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 thicknessd0and clear aperture Dp. 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 velocitiesωr1,ωr2andωr3, respectively, where clockwise rotation angle is defined as the positive angle. All prism surfaces are marked aswith 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 that12,21and32are plane surfaces exactly perpendicular to Z-axis, whereas11,22and31are 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,D1defines the spatial distance from prism 1 to prism 2,D2defines the one from prism 2 to prism 3, andD3is that from prism 3 to the screen P.

 figure: Fig. 1

Fig. 1 Theoretical model of the three-Risley-prism system.

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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 byθr1(t)for prism 1,θr2(t)for prism 2 andθr3(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:

N11=(cosθr1sinα,sinθr1sinα,cosα)T,N12=(0,0,1)T,N21=(0,0,1)T,N22=(cosθr2sinα,sinθr2sinα,cosα)T,N31=(cosθr3sinα,sinθr3sinα,cosα)T,N32=(0,0,1)T.

Provided that the incident beam to prism 1 is specified by the unit vectorA0=(0,0,1)T, the refracted beam vectors at other prism surfaces can be deduced according to the vector refraction theorem:

Ai=(ni1ni)Ai-1+{1(ni1ni)2[1(Ai-1N)2](ni1ni)Ai-1N}N=(xi,yi,zi)T.
whereAi-1andAi(i = 1, 2, 3…6) are, respectively, the incident and refracted beam vectors at a specific prism surface described by the normal vector N, and the prism surface lies between homogeneous media with different refractive indices, ni-1 and ni. For instance, as the beam propagates from air into prism 1, the individual unit vectors are given byAi-1=A0,Ai=A1andN=N11, and the refractive indices should be substituted withni1=1andni=n.

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 pointPr, 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 angleα=10°, refractive indexn=1.517, thinnest-end thicknessd0=5mmand clear apertureDp=80mm, and the defined spatial distances among the prisms and screen P are equal toD1=D2=D3=100mm. By introducing differences to the angular velocity ratio of three prisms, written asωr1:ωr2:ωr3, 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.

 figure: Fig. 2

Fig. 2 Beam scan trajectories under different angular velocity ratios of three prisms, whereωr1:ωr2:ωr3equals to (a)1:1:1, (b)1:2:1.5, (c)1:2:1.5, (d)1:1.5:1.5, (e)1:1.5:1.5 and (f)1:1.5:1, respectively.

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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 D1c and the distance from the second prism to the observation plane is below another threshold D2c. But the threshold D2c 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.

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 D0. Unfortunately, the distance value D0 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 D0. 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 anglesθr1=0°andθr2=θr3=180°, 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 pointSr, which would introduce a circular blind zone with the radius of |OPSr|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 asD3Dc, whereD3denotes the spatial distance from prism 3 to screen P andDcis that from prism 3 to point C.

 figure: Fig. 3

Fig. 3 Schematic diagram illustrating the critical condition for a fully eliminated blind zone.

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The beam refractions at all prism surfaces can be quantitatively specified with the incident and emergent angles labeled fromδ0toδ11in Fig. 3. Given the incident angle at the wedge surface11asδ0=α, 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 asJr,Mr,QrandPrin turn. Thus the beam deflections imparted by three prisms can be, respectively, described by the X-separations fromJrtoMr, fromMrtoQrand fromQrtoPr, expressed as

r1=dtanδ2+(D12d)tanδ3,
r2=dr1tanαcotδ5+tanα,
r3=[d(r1+r2)tanα]tanδ10+D3tanδ11.
whered=d0+(Dp/2)tanαis the central-axis thickness for each prism, and the involved angles are given byδ2=αarcsin(sinα/n),δ3=arcsin(nsinδ2),δ5=arcsin(sinδ3/n),δ10=δ2andδ11=δ3.

WhenD3Dc, the beam deflections contribute to the radiusRbzof the scan blind zone, as follows:

Rbz=r1+r2r3.

