Design and Study of a Two-Dimensional (2D) All-Optical Spatial Mapping Module

Sequentially timed all-optical mapping photography is one of the main emerging ultra-fast detection technologies that can be widely applicable to ultra-fast detection at the picosecond level in fields such as materials and life sciences. We propose a new optical structure for an all-optical spatial mapping module that can control the optical field of two-dimensional imaging while improving spectral resolution and detector sensor utilization. The model of optical parameters based on geometrical optics theory for the given structure has been established, and the theoretical analysis of the inter-frame energy crosstalk caused by incident beam spot width, chromatic aberration, and main errors of the periscope array has been conducted. The optical design of the two-dimensional (2D) all-optical spatial mapping module was finally completed using ZEMAX OpticStudio 2018 software. The results show that our optical module can realize targets of 16 frames and 1.25 nm spectral resolution.


Introduction
All-optical mapping photography [1][2][3] has the advantages of high time resolution, high luminous flux, and anti-radiation interference and can effectively overcome the shortcomings of the pump-probe technique [4,5], which is unable to perform non-repeatable or difficult-to-replicate event observation.Meanwhile, it can also avoid the drawbacks of low spatial resolution and complex image reconstruction from compressed ultra-fast spectroscopy [6,7] and multi-spectral tomography [8].This technology has significant application prospects in the fields of observation and analysis of events at the picosecond and even femtosecond scale, such as shockwave dynamics, engine tail flame detection, superconducting electronic states, structural dynamics of chemical reactions, protein folding, and the process of photosynthesis [9][10][11][12][13][14].
Amplitude-modulated all-optical mapping photography uses grating and amplitude modulation elements to control the optical field.It is a little more complex in optical structure and has a few frames and low utilization of detection sensor, which is limited by the one-dimensional control mode.However, it has the advantages of high energy utilization, high amplitude consistency, low component fabrication challenge, spatial distribution consistency (image position, spacing, and size), and flexible change of spectral range, which has gradually made it a research hotspot.
Nakagawa et al. first proposed sequentially timed all-optical mapping photography; they introduced the composition of the whole system, which used a one-dimensional periscope array in the spatial mapping module.The principal system was built, and the observation of a 200-fs ultra-fast event was realized, which was published in the highimpact journal Nature Photonics [1].On this basis, they applied this system to observe plasma and phonon polarization dynamics [15].Subsequently, temporal characteristics, such as exposure time and frame rate, were investigated [16].Although the principal system has successfully achieved hundred-femtosecond-scale observation, there are problems, such as a small number of frames and low utilization of detectors.Later, diffuse optical elements that may achieve 25-frame burst imaging were proposed for sequentially timed alloptical mapping photography using a spectral filtering technique [17,18].This system has the advantages of fewer optical elements and a compact structure; however, the fabrication of diffractive optical elements was challenging and expensive, and lower energy utilization and poor consistency of 2D diffraction efficiency limit the further development of this system.Saiki et al. used slicing mirrors instead of the periscope array for complete optical field control [19], which was then developed by Yuan et al. [2,20,21].This can effectively increase the frame and detector utilization of the instrument; however, there are problems, such as the high precision of the grating and the slice mirror and the existence of chromatic aberration that reduces the quality of imaging.
A higher spectral resolution represents a shorter frame interval, and a higher number of frames means a higher number of observation windows.Furthermore, consistency in frame quality can ensure that all viewing windows are consistent.While amplitudemodulated examples have some inherent advantages, the one-dimensional structure limits the number of increased frames, and wastes another dimension of the ultra-fast image sensor.In summary, we select the periscope array, which is much less challenging to fabricate by combining conditions, as an amplitude modulation element.In order to solve the problems of fewer frames, lower spectral resolution, and lower detector sensor utilization, an optical structure for a two-dimensional (2D) all-optical spatial mapping module is proposed.A parametric model of our optical structure is established verified by Zemax OpticStudio 2018 software.A numerical theoretical calculation is conducted for the inter-frame energy crosstalk, chromatic aberration, and main errors of the periscope array, which cannot be obtained in a non-sequential design.

