1. Introduction

Risley prism beam steering devices have the advantages of compact structure, small moment of inertia, good dynamic performance, large field of view (FOV) and high reliability [1–3]. The basic configuration with two identical Risley prisms rotating coaxially are widely used in free space optical communication, optical interconnection, photoelectric reconnaissance, interference measurement, and so on [4–6].

In 1960, Rosell [1] firstly proposed two Risley prisms to implement various beam scanning patterns. Then many researchers have contributed to the theoretical and applied research on Risley prisms, including inverse solutions, beam scan patterns, scan errors, blind zone, singularity problems, and Risley prism applications. For inverse solutions, Amirault and DiMarzio [7] put forward a two-step method, which simplified the variables during the inverse derivation. Tao et al. [8] established a numerical iterative inverse solution based on the damped least squares method and ensured the theoretical tracking errors less than 1 μm in both x and y directions. In 1999, Marshall’s seminal study [9] focused on the beam scan patterns through Risley prisms. By employing a first-order approximation method, Marshall simulated some interesting patterns under different wedge angles, angular rotation speeds or initial phases between prisms. Furtherly, in 2013, the exact scan patterns were presented by Schitea et al [10] with a mechanical design program, CATIA V5R20. In 2016, we [11] exhibited the blind zone resulting from the structural parameters and the singularity problems of the Risley prism system. As for the applications, researchers has developed many Risley-prism-based devices in recent years, such as confocal reflectance microscope, laser Doppler vibrometry, explosive detector, and so on [6,12,13].

In 2005, we reported the principle of tilting double-prism scanner used into free-space laser communication test [14]. The proposed scanner moves with two orthogonal tilting drive pairs [14–16], which renews the traditional Risley prism model in principle [1–4]. Because the reduction ratio from the tilting angles of the prisms to the beam deviation angle can reach a hundredfold magnitude, a common mechanical device can realize the scanning precision of submicroradian order, which is superior to the traditional rotating Risley prisms. So the tilting double-prism scanning method shows a good application potential in precision pointing fields.

In previous studies, we mainly focused on the directed function of the emergent beam, and investigated the beam scanning field, scanning precision, inverse solutions, and driving method under the hypothesis of paraxial approximation. However, some underlying technical problems should be further explored. First of all, some system parameters which haven’t been overall considered previously, may influence the beam steering performances to some extent, such as beam scanning field and exact beam scanning trajectory. And during the scanning process, the nonlinear beam scanning characteristics of the prisms has to be quantitatively described. In addition, for a typical refractive beam steering system, the beam distortion should be studied to reveal the laser energy distribution on the tracking targets. In this study, we focus on the above beam scanning problems and draw some valuable conclusions by the analysis method of multiple parameter combinations, which can provide a theoretical basis for the system design and application of tilting double-prism scanner.

2. Scanning range and scanning precision of tilting double prisms

2.1 Modelling of tilting double-prism system

A rectangular coordinate system XYZ is set as shown in Fig. 1 [17]. The beam steering system is composed of two identical circular wedge prisms with wedge angles α, refractive indexes n and the thickness of the thinnest end d₀. Two prisms are sequentially named as the prism П₁ and the prism П₂ along the positive direction of the Z axis, respectively. At the initial state, the tilting angles of the prisms are θ_t1 = θ_t2 = 0, and the plane sides of two prisms are situated outward, which is one of the four configurations of a double-prism system proposed by Li [18]. The prism П₁ can tilt around the horizontal axis L₁, which is vertical to the principal cross section of prism П₁ and through the center point O. Similarly, the prism П₂ can tilt around the vertical axis L₂, which is vertical to the principal cross section of prism П₂ and through the center point O₂. The distance between O and O₂ is D₁, and that between O₂ and the screen P is D₂.

 

Fig. 1 Schematic diagram illustrating the tracking principle of tilting orthogonal double prisms.

