## Abstract

Scan blind zone and control singularity are two adverse issues for the beam scanning performance in double-prism Risley systems. In this paper, a theoretical model which introduces a third prism is developed. The critical condition for a fully eliminated scan blind zone is determined through a geometric derivation, providing several useful formulae for three-Risley-prism system design. Moreover, inverse solutions for a three-prism system are established, based on the damped least-squares iterative refinement by a forward ray tracing method. It is shown that the efficiency of this iterative calculation of the inverse solutions can be greatly enhanced by a numerical differentiation method. In order to overcome the control singularity problem, the motion law of any one prism in a three-prism system needs to be conditioned, resulting in continuous and steady motion profiles for the other two prisms.

© 2017 Optical Society of America

## 1. Introduction

The double-prism beam scanner generally comprises a matched Risley-prism pair capable of coaxial and independent rotation, which can produce diverse scan patterns within a certain field-of-view (FOV) to perform beam steering or boresight adjustment [1,2]. Such a beam scanner is advantageous over other alternatives because it offers compact structure, rapid response, superior precision, large FOV and impressive adaptability to working environment. These merits make the double-prism system attractive in the fields of target searching and recognition [3], large-scale imaging [4], biomedical observation [5], laser communication [6], vision based microassembly [7] and so on.

However, there are several problems encountered in further applications of the double-prism beam scanner. Some problems related to optical design have caught much attention of the researchers, such as beam shape distortion [8], chromatic dispersion [9] and wavefront quality [10]. Other problems, especially a scan blind zone and control singularities, are also prevalent, but are seldom considered in studies of the beam steering process. The blind zone is inherent in the central scan region owing to the structural parameters of Risley prisms, which has detrimental effects on the beam coverage [11]. Tilting the prisms or using prisms with different wedge angles or refractive indices appears to be effective to overcome the scan blind zone, but this method fails once the distance from the system to the observation plane is not specified. Besides, the double-prism system may suffer from control singularities while steering the emergent beam through the center of FOV, where challenging step speeds of the prism rotations are required [12]. Our previous paper [13] has reported that the spatial distance between the second prism and the observation plane should be selected to avoid control singularities, which is not applicable to optical occasions where the distance must be changed frequently. Actually, the use of a third prism has proved to be flexible as well as practical enough to overcome both scan blind zone and control singularities [14, 15]. But only a vector approximation method with limited precision has ever been introduced to help illustrate the power of the third prism in eliminating blind spots and control singularities, and the motion control law for each prism was expressed in an undefined form [16]. In many specific applications, it is desirable to evaluate the scan blind zone with a quantitative method involving all three prisms, or to formulate detailed control strategies to determine the steady motion processes. It is these complications that revealed to us that more effort needed be concentrated on integrating both forward and inverse solutions for the three-prism beam scanner to achieve satisfactory precision with high efficiency.

This paper is outlined as follows. In Section 2, the beam scan model for a three-Risley-prism system is illuminated as the foundation to investigate various beam scan modes and the scan blind zone with manageable area. Section 3 summarizes an inverse algorithm aimed at continuous and smooth motion profiles of three prisms, the efficiency of which can be greatly enhanced by a numerical differentiation method. Conclusions are finally drawn in Section 4.

## 2. Forward solution for three-prism beam scanners

#### 2.1 Modelling of three-prism system

Under Cartesian coordinate frame *OXYZ* established in Fig. 1, the three-prism system consists of three identical Risley prisms, each of which is prescribed with refractive index *n*, wedge angle *α*, thinnest-end thickness${d}_{0}$and clear aperture *D _{p}*. The prisms sequentially named as prism 1, prism 2 and prism 3 in the positive

*Z*-direction can rotate independently around

*Z*-axis at arbitrary angular velocities${\omega}_{r1}$,${\omega}_{r2}$and${\omega}_{r3}$, respectively, where clockwise rotation angle is defined as the positive angle. All prism surfaces are marked as$\sum $with the first subscript “1”, “2” or “3” to distinguish each prism and the second subscript “1” or “2” indicating the incident or emergent surface of the prism. It is notable that${\sum}_{12}$,${\sum}_{21}$and${\sum}_{32}$are plane surfaces exactly perpendicular to

*Z*-axis, whereas${\sum}_{11}$,${\sum}_{22}$and${\sum}_{31}$are wedge surfaces inclined to those plane surfaces, respectively. Such an arrangement of three prisms is chosen, by way of example, from many possible configurations. In addition,${D}_{1}$defines the spatial distance from prism 1 to prism 2,${D}_{2}$defines the one from prism 2 to prism 3, and${D}_{3}$is that from prism 3 to the screen P.