Note thatD3=DcandRbz=0when the scan blind zone is just eliminated. Upon substituting these two equations into Eq. (6), a concise expression for the critical distanceDcis obtained as

Dc={(r1+r2)[d(r1+r2)tanα]tanδ2}cotδ3.

Unfortunately, designers are not always allowed to simply increase the distanceD3up toDcin 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 distancesD1,D2andD3should be carefully considered on the basis of Eq. (7). Assume thatD1+D2+D3=BandD1:D2:D3=λ1:λ2:λ3, where B is a constant andλ1+λ2+λ3=1. The coefficientsλ1,λ2andλ3are required as follows to manage the blind zone:

λ21(m1+1)λ1m2,
λ3m1λ1+m2.
wherem1=tanαtanδ2+1tanαtanδ5+1 and m2=d[m1(tanδ22tanδ3+tanδ5)tanδ2]Btanδ3 are both constants once each of the prisms is designated with specific geometrical parameters.

In simulation, geometrical parameters for each prism are consistent with aforementioned ones, and the total distance from prism 1 to screen P is constant atB=300mm. Whenλ1=1/3, the individual critical values of λ2=0.3873andλ3=0.2794can be obtained from Eq. (8) and Eq. (9). Therefore, the total distance B is resolved into three components, i.e.,D1=100mm,D2=116.1794mmandD3=83.8206mm. 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 areθr1=θr2=0°and prism 3 is absent. Comparatively, the distortion ratio is reduced to 1.0559% when prism 3 is present with the rotation angleθr3=180°, but the ratio can increase to 1.3091% whenθr3=0°.

 figure: Fig. 4

Fig. 4 Comparison between beam scan regions (a) in the absence of prism 3 and (b) in the presence of prism 3.

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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 S1 comprising prism 1 and a double-prism subsystem S2 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 pointOPof screen P to the target pointPr. Therefore, the elementary motion law is written as

θr1=φ180°.
whereφrepresents the azimuth angle for the final emergent beam, given by
φ={arccos(x6x62+y62),y602πarccos(x6x62+y62),y6<0
wherex6andy6are X- and Y-components of the emergent beam vectorA6, respectively.

 figure: Fig. 5

Fig. 5 The three-prism system separated into single-prism and double-prism subsystems.

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With the knowledge of the motion law of prism 1, the beam vectorA2emerging from the subsystem S1, which also performs as the incident beam vector to the subsystem S2, can be easily specified through a forward ray tracing process. Accordingly, the coordinates(xd,yd)of the target pointPrare both functions dependent on the prism orientationsθr2andθr3:

[xpyp]=[fx(θr2,θr3)fy(θr2,θr3)]=F(θ).
whereθdenotes a joint variable of θr2andθr3.

The next step is to find inverse solutions for the double-prism subsystem S2, 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

J=[fxθr2fxθr3fyθr2fyθr3]

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 byPd=(xd,yd).
  • 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 variableθin Eq. (12) to beθ0=(θ20,θ30).
  • Step 5. Calculate the Jacobian matrixJwhenθ=θiat the ith iteration.
  • Step 6. Updateθi+1using the equationθi+1=θi+ζJ+[PdF(θi)], whereζdenotes a gain factor, andJ+=JT(JJT+δI)1. Andδis a damping factor given byδ=δ0(1ω/ω0)2ifω=det(JJT)is below a thresholdω0, otherwiseδ=0.
  • Step 7. Compute the absolute error between the target point and the actual beam scan point according toΔ=|PsF(θi+1)|. IfΔis below an error thresholdε, θi+1is accepted as the final solution. Otherwise seti=i+1and 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 areD1=D2=D3=100mm, and the error threshold remains atε=0.001mm.

 figure: Fig. 6

Fig. 6 Prism motion profiles for (a) a given heart-shaped trajectory, including (b) the first set and (c) the second set of inverse solutions.

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 figure: Fig. 7

Fig. 7 Prism motion profiles for (a) a given pentacle trajectory, including (b) the first set and (c) the second set of inverse solutions.