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As shown in Fig. 1, $\overrightarrow{A_{t0}}={\left(0,0,1\right)}^{\text{T}}$ is the incident beam of the prism П₁, $\overrightarrow{A_{t\text{1}}}$, $\overrightarrow{A_{t2}}$, $\overrightarrow{A_{t\text{3}}}$ and $\overrightarrow{A_{tf}}$ are the refracted beams after surface 11, surface 12, surface 21 and surface 22, respectively. Suppose that the tilting angles of two prisms are θ_t1 and θ_t2, the beam hits each surface of two prisms and screen P at intersection points O, K_t, N_t, M_t and P_t sequentially after refracted by two prisms. The vertical field angle ρ_V is defined as the angle between the projection of the emergent beam in the XOZ plane and the positive direction of Z axis, and the horizontal field angle ρ_H represents the angle between the projection of the emergent beam in the YOZ plane and the positive direction of Z axis.

The normal vectors of each surface of two prisms can be written as $\overrightarrow{N_{11}}$, $\overrightarrow{N_{12}}$, $\overrightarrow{N_{21}}$ and $\overrightarrow{N_{22}}$ in sequence, given by

Then according to Snell’s law [18], each refracted beam can be expressed as $\overrightarrow{A_{ti}}={\left(x_{ti},y_{ti}.z_{ti}\right)}^T(i=1,2,3,f)$. In our previous works [15,17,19], the derivation was presented in detail, and here only the result of the emergent beam $\overrightarrow{A_{tf}}$ is presented as follows.

The vertical field angle ρ_V and the horizontal field angle ρ_H can be expressed as

However, considering the structural parameters d₀, D₁ and D₂ of the tilting double-prism system, an exact expression should be provided to describe the scanning point P_t. According to Eq. (1) and the vectors of refracted beams, the equations of each surface and each refracted beam can be solved out, and then the intersection points K_t (x_tk, y_tk, z_tk), N_t (x_tn, y_tn, z_tn), M_t (x_tm, y_tm, z_tm) and P_t (x_tp, y_tp, z_tp) can be given by

2.2 Scanning range of the emergent beam

Based on Eqs. (2-4) given in Section 2.1, the scanning range of the tilting double-prism system can be analyzed. In earlier researches [14, 15], the deviation angle of the emergent beam were calculated within the monotonous zone of θ_t1∈[0°,5°] and θ_t2∈[0°,8°], with no regard to other possible tilting angle ranges. According to Eq. (3) and Eq. (4), the functional relations between the vertical field angle ρ_V, the horizontal field angle ρ_H, and the tilting angles θ_t1, θ_t2 are drawn in Fig. 2, taking the situation where n = 1.517, θ_t1∈[−45°,45°] and θ_t2∈[−45°,45°] as an example. Figure 2 shows that the tilting angle θ_t1 mainly influences the vertical field angle ρ_V rather than the horizontal field angle ρ_H, while the tilting angle θ_t2 almost only influences the horizontal field angle ρ_H. Obviously, with the increase of the wedge angles α, the scanning range is greatly enlarged and moves away from the optical axis in both the vertical and the horizontal direction.

 

Fig. 2 The field angle of the emergent beam. (a) the vertical field angle ρ_V; (b) the horizontal field angle ρ_H.

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As shown in Fig. 2, the case of α = 5° is taken as an example. Providing that θ_t2 is a constant, ρ_V increases to the maximum and then decreases along with the increase of θ_t1 within [−45°, 45°]. Similarly, when θ_t1 is a constant, ρ_H increases to the maximum and then decreases along with the increase of θ_t2 within [−45°, 45°]. Interestingly, two maxima, namely ρ_Vmax = −2.5908° and ρ_Hmax = −2.5925°, are both obtained at θ_t1 = −3.78° and θ_t2 = −1.17°. And the minima of ρ_V and ρ_H are both found at θ_t1 = θ_t2 = 45°, where ρ_Vmin = −5.2406° and ρ_Hmin = −4.5602°. So the theoretical scanning ranges are ∆ρ_V = ρ_Vmax−ρ_Vmin = 2.6498°, and ∆ρ_H = ρ_Hmax−ρ_Hmin = 1.9677°. In other two cases of α = 10° and 13°, the different scanning ranges with similar change trends can be obtained.

