Investigation of scan errors in the three-element Risley prism pair

Anhu Li; Wei Gong; Yang Zhang; Xingsheng Liu

doi:10.1364/OE.26.025322

1. Introduction

Double-prism scanners [1], generally composed of a matched Risley-prism pair, have been pursued by scientists and engineers for many decades, because of its compact structure, vibration insensitivity, outstanding dynamic performance, and excellent adaptability [2–4]. Compelling results have been demonstrated in applications such as intersatellite lasercom [5], infrared countermeasure [6], laser tracking [7, 8] and others, to name a few.

However, several problems still exist in the double-prism scanner, such as distortion [9], chromatic dispersion [10] and so on. Among them, scan blind zone and control singularity are two prevalent adverse issues for the beam scan performance in double-prism Risley systems [11]. Besides, complete coverage of all pointing angles can only be achieved when the pair of prisms is exactly matched [12]. Thanks to the advances in blind-zone elimination [13], control singularity elimination [12], and field of view (FOV) expansion [14], the introduction of a third prism has been verified for favorable adaptability and flexibility, characterized by wide scan range, high accuracy and diverse scan trajectories in the three-prism scanner.

Our previous paper [11] has established inverse solutions for a three-prism system, and eliminated the scan blind zone and control singularity by a multi-mode beam scan pattern. Whereas, the research is incomplete for lack of the error analysis, which is a key index for the beam scan performance. The error sources of Risley prisms mainly include component errors (i. e. the errors in wedge angle and refractive index), prism orientation errors and assembly errors. Among them, the prism orientation errors belong to random errors, while component and assembly errors are systematic errors. Research results about some error sources have been achieved by this decade. In 2012, J. S. Horng and Y. Li [15] elucidated different types of error sources and their impact on the performance of dual-wedge beam steering systems. In 2014, the first-order approximations of the error analysis were derived and compared with the exact results [16]. In the same year, D. Fan et al. [17] elucidated the pointing error distributions in the field of regard and evaluated the allowances of the error sources for a given pointing accuracy. The attention of these papers is concentrated on errors in component alignment, such as wedge prism tilt and bearing tilt, while other errors, like the component errors, prism orientation errors and so forth, are rarely mentioned. It is not comprehensive for the whole system. Recently, having considered the effects of the prism thickness and the separation between prisms, B. Bravo-Medina et al. [18] proposed a paraxial approximation method to compensate typical alignment errors. However, the research mainly presented analytical and numerical results that elucidate the greater impact of errors introduced by misalignment than the approximation errors, without mention of the impact of each error source.

Due to the additional degree of freedom, uncertainty and complexity, the error analysis of the three-element Risley-prism scan system (TRSS) has become a heavy work. To simplify the analysis process, introducing intelligent optimization algorithm such as genetic algorithm [19] may have better effects. To my knowledge, no relevant research has been reported yet.

In this paper, categories of error sources are redefined and their impacts on the pointing accuracy are graphically presented with analytical and numerical results. For thin prisms, error compensation algorithm for the TRSS is proposed to achieve high-accuracy beam scanning. The paper is outlined as follows. In Section 2, the theoretical model for the TRSS is illuminated as the foundation to investigate various scan errors. In Section 3, error sources and their impact on the beam pointing accuracy is illustrated with simulation results. The forward algorithm for error compensation is established in Section 4. Conclusions are finally drawn in Section 5.

2. Theoretical model

Figure 1 shows a schematic diagram illustrating the configuration of the TRSS. Three prisms, named prism 1, prism 2 and prism 3, are sequentially arranged along the Z-axis, and each prism is prescribed with refractive index n, wedge angle α, thinnest-end thickness d₀ and clear aperture D_p. The distances between the adjacent surfaces of the plane surface of prism 1, prism 2, prism 3 and the receiving screen P are denoted as D₁, D₂ and D₃, sequentially. Three prisms can rotate independently around the Z-axis, with rotation angles defined by θ_r₁, θ_r₂ and θ_r₃, respectively, where the clockwise rotation angle is prescribed as the positive angle. Supposing that the incident beam propagates along the direction specified by the ray vector A_r₀ = (0,0,1) ^T and passes the center of the incident surface of prism 1, it finally arrives at the receiving screen P after refractions at each surface of three prisms.