Composition of All-Optical Spatial Mapping Module
The schematic diagram of a two-dimensional (2D) all-optical spatial mapping module is shown in Figure 1.The whole device consists of two orthogonal modulation modules, which include a one-dimensional diffraction grating, a cylindrical mirror, and a periscope array.
Sensors 2024, 24, x FOR PEER REVIEW 2 of 14 phonon polarization dynamics [15].Subsequently, temporal characteristics, such as exposure time and frame rate, were investigated [16].Although the principal system has successfully achieved hundred-femtosecond-scale observation, there are problems, such as a small number of frames and low utilization of detectors.Later, diffuse optical elements that may achieve 25-frame burst imaging were proposed for sequentially timed all-optical mapping photography using a spectral filtering technique [17,18].This system has the advantages of fewer optical elements and a compact structure; however, the fabrication of diffractive optical elements was challenging and expensive, and lower energy utilization and poor consistency of 2D diffraction efficiency limit the further development of this system.Saiki et al. used slicing mirrors instead of the periscope array for complete optical field control [19], which was then developed by Yuan et al. [2,20,21].This can effectively increase the frame and detector utilization of the instrument; however, there are problems, such as the high precision of the grating and the slice mirror and the existence of chromatic aberration that reduces the quality of imaging.A higher spectral resolution represents a shorter frame interval, and a higher number of frames means a higher number of observation windows.Furthermore, consistency in frame quality can ensure that all viewing windows are consistent.While amplitude-modulated examples have some inherent advantages, the one-dimensional structure limits the number of increased frames, and wastes another dimension of the ultra-fast image sensor.In summary, we select the periscope array, which is much less challenging to fabricate by combining conditions, as an amplitude modulation element.In order to solve the problems of fewer frames, lower spectral resolution, and lower detector sensor utilization, an optical structure for a two-dimensional (2D) all-optical spatial mapping module is proposed.A parametric model of our optical structure is established verified by Zemax Op-ticStudio 2018 software.A numerical theoretical calculation is conducted for the interframe energy crosstalk, chromatic aberration, and main errors of the periscope array, which cannot be obtained in a non-sequential design.

Composition of All-Optical Spatial Mapping Module
The schematic diagram of a two-dimensional (2D) all-optical spatial mapping module is shown in Figure 1.The whole device consists of two orthogonal modulation modules, which include a one-dimensional diffraction grating, a cylindrical mirror, and a periscope array.When the incident pulse is diffracted by grating dispersion and reflected by a cylindrical mirror, a series of transverse λ 1 -λ 4 spots is formed.The No. 1 periscope array shown in Figure 2 is composed of several array elements fixed together, and each single array is fixed on a flat substrate by a pair of triangular mirrors (with increasing spacing).The difference in thickness of the flat substrate is used to compensate for the optical path variation caused by the difference in distance of the triangular mirrors on different arrays.The λ 1 -λ 4 spot is modulated by the No. 1 periscope array and then returned to the diffraction grating via the cylindrical mirror.In the incident direction of the original grating, there is the transverse spatially separated λ 1 -λ 4 frame, and then it is incident to another group of the orthogonal spatial mapping module.The No. 2 periscope array is a modular splice of the No. 1 periscope array, which is orthogonal.In the same way, λ 1 is subdivided into λ 11 , λ 12 , λ 13 , λ 14 by grating dispersion and then modulated by the first module of the No. 2 periscope array, and the rest λ 2 , λ 3 , λ 4 are also modulated in the corresponding module.Finally, a 4 × 4 two-dimensional frame image is formed at the detector by the cylindrical mirror reflection and diffraction grating dispersion combining beam.When the incident pulse is diffracted by grating dispersion and reflected by a cylindrical mirror, a series of transverse λ1-λ4 spots is formed.The No. 1 periscope array shown in Figure 2 is composed of several array elements fixed together, and each single array is fixed on a flat substrate by a pair of triangular mirrors (with increasing spacing).The difference in thickness of the flat substrate is used to compensate for the optical path variation caused by the difference in distance of the triangular mirrors on different arrays.The λ1-λ4 spot is modulated by the No. 1 periscope array and then returned to the diffraction grating via the cylindrical mirror.In the incident direction of the original grating, there is the transverse spatially separated λ1-λ4 frame, and then it is incident to another group of the orthogonal spatial mapping module.The No. 2 periscope array is a modular splice of the No. 1 periscope array, which is orthogonal.In the same way, λ1 is subdivided into λ11, λ12, λ13, λ14 by grating dispersion and then modulated by the first module of the No. 2 periscope array, and the rest λ2, λ3, λ4 are also modulated in the corresponding module.Finally, a 4 × 4 two-dimensional frame image is formed at the detector by the cylindrical mirror reflection and diffraction grating dispersion combining beam.