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Case 1. A heart-shaped beam scan trajectory given by{x=10cost5cos(2t)y=10sint5sin(2t),t[0,2π].

Case 2.A pentacle beam scan trajectory given by{x=4cost+8cos(2t/3)y=4sint8sin(2t/3),t[0,6π].

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 polynomialLn(θ)is regarded as the substitution for the functional relationF(θ)between prism orientations and target coordinates. Thus it is acceptable that the partial differentiation functions of F(θ)are approximate to those of Ln(θ). In this way, the partial differentials of Ln(θ)relative to the variableθr2are obtained at a specific node whereθr2=θs. A similar approach can be applied to deduce the partial differentials of F(θ)with respect to the other variableθr3, which is omitted here.

During the numerical differentiation process of F(θ)relative toθr2, the other variableθr3is held constant, which accounts for the functional relation transformed asF(θ)=g(θr2). For necessity, other four discrete nodes, includingθs2=θs2h,θs1=θsh,θs+1=θs+handθs+2=θs+2h, are taken in the neighborhood of the specific nodeθs. Here h is the distance between any two adjacent nodes, which depends on the expected precision. Since the function values of g(θr2)at these nodes are available from Eq. (12), a fourth-order Lagrange polynomial approximate tog(θr2)can be obtained from

L4(θr2)=i=s2s+2[j=s2,jis+2(θr2θjθiθj)g(θi)].

Then the interpolation remainder defined byR4(θr2)=g(θr2)L4(θr2)is differentiated in order to derive the numerical differential of g(θr2)at the node whereθr2=θs, written as

g'(θs)=112h[g(θs2)8g(θs1)+8g(θs+1)g(θs+2)]h430g(5)(ξ)
whereξ(θs2,θs+2). The last term is relatively small and can be neglected in approximation.

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 to0.001mm.

Tables Icon

Table 1. Operation time for the inverse solutions to given beam scan trajectories

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 asD1=D2=D3=100mm.

 figure: Fig. 8

Fig. 8 Control singularities caused by (a) a linear scan trajectory passing through the center of scan region, which exist in both (b) the first set and (c) the second set of inverse solutions.

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Case 3. A linear beam scan trajectory given by{x=ty=t,t[10,10].

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:

θr1=(φ180°)±f(σ).
whereσrepresents the distance from beam scan pointPrto the center pointOPof scan region, and the functionf(σ)is under the limits of f(0)=fmaxandf(σmax)=0°. Importantly,f(σ)is appended here to provide the rotation angleθr1with a variable offset that depends on the actual position of pointPr. Whenσ=σmax, there is usually no sudden change inθr1, so the offset fromf(σ)is minimized. Asσdecreases to 0, the offset becomes more significant and finally reaches the maximumfmax.

It is noticed that the previous literature suggests the maximized offsetfmaxto be constant at 90° [16]. But in this paper,fmaxis dominated by the variation of θr1at 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 byφ(k), and the distance between this specific point and pointOPis written asσ(k). Supposing that the sample point which acts as a singularity point is marked by another variable p, the maximized offset can be obtained fromfmax=[φ(p1)φ(p+1)]/2. Therefore, Eq. (16) is further specified by

θr1(k)={[φ(k)180°]fmax90°arccos[σ(k)σmax],1k<p[φ(k)180°]+fmax90°arccos[σ(k)σmax],p<kqθr1(k1)+θr1(k+1)2,k=p

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].

 figure: Fig. 9

Fig. 9 The continuous motion profiles of three prisms for (a) the given linear scan trajectory, including (b) the first set and (c) the second set of inverse solutions.

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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).