Fig. 1 Schematic diagram of the TRSS. All prism surfaces are marked as ∑ with the first subscript “1”, “2” or “3” to distinguish each prism and the second subscript “1” or “2” indicating the incident or emergent surface of the prism. It is notable that ∑₁₁, ∑₂₂ and ∑₃₂ are plane surfaces exactly perpendicular to Z-axis, whereas ∑₁₂, ∑₂₁ and ∑₃₁ are wedge surfaces inclined to those plane surfaces, respectively.

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Each prism possesses two types of configurations, and there are 8 possible configurations in the TRRS. For convenience of description, all prism surfaces are marked as A and B, indicating the plane facet and the wedge facet of the prism, sequentially. Then the 8 possible configurations can be expressed as AB-AB-AB, AB-AB-BA, AB-BA-BA, AB-BA-AB, BA-AB-AB, BA-AB-BA, BA-BA-BA and BA-BA-AB, respectively. The configuration AB-BA-BA is chosen in this paper as an example from 8 possible configurations.

In the initial state, the principal section of each prism is located in the XOZ plane with the thinnest end towards the positive X-direction, namely, the rotation angle of each prism is equal to 0°. Only by determining the principal section of three prisms, the initial orientation of each prism can be accurately calibrated and defined as the reference for calculating prism rotation angle. A conventional method to determine the principal section of prism is the reticle mark in manufacturing process, which can only be utilized as an approximation approach. In practical applications, we can readjust the positions of principal sections by means of autocollimation tool [24]. Position error in the principal section of each prism may account for the actual prism orientation ahead or behind the theoretical one, which leads to the degradation of the pointing accuracy of laser beam.

2.1 Beam pointing vectors

The prism surfaces on the beam path are sequentially described by unit normal vectors:

\begin{array}{l} N_{11} = {(0, 0, 1)}^{T}, N_{12} = {(\cos θ_{r 1} \sin α, \sin θ_{r 1} \sin α, \cos α)}^{T}, \\ N_{21} = {(- \cos θ_{r 2} \sin α, - \sin θ_{r 2} \sin α, \cos α)}^{T}, N_{22} = {(0, 0, 1)}^{T}, \\ N_{31} = {(- \cos θ_{r 3} \sin α, - \sin θ_{r 3} \sin α, \cos α)}^{T}, N_{32} = {(0, 0, 1)}^{T} . \end{array}

According to Snell’s law [20], the vector of each refracted beam is obtained from

A_{r j + 1} = \frac{n_{j}}{n_{j + 1}} [A_{r j} - (N \cdot A_{r j}) N] + N \sqrt{1 - {(\frac{n_{j}}{n_{j + 1}})}^{2} [1 - {(N \cdot A_{r j})}^{2}]} .

where A_rj and A_rj₊₁ (j = 0,1, 2…5) represent the unit vectors of the incident beam and the refracted beam at each surface, respectively. N is defined as the unit normal vector of the surface. n_j and n_j₊₁ (j = 0,1, 2…5) are refractive indices of two homogeneous mediums on both sides of each refracted surface. According to Eq. (2), the direction vector of each refracted beam can be derived one by one from the incident beam vector A_r₀.

2.2 Analysis of error sources

The differential of Eq. (2) is [21]:

\begin{array}{l} Δ A_{r j + 1} = \frac{n_{j}}{n_{j + 1}} Δ A_{r j} + k_{j + 1} Δ N + h_{j + 1} (Δ N \cdot A_{r j} + N \cdot Δ A_{r j}) N \\ where k_{j + 1} = \sqrt{1 - {(\frac{n_{j}}{n_{j + 1}})}^{2} [1 - {(N \cdot A_{r j})}^{2}]} - \frac{n_{j}}{n_{j + 1}} (N \cdot A_{r j}) \\ and h_{j + 1} = \frac{{(\frac{n_{j}}{n_{j + 1}})}^{2} (N \cdot A_{r j})}{\sqrt{1 - {(\frac{n_{j}}{n_{j + 1}})}^{2} [1 - {(N \cdot A_{r j})}^{2}]}} - \frac{n_{j}}{n_{j + 1}} . \end{array}

Equation (3) describes the direction deviation of refracted beam at one prism surface. ΔA_rj and ΔA_rj₊₁ are the direction deviations of the incident beam and refracted beam, respectively, and ΔN is the normal direction deviation caused by the deviation of incident beam direction or the error in the orientation of prism surfaces (we neglect the changes in refractive indices here) [21]. Therefore, all the factors impacting the incident beam direction or the orientations of prism surfaces are the error sources of the TRSS, which will be expounded later in Section 3.