Theoretical Design of Two-Dimensional (2D) All-Optical Spatial Mapping Module
The schematic diagram of the one-dimensional spatial mapping module is shown in Figure 3.The cylindrical mirror is equivalent to the transmissive mirror, and separation points P1, P2, P3, and P4 on the periscope array correspond to the wavelengths λ1-λ4 dispersed by diffraction grating.According to the grating equation, the diffraction angle θ0 can be expressed as arcsin sin

Theoretical Design of Two-Dimensional (2D) All-Optical Spatial Mapping Module
The schematic diagram of the one-dimensional spatial mapping module is shown in Figure 3.The cylindrical mirror is equivalent to the transmissive mirror, and separation points P 1 , P 2 , P 3 , and P 4 on the periscope array correspond to the wavelengths λ 1 -λ 4 dispersed by diffraction grating.When the incident pulse is diffracted by grating dispersion and reflected by a cylindrical mirror, a series of transverse λ1-λ4 spots is formed.The No. 1 periscope array shown in Figure 2 is composed of several array elements fixed together, and each single array is fixed on a flat substrate by a pair of triangular mirrors (with increasing spacing).The difference in thickness of the flat substrate is used to compensate for the optical path variation caused by the difference in distance of the triangular mirrors on different arrays.The λ1-λ4 spot is modulated by the No. 1 periscope array and then returned to the diffraction grating via the cylindrical mirror.In the incident direction of the original grating, there is the transverse spatially separated λ1-λ4 frame, and then it is incident to another group of the orthogonal spatial mapping module.The No. 2 periscope array is a modular splice of the No. 1 periscope array, which is orthogonal.In the same way, λ1 is subdivided into λ11, λ12, λ13, λ14 by grating dispersion and then modulated by the first module of the No. 2 periscope array, and the rest λ2, λ3, λ4 are also modulated in the corresponding module.Finally, a 4 × 4 two-dimensional frame image is formed at the detector by the cylindrical mirror reflection and diffraction grating dispersion combining beam.