Then an experimental setup shown in Fig. 3 is developed to validate the scanning range. The parameters of the tilting double prisms are as follows: the wedge angles α = 5°, the refractive indexes n = 1.517, the clear apertures D_p = 80 mm, the thicknesses of the thinnest end d₀ = 5 mm, the distance between two prisms D₁ = 120 mm, and the distance between the surface 22 of prism Π₂ and the receiving screen D₂ = 2000 mm. The laser source with 650 nm wavelength is adjusted to be coaxial with the optical axis. The laser beam is deflected by two prisms and hits the receiving screen with two dimension grids. Then the scanning point on the grid can be measured and thus the vertical and horizontal field angle ρ_V and ρ_H can be obtained. It is noteworthy that the tilting angles are limited to θ_t1∈[−45°,45°] and θ_t2∈[−45°,45°] in the experiment.

 

Fig. 3 Experimental setup, mainly including tilting double prism scanner, receiving screen, prism controller, laser source, laser controller and guide.

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Considering the structural parameters of the tilting double prism scanner, the exiting point from prism Π₂ is not the center of the surface 22. However, because the distance D₂ is much longer than D₁, the deviation from the center of the surface 22 to the actual exiting point can be neglected. And then the vertical field ρ_V and the horizontal field angle ρ_H can be approximately calculated according to the location $P_t\text{'}\left(x,y\right)$ of the scanning point on the receiving screen in the form of

In the experiment, two controllable prisms are rotated to θ_t1 = −3.78° and θ_t2 = −1.17° firstly and θ_t1 = θ_t2 = 45° secondly, where the deviation angle of the beam can reach the minimum or the maximum value in theory. Then the positions of the laser beam on the receiving screen are measured. Consequently, ${{\rho }^{\prime }}_{V\max }=-2.\text{3244}°$ and ${{\rho }^{\prime }}_{H\max }=-2.7\text{709}°$ corresponding to θ_t1 = −3.78° and θ_t2 = −1.17°. And ${{\rho }^{\prime }}_{V\min }=-4.9\text{807}°$ and ${{\rho }^{\prime }}_{H\min }=-4.\text{7361}°$ corresponding to θ_t1 = θ_t2 = 45°. Hence, the measured scanning ranges are $\Delta {{\rho }^{\prime }}_V={{\rho }^{\prime }}_{V\max }-{{\rho }^{\prime }}_{V\min }\text{=}2.6\text{563}°$, and $\Delta {{\rho }^{\prime }}_H={{\rho }^{\prime }}_{H\max }-{{\rho }^{\prime }}_{H\min }\text{=1}\text{.9652}°$.Comparing the experimental results with the theoretical ones, the relative errors of $\Delta {{\rho }^{\prime }}_V$ to $\Delta {\rho }_V$ and $\Delta {{\rho }^{\prime }}_H$ to $\Delta {\rho }_H$are less than 0.245% and 0.127% respectively, which well validate the scanning range of the tilting double prisms. However, the relatively small FOV also reveals the shortcoming of the tilting double prisms with comparison to the rotating ones.

As shown in Fig. 2, ρ_V and ρ_H are non-monotonous functions of θ_t1 and θ_t2 in the ranges of θ_t1∈[−45°, 45°] and θ_t2∈[−45°, 45°]. However, for the easy control of the tilting double prisms and to avoid multi-group inverse solutions of θ_t1 and θ_t2, the range of the tilting angles θ_t1∈[−45°, 45°] and θ_t2∈[−45°, 45°] are not appropriate. With the overall consideration of the mechanical structure, the tilting range of the prisms should be set to [0°, 45°], where functions of ρ_V and ρ_H belong to monotonic functions.

