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Abstract
A fore optics for the hyperspectral spectrometer is designed, manufactured, assembled, and aligned. The optics has a telecentric off-axis three-mirror configuration with a field of view wider than 14 degrees and an f-number as small as 2.3. The primary mirror (M1) and the secondary mirror (M2) are axially symmetric aspheric surfaces to minimize the sensitivity. The tertiary mirror (M3) is a decentered aspheric surface to minimize the coma and astigmatism aberration. The M2 also has a hole for the slit to maintain the optical performance while maximizing the telecentricity. To ensure the spatial resolution performance of the optical system, an alignment procedure is established to assemble and align the entrance slit of the spectrometer to the rear end of the fore optics. It has a great advantage to confirm and maintain the alignment integrity of the fore optics module throughout the alignment procedure. To perform the alignment procedure successfully, the precision movement control requirements are calculated and applied. As a result, the alignment goal of the RMS wave front error (WFE) to be smaller than 90 nm at all fields is achieved.
© 2017 Optical Society of America
1. Introduction
Airborne hyperspectral imagers can obtain spectrally divided images to categorize the characteristics of the objectives [1]. The hyperspectral imagers such as MODIS, Hyperion, Hymap and CASI have been widely utilized in many applications related to the remote sensing such as mineral survey, ecosystem monitoring, gas measuring and vegetation mapping [2–5]. Since the spectrometer channel of the hyperspectral imagers reimage the light pass through the slit, the system can be designed and aligned separately in the sub-system level to be assembled in the system level [6].
Since the hyperspectral imager has more spectral bands than the multispectral imager, a fore optics for the hyperspectral imager requires tighter optical performances such as telecentricity, focus and lateral color [7]. A three-mirror reflective telecentric optics is designed to work as a fore optics in the hyperspectral imager to convey and focus the rays on the slit efficiently with minimum aberrations [7]. As a result, the designed fore optics has advantages in the transmittance and the chromatic aberration when the spectral channel has a wide range of spectral coverage from visible and near infrared (VNIR) to shortwave infrared (SWIR) [7]. To simplify and minimize the system volume, the mechanical athermal compensator is not included. Instead, the mechanical shape of each mirror is axially symmetric and all optical and mechanical structures are manufactured in the same material [8,9].
Although the established optical layout can ensure the performance of the telecentricity of 0.01 degrees in all fields with wide field of view (FOV), flatness of the field image and the small F-number, the residual aberration was not acceptable in the system level. To remove the residual aberration, two alternatives were proposed. One of the solution in the aspect of the optical design is to make all the mirrors freeform surfaces [10]. However, it has disadvantage that it is hard to manufacture the freeform surface mirror and align the optical system with freeform surfaces [11]. Also, there’s a possibility to cause a non-linear effect according to the ambient thermal variation [12]. The other solution is to make the M3 a slightly decentered aspheric surface and to make a hole in the middle of the M2. Although M1 and M2 are axially symmetric aspheric surfaces in this configuration, it has advantages of manufacturability and assemblability despite the possibilities of making the stray light paths due to the hole in the middle of the second mirror.
The pros and cons between the two proposed layouts are compared as shown in the Table 1. As shown in the table, the layout without the M2 hole requires higher level of the fabrication capability of the optical components, which leads to the higher risk of the assembly and alignment. On the other hand, the layout with the M2 hole has more advantages and lower risks. It also has higher axial symmetry of each optical surface and lower sensitivity, which can minimize the burden of the assembly and alignment. The final optical layout was selected as the one with the M2 hole and optimized to have a layout with the aspheric M1 and M2, decentered aspheric the M3 and the hole in the M2.
Table 1. Advantages and disadvantages of the layouts with or without M2 Hole
As the optical layout is set, a sensitivity analysis is performed to establish the assembly and alignment procedure. Since the fore optics channel and the spectrometer channel are designed and assembled separately in the sub-system level, the slit can be either on the rear end of the fore optics channel or the front end of the spectrometer channel. The slit is a component of the spectrometer which works as a stop and usually called as an entrance slit [13]. It need to be aligned to the dispersive optical component and the detector of the spectrometer [13]. It is easy to maintain the integrity of the alignment of the spectrometer by assembling these components in a single rigid mechanical structure in a sub system level [14]. It has great advantages that the optimum spectral resolution is fixed and maintained until the alignment process completes despite the disadvantage of being hard to adjust the fore optics module to achieve the maximum spatial resolution with this scheme. This disadvantage can be significant since it is getting harder to find the optimal position of the fore optics module when the depth of focus becomes smaller as the F-number becomes smaller. Also, it is even harder to measure the wave front error (WFE) in the spectrometer channel as a system performance because it may require to disassemble the spectrometer channel and make a measurement setup with interferometer solely dedicated to the WFE performance, which may degrade the spectral resolution of the spectrometer channel.
