the calibration of the observation data. This calibration method provides a stable calibration target, realizes multiple calibrations to reduce errors and improve accuracy, and meets the requirements of multi-spectral calibration.

There are many difficulties in arranging the observation system at the L1 point of the Earth–Moon. The position of this point is unstable, and a correction model must be constructed to continuously correct the position [23,24]. The radiation intensity of the universe is far greater than that of the radiation of the Earth. The high-energy particles in the strong radiation environment will damage the detector and electronic equipment, resulting in decreased performance and a shortened life. Therefore, it is necessary to use anti-radiation materials and implement anti-radiation reinforcement of the electronic equipment to protect it [25,26]. Moreover, due to the direct exposure of the system to space, the temperature change is extremely severe, which poses a great challenge to the thermal control design. It is necessary to use efficient thermal control materials and complex thermal management systems to ensure that the working temperature of the instrument and equipment is appropriate. In addition, when the Sun enters the field of view, the instrument will also suffer from complex and strong stray radiation interference, which aggravates the difficulty of the observation task. In addition to the above problems, in the future iteration, it is necessary to focus on solving the calibration problem of the instrument before or after the task starts, optimizing the mechanical design and improving its data processing capability.

6. Conclusions

Polarization measurement at the L1 point of the Earth and the Moon can directly obtain the polarization information of the Earth and the Moon, which can avoid the fitting error caused by the fitting of data from different time series observations. In order to solve the problem that the current polarimeter mainly uses time-sharing to obtain the same target or uses multiple detectors to measure the target, this paper innovatively proposes the concept of a Multi-Spectral Polarization Imager, as shown in Figure 4, which uses a unique method of combining detector surface pixel coating with aperture division polarization information. A complete imaging function for nine spectral bands of 400–420 nm, 433–453 nm, 460–480 nm, 480–500 nm, 522–542 nm, 545–565 nm, 590–610 nm, 660–680 nm and 845–885 nm on a single detector is achieved, which is designed to serve the Earth scene recognition application. The system integrates a four-channel aperture dividing system at the back end of the common aperture telescope system with a focal length of $f=1016$ mm. In order to achieve a 2.5° field of view for the purpose of global observation of the Earth, the focal length of the four-channel lens is set to $f=200$ mm, corresponding to a spatial resolution of 10 km. While meeting the requirements of lightweight and miniaturization, the MEPI can obtain the Earth’s shape, polarization information and spectral information. Among them, geosynchronous polarization measurement can effectively improve the real-time performance of Earth polarization measurement. The optical design results showed that the MEPI has enough image quality to support scene recognition. The spatial resolution of each channel is 10 km. The RMS radius of the system is less than 9.6 μm in the full field of view. At 52 cycles/mm, the minimum MTF of each channel of the system is higher than 0.5. The images of the four channels can be collected synchronously on a single 53 mm × 29.4 mm sensor.

It is of great significance to regard the Earth as an exoplanet for global synchronous polarization observation. The atmospheric composition is affected by human activities, and the dynamic fluctuation of concentration profoundly changes the radiation balance. Aerosols are derived from volcanoes, dust, and anthropogenic emissions, and the uneven spatial and temporal distribution interferes with light propagation. The structure of ice crystals and water droplets in the cloud layer is variable, and the difference in optical properties significantly dominates the radiation transmission. Each link of the water cycle is linked to regulate the surface, dry and wet. Through polarization observation technology, we can capture the differences and changes of these elements on the second to day scale, and provide key data for revealing the internal mechanism of the Earth system, predicting climate change and monitoring natural disasters. In view of the highly dynamic nature of the Earth system and the close coupling among the various components, the simultaneous observation of the Earth can not only capture instantaneous phenomena, but also construct long-term time series data sets, open up new horizons for research in the field of earth science, and promote the in-depth exploration of the unique life maintenance mechanism and environmental change law of this ’exoplanet’.