3. Impact of each error on pointing accuracy

3.1 Incident beam direction deviation

Figure 2 shows δ_Z, defined as the angle of the incident beam relative to the Z-axis, and δ_Y, defined as the angle of the projection line of incident beam in the XOY plane relative to the Y-axis when the incident beam deviates from the optical axis.

Fig. 2 Schematic of the incident beam direction deviation.

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The ideal incident beam direction is specified by A_r₀, while the real direction is expressed as

A_{r 0}^{'} = {(\sin δ_{Z} \sin δ_{Y}, \sin δ_{Z} \cos δ_{Y}, \sqrt{1 - {(\sin δ_{Z} \sin δ_{Y})}^{2} - {(\sin δ_{Z} \cos δ_{Y})}^{2}})}^{T} .

Therefore, the incident beam direction deviation is ΔA_r₀ = A'_r₀ – A_r₀. Supposing three prisms are assembled perfectly or the errors are negligibly small, the deviation of each refracted beam vector is derived successively according to above-mentioned formulas, and ultimately ΔA_r₆, regarded as the pointing deviation of the emergent beam vector from prism 3, can be achieved.

To investigate the effect of the incident beam direction deviation on the pointing accuracy of the TRSS, the amplitude response function of ΔA_r₆ to ΔA_r₀ is defined as

S_{Δ A_{r 0}} = \frac{| Δ A_{r 6} |}{| Δ A_{r 0} |} .

where |ΔA_r₆| and |ΔA_r₀| are defined as the vector magnitude of ΔA_r₆ and ΔA_r₀, respectively. In simulation, geometrical parameters of each prism are wedge angle α = 10°, refractive index n = 1.517. Provided that the deviation angles of incident beam are δ_Z = 0.2 arcsec, δ_Y = 90°. To obtain the maximum value of S_Δ_Ar₀, the relative rotation angle of the first two prisms is kept as a specific value, defined as Δθ₁₂ = θ_r₂-θ_r₁, and prism 3 is synchronized with prism 2 for one cycle (i. e. Δθ₁₃ = θ_r₃-θ_r₁ = Δθ₁₂). Similarly, when the relative rotation angle Δθ₂₃ of the latter two prisms is set as a specific value, prism 1 should be rotated simultaneously with prism 2. Figure 3(a) shows the simulation result of S_Δ_A_r₀ as a function of Δθ₁₂ and Δθ₂₃. By introducing differences to the wedge angle, written as α = 3°, α = 6° and α = 10°, the impact of the wedge angle on pointing accuracy is shown in Fig. 3(b). It can be seen that the effect of incident beam direction deviation on the pointing accuracy is largest with Δθ₁₂ = Δθ₂₃ = 0°, while it is smallest with Δθ₁₂ = Δθ₂₃ = ± 180°, and the consequences are enhanced gradually with the increase of wedge angle.

Fig. 3 Impact of direction deviation of the incident beam on the pointing accuracy. (a) S_Δ_A_r0 as a function of Δθ₁₂ and Δθ₂₃; (b) S_Δ_A_r0 as a function of Δθ₁₂ and Δθ₂₃ with different wedge angles.

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3.2 Assembly error

The assembly error of prisms can be summarized as the assembly error of the prism and bearing. Deviation of the ray caused by prism translation is not discussed here, which is produced by the change in the length of the optical path through the prism [18], and it exerts few effect on the direction vector of emergent beam. The refracted and emergent beam propagating through the TRSS will deviate from the theoretical position, leading to the decline of beam pointing accuracy, when the prism or bearing is unaligned [22]. In this paper, the assembly errors of the prism and bearing are divided into the tilt deviation around the X-axis and the Y-axis.