On the other hand, it is easy to confirm whether the system performance is optimized in the point of the spatial and spectral resolution at the same time when the slit is assembled at the rear end of the fore optics. First of all, the spatial performance can be measured on the slit and be kept its integrity till the system assembly and align is finished. After the fore optics alignment is complete, the maximum spectral resolution can be achieved and measured instantaneously by adjusting the spectrometer channel. It won’t need any break of the aligned integrity of either channel to adjust or measure any of the system performance when the spectrometer is assembled aligned properly.
In this paper, we present a different approach of the aligning and assembling the slit at the rear end of the fore optics with the precision movement stage with the sub-μm accuracy and repeatability to maximize the spatial resolution in the fore optics channel level. This alignment method has rarely been proposed since the slit is considered to be a component of the spectrometer. Although the high precision movement is required to establish the alignment method adjusting the slit, the current development of the precision movement control technology made it possible.
2. Optical design
2.1 Optical design and sensitivity analysis
The purpose of the fore optics is to collect and focus the incoming rays to the slit. To ensure the signal to noise (SNR) performance, the following specifications were assigned to the fore optics as shown in the Table 2.
Table 2. System specification assigned to the fore optics module.
The optical layout has four mirrors of M1, M2, M3, FM and a window as shown in the Fig. 1. For the reference axis of coordinates, + X axis is along the slit width direction and + Z along the slit direction as illustrated in the Fig. 1.
The optics performance shows the f-number of 2.3, FOV of over 14 ° in spatial direction and telecentricity of 0.01° in all fields. The specification of the optical system is shown in the Table 3. The M1 and M2 are axially symmetric aspheric surfaces. The M3, on the other hand, has decentered aspheric surface to compensate the residual aberration and to maintain the goal of telecentricity and the WFE. It is optimized to have a minimum decenter of 6 mm to minimize the asymmetry of the optical path. As a result, the ray path to the slit is optimized to pass through the M2 as shown in the Fig. 1. It is slightly decentered by 11.5 mm form the optimum performance position, the vertex of the M2, for the manufacturing and alignment purpose. To put the slit on the rear of M2, a rectangular hole was made in the M2. The obscuration ratio induced by the hole is about 0.9 so the effective f-number including the obscuration in the aspect of the radiometric transfer to be 2.5.
Table 3. Optical Specification
The sensitivity analysis was performed to select a proper compensator for the alignment. The sensitivity analysis is based on the change of the Zernike polynomial coefficient according to the movement of the optical component as shown in the Table 4 [15]. The most dominant aberration is the defocus aberration () and it is followed by the astigmatism aberration (), coma aberration () and spherical aberration. The slit is selected as the defocus aberration compensator since it is the most sensitive component to the defocus and independent of generating other aberration. Likewise, the M1 is selected for the astigmatism aberration compensator since its astigmatism aberration is the most sensitive term. The coma aberration can be minimized by tightening the Y axis tolerance of the M2 and the M3 and the residual coma aberration is well below the allowable level. The spherical aberration is small enough to neglect.
Table 4. Sensitivity analysis result according to the movement of each component
To ensure the integrity of the layout with the M2 hole, a brief stray light analysis was performed and all the rays pass through the slit can be blocked with additional mask in the spectrometer channel.
2.2 Accuracy of the slit position and orientation
The slit is assembled to the fore optics to measure the performance of the fore optics in a WFE point of view and maintain its integrity during the module assembly and alignment in the system level later. Since the slit works as a detector in the fore optics and object in the spectrometer channels at the same time, the assembled positional error of the slit to the fore optics module affects directly the accuracy required for the fore optics module to be aligned with the spectrometer channels.
As the slit actually cuts the rays passing through it in the finite length and width, the positional error of the slit brings about the obscuration and skewed image. When the obscuration occurs, the amount of the rays passes through the slit decreases, which leads to the lower SNR along the spectral axis and the lower spatial resolution along slit axis. When the slit is assembled in a skewed position about the center field, the acquired image is also skewed in a same manner, and requires an additional correction of this skewed image.
To assemble the slit not occurring the obscuration of the desired field rays, the allowable positional error was calculated. The slit can be assembled decentered and tilted about its center. The slit is illustrated in the Fig. 2 so as the 0 field on the slit to coincide with the origin of the reference coordinate system. The center of the slit is expressed as (a, b, c) and the tilt angle about X axis is α, about Y axis is β and about Z axis is γ as shown in the Fig. 2. β can be neglected since rotation along Y axis has small effect compare to α and γ.