References and links

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10. L. Wang, L. Liu, L. Zhu, J. Sun, Y. Zhou, and D. Liu, “The mechanical design of the large-optics double-shearing interferometer for the test of the diffraction-limited wavefront,” Proc. SPIE 7091, 70910S (2008). [CrossRef]  

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12. Y. Zhou, Y. Lu, M. Hei, G. Liu, and D. Fan, “Motion control of the wedge prisms in Risley-prism-based beam steering system for precise target tracking,” Appl. Opt. 52(12), 2849–2857 (2013). [CrossRef]   [PubMed]  

13. A. Li, W. Sun, W. Yi, and Q. Zuo, “Investigation of beam steering performances in rotation Risley-prism scanner,” Opt. Express 24(12), 12840–12850 (2016). [CrossRef]   [PubMed]  

14. M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006). [CrossRef]  

15. P. J. Bos, H. Garcia, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasures and imaging applications: solving the singularity problem in the two-prism design,” Opt. Eng. 46(11), 113001 (2007). [CrossRef]  

16. M. Sánchez and D. Gutow, “Control laws for a 3-element Risley prism optical beam pointer,” Proc. SPIE 6304, 630403 (2006). [CrossRef]  

17. A. Li, Q. Zuo, W. Sun, and W. Yi, “Beam distortion of rotation double prisms with an arbitrary incident angle,” Appl. Opt. 55(19), 5164–5171 (2016). [CrossRef]   [PubMed]  

18. C. T. Amirault and C. A. Dimarzio, “Precision pointing using a dual-wedge scanner,” Appl. Opt. 24(9), 1302–1308 (1985). [CrossRef]   [PubMed]  

19. Y. Li, “Closed form analytical inverse solutions for Risley-prism-based beam steering systems in different configurations,” Appl. Opt. 50(22), 4302–4309 (2011). [CrossRef]   [PubMed]  

20. A. Li, X. Gao, W. Sun, W. Yi, Y. Bian, H. Liu, and L. Liu, “Inverse solutions for a Risley prism scanner with iterative refinement by a forward solution,” Appl. Opt. 54(33), 9981–9989 (2015). [CrossRef]   [PubMed]  

21. X. Tao, H. Cho, and F. Janabi-Sharifi, “Active optical system for variable view imaging of micro objects with emphasis on kinematic analysis,” Appl. Opt. 47(22), 4121–4132 (2008). [CrossRef]   [PubMed]  