3.2.1 Prism tilt deviation

The tilt deviation of the prism is expressed as the tilt of the prism around the Y- or X-axis at the initial position, as illustrated in Fig. 4(a) and Fig. 4(b), while the rotation axis is parallel to the optical axis (i.e., the Z-axis). X'OZ' (Y'OZ') is the coordinate system of the prism after prism tilt. The magnitude of tilt deviation can be described by the separated angle δ_Y (δ_X) between the theoretical prism position and the tilted prism position.

Fig. 4 Schematic of the assembly error of the prism. (a) Prism tilt deviation around the Y-axis. The tilt angle δ_Y is the separated angle of the tilted principle section relative to the theoretical one in the XOZ plane. (b) Prism tilt deviation around the X-axis. The tilt angle δ_X is the separated angle of the position of the tilted prism relative to the theoretical one in the YOZ plane. (taking prism 1 as an example)

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Supposed that only prism 1 tilts around the Y-axis and the other two prisms are assembled perfectly, the normal vectors of prism 1 are expressed as

N_{11}^{'} = R o t (Z, θ_{r 1}) \cdot R o t (Y, δ_{Y}) \cdot {(0, 0, 1)}^{T},

N_{12}^{'} = R o t (Z, θ_{r 1}) \cdot R o t (Y, δ_{Y}) \cdot {(\sin α, 0, \cos α)}^{T} .

where Rot (Z, θ_r₁) and Rot (Y, δ_Y) are the rotation matrix around Z-axis and Y-axis, with the rotation angle of θ_r₁ and δ_Y, respectively.

Therefore, normal vector deviations induced by the tilt of prism 1 are $Δ N_{11} = N_{11}^{'} - N_{11}$ and $Δ N_{12} = N_{12}^{'} - N_{12}$ . The deviation of each refracted beam vector can be derived by substituting the unit normal vectors in Eq. (1) and the refracted beam vectors in Eq. (2) into Eq. (3), successively.

Similarly, the amplitude response function of ΔA_r₆ to δ_X or δ_Y is defined as

S_{δ_{i}} = \frac{| Δ A_{r 6} |}{| δ_{i} |}, = X, Y .

where |δ_i| is the length of δ_i, i = X or Y. The solution process of prism tilt around the X-axis, and geometrical parameters for each prism are consistent with aforementioned ones. Provided that the tilt angle of the prism is δ_X = 0.2 arcsec or δ_Y = 0.2 arcsec. Figure 5 shows the simulation result of S_δi as a function of Δθ₁₂ and Δθ₂₃. It can be seen that in the case with the same error margin of 0.2 arcsec, the beam pointing deviation caused by prism tilt around the Y-axis is relatively larger than that around the X-axis. Comparing Figs. 5(a)-(c), the maximum value of S_δX for prism 3 is 0.021, while it is 0.004 for prism 1 and 0.013 for prism 2. Similarly, comparing Figs. 5(d)-(f), the maximum value of S_δY for prism 3 is 0.035, while it is 0.021 for prism 1 and 0.020 for prism 2. Therefore, it can also be concluded that the tilt deviation of prism 3 has the greatest influence on the pointing accuracy. This comparative analysis method is also applied in Section 3.2.2 as well as Section 3.3.

Fig. 5 Impact of prism tilt deviation on the pointing accuracy. (a) S_δX for prism 1 tilting around the X-axis, (b) S_δX for prism 2 tilting around the X-axis, (c) S_δX for prism 3 tilting around the X-axis, (d) S_δY for prism 1 tilting around the Y-axis, (e) S_δY for prism 2 tilting around the Y-axis and (f) S_δY for prism 3 tilting around the Y-axis.

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3.2.2 Bearing tilt deviation

The bearing tilt deviation is expressed by the rotation axis tilting around the Y- or X-axis, without prism tilt deviation at the initial position. That is, there is a certain tilt deviation between the rotation axis and the optical axis (i.e., the Z-axis). Figure 6 shows the tilted bearing of the Risley prism, expressed by the tilt of bearing axis [22]. Similar to the case of prism tilt deviation, X'OZ' (Y'OZ') is the coordinate system of the prism after bearing tilt. The magnitude of tilt deviation is the separated angle ξ_Y (ξ_X) between the theoretical position and the tilted position of bearing axis. For any tilt orientation of the bearing, the tilt deviation can be synthetically expressed by the tilt angles ξ_Y and ξ_X.