When the desired focal point of each field on the slit lies along Y axis, the cross section of the incident ray bundle of arbitrary field into the spectral channels can be illustrated as shown in the Fig. 3(a) and 3(b).
When f-number is fno, semi-diameter of the ray bundle’s cross section along X axis is r, and the distance from the desired focal point to the slit along Z axis is d, r can be expressed as Eq. (1).
Since α, a, b, and c can be positive or negative, they must be included in the equations in the form of absolute. When the distance of the desired focal point from the origin is h, d-c and h-b can be calculated according to the tilt angel of the slit about X axis α as shown in the Eq. (2).Transforming Eq. (2) about and substitute d with the Eq. (1) as shown in the Eq. (3), tanα can be expressed as the Eq. (4). Then, α can be expressed as the Eq. (5) from Eq. (4).When seen from the + Z axis, the cross section of the ray bundle can be illustrated as the Fig. 3 (b). When airy disk diameter is AD, the maximum r, is limited by the slit width 24 μm and AD as shown in the Eq. (6).When the slit is decentered by a along X axis and b along Y axis, the allowable r is limited by a and from . The elliptical cross section of the ray bundle must be reshaped according to the tangent of the slanted slit by γ about Z axis and r must be calculated accordingly, γ is small enough to approximate r as shown in the Eq. (7).Substituting r in Eq. (5) with Eq. (7), α can be expressed as the Eq. (7).As a, b and c are decided by the initial position of the slit, they can be expressed otherwise as the positional accuracy of the slit. When a, b and c are equally 0, the Eq. (8) can be simplified as the following Eq. (9).Since the constants in the Eq. (9) into A and B as shown in the Eq. (10), the Eq. (9) can be even simplified as Eq. (11). As α can be expressed a function of γ as shown in the Eq. (11), its tendency according to α can be illustrated as shown in the Fig. 4.The illustrated 4 lines in the Fig. 4 are the relationships between α and γ when (a, b, c) are (0, 0, 0), (2, 0, 0), (0, 0, 2) and (2, 2, 2) in μm. (0, 2, 0) can be neglected since it has smaller effect to the relationship. When the positional accuracy of the slit decreases, the curve tends to be pulled down from the one that (a, b, c) is (0, 0, 0). As the curve is pulled down, the allowable values of α and γ are limited in a smaller region. When the positional accuracy of the central coordinates of the slit reaches ± 2 μm along each axis, the angular accuracy is limited in the region of A as shown in the Fig. 4. As a, b and c increase by 1 μm, γ decreases by 0.01°. If the number of the total steps along positive γ from 0° to 0.1° in the region A is 10, the minimum increment of the tool controlling the slit must be lower than 0.01°. When the positional error of the slit can be allowed up to ± 2 μm along each axis, the minimum increment can be estimated as 1/10 of the positional error in each axis.
The estimated requirements for the slit alignment in order to avoid the obscuration are shown in the Table 5. The travel range is derived from the required travel distance to put the slit in and out of the optical path mechanically during the alignment procedure. As the allowable γ is about ± 0.1° in the Fig. 4, the minimum increment of θ in each direction is to be 0.01°, 1/10 of the maximum γ in one direction. As the increment along each axis corresponding to 0.01° in the Fig. 4 is about 1 μm, the minimum increment in each axis is to be 1 μm. The repeatability can be loosened in the same level of the minimum increment in each direction because the minimum increment is already assigned to be 1/10 of the required position and the same level of the repeatability only loses one step of the increment in each direction.
Table 5. Requirements for the slit alignment
3. Assemble and alignment
3.1 Assemble and alignment procedure
As stated in the previous chapter of the sensitivity analysis results, astigmatism can be compensated by M1, coma by M3 and defocus by the slit and M1. When the optical components are fabricated according to each component’s tolerance requirements, coma can be neglected after the alignment is finished. Since the tolerance requirements to each optical and mechanical component are allocated accordingly, M1 is assigned as the compensator for astigmatism and the slit for the defocus.
The installing of optical components including slit requires the precise measurements of the manufactured components. Since the whole alignment procedure neglects the residual coma aberration, initial assemble error has an effect on the difficulty of the rest of the procedure. When the WFE goal is reached in all fields with the compensator adjustment, the defocus compensation amount in each filed must be measured in terms of the movement of the slit. If the amount of the defocus compensation exceeds 100 μm, in one direction, the system integration procedure with the spectrometer channels must be reconsidered.