On the initial state, the principal section of each prism is located in the *XOZ* plane with the thinnest end towards the positive *X*-direction. The rotation angles of three prisms are equal to 0°, which act as time-dependent variables denoted by${\theta}_{r1}(t)$for prism 1,${\theta}_{r2}(t)$for prism 2 and${\theta}_{r3}(t)$for prism 3. As shown in Fig. 1, the incident beam aligned with the positive *Z*-direction is refracted at each prism surface and eventually reaches the screen P. Here the pitch angle *ρ* represents the deviation angle of the emergent beam with respect to the positive *Z*-direction, and the azimuth angle *φ* is the angle between the emergent beam projection on screen P and the positive *X*-direction.

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

Provided that the incident beam to prism 1 is specified by the unit vector${\text{A}}_{\text{0}}={\left(0,0,1\right)}^{\text{T}}$, the refracted beam vectors at other prism surfaces can be deduced according to the vector refraction theorem:

*i*= 1, 2, 3…6) are, respectively, the incident and refracted beam vectors at a specific prism surface described by the normal vector

**, and the prism surface lies between homogeneous media with different refractive indices,**

*N**n*

_{i}_{-1}and

*n*. For instance, as the beam propagates from air into prism 1, the individual unit vectors are given by${\text{A}}_{\text{i}\text{-1}}={\text{A}}_{\text{0}},{\text{A}}_{\text{i}}={\text{A}}_{\text{1}}$and$\text{N}={\text{N}}_{\text{11}}$, and the refractive indices should be substituted with${n}_{i-1}=1$and${n}_{i}=n$.

_{i}After a tedious mathematical development to combine Eq. (2) with the analytic equations for prism surfaces, which is omitted here, the intersection point of the beam path and each interface can be found, including the beam scan point${P}_{r}$, which is of particular significance.

The forward procedure is usually performed to predict beam scan trajectories that result from a beam scanner with three prisms rotating at arbitrary speeds. In simulation, geometrical parameters for each of the prisms are wedge angle$\alpha =10\xb0$, refractive index$n=1.517$, thinnest-end thickness${d}_{0}\text{=5mm}$and clear aperture${D}_{p}=80\text{mm}$, and the defined spatial distances among the prisms and screen P are equal to${D}_{1}={D}_{2}={D}_{3}=100\text{mm}$. By introducing differences to the angular velocity ratio of three prisms, written as${\omega}_{r1}:{\omega}_{r2}:{\omega}_{r3}$, various beam scan patterns may be produced on screen P, as shown in Fig. 2. There are many possible scan patterns because the third prism increases the diversity in matching prism orientations. It is worth mentioning that Fig. 2(f) displays a scan trajectory passing through the center of scan region, which underlines the probability to eliminate central blind spots under some particular conditions.

#### 2.2 Scan blind zone

The scan blind zone of any conventional double-prism system is intrinsically attributed to the identical Risley prisms that may cancel beam deflections imparted by each other. Thus the area of scan blind zone is jointly dominated by the pitch angle of the emergent beam and the beam exiting position from the second prism [13]. As illustrated in reference [13], the beam exiting position can affect the area of scan blind zone under two conditions, i.e., the distance between two prisms is over a threshold *D*_{1}* _{c}* and the distance from the second prism to the observation plane is below another threshold

*D*

_{2}

*. But the threshold*

_{c}*D*

_{2}

*is too small with respect to the size of the mechanical structure in which the prisms are mounted. Hence, it is usually reasonable to neglect the effects of the beam exiting position on the area of scan blind zone.*