2.3 Scanning precision of the emergent beam

Based on the total deviation angle of the emergent beam, it has been theoretically confirmed that a high scanning precision can be achieved for a titling double-prism system with wedge angles α = 5° or with wedge angles α = 10° [14,15,20]. Furthermore, an experiment was conducted and strongly validated the conclusions [15]. However, considering the motion of the target, the scanning precision in the vertical and horizontal direction should be further explored. And a larger range of the tilting angle of each prism should be taken into consideration to determine an appropriate range with a relative higher scanning precision.

The scanning error of the emergent beam consists of error terms induced by the tilting error, the wedge angle error and the refraction index error of the prisms, which can be expressed as

Suppose that α = 10° and n = 1.517, then the relations between the directions of the emergent beam and the tilting angles of the prisms can be drawn in Fig. 4. Figures 4(a) and 4(b) show the change laws of the partial derivatives expressed by $\frac{\partial {\rho }_V}{\partial {\theta }_{t1}}$ and $\frac{\partial {\rho }_H}{\partial {\theta }_{t1}}$ when the prism П₂ is fixed. Similarly, Figs. 4(c) and 4(d) show the change laws of the partial derivatives expressed by $\frac{\partial {\rho }_V}{\partial {\theta }_{t2}}$ and $\frac{\partial {\rho }_H}{\partial {\theta }_{t2}}$ when the prism П₁ is fixed.

 

Fig. 4 The change laws of the partial derivatives between the field angles and the tilting angles. (a) and (b) represent the influences of θ_t1 on $\frac{\partial {\rho }_V}{\partial {\theta }_{t1}}$ and $\frac{\partial {\rho }_H}{\partial {\theta }_{t1}}$; (c) and (d) represent the influences of θ_t2 on $\frac{\partial {\rho }_V}{\partial {\theta }_{t2}}$ and $\frac{\partial {\rho }_H}{\partial {\theta }_{t2}}$, respectively.

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It can be concluded from Fig. 2 and Fig. 4 that with the increase of the tilting amplitudes of prisms in the ranges of [0, 45°], the corresponding field angles of the emergent beam are enlarged, while the absolute values of the change rate of the field angles increase, which means the scanning precision of the emergent beam reduces to a certain extent. Specially, when θ_t1∈[20°,45°] and θ_t2∈[20°,45°], the scanning precision drops sharply with the increase of θ_t1 and θ_t2. Therefore, in order to ensure a high beam steering precision, the tilting angles are limited within θ_t1∈[0°,10°] and θ_t2∈[0°,10°], where the vertical and horizontal field angles vary from −5.66° to −5.29° and −5.44° to −5.24°.

Under the above conditions, the maxima of $\left|\frac{\partial {\rho }_V}{\partial \alpha }\right|$ and $\left|\frac{\partial {\rho }_H}{\partial \alpha }\right|$ can both be found at θ_t1 = θ_t2 = 10°, and the values are 0.621 and 0.567, respectively. Hence, the maximal errors of ρ_V and ρ_H caused by α are 3.01 μrad and 2.75 μrad, with the hypothesis that the manufacturing precision of the wedge angle is up to 1″. Similarly, at θ_t1 = θ_t2 = 10°, $\left|\frac{\partial {\rho }_V}{\partial n}\right|$ and $\left|\frac{\partial {\rho }_H}{\partial n}\right|$ reach the maxima, of which the values are 0.196 μrad and 0.182 μrad, respectively. So the maximal errors of ρ_V and ρ_H caused by n are 1.960 μrad and 1.820 μrad, with the hypothesis that the refractive index error caused by the inhomogeneity of the optical glass is ± 1 × 10⁻⁵. Both the wedge angle error and refractive index error belong to systematic error terms, which can be corrected by the optics testing and calibration. Therefore, the random errors of the tilting mechanisms are the primary error sources that determine the scanning field angle errors, which are analyzed in detail as follows.