The procedure requires a high precision movement of the slit with the resolution smaller than 0.1 μm. To perform the procedure properly, the 6 axis precision movement control stages were considered to utilize by the requirements of the slit and the compensator control. Total two precision movement control stages were required for the alignment process and the requirements of the 6 axis precision movement control stages for the slit and the compensator control are as shown in the Table 6.
Table 6. Brief Specifications of the 6 axis precision movement control stage
3.2 Manufactured optical components
The mirrors are fabricated according to the tolerance allocated. The manufactured mirrors’ measurements of RMS WFE and height difference of each leg are as shown in the Table 7. The RMS WFE are measured with computer generated hologram (CGH) and interferometer, the height difference with the coordinate measuring machine (CMM). The measurement result implies that the established alignment procedure is valid. The RMS WFE of each mirror surface is below 30 nm satisfying goal of the tolerance allocated to each optical component. Each mirror has three legs to be attached on the mechanical frame. The reference surface of each leg which will contact the corresponding reference surface of the mechanical frame must be in a single plane with the displacement tolerance of less than 20 μm so as to neglect the residual aberration induced by the assembly. The height difference of the attaching surfaces of the legs in each mirror are reached below 9.6 μm and even to 6.5 μm, which are well below the goal of 20 nm.
Table 7. Manufactured Mirrors
The mechanical slit adapter has two reference planes for the slit to be assembled. As the slit hole was manufactured according to the drawing, some mechanical lengths of the slit associated with the mechanical adapter must be measured and verified before they are assembled together. The detailed measurements of the slit are shown in the Table 8. Due to the detailed measurements of the slit, the tolerance of the assembling the slit and its adaptor was achieved smaller than 10 μm.
Table 8. Manufactured Slit and its measurements
3.3 WFE performance
The overall mechanical structure is as shown in the Fig. 5. The reference axis shown in the Fig. 5 coincides with the one in Fig. 1 and the installed 6 axis precision movement control stages which were attached to M1 and the slit. The slit is assembled behind the M2.
To measure the initial WFE, the reference source of the interferometer was aligned according to the initial mechanical position of the slit. Then, + 1 and −1 field were measured. When the mirrors are not aligned properly, the incoming rays are often blocked by the slit. To speed up the alignment process, the slit was removed from the line of sight (LOS) temporarily by the 6 axis precision movement control stages.
The initial WFE measurements are as shown in the Table 9. The first WFE measurements of each field are about 0.7 λ exceeding expectation. When broken down into Zernike terms, the astigmatism aberration took the most amount of the WFE. It also implies that the manufacturing tolerance was allocated properly and manufactured accordingly since the coma aberrations in all three fields were in allowable level.
Table 9. Initial WFE measurement
After the initial measurement, M1 is moved to compensate the astigmatism according to the alignment procedure. When the astigmatism aberration removed, the dominant residual aberration is coma and defocus. The residual coma aberration is allowable level in all fields while defocus aberration is exceeding allowable level and must be eliminated. In case that the image plane of the optics on the slit surface is curved in any direction, it may be highly tricky to balance the defocus aberrations in all three fields evenly.
As shown in the second column of the Table 10, the movement of all slit is fairly linear. To balance the defocus aberration in each field, the possibility to flatten the field curvature along Y direction was surveyed. As a result, the movement was rebalanced as shown in the third column in the Table 10. After putting the slit back to the initial position of the optical path, the movements of each field in the third column of the Table 10 are applied. The WFE in each field was measured after the slit is installed and the defocus is compensated as shown in the Table 11. As shown in the Table 11, the WFEs are balanced in each field satisfying the goal of RMS WFE 90 nm in all fields.
Table 10. Movement to the best focus position of each field
Table 11. Final WFE measurement after slit assembled
4. Conclusion
A three-mirror telecentric optical system is designed, manufactured, assembled and aligned. The M2 has a hole for the slit in the middle to make the manufacturing process easier and minimize the sensitivity. After the stray light analysis and the sensitivity analysis, a proper alignment procedure utilizing a 6 axis precision movement control stages is established. The slit is positioned in the rear end of the fore optics and assigned as a final defocus compensator. With this assemble and aligning scheme, the maximum spatial resolution is achieved and confirmed with the interferometer.
The final RMS WFE performance is well below 90 nm at all fields satisfying the system requirement. Due to the movement resolution and repeatability of a precision 6-axis stage, an adventurous optical layout could have been proposed. The proposed alignment method is proven to have been optimized for the fore optics of the hyperspectral imager when the slit is assembled to the rear end of the fore optics to maximize the spatial resolution.
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