_{c}If two prisms employed in a double-prism system have different parameters, such as different wedge angles or refractive indices, or the prisms are tilted with individual angles, the emergent beam can be steered towards the optical axis. Specially, the scan blind zone disappears when the distance from the second prism to the receiving screen is specified, defined as *D*_{0}. Unfortunately, the distance value *D*_{0} is unique for designed double prisms with different parameters, which can restrict other parameter matching of the double-prism system, such as the distance between two prisms. Moreover, this method cannot completely eliminate the scan blind zone, and the scan blind zone can occur again when the receiving screen is located away from or near the distance *D*_{0}. Comparatively, it has been asserted that the use of a third prism can eliminate the scan blind zone as long as the deflection imparted by the third prism is greater than the resultant deflection imparted by the original double-prism system [16]. To determine the critical condition for a fully eliminated scan blind zone, the prisms are orientated with the rotation angles${\theta}_{r1}=0\xb0$and${\theta}_{r2}={\theta}_{r3}=180\xb0$, respectively. Figure 3 shows that the emergent beam from prism 2 is parallel to the *Z*-axis and its projection intersects the screen P at point${S}_{r}$, which would introduce a circular blind zone with the radius of $\left|{O}_{P}{S}_{r}\right|$in the absence of prism 3. But in fact, the beam emerging from prism 2 is then refracted by prism 3, and as a result, the final emergent beam intersects *Z*-axis at the point C. Hence, it is deduced with ease that the scan blind zone will disappear when the receiving screen P is located at or behind the point C. In other words, the critical condition is expressed as${D}_{3}\ge {D}_{c}$, where${D}_{3}$denotes the spatial distance from prism 3 to screen P and${D}_{c}$is that from prism 3 to point C.

The beam refractions at all prism surfaces can be quantitatively specified with the incident and emergent angles labeled from${\delta}_{0}$to${\delta}_{11}$in Fig. 3. Given the incident angle at the wedge surface${\sum}_{11}$as${\delta}_{0}=\alpha $, any other angle is available in terms of Snell’s law. In addition, the beam path intersects each prism and the screen P at the points marked as${J}_{r},{M}_{r},{Q}_{r}$and${P}_{r}$in turn. Thus the beam deflections imparted by three prisms can be, respectively, described by the *X*-separations from${J}_{r}$to${M}_{r}$, from${M}_{r}$to${Q}_{r}$and from${Q}_{r}$to${P}_{r}$, expressed as

When${D}_{3}\le {D}_{c}$, the beam deflections contribute to the radius${R}_{bz}$of the scan blind zone, as follows:

Note that${D}_{3}={D}_{c}$and${R}_{bz}=0$when the scan blind zone is just eliminated. Upon substituting these two equations into Eq. (6), a concise expression for the critical distance${D}_{c}$is obtained as

Unfortunately, designers are not always allowed to simply increase the distance${D}_{3}$up to${D}_{c}$in order to achieve beam scanning across the FOV of the system, since the size of the three-prism beam scanner may have to be limited. Thus the matching relation among the distances${D}_{1}$,${D}_{2}$and${D}_{3}$should be carefully considered on the basis of Eq. (7). Assume that${D}_{1}+{D}_{2}+{D}_{3}=B$and${D}_{1}:{D}_{2}:{D}_{3}={\lambda}_{1}:{\lambda}_{2}:{\lambda}_{3}$, where *B* is a constant and${\lambda}_{1}+{\lambda}_{2}+{\lambda}_{3}=1$. The coefficients${\lambda}_{1}$,${\lambda}_{2}$and${\lambda}_{3}$are required as follows to manage the blind zone:

In simulation, geometrical parameters for each prism are consistent with aforementioned ones, and the total distance from prism 1 to screen P is constant at$B=300\text{mm}$. When${\lambda}_{1}=1/3$, the individual critical values of ${\lambda}_{2}=0.3873$and${\lambda}_{3}=0.2794$can be obtained from Eq. (8) and Eq. (9). Therefore, the total distance *B* is resolved into three components, i.e.,${D}_{1}=100\text{mm}$,${D}_{2}=116.1794\text{mm}$and${D}_{3}=83.8206\text{mm}$. Once these details are accounted for, two beam scan regions in the absence and presence of prism 3, respectively, can be displayed in Fig. 4. It is seen that the use of prism 3 cannot only be a feasible way to overcome the scan blind zone in nature, but also expand the beam scan region to some extent. For example, the area of scan region in Fig. 4(b) is enlarged by 43.33% compared to that in Fig. 4(a). Another effect due to prism 3 is that the distortion of beam shape may become better or worse, depending on the orientations of three prisms. Illustrated with examples, the distortion ratio, defined as the maximal stretching or squeezing ratio of the beam diameter [17], is determined to be 1.2442% when the rotation angles of prism 1 and prism 2 are${\theta}_{r1}={\theta}_{r2}=0\xb0$and prism 3 is absent. Comparatively, the distortion ratio is reduced to 1.0559% when prism 3 is present with the rotation angle${\theta}_{r3}=180\xb0$, but the ratio can increase to 1.3091% when${\theta}_{r3}=0\xb0$.

## 3. Inverse solutions for three-prism beam scanners

#### 3.1 Inverse algorithm

A target coordinate or desired pointing angle usually provides two independent constraints, whereas there are three degrees of freedom to determine in a three-prism beam scanner, which implies countless possible solutions to any inverse problem [16]. In order to simplify the general procedure which leads to inverse solutions with high precision, the motion law of one prism can be pre-determined as another independent constraint.

The three-prism system is separated into a single-prism subsystem S_{1} comprising prism 1 and a double-prism subsystem S_{2} composed of prism 2 and prism 3, as shown in Fig. 5. Here the single prism is the one chosen to follow a given motion law. For the minimized loss of beam scan region, prism 1 should always be orientated with its thickest end towards the direction from the center point${O}_{P}$of screen P to the target point${P}_{r}$. Therefore, the elementary motion law is written as

*X*- and

*Y*-components of the emergent beam vector${\text{A}}_{\text{6}}$, respectively.

With the knowledge of the motion law of prism 1, the beam vector${\text{A}}_{\text{2}}$emerging from the subsystem S_{1}, which also performs as the incident beam vector to the subsystem S_{2}, can be easily specified through a forward ray tracing process. Accordingly, the coordinates$({x}_{d},{y}_{d})$of the target point${P}_{r}$are both functions dependent on the prism orientations${\theta}_{r2}$and${\theta}_{r3}$:

The next step is to find inverse solutions for the double-prism subsystem S_{2}, the incident beam to which is inclined rather than parallel to *Z*-axis as usual. Considering that previous inverse algorithms based on the two-step method or iterative method are mostly confined to double-prism systems with the incident beam in the positive *Z*-direction [18–20], the damped least-squares iterative method proposed by Tao appears to be the most appropriate approach [21]. The adopted method relies on partial differentials of Eq. (12) to build a Jacobian matrix written as

Hereby, a combined algorithm aimed at numerical inverse solutions for the three-prism beam scanner is summarized as follows.

- Step 1. The target point is given by${P}_{d}=({x}_{d},{y}_{d})$.
- Step 2. Pre-determine the motion law of prism 1 in terms of Eq. (10).
- Step 3. Derive the function relationship in the form as Eq. (12).
- Step 4. Initialize the joint variable$\theta $in Eq. (12) to be${\theta}_{0}=({\theta}_{20},{\theta}_{30})$.
- Step 5. Calculate the Jacobian matrix$\text{J}$when$\theta ={\theta}_{i}$at the
*i*th iteration. - Step 6. Update${\theta}_{i+1}$using the equation${\theta}_{i+1}\text{=}{\theta}_{i}+\zeta {\text{J}}^{\text{+}}\left[{P}_{d}-F\left({\theta}_{i}\right)\right]$, where$\zeta $denotes a gain factor, and${\text{J}}^{\text{+}}={\text{J}}^{\text{T}}{\left(\text{J}{\text{J}}^{\text{T}}+\delta \text{I}\right)}^{-1}$. And$\delta $is a damping factor given by$\delta ={\delta}_{0}{\left(1-\omega /{\omega}_{0}\right)}^{2}$if$\omega =\sqrt{\mathrm{det}\left(\text{J}{\text{J}}^{\text{T}}\right)}$is below a threshold${\omega}_{0}$, otherwise$\delta =0$.
- Step 7. Compute the absolute error between the target point and the actual beam scan point according to$\Delta =\left|{P}_{s}-F\left({\theta}_{i+1}\right)\right|$. If$\Delta $is below an error threshold$\epsilon $, ${\theta}_{i+1}$is accepted as the final solution. Otherwise set$i=i+1$and then return to Step 5.