According to Fig. 4, $\left|\frac{\partial {\rho }_V}{\partial {\theta }_{t1}}\right|$, $\left|\frac{\partial {\rho }_V}{\partial {\theta }_{t2}}\right|$, $\left|\frac{\partial {\rho }_H}{\partial {\theta }_{t1}}\right|$ and $\left|\frac{\partial {\rho }_H}{\partial {\theta }_{t2}}\right|$ get their maximal values at θ_t1 = θ_t2 = 10°, and the values are 5.42 × 10⁻², 3.17 × 10⁻⁴, 8.22 × 10⁻⁴ and 0.339 × 10⁻², respectively. That is to say, the maximal errors of the field angles of the emergent beam are

3. Analysis of beam scanning

3.1 Nonlinear problems of beam scanning

According to Fig. 2, the vertical field angle of the final emergent beam are mainly affected by θ_t1, and the horizontal field angle are mainly affected by θ_t2. Actually, a complicated nonlinear relation generally exists between the deviation angle of the emergent beam and the tilting angles of the prisms when θ_t1∈[0°,10°] and θ_t2∈[0°,10°], as mentioned in our previous work [17,19]. Suppose that one prism do not tilt, the other prism tilt individually, then the above nonlinear relations can be quantitatively described as follows. Based on the equations given in Section 2.1, Fig. 5 shows the relations between θ_t1, θ_t2 and the deviation angle ρ of the final emergent beam (including the vertical field angle ρ_V and the horizontal field angle ρ_H) with the wedge angle α = 10° and the refractive index n = 1.517.

 

Fig. 5 The nonlinear relations between the tilting angles θ_t1 and θ_t2 of two prisms and the deviation angle ρ of the final emergent beam. (a) θ_t1 corresponding to ρ_V and ρ_H; (b) θ_t2 corresponding to ρ_V and ρ_H.

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As shown in Fig. 5, there are two sets of inverse tilting angle solutions for one specific combination of deviation angles ρ_V and ρ_H. But within the aforementioned chosen range of θ_t1∈[0°,10°] and θ_t2∈[0°,10°], the tilting angles θ_t1 and θ_t2 are strictly one-to-one mapped to the deviation angles ρ_V and ρ_H of the final emergent beam, so there is a unique set of the tilting angle solutions of prisms corresponding to the vertical field angle and the horizontal field angle of the final emergent beam. However, due to the nonlinear function relations between θ_t1 and ρ_V, and θ_t2 and ρ_H, the inverse solutions are preferably solved by a numerical solution method rather than the analytic ones [17, 19].

Similarly, Fig. 6 shows the nonlinear relation between the angular velocities of prisms and the change rate of deviation angle. As we can see, within the beam scanning range, the change rate of tilting angle θ_t1 to the deviation angle ρ_V nonlinearly decreases with the increase of ρ_V, and that of θ_t2 to ρ_H nonlinearly decreases with the increase of ρ_H, too. In other words, the closer the emergent beam is to the system optical axis, the faster the prisms tilt. Hence for the esay control of the prisms, during system parameter selection, an appropriate scanning range should be designed on the basis of target position estimation. In addition, the change rates of θ_t2 to ρ_V and θ_t1 to ρ_H are near zero, which in another way proves the conclusion drawn from Section 2.2 that the tilting angle θ_t1 mainly influences the vertical field angle ρ_V rather than the horizontal field angle ρ_H, while the tilting angle θ_t2 almost only influences the horizontal field angle ρ_H.

 

Fig. 6 The nonlinear relation between the tilting angular velocities of prisms and the change rate of deviation angle dθ/dρ. (a) ρ_V corresponding to dθ_t1/dρ_V and dθ_t2/dρ_V; (b) ρ_H corresponding to dθ_t1/dρ_H and dθ_t2/dρ_H.