Fig. 6 Schematic of the assembly error of the bearing. (a) Bearing axis tilting around the Y-axis with the magnitude of ξ_Y, which is the separated angle of the tilted bearing axis relative to the theoretical one in the XOZ plane. (b) Bearing axis tilting around the X-axis with the magnitude of ξ_X, which is the separated angle of the tilted bearing axis relative to the theoretical one in the YOZ plane. (taking prism 1 as an example)

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Supposed that only the bearing of prism 1 is tilted and the other two prisms are assembled perfectly, the direction vector of the bearing axis of prism 1 can be expressed by

\begin{array}{l} (^{u_{x}^{'}, u_{y}^{'}, u_{z}^{'}) T} = R o t (Y, ξ_{Y}) \cdot R o t (X, ξ_{X}) \cdot (^{u_{x}, u_{y}, u_{z}) T} \\ = (^{\sin ξ_{Y} \cos ξ_{X}, - \sin ξ_{X}, \cos ξ_{Y} \cos ξ_{X}) T} . \end{array}

where Rot (Y, ξ_Y) and Rot (X, ξ_X) are the rotation matrix around Y-axis and X-axis, with the rotation angle of ξ_Y and ξ_X.

{(u_{x}, u_{y}, u_{z})}^{T} = {(0, 0, 1)}^{T}

, which is the direction vector of the bearing axis of prism 1 before tilting. According to the Rodrigues rotation formula [23], the rotation matrix of normal vectors on the incident surface and the emergent surface of the prism, denoted as M_b, is defined as

M_{b} = A_{b} + \cos θ_{r 1} \cdot (I - A_{b}) + \sin θ_{r 1} \cdot B_{b} .

where I is a three-order unit matrix, A_b and B_b are specified in the following form

A_{b} = [\begin{matrix} u_{x}^{' 2} & u_{x}^{'} u_{y}^{'} & u_{x}^{'} u_{z}^{'} \\ u_{y}^{'} u_{x}^{'} & u_{y}^{' 2} & u_{y}^{'} u_{z}^{'} \\ u_{z}^{'} u_{x}^{'} & u_{z}^{'} u_{y}^{'} & u_{z}^{' 2} \end{matrix}],

B_{b} = [\begin{matrix} 0 & - u_{z}^{'} & u_{y}^{'} \\ u_{z}^{'} & 0 & - u_{x}^{'} \\ - u_{y}^{'} & u_{x}^{'} & 0 \end{matrix}] .

According to the rotation transformation matrix defined in Eq. (9), the normal vectors of prism 1 can be expressed as

N_{11}^{'} = M_{b} \cdot {(0, 0, 1)}^{T}

N_{12}^{'} = M_{b} \cdot {(\sin α, 0, \cos α)}^{T}

The pointing deviation ΔA_r₆ can be calculated by applying Eqs. (9) and (10) into Eqs. (1)-(3) successively, and the amplitude response function of ΔA_r₆ to ξ_X or ξ_Y is defined as

S_{ξ_{i}} = \frac{| Δ A_{r 6} |}{| ξ_{X} | + | ξ_{Y} |} .

where |ξ_X| and |ξ_Y| are the length of ξ_X and ξ_Y. Case examples are given to elucidate the impact of the bearing tilt deviation on the pointing accuracy, which is ξ_X = 0.2arcsec or ξ_Y = 0.2arcsec. Figure 7 shows the simulation result of S_ξi as a function of Δθ₁₂ and Δθ₂₃. It is worth noting that in the case of the same error margin of 0.2 arcsec, pointing accuracy is more sensitive to the bearing tilt deviation around the Y-axis than that around the X-axis, and the deviation in prism 3 will result in more remarkable pointing errors in contrast with the other two prisms.