Theoretical Design of Two-Dimensional (2D) All-Optical Spatial Mapping Module
The schematic diagram of the one-dimensional spatial mapping module is shown in Figure 3.The cylindrical mirror is equivalent to the transmissive mirror, and separation points P1, P2, P3, and P4 on the periscope array correspond to the wavelengths λ1-λ4 dispersed by diffraction grating.According to the grating equation, the diffraction angle θ0 can be expressed as arcsin sin According to the grating equation, the diffraction angle θ 0 can be expressed as In Equation ( 1), where d is the grating constant, θ i is the angle of incidence, m is the diffraction level, λ is the wavelength, and f is the focal length of the cylindrical mirror, then the spatial distance between two neighboring wavelengths on the periscope array is: Let δ = w.The relationship between the cylindrical mirror and the grating parameters can be expressed as follows: According to Equation ( 3), the grating constant d, the angle of incidence θ i , and the relationship with the focal length of the cylindrical mirror f can be determined.Let w 0 = 6 mm, m = 1, λ 1 = 790 nm, λ 2 = 791.25 nm, and set the groove density of grating in the range of 500-2500 g/mm, and angle of incidence in the range of 20 • -45 • .The relation curve is plotted as shown in Figure 4, where the horizontal coordinate is the density of the grating lines, the vertical coordinate is the grating angle of incidence, and the contour line indicates the focal length of the cylindrical mirror.As can be seen from Figure 4, the focal length along with an increased density of grating lines decreases under the same grating angle of incidence, while the focal length along with an increased grating angle of incidence increases under the same density of grating lines.
Sensors 2024, 24, x FOR PEER REVIEW 4 of 14 In Equation ( 1), where d is the grating constant, θi is the angle of incidence, m is the diffraction level, λ is the wavelength, and f is the focal length of the cylindrical mirror, then the spatial distance between two neighboring wavelengths on the periscope array is: Let δ = w.The relationship between the cylindrical mirror and the grating parameters can be expressed as follows: According to Equation ( 3), the grating constant d, the angle of incidence θi, and the relationship with the focal length of the cylindrical mirror f can be determined.Let w0 = 6 mm, m = 1, λ1 = 790 nm, λ2 = 791.25 nm, and set the groove density of grating in the range of 500-2500 g/mm, and angle of incidence in the range of 20°-45°.The relation curve is plo ed as shown in Figure 4, where the horizontal coordinate is the density of the grating lines, the vertical coordinate is the grating angle of incidence, and the contour line indicates the focal length of the cylindrical mirror.As can be seen from Figure 4, the focal length along with an increased density of grating lines decreases under the same grating angle of incidence, while the focal length along with an increased grating angle of incidence increases under the same density of grating lines.The value of the width b of a single mirror on the periscope array can be expressed as: By using the grating and cylindrical mirror parameters as in Table 1, the b values of the No. 1 and No. 2 periscope array can be calculated as shown in Table 2 according to Equations ( 2) and (4): When there is a width of incident spot rather than a point, inter-frame energy crosstalk will affect the imaging quality and spectral resolution of the module.The principal schematic is shown in Figure 5.
Sensors 2024, 24, x FOR PEER REVIEW 5 o The value of the width b of a single mirror on the periscope array can be expres as:

2
N N b   By using the grating and cylindrical mirror parameters as in Table 1, the b value the No. 1 and No. 2 periscope array can be calculated as shown in Table 2 according Equations ( 2) and (4): When there is a width of incident spot rather than a point, inter-frame energy cr talk will affect the imaging quality and spectral resolution of the module.The princ schematic is shown in Figure 5.As shown in Figure 5a, in the plane formed by the direction of the grating line the grating normal, the incident spot is projected as a line segment with projection size and the two endpoints are Pa and Pb.The parameters of the grating and the cylindr mirrors above are designed based on the point of Pa, and the P1, P2, P3, and P4 points the periscope array correspond to the wavelengths of λ1, λ2, λ3, and λ4, which are based the dispersion of Pa.The wavelengths λ1-λ4 no longer correspond to P1-P4 points on periscope array for the other endpoint Pb.
The equivalent optical structure is shown in Figure 5b, where w' represents offse any wavelength.We set the offset w' equal to w, and then the energy distribution of e frame is shown in Figure 6a, where the groove density of the grating is 1800 g/mm, As shown in Figure 5a, in the plane formed by the direction of the grating line and the grating normal, the incident spot is projected as a line segment with projection size w, and the two endpoints are P a and P b .The parameters of the grating and the cylindrical mirrors above are designed based on the point of P a , and the P 1 , P 2 , P 3 , and P 4 points on the periscope array correspond to the wavelengths of λ 1 , λ 2 , λ 3 , and λ 4 , which are based on the dispersion of P a .The wavelengths λ 1 -λ 4 no longer correspond to P 1 -P 4 points on the periscope array for the other endpoint P b .
The equivalent optical structure is shown in Figure 5b, where w ′ represents offset at any wavelength.We set the offset w ′ equal to w, and then the energy distribution of each frame is shown in Figure 6a, where the groove density of the grating is 1800 g/mm, the angle of incidence is 30 • , and the focal length of the cylindrical mirror is 1067.5 mm.The analysis above ignores the diffraction efficiency of the grating and the reflectivity of each Sensors 2024, 24, 2219 6 of 14 mirror surface.The horizontal coordinates represent wavelength, the vertical coordinates represent normalized energy, and the curves of different colors represent the normalized energy of each wavelength in a different frame.The flat top indicates the wavelength range without crosstalk; in Figure 6a, its ratio to the design bandwidth (1.25 nm) is between 56% and 72%.We believe the ideal situation is when the ratios are all greater than 85%.While the offset w ′ is equal to w/6, the energy distribution of each frame is shown in Figure 6b, and the ratio is between 88% and 96%, which can better eliminate the effect of inter-crosstalk.
Sensors 2024, 24, x FOR PEER REVIEW 6 of 14 angle of incidence is 30°, and the focal length of the cylindrical mirror is 1067.5 mm.The analysis above ignores the diffraction efficiency of the grating and the reflectivity of each mirror surface.The horizontal coordinates represent wavelength, the vertical coordinates represent normalized energy, and the curves of different colors represent the normalized energy of each wavelength in a different frame.The flat top indicates the wavelength range without crosstalk; in Figure 6a, its ratio to the design bandwidth (1.25 nm) is between 56% and 72%.We believe the ideal situation is when the ratios are all greater than 85%.While the offset w' is equal to w/6, the energy distribution of each frame is shown in Figure 6b, and the ratio is between 88% and 96%, which can be er eliminate the effect of inter-crosstalk.It should be additionally pointed out that, when w' = 0, the wavelengths λ corresponding to the spots Pa to Pb are all incident on the same position of the periscope array.At this time, the positions of the incident spot, the cylindrical mirror, and the periscope array satisfy the 4f condition, in which inter-frame crosstalk no longer exists.