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Faced with the above nonlinear problems, some appropriate strategies must be taken for the real-time nonlinear control. Last year, we proposed a nonlinear control scheme to solve this problem [19]. A tilting double-prism scanner driven by cam-based mechanism was designed to transform the complicated control procedure of the drive motor to the profile curve design of the cam. Compared to the reciprocating rotational motor movement in earlier rotating or tilting Risley prism scanners [5–7,14–16], this cam-based continuous rotational motor driving method is easier to be performed, and could achieve a sufficiently high oscillatory frequency. This method was confirmed to be feasible for a steady scanning trajectory.

3.2 Beam scanning trajectory

Suppose that the incident beam vector is $\overrightarrow{A_{t0}}={\left(0,0,1\right)}^{\text{T}}$, according to Eqs. (5-9), the various scanning trajectories can be drawn when the prisms tilt at different angular velocity combinations within the range of 0°~10° (In the following part of the paper, the ranges of the tilting angles are set to 0°~10°without special notes).

In Fig. 7(a), the beam scanning trajectory curve is viewed when the tilting angular velocities of the prisms keep uniform and ω_t2 = 2ω_t1. The tilting angles of the prisms are ${\theta }_{t1}=-\left|t-10\right|+10$ and ${\theta }_{t2}=-\left|2t-10\right|+10$, respectively. As we can see, the trajectory is a closed curve, and the scanning point moves along the curve circularly. Similarly, Figs. 7(b) and 7(c) show the beam scanning trajectories with ω_t2 = 4ω_t1, and the tilting angles of the prisms are ${\theta }_{t1}=-\left|t-10\right|+10$ and ${\theta }_{t2}=-\left|4t-10\right|+10$, respectively. During the scanning process, the scanning point firstly moves along the path shown in Fig. 7(b), basically toward the positive direction of the X axis, and then returns to the starting point along the identical path shown in Fig. 7(c). Besides the above uniform tilting angular velocities, we simulate several scanning curves with non-uniform velocities according to the sine or cosine function laws. In Figs. 7(d) and 7(e), the tilting angles θ_t1 are the same, and ${\theta }_{t1}=5\sin (\pi t/5)+5$, while ${\theta }_{t2}=-\left|2t-10\right|+10$ and ${\theta }_{t2}=5\cos (\pi t/5)+5$, respectively. The scanning trajectories in Figs. 7(d) and 7(e) are two closed curves, and the scanning points move along the curve circularly. It is noteworthy that the former has sharp points on the trajectory, while the latter is a relatively smooth curve over a full cycle due to sine function for ω_t1 and cosine function for ω_t2. In general, the inflection points on these scanning trajectories as shown in Figs. 7(a)-7(d) reflect the discontinuous differentiability of the tilting angle function, and the closed trajectory curves indicate the beam scanning periodicity caused by the different tilting cycles within the limited tilting angle range of 0°~10°.

 

Fig. 7 Beam scanning trajectories with different tilting angular velocities, where the red arrows show the moving direction of the scanning point. These tilting angular velocities are chosen arbitrarily, while in practice the tilting angles and velocities should be determined based on specific applications. (a) uniform ω_t1, uniform ω_t2 and ω_t2 = 2ω_t1; (b) and (c) uniform ω_t1, uniform ω_t2 and ω_t2 = 4ω_t1; (d) sine function for ω_t1 and uniform ω_t2; (e) sine function for ω_t1 and cosine function for ω_t2.

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It can be seen from Fig. 7 that two prisms with different angular velocity combinations can produce various trajectory curves. In fact, according to the given target trajectory, the corresponding tilting angle curves of prisms can be obtained using an inverse solution method [19]. Through the precise nonlinear control or the mechanism transmission design, any desired beam scanning trajectories within the scanning field can be generated in theory.

3.3 Influences of system structure parameters on scanning field

In the design of the tilting double-prism system, besides the scanning precision, the position and size of the scanning field on the receiving device are also significant indexes. In Section 2.2, we discuss the scanning range of the emergent beam to illustrate the vertical field angle ρ_V and the horizontal field angle ρ_H. However, with the change of some structure parameters, such as D₁, d₀ and D₂, the scanning point position on the receiving device can make different shifts, while the emergent beam direction has no changes. Hence in this section, we will discuss the influences of D₁, d₀ and D₂ on the scanning field with the wedge angles α = 10°, the refractive indexes n = 1.517, and the clear apertures D_p = 80 mm, respectively.