Fig. 7 Impact of bearing tilt deviation on the pointing accuracy. (a) S_ξX for the bearing axis of prism 1 tilting around the X-axis, (b) S_ξX for the bearing axis of prism 2 tilting around the X-axis, (c) S_ξX for the bearing axis of prism 3 tilting around the X-axis, (d) S_ξY for the bearing axis of prism 1 tilting around the Y-axis, (e) S_ξY for the bearing axis of prism 2 tilting around the Y-axis and (f) S_ξY for the bearing axis of prism 3 tilting around the Y-axis.

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3.3 Prism rotational error

Prisms rotational errors mainly come from errors of prisms position sensors as well as prisms position tracking errors [3]. Supposed that only prism 1 has the rotational error Δθ_r₁ and the other two prisms are rotated faultlessly, the deviation of the refracted beam through prism 1 is ΔA_r₁ = 0, and the normal vector of the emergent surface on prism 1 is

N_{12}^{'} = {[\cos (θ_{r 1} + Δ θ_{r 1}) \sin α, \sin (θ_{r 1} + Δ θ_{r 1}) \sin α, \cos α]}^{T} .

The deviation of the normal vector N₁₂ is

Δ N_{12} = N_{12}' - N_{12} \approx Δ θ_{r 1} {(- \sin θ_{r 1} \sin α, \cos θ_{r 1} \sin α, 0)}^{T} .

Combining the vector deviation in Eq. (12) and the refracted vectors in Eq. (2), a nonparaxial ray tracing analysis can be performed, and the pointing deviation can then be derived from Eq. (3). The impact of prisms rotational errors on the pointing accuracy of the TRSS, is expressed as the amplitude response function of ΔA_r₆ to Δθ_ri

S_{Δ θ_{r i}} = \frac{| Δ A_{r 6} |}{| Δ θ_{r i} |}, i = 1, 2, 3.

where |Δθ_ri| is the length of Δθ_ri (i = 1, 2, 3), which is set as 0.2 arcsec. Figure 8 shows the simulation result of S_Δ_θri as a function of Δθ₁₂ and Δθ₂₃. It's worth mentioned that in the case of the same error margin of 0.2 arcsec, the pointing deviation induced by rotational error of prism 3 is more notable in contrast with the other two prisms.

Fig. 8 Impact of prisms rotational error on the pointing accuracy. (a) S_Δ_θr₁ for rotational error of prism 1, (b) S_Δ_θr₂ for rotational error of prism 2 and (c) S_Δ_θr₃ for rotational error of prism 3.

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3.4 Systematic error

Systematic errors, including errors in refractive index Δn and wedge angle Δα, should not be neglected due to inevitable matching errors and changeable working environments. It has been investigated in Ref [17]. that the error allowance decreases rapidly as Δn or Δα increases. If the pointing accuracy is planned, the upper limit of the index error or wedge angle error can therefore be estimated. One can readily confirm that the large refractive index and wedge angle pose significant challenges for prism fabrication.

To achieve a given pointing accuracy, it is suggested to heavily suppress these systematic errors well within the corresponding tolerance under strict fabrication conditions [15–19].

3.5 Summary

According to the graphical and analytical results, in the case of the same error margin of 0.2 arcsec, the amplitude response function S of the incident beam direction deviation is larger than that of any other error sources, which indicates that the incident beam direction deviation plays a major role in affecting the pointing accuracy of the TRSS. To correct the angular error of the incident beam, Ref [25]. presented a two-degrees-of-freedom flexible fine-tuning mechanism based on the principle of cantilever beam type, which can suppress the angular error lower than 0.05 deg.

Based on the requirements of the manufacturing and assembly accuracy, the errors of wedge angle and refractive index are both suppressed within a reasonable range. Therefore, the main error sources considered in practical applications include deviation errors of incident beam direction, prism rotational errors and assembly errors, arranged in a descending order of importance. Particularly, the impact of deviation errors of incident beam direction can be negligible for a thin prism system. With the increase of wedge angle, the pointing accuracy of the system becomes more dependent on the deviation error of incident beam direction.

4. Error compensation

In simulation, geometrical parameters for each prism are wedge angle α = 3°, refractive index n = 1.517, thinnest-end thickness d₀ = 5mm and clear aperture D_p = 80mm, and the defined spatial distances among the prisms and screen P are equal to D₁ = D₂ = 100mm and D₃ = 300mm. By introducing differences to the angular velocity ratio of three prisms, written as ω₁: ω₂: ω₃, various beam scan patterns may be produced on the screen P, as shown in Fig. 9.