For any desired beam scan trajectory, the inverse algorithm can be employed to obtain the consequent motion profiles of three prisms. Two implementation cases are, respectively, shown in Figs. 6 and 7, where geometrical parameters for each prism are still known as before, the spatial distances are${D}_{1}={D}_{2}={D}_{3}=100\text{mm}$, and the error threshold remains at$\epsilon =0.001\text{mm}$.

Case 1. A heart-shaped beam scan trajectory given by$\{\begin{array}{l}x=10\mathrm{cos}t-5\mathrm{cos}\left(2t\right)\\ y=10\mathrm{sin}t-5\mathrm{sin}\left(2t\right)\end{array},t\in \left[0,2\pi \right]$.

Case 2.A pentacle beam scan trajectory given by$\{\begin{array}{l}x=4\mathrm{cos}t+8\mathrm{cos}\left(2t/3\right)\\ y=4\mathrm{sin}t-8\mathrm{sin}\left(2t/3\right)\end{array},t\in \left[0,6\pi \right]$.

#### 3.2 Improvement

The disadvantage of using the inverse algorithm is that the iterative process is too lengthy for real-time beam scanning operation. As the above cases indicate, much of the operation time is spent on finding the Jacobian matrix, which requires the onerous symbolic calculation of four partial differentials at each iteration. Therefore, a numerical differentiation method is involved to provide approximate partial differentials with high efficiency.

According to this numerical differentiation method, an *n*-order Lagrange interpolation polynomial${L}_{n}(\theta )$is regarded as the substitution for the functional relation$F(\theta )$between prism orientations and target coordinates. Thus it is acceptable that the partial differentiation functions of $F(\theta )$are approximate to those of ${L}_{n}(\theta )$. In this way, the partial differentials of ${L}_{n}(\theta )$relative to the variable${\theta}_{r2}$are obtained at a specific node where${\theta}_{r2}={\theta}_{s}$. A similar approach can be applied to deduce the partial differentials of $F(\theta )$with respect to the other variable${\theta}_{r3}$, which is omitted here.

During the numerical differentiation process of $F(\theta )$relative to${\theta}_{r2}$, the other variable${\theta}_{r3}$is held constant, which accounts for the functional relation transformed as$F(\theta )=g({\theta}_{r2})$. For necessity, other four discrete nodes, including${\theta}_{s-2}={\theta}_{s}-2h$,${\theta}_{s-1}={\theta}_{s}-h$,${\theta}_{s+1}={\theta}_{s}+h$and${\theta}_{s+2}={\theta}_{s}+2h$, are taken in the neighborhood of the specific node${\theta}_{s}$. Here *h* is the distance between any two adjacent nodes, which depends on the expected precision. Since the function values of $g({\theta}_{r2})$at these nodes are available from Eq. (12), a fourth-order Lagrange polynomial approximate to$g({\theta}_{r2})$can be obtained from

Then the interpolation remainder defined by${R}_{4}({\theta}_{r2})=g({\theta}_{r2})-{L}_{4}({\theta}_{r2})$is differentiated in order to derive the numerical differential of $g({\theta}_{r2})$at the node where${\theta}_{r2}={\theta}_{s}$, written as

Given this relationship, it is feasible to establish a Jacobian matrix which contains four numerical differentials at each iteration. The improved algorithm is also applied to the cases in Figs. 6 and 7 so as to evaluate the operation efficiency. It turns out that the inverse algorithm after improvement can provide exactly the same solutions as before, but in much less operation time. To be specific, Table 1 has listed the total time for overall operations before and after improvement, along with the average time for one sample point. The comparison results show that this numerical differentiation method may reduce the operation time by about two orders of magnitude when the inverse solutions to be obtained are required with precision superior to$0.001\text{mm}$.