Figure 8 shows the scanning field with structure parameters of D₁ = 100 mm, D₂ = 400 mm and d₀ = 5 mm. The real scanning field is a parallelogram-like area with the angles between each adjacent edges very close to a right angle. For convenience, the biggest rectangle area in the real scanning field is taken as the scanning field. In this example, the scanning field locates at the area of x_tp∈[−47.17, −44.82] and y_tp∈[−39.63, −37.44], the scanning area A = 2.35 mm × 2.19 mm = 5.15 mm² and the center point coordinate is (−45.99, −38.53) mm.

 

Fig. 8 The scanning field of a tilting double-prism scanner. The area enclosed by the red rectangle, which is the largest rectangle within the parallelogram-like area, is taken as the scanning field of this scanner.

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In Table 1, the scanning field parameters are viewed under the different separation distance D₁ between two prisms when D₂ = 400 mm and d₀ = 5 mm. Suppose that D₂ is constant, the change of D₁ only affects the coordinate range of the scanning field in the X direction, while that in the Y direction remains unchanged, as shown in Table 1. The reason is that the deviation of the final emergent beam in Y direction is mainly caused by the tilt of the prism П₂, and the optical path after surface 21 has nothing to do with the distance between two prisms D₁. With the increase of D₁, the scanning range in the X direction increases, the area of the scanning field increases and the center point of the scanning field moves along the negative direction of the X axis.

Table 1. Parameters of Scanning Field under Different D₁ (mm)

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Similarly, Table 2 shows the parameters of scanning field which are obtained under different d₀ when D₁ = 100 mm and D₂ = 400 mm. When D₁ and D₂ keep invariant, with the increase of d₀, the coordinate range of the scanning field in the X direction decreases, while that in the Y direction increases. Correspondingly, the X coordinate value of the center point of the scanning field increases but the Y coordinate value decreases.

Table 2. Parameters of Scanning Field under Different d₀ (mm)

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In addition, if D₁ and d₀ keep invariant while D₂ changes, it is easy to conclude that the farther the receiving screen is, the larger the scanning area becomes, and the farther the center point of the scanning field deviates from the origin of the screen.

4. Beam distortion analysis

In general, for a refractive optical system, the beam distortion should be taken into consideration in practical applications. In previous researches, the beam distortion of rotation Risley prisms was studied using the vector refraction theorem [21, 22]. In 2016, we further presented a detailed analysis about it considering the effects of system parameters and prism configurations [23]. The above works shows that the distortion effect on the beam shape is relatively weak in early double-prism systems with shallow wedge angles and small refractive indexes. But with the larger wedge angles and higher refractive indexes applied in the prism scanners, the distortion degree become too severe to be neglected anymore. However, as for a tilting double-prism system, the beam distortion is also an important issue in some application situations, and a comprehensive analysis should be performed to reveal the beam distortion laws.

In this section, a ray-tracing method is used to analyze the beam shape distortion. The incident beam $\overrightarrow{A_{t\text{0}}}$ parallel to the Z axis is prescribed to be circular with radius r = 20 mm, and the central axis of the incident beam intersects with the surface 11 at the point (10, 0, 0). Some necessary system parameters defined in Section 2.1 are reset as below: D₁ = 400 mm, D₂ = 400 mm, n = 3, α = 10°, d₀ = 10 mm and D_p = 400 mm. Therefore, the beam edge on the prism surface 11 can be written as (unit is millimeters)

By utilizing the ray-tracing method and the vector refraction theorem, the intersection lines between the beam edge and each prism surfaces can be expressed by coordinate equations sequentially. This process is very similar to the forward derivation of the beam propagation [15]. Here the beam distortion degree ε is defined as the ratio of the maximal distortion value of the emergent beam in radial direction to the initial diameter of the incident beam.