References

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  1. F. A. Rosell, “Prism scanner,” J. Opt. Soc. Am. 50(6), 521–526 (1960).
    [Crossref]
  2. G. F. Marshall, “Risley prism scan patterns,” Proc. SPIE 3787, 74–86 (1999).
    [Crossref]
  3. F. Souvestre, M. Hafez, and S. Régnier, “DMD-based multi-target laser tracking for motion capturing,” Proc. SPIE 7596, 75960B (2010).
    [Crossref]
  4. V. Lavigne and B. Richard, “Step-stare image gathering for high-resolution targeting,” RTO-MP-SET 092, 17 (2005).
  5. W. C. Warger and C. A. DiMarzio, “Dual-wedge scanning confocal reflectance microscope,” Opt. Lett. 32(15), 2140–2142 (2007).
    [Crossref] [PubMed]
  6. L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
    [Crossref]
  7. X. Tao and H. Cho, “Variable view imaging system and its application in vision based microassembly,” Proc. SPIE 6719, 67190L (2007).
    [Crossref]
  8. J. Sun, L. Liu, M. Yun, L. Wang, and M. Zhang, “The effect of the rotating double-prism wide-angle laser beam scanner on the beam shape,” Optik (Stuttg.) 116(12), 553–556 (2005).
    [Crossref]
  9. B. D. Duncan, P. J. Bos, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasure applications,” Opt. Eng. 42(4), 1038–1047 (2003).
    [Crossref]
  10. L. Wang, L. Liu, L. Zhu, J. Sun, Y. Zhou, and D. Liu, “The mechanical design of the large-optics double-shearing interferometer for the test of the diffraction-limited wavefront,” Proc. SPIE 7091, 70910S (2008).
    [Crossref]
  11. A. Li, X. Gao, and Y. Ding, “Comparison of refractive rotating dual-prism scanner used in near and far field,” Proc. SPIE 9192, 919216 (2014).
    [Crossref]
  12. Y. Zhou, Y. Lu, M. Hei, G. Liu, and D. Fan, “Motion control of the wedge prisms in Risley-prism-based beam steering system for precise target tracking,” Appl. Opt. 52(12), 2849–2857 (2013).
    [Crossref] [PubMed]
  13. A. Li, W. Sun, W. Yi, and Q. Zuo, “Investigation of beam steering performances in rotation Risley-prism scanner,” Opt. Express 24(12), 12840–12850 (2016).
    [Crossref] [PubMed]
  14. M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
    [Crossref]
  15. P. J. Bos, H. Garcia, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasures and imaging applications: solving the singularity problem in the two-prism design,” Opt. Eng. 46(11), 113001 (2007).
    [Crossref]
  16. M. Sánchez and D. Gutow, “Control laws for a 3-element Risley prism optical beam pointer,” Proc. SPIE 6304, 630403 (2006).
    [Crossref]
  17. A. Li, Q. Zuo, W. Sun, and W. Yi, “Beam distortion of rotation double prisms with an arbitrary incident angle,” Appl. Opt. 55(19), 5164–5171 (2016).
    [Crossref] [PubMed]
  18. C. T. Amirault and C. A. Dimarzio, “Precision pointing using a dual-wedge scanner,” Appl. Opt. 24(9), 1302–1308 (1985).
    [Crossref] [PubMed]
  19. Y. Li, “Closed form analytical inverse solutions for Risley-prism-based beam steering systems in different configurations,” Appl. Opt. 50(22), 4302–4309 (2011).
    [Crossref] [PubMed]
  20. A. Li, X. Gao, W. Sun, W. Yi, Y. Bian, H. Liu, and L. Liu, “Inverse solutions for a Risley prism scanner with iterative refinement by a forward solution,” Appl. Opt. 54(33), 9981–9989 (2015).
    [Crossref] [PubMed]
  21. X. Tao, H. Cho, and F. Janabi-Sharifi, “Active optical system for variable view imaging of micro objects with emphasis on kinematic analysis,” Appl. Opt. 47(22), 4121–4132 (2008).
    [Crossref] [PubMed]

2016 (2)

2015 (1)

2014 (1)

A. Li, X. Gao, and Y. Ding, “Comparison of refractive rotating dual-prism scanner used in near and far field,” Proc. SPIE 9192, 919216 (2014).
[Crossref]

2013 (1)

2011 (1)

2010 (1)

F. Souvestre, M. Hafez, and S. Régnier, “DMD-based multi-target laser tracking for motion capturing,” Proc. SPIE 7596, 75960B (2010).
[Crossref]

2008 (2)

L. Wang, L. Liu, L. Zhu, J. Sun, Y. Zhou, and D. Liu, “The mechanical design of the large-optics double-shearing interferometer for the test of the diffraction-limited wavefront,” Proc. SPIE 7091, 70910S (2008).
[Crossref]

X. Tao, H. Cho, and F. Janabi-Sharifi, “Active optical system for variable view imaging of micro objects with emphasis on kinematic analysis,” Appl. Opt. 47(22), 4121–4132 (2008).
[Crossref] [PubMed]

2007 (4)

W. C. Warger and C. A. DiMarzio, “Dual-wedge scanning confocal reflectance microscope,” Opt. Lett. 32(15), 2140–2142 (2007).
[Crossref] [PubMed]

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

X. Tao and H. Cho, “Variable view imaging system and its application in vision based microassembly,” Proc. SPIE 6719, 67190L (2007).
[Crossref]

P. J. Bos, H. Garcia, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasures and imaging applications: solving the singularity problem in the two-prism design,” Opt. Eng. 46(11), 113001 (2007).
[Crossref]

2006 (2)

M. Sánchez and D. Gutow, “Control laws for a 3-element Risley prism optical beam pointer,” Proc. SPIE 6304, 630403 (2006).
[Crossref]