Fig. 9 Beam scan trajectories under different angular velocity ratios of three prisms, where ω₁: ω₂: ω₃ equals to (a) 1: 2: 1, (b) 1: 1.5: 2, (c) 1: −1.5: 1 and (d) 1: 2: −1.5.

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It is worth mentioning that the third prism can increase the diversity in matching prism orientations and achieve more abundant scan patterns. Meanwhile, the error analysis becomes more complex because of the accumulated coupling relationships between various errors of three prisms. Therefore, the corresponding error compensation method must be adopted to mitigate the impact of each error on the pointing accuracy.

In previous research on the error analysis of rotating Risley-prism scan systems, an error compensation method was proposed by B. Bravo-Medina et al. [18]. In this paper, we have developed and applied it into the TRSS, describing the beam pointing system with the error-compensation model, as illustrated in Fig. 10.

Fig. 10 Schematic of error-compensation model for the TRSS, where the stationary orientations are θ'_r₁ = 90°, θ'_r₂ = 0° and θ'_r₃ = 30°. The position of each circle center depends on the stationary orientations of other two prisms. d₁₀, d₂₀ and d₃₀ represent the separate deviations produced by three prisms at their initial positions, which are defined as the vectors that connect the three circle centers with P₁, respectively.

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Here, we introduce four vectors d₁, d₂, d₃ and d_E that contain information about the displacements and changes in the deviation angle. d₁ denotes the deviation produced by prism 1 and the errors that depend on the orientation of prism 1. Similarly, d₂ and d₃ represent the deviation caused by prism 2 and 3, respectively. Additionally, we donate the deviation induced by remaining errors, including deviation errors of the incident beam deviation and systematic errors, as the vector d_E. Consequently, the position of the beam on the receiving screen can be described as the sum of these four vectors:

d_{T} = d_{1} + d_{2} + d_{3} + d_{E} .

where d_T is the total deviation of the emergent beam. First, we assume that prism 2 and prism 3 remain stationary as θ'_r₂ and θ'_r₃, respectively, and prism 1 rotates from 0° to 360°, to obtain a circumference of a circle around an as-yet unknown point O₁ on the screen. Then, a circle around another as-yet unknown point O₂ can be obtained on the screen when prism 1 and prism 3 remain stationary as θ'_r₁ and θ'_r₃, and prism 2 rotates from 0° to 360°. Finally, the third circle around O₃ is painted on the screen when prism 1 and prism 2 remain stationary as θ'_r₁ and θ'_r₂, and prism 3 rotates from 0° to 360°. The stationary orientation angles of three prisms can be arbitrarily selected and the graphics construction of three circles is illustrated in Fig. 10. The three circles intersect at a point P₁, namely, when stationary orientation angles of three prisms are θ'_r₁, θ'_r₂ and θ'_r₃, the point P₁ (x₁, y₁) will be painted on the screen.

The coordinates of several equidistant points on each circle are measured to determine the position of the center point O_i (i = 1, 2 or 3). For each circle, we select four points on the circumference with an angular increment of π/2 rad, marked as P₁, P₃_i_-1, P₃_i and P₃_i₊₁, of which the arithmetic mean value of the four coordinates is the coordinate of the center point O_i. Afterwards, we denote d₁₀, d₂₀ and d₃₀ as the vector from O_i to the intersection point P₁, respectively, which is the initial position of d₁, d₂ and d₃. As shown in Fig. 10, the error compensation vector d_E can be expressed as

d_{E} = d_{V} - d_{120} = d_{V} - d_{10} - d_{20} .

where d₁₂₀ represents the sum of d₁₀ and d₂₀; d_V is the vector which is pointing from O to O₃. It is worth noting that if the magnitude of d_i is calculated as the distance from O_i to any point on the circumference, which is actually an approximately circular ellipse, it will bring out relatively large errors. Here, we propose two solutions to avoid this situation: (1) the average value of 4 radii of the circumference can be taken to improve the pointing accuracy of the system; (2) the parameter values of the circle can be optimized by genetic algorithm. The simulation improvement of pointing accuracy for each method is shown in Fig. 11, Fig. 12 and Table 1.