Figure 9 indicates that the distortion degree ε reaches 8.46%, 9.52%, 13.01% and 20.30% respectively under the different tilting angle combinations of two prisms, namely θ_t1 = θ_t2 = 0°, θ_t1 = 0° and θ_t2 = 5°, θ_t1 = 0° and θ_t2 = 10°, θ_t1 = 10° and θ_t2 = 10°. Obviously, the beam exiting from the tilting double prisms is squeezed in one direction but stretched in the mutual perpendicular direction, which is similar to the results in rotation Risley prisms [23]. It is also evident that the distortion degree ε of the emergent beam depends on the tilting angles of two prisms.

 

Fig. 9 The beam shape distortion induced by double prisms with different tilting angle combinations. Red curves with “°” signs and blue curves with “×” signs represent the original beam shapes and the distorted ones, respectively. (a) θ_t1 = 0°, θ_t2 = 0°; (b) θ_t1 = 0°, θ_t2 = 5°; (c) θ_t1 = 0°, θ_t2 = 10°; (d) θ_t1 = 10°, θ_t2 = 10°.

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Furthermore, when the wedge angle α, the refractive index n, the thickness of the thinnest end d₀ and the distance D₁ change in sequence, the distortion laws of the beam can be observed in comparison with the initial circular shape. Providing the tilting angles of prisms are θ_t1 = 0° and θ_t2 = 0°, θ_t1 = 0° and θ_t2 = 5°, θ_t1 = 0° and θ_t2 = 10°, θ_t1 = 10° and θ_t2 = 10° respectively, the corresponding beam distortion degree ε under different α, n, d₀ and D₁ can be calculated, and then the distortion laws are clearly revealed in Fig. 10.

 

Fig. 10 The influence of system parameters on beam distortion. (a) refractive index n; (b) wedge angle α; (c) thickness of the thinnest end d₀; (d) distance D₁ between O and O₂.

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Seen from Fig. 10, we can draw some valuable conclusions. Both the higher refractive index and the larger wedge angle can aggravate the distortion degree on beam shape, while the thicknesses of thinnest end and the distance between double prisms are verified to be nearly irrelevant factor. And in general, the distortion degree increases with the tilting angles. In the following comparisons, θ_t1 = 0° and θ_t2 = 5°. Let α = 15°, d₀ = 10 mm and D₁ = 400 mm, when n increases from 2 to 3, ε correspondingly increase from 4.42% to 7.63%. Similarly, let n = 3, d₀ = 10 mm and D₁ = 400 mm, ε equals to 3.18%, 7.63% and 23.04%, in response to the incremental α values of 5°, 10° and 15°. As for the distance D₁ between two prisms, the distortion degree ε keeps uniform no matter how D₁ varies. Moreover, the influence of d₀ on the distortion degree ε can be neglected. Generally, there is a contradiction between wider FOV with higher or larger α and less beam distortion with lower n or smaller α, and a balance between these requirements has to be comprehensively considered.

5. Conclusions

A pair of tilting double prisms, as one of the most promising scanning methods, can realize much higher beam-steering accuracy than the traditional rotation Risley-prism scanning systems. The beam scanning range and scanning precision, as well as the beam distortion, are the basic beam performance indexes which condition the use fields of this scanning system. In the paper, we discuss the influences of key system parameters on the beam scanning properties, including the wedge angle, the thickness and the refractive index of the prism, the distance between two prisms, and the distance between the scanner and the receiving device. As for the specific scanning field on the receiving screen, its size and position are determined by the change of the above parameters. In order to regularize the motion control, the nonlinear relation between the tilting angles of the prisms and the deviation angle of the emergent beam is thoroughly investigated, and some possible solutions are proposed. Moreover, the beam distortion is irrelevant to the distance between two prisms, while is greatly influenced by the wedge angle and the refractive index. These results can provide the foundation for the design and application of the tilting double-prism scanning system.