Analysis of Chromatic Aberration due to Incident Spot Width
Considering the characteristics of the periscope array, off-axis point imaging will cause object-image flip, which results in the spatial position of the image in each frame, It should be additionally pointed out that, when w ′ = 0, the wavelengths λ corresponding to the spots P a to P b are all incident on the same position of the periscope array.At this time, the positions of the incident spot, the cylindrical mirror, and the periscope array satisfy the 4f condition, in which inter-frame crosstalk no longer exists.

Analysis of Chromatic Aberration due to Incident Spot Width
Considering the characteristics of the periscope array, off-axis point imaging will cause object-image flip, which results in the spatial position of the image in each frame, so that the image converging at the grating again does not satisfy the achromatic condition.The actual optical path from the grating and the cylindrical mirrors to the periscope array is shown in Figure 7.
Sensors 2024, 24, x FOR PEER REVIEW so that the image converging at the grating again does not satisfy the achromati tion.The actual optical path from the grating and the cylindrical mirrors to the pe array is shown in Figure 7.The distance of PP' is the dispersion direction shift on the focal plane, which be y, and the distance of AB is set to be x.Then we have: in which: The relationship between θ1 and θ2 is: in which: where L1 is the offset distance in vertical direction after reflection by the periscop and α is the angle between light ray and the normal direction of the periscope arra we can obtain: The distance of PP' is the dispersion direction shift on the focal plane, which is set to be y, and the distance of AB is set to be x.Then we have: in which: Coupling gives rise to: The relationship between θ 1 and θ 2 is: in which: The distance between the B ′ point position and point A can be determined using Equation (11), when the focal length f 2 » L 1 x, the outgoing and incident light rays satisfy the grating equation at the same time, and there is no chromatic aberration in our module.We can obtain: where λ 800 nm is the central wavelength, and for the other wavelengths: From connective Equations ( 8) and ( 13), we can obtain: Let the grooving density of the grating 1/d = 1800 g/mm, angle of incidence θ i = 30 • , L 1 = 8 mm, and x = 3 mm.Different wavelengths are diffracted by the grating and reflected back by the periscope array and cylindrical mirror.Its angle aberration after beam combination (∆θ) curve, between the outgoing angle after beam combination and the angle of incidence, is shown in Figure 8.
where λ800 nm is the central wavelength, and for the other wavelengths: From connective Equations ( 8) and ( 13), we can obtain: Let the grooving density of the grating 1/d = 1800 g/mm, angle of incidence L1 = 8 mm, and x = 3 mm.Different wavelengths are diffracted by the grating and r back by the periscope array and cylindrical mirror.Its angle aberration after beam nation (Δθ) curve, between the outgoing angle after beam combination and the incidence, is shown in Figure 8.As can be seen from Figure 8, the angular aberration ∆θ is within 0.0002057 • under condition of 1075 mm focal length, so chromatic aberration can be expressed as: where D is the total optical path, which is approximately equal to 4f, and finally we can compute the maximum chromatic aberration, which is equal to 15.44 um.