Fig. 11 Take the average of four radii of the circle to improve accuracy, where Δxy is the distance difference between the actual point and the theoretical one.

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Fig. 12 Optimized radius by the genetic algorithm to improve the pointing accuracy.

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Table 1. Simulation result of pointing error for each method.

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The magnitudes of d₁, d₂ and d₃, optimized by the above method, are |d₁|, |d₂| and |d₃|, respectively. Then the initial angular azimuth angles ω₁₀, ω₂₀ and ω₃₀ of prism 1, 2 and 3 with respect to the horizontal axis, are determined from the respective right triangle

ω_{i 0} = {\begin{cases} arct \tan (\frac{y_{1} - y_{O_{i}}}{x_{1} - x_{O_{i}}}), x_{1} - x_{O_{i}} \geq 0 \\ π + arc \tan (\frac{y_{1} - y_{O_{i}}}{x_{1} - x_{O_{i}}}), x_{1} - x_{O_{i}} < 0 \end{cases} (i = 1, 2, 3)

The relative rotation angle of the prism ϕ_i is given by ϕ_i = ω_i - ω_i₀, i = 1,2 or 3. For forward solutions, the deviation produced by the prism can be obtained as

d_{i} = [| d_{i} | \cos (ϕ_{i} - ω_{i 0}), | d_{i} | \sin (ϕ_{i} - ω_{i 0})], i = 1, 2, 3.

Different from the exact ray trace, the aforementioned error compensation-based forward algorithm itself has a certain amount of error. Before applying this method, we need to make a discussion on its effect on improving the pointing accuracy, in consideration of the assembly errors of the prism. In simulation, geometrical parameters for each prism are consistent with aforementioned ones. Figure 13 simulates the variation of the distance Δ_xy between the actual spot position and the theoretical one before and after error compensation. It can be seen that the error compensation method can improve the pointing accuracy in the case of a few errors. The Root Mean Square (RMS) value of the position error can be reduced from 20.7μm to 10.5μm, which can be maintained within the range of 15μm.

Fig. 13 Improvement of error compensation method for pointing accuracy.

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5. Conclusion

In order to make clear the impact of various errors on the beam pointing accuracy of the TRSS, we proposed a three-prism error analysis model based on the beam refraction deviation. The analytical and numerical results show that: (1) the error sources from greatest to smallest, are incident beam direction deviation, prism rotational error, tilt deviation of the bearing axis and prism. (2) the beam deviation caused by prism 3 has the greatest influence on the pointing accuracy in the same case. The derived formulas can be implemented to analyze beam pointing errors of any three-prism scanner with a similar configuration. Besides, if we want to obtain a certain accuracy, the quantitative analysis method can guide us to adjust the allowances of the error sources for a given pointing accuracy. To achieve higher pointing accuracy, a forward algorithm based on error compensation is developed, and simulation results show that the beam pointing accuracy can be well improved. It is worth mentioning that this method is only applicable for thin prisms, and the error compensation for thick prisms needs further research.

Funding

National Natural Science Foundation of China (NSFC) (61675155); Fundamental Research Funds for the Central Universities (kx0100020172644).

Acknowledgments

The authors gratefully acknowledge Wansong Sun, Shengze Zhong, Qiao Li, School of Mechanical Engineering, Tongji University, for assistance with discussions and critical reviews.

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Method	Error/μm
Method	Minimum	Maximum	RMS
Arbitrary radius	0.7	18.8	11.2
Average of 4 radii	1.3	16.4	10.9
GA optimization	0.9	13.6	9.7

Investigation of scan errors in the three-element Risley prism pair

Abstract

1. Introduction

2. Theoretical model

2.1 Beam pointing vectors

2.2 Analysis of error sources

3. Impact of each error on pointing accuracy

3.1 Incident beam direction deviation

3.2 Assembly error

3.2.1 Prism tilt deviation

3.2.2 Bearing tilt deviation

3.3 Prism rotational error

3.4 Systematic error

3.5 Summary

4. Error compensation

5. Conclusion

Funding

Acknowledgments

References

Cited By

Figures (13)

Tables (1)

Equations (22)

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