#### 3.3 Control singularity

Attention is now turned to motion control strategies that can remove the problem of central control singularities. Similar to a double-prism system, the three-prism system suffering from control singularities cannot achieve continuous and steady beam scanning across the FOV of the system, which may cause the loss of target in optical tracking applications [13]. The control singularity problem still occurs because the rotation angle of prism 1 is directly associated with the azimuth angle, which switches to a dramatic step speed as the target approaches the center of scan region. For example shown in Fig. 8, since the linear scan trajectory passes through the central scan region, there are some singularity points existent in the consequent inverse solutions. Here geometrical parameters for each prism remain unchanged, and the defined spatial distances are also equal as${D}_{1}={D}_{2}={D}_{3}=100\text{mm}$.

Case 3. A linear beam scan trajectory given by$\{\begin{array}{l}x=t\\ y=t\end{array}\text{,}t\in \left[-10,10\right]$.

Fortunately, the use of a third prism offers the flexibility to modify the pre-determined motion law of prism 1. Once a smooth motion profile is generated for prism 1, the step speed switching of the other two prisms will become smooth and continuous as well. The following motion law of prism 1 is chosen from many possibilities:

where$\sigma $represents the distance from beam scan point${P}_{r}$to the center point${O}_{P}$of scan region, and the function$f(\sigma )$is under the limits of $f(0)={f}_{\mathrm{max}}$and$f({\sigma}_{\mathrm{max}})=0\xb0$. Importantly,$f(\sigma )$is appended here to provide the rotation angle${\theta}_{r1}$with a variable offset that depends on the actual position of point${P}_{r}$. When$\sigma ={\sigma}_{\mathrm{max}}$, there is usually no sudden change in${\theta}_{r1}$, so the offset from$f(\sigma )$is minimized. As$\sigma $decreases to 0, the offset becomes more significant and finally reaches the maximum${f}_{\mathrm{max}}$.It is noticed that the previous literature suggests the maximized offset${f}_{\mathrm{max}}$to be constant at 90° [16]. But in this paper,${f}_{\mathrm{max}}$is dominated by the variation of ${\theta}_{r1}$at the singularity point in order to adapt to more optical applications. For convenience, all sample points in an arbitrary beam scan trajectory are numbered with a variable *k*, where *k* ranges from 1 to *q*. Then the azimuth angle for a specific sample point is denoted by$\phi (k)$, and the distance between this specific point and point${O}_{P}$is written as$\sigma (k)$. Supposing that the sample point which acts as a singularity point is marked by another variable *p*, the maximized offset can be obtained from${f}_{\mathrm{max}}=[\phi (p-1)-\phi (p+1)]/2$. Therefore, Eq. (16) is further specified by

Due to the offset function which reduces the impact of azimuth angle, the motion law can produce a continuous and steady motion profile for prism 1. The motion profiles for other prisms should be obtained from the inverse algorithm incorporated with the motion law. Through the procedure mentioned before, the consequent inverse solutions for the linear scan trajectory given by Case 3 are determined as Fig. 9 shows. Now that the motion profile for each prism is displayed with satisfactory continuity, the motion control of the overall beam scanner tends to be much easier compared to former experiences. Different from three-prism systems, the control singularities of double-prism systems can be effectively solved by interchanging two sets of inverse solutions, which needs be performed by targeted motion control strategies [12, 13].

Case 4. The improved inverse solutions for the same beam scan trajectory as in Case 3.

## 4. Conclusions

A systematic model for the three-prism system is firstly presented as the basis for further developments in this paper. A ray tracing model of the system is used to demonstrate forward and inverse solutions based on the structural parameters of the three prisms. To provide useful guidance for system designers, a quantitative analysis method is also introduced to derive the critical condition for elimination of a scan blind zone. Moreover, a damped least-squares method with iterative refinement is proposed for the inverse solutions, the efficiency of which can be significantly enhanced by a numerical differentiation method. Since the discontinuous motion profiles of three prisms can lead to control singularities, the motion law of one of the prisms is specially conditioned to produce continuous and steady motion profiles of all three prisms. Therefore, the three-prism beam scanner can offer great flexibility to various optical scanning applications, producing a multi-mode beam scan pattern without a scan blind zone or control singularity.

## Funding

National Natural Science Foundation of China (NSFC) (51375347, 61675155).

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