Analysis of Main Errors of the Periscope Array
Because of the long focal length of the module, any error of the periscope array may be more stringent, and the following analysis is carried out for the main errors of the periscope array fabrication.

•
Substrate angle error of the periscope array The spot is shifted when there is a substrate angle error on the periscope array, and Figure 9 shows a schematic diagram of a substrate angle error: where L represents the lateral distance from the midpoint of the step reflection unit to the rightmost side of the step, and η 1 is the angular deviation as in Figure 9.

Substrate angle error of the periscope array
The spot is shifted when there is a substrate angle error on Figure 9 shows a schematic diagram of a substrate angle error:  • Triangular reflector traverse error of the periscope array As shown in Figure 10a,b, the periscope array has a triangular reflector traverse error, which can lead to the transverse shift of the spot.

Analysis of Main Errors of the Periscope Array
Because of the long focal length of the module, any error of the periscope arr be more stringent, and the following analysis is carried out for the main errors of t iscope array fabrication.

Substrate angle error of the periscope array
The spot is shifted when there is a substrate angle error on the periscope arr Figure 9 shows a schematic diagram of a substrate angle error: where L represents the lateral distance from the midpoint of the step reflection uni rightmost side of the step, and η1 is the angular deviation as in Figure 9.Let h be the side length of the isosceles right triangle and the spot size set to 6 mm.The spot transverse displacement can be expressed as: • Triangular reflector angle error of the periscope array As shown in Figure 11a, if the reflector of the periscope array has an angle error that is around the normal, then the spot offset can be expressed as: Sensors 2024, 24, x FOR PEER REVIEW Let h be the side length of the isosceles right triangle and the spot size set to The spot transverse displacement can be expressed as: As shown in Figure 11a, if the reflector of the periscope array has an angle er is around the normal, then the spot offset can be expressed as: Figure 11b shows the schematic diagram of the top-view principal of a tria reflector, and the spot offset can be expressed as:

  
The total error can be expressed as: We use the distance between adjacent frame boundaries (1.5 mm) as the tot ΔY constraint and, with comprehensive consideration of the sensitivity of the er the level of fabrication, the error is distributed and tightened as shown in Table 3.  Figure 11b shows the schematic diagram of the top-view principal of a triangular reflector, and the spot offset can be expressed as: The total error can be expressed as: We use the distance between adjacent frame boundaries (1.5 mm) as the total error ∆Y constraint and, with comprehensive consideration of the sensitivity of the error and the level of fabrication, the error is distributed and tightened as shown in Table 3.

Results
The technical parameters, such as in Tables 1 and 2, are selected according to the above theoretical model, and the two-dimensional (2D) all-optical spatial mapping module is designed using ZEMAX software.A four-frame transverse module is constructed as shown in Figure 12a, and the MTF of the transverse module in sequence mode is shown in Figure 12b.It reaches 0.64 at 50 lp/mm, so the 50 lp/mm MTF of a two-dimensional module can be estimated to be above 0.3.The complete optical structure is shown in Figure 12c, and the spot distribution on the image plane of the complete optical structure is shown in Figure 12d.The design results show 1.25 nm spectral resolution and 16 wavelengths with the 4 × 4 frame arrangement on the detector.It should be explained that the unequal spacing of each frame is due to the non-linearity of grating dispersion.Fine tuning of the angle of the corresponding glass substrate of the periscope array allows for an approximate equidistant arrangement, as shown in Figure 12e.

Results
The technical parameters, such as in Tables 1 and 2, are selected according to the above theoretical model, and the two-dimensional (2D) all-optical spatial mapping module is designed using ZEMAX software.A four-frame transverse module is constructed as shown in Figure 12a, and the MTF of the transverse module in sequence mode is shown in Figure 12b.It reaches 0.64 at 50 lp/mm, so the 50 lp/mm MTF of a two-dimensional module can be estimated to be above 0.

Figure 2 .
Figure 2. Schematic diagram of the periscope array with top view and front view.

Figure 3 .
Figure 3. Schematic diagram of one-dimensional spatial mapping module.

Figure 2 .
Figure 2. Schematic diagram of the periscope array with top view and front view.

Figure 2 .
Figure 2. Schematic diagram of the periscope array with top view and front view.

Figure 3 .
Figure 3. Schematic diagram of one-dimensional spatial mapping module.

Figure 3 .
Figure 3. Schematic diagram of one-dimensional spatial mapping module.

Figure 4 .
Figure 4. Relation curve of grooving density of grating 1/d, angle of incidence θi and focal length f of cylindrical mirror.

Figure 4 .
Figure 4. Relation curve of grooving density of grating 1/d, angle of incidence θ i and focal length f of cylindrical mirror.The colored part in Figure4can meet the 6 mm line dispersion requirement and, considering the factors of fabrication difficulty, grating diffraction efficiency and component position interference, the final selected parameters are shown in Table1:

Figure 5 .
Figure 5. (a) Schematic diagram of energy crosstalk at the vertical optical path cross-section schematic diagram of energy crosstalk at the cross-section of the optical path.

Figure 5 .
Figure 5. (a) Schematic diagram of energy crosstalk at the vertical optical path cross-section; (b) schematic diagram of energy crosstalk at the cross-section of the optical path.

Figure 6 .
Figure 6.(a) Energy distribution of each frame when w' = w; (b) energy distribution of each frame when w' = w/6.

Figure 6 .
Figure 6.(a) Energy distribution of each frame when w ′ = w; (b) energy distribution of each frame when w ′ = w/6.

Figure 7 .
Figure 7. Schematic diagram of imaging characteristics of off-axis points of incident light s

Figure 7 .
Figure 7. Schematic diagram of imaging characteristics of off-axis points of incident light spots.

Figure 8 .
Figure 8. Relation curve of wavelength λ and angle aberration after beam combination Δθ.

Figure 8 .
Figure 8. Relation curve of wavelength λ and angle aberration after beam combination ∆θ.

Figure 9 .Figure 10 .
Figure 9. Schematic diagram of substrate angle error of the periscope a the direction of light propagation.The dashed line represents the ideal the solid line represents the actual propagation direction when there is

Figure 9 .
Figure 9. Schematic diagram of substrate angle error of the periscope array.The arrow indicates the direction of light propagation.The dashed line represents the ideal propagation direction, and the solid line represents the actual propagation direction when there is an error.

Figure 9 .Figure 10 .
Figure 9. Schematic diagram of substrate angle error of the periscope array.The arrow indi the direction of light propagation.The dashed line represents the ideal propagation directio the solid line represents the actual propagation direction when there is an error.

Figure 10 .
Figure 10.(a) Schematic diagram of the triangular reflector traverse error of the periscope array in the direction of the outgoing light; (b) Schematic diagram of the triangular reflector traverse error of the periscope array in the vertical direction of the outgoing light.


Triangular reflector angle error of the periscope array

Figure 11 .
Figure 11.(a) Schematic diagram of pitch angle error of the triangular reflector.The arrow i the direction of light propagation.The dashed line represents the ideal propagation direct the solid line represents the actual propagation direction when there is an error; (b) schem gram of azimuth angle error of the triangular reflector, 1-4 represent different triangular re and A-D represent corresponding triangular reflectors, where C triangular reflector have muth angle error.

Figure 11 .
Figure 11.(a) Schematic diagram of pitch angle error of the triangular reflector.The arrow indicates the direction of light propagation.The dashed line represents the ideal propagation direction, and the solid line represents the actual propagation direction when there is an error; (b) schematic diagram of azimuth angle error of the triangular reflector, 1-4 represent different triangular reflectors, and A-D represent corresponding triangular reflectors, where C triangular reflector have an azimuth angle error.

Table 1 .
1: Main parameters of grating and cylindrical mirror.

Table 1 .
1: Main parameters of grating and cylindrical mirror.

Table 2 .
Value of b for the two periscope array.

Table 2 .
Value of b for the two periscope array.

Table 3 .
Final numerical value of errors.