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[Crossref]

2005 (2)

J. Sun, L. Liu, M. Yun, L. Wang, and M. Zhang, “The effect of the rotating double-prism wide-angle laser beam scanner on the beam shape,” Optik (Stuttg.) 116(12), 553–556 (2005).
[Crossref]

V. Lavigne and B. Richard, “Step-stare image gathering for high-resolution targeting,” RTO-MP-SET 092, 17 (2005).

2003 (1)

B. D. Duncan, P. J. Bos, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasure applications,” Opt. Eng. 42(4), 1038–1047 (2003).
[Crossref]

1999 (1)

G. F. Marshall, “Risley prism scan patterns,” Proc. SPIE 3787, 74–86 (1999).
[Crossref]

1985 (1)

1960 (1)

Amirault, C. T.

Bian, Y.

Bos, P. J.

P. J. Bos, H. Garcia, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasures and imaging applications: solving the singularity problem in the two-prism design,” Opt. Eng. 46(11), 113001 (2007).
[Crossref]

B. D. Duncan, P. J. Bos, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasure applications,” Opt. Eng. 42(4), 1038–1047 (2003).
[Crossref]

Cho, H.

DiMarzio, C. A.

Ding, Y.

A. Li, X. Gao, and Y. Ding, “Comparison of refractive rotating dual-prism scanner used in near and far field,” Proc. SPIE 9192, 919216 (2014).
[Crossref]

Doughty, N.

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[Crossref]

Duncan, B. D.

B. D. Duncan, P. J. Bos, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasure applications,” Opt. Eng. 42(4), 1038–1047 (2003).
[Crossref]

Fan, D.

Gao, X.

Garcia, H.

P. J. Bos, H. Garcia, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasures and imaging applications: solving the singularity problem in the two-prism design,” Opt. Eng. 46(11), 113001 (2007).
[Crossref]

Gutow, D.

M. Sánchez and D. Gutow, “Control laws for a 3-element Risley prism optical beam pointer,” Proc. SPIE 6304, 630403 (2006).
[Crossref]

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[Crossref]

Hafez, M.

F. Souvestre, M. Hafez, and S. Régnier, “DMD-based multi-target laser tracking for motion capturing,” Proc. SPIE 7596, 75960B (2010).
[Crossref]

Harford, S.

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[Crossref]

Hei, M.

Hoffman, C.

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[Crossref]

Janabi-Sharifi, F.

Lavigne, V.

V. Lavigne and B. Richard, “Step-stare image gathering for high-resolution targeting,” RTO-MP-SET 092, 17 (2005).

Li, A.

Li, Y.

Liu, D.

L. Wang, L. Liu, L. Zhu, J. Sun, Y. Zhou, and D. Liu, “The mechanical design of the large-optics double-shearing interferometer for the test of the diffraction-limited wavefront,” Proc. SPIE 7091, 70910S (2008).
[Crossref]

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

Liu, G.

Liu, H.

Liu, L.

A. Li, X. Gao, W. Sun, W. Yi, Y. Bian, H. Liu, and L. Liu, “Inverse solutions for a Risley prism scanner with iterative refinement by a forward solution,” Appl. Opt. 54(33), 9981–9989 (2015).
[Crossref] [PubMed]

L. Wang, L. Liu, L. Zhu, J. Sun, Y. Zhou, and D. Liu, “The mechanical design of the large-optics double-shearing interferometer for the test of the diffraction-limited wavefront,” Proc. SPIE 7091, 70910S (2008).
[Crossref]

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

J. Sun, L. Liu, M. Yun, L. Wang, and M. Zhang, “The effect of the rotating double-prism wide-angle laser beam scanner on the beam shape,” Optik (Stuttg.) 116(12), 553–556 (2005).
[Crossref]

Lu, Y.

Luan, Z.

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

Marshall, G. F.

G. F. Marshall, “Risley prism scan patterns,” Proc. SPIE 3787, 74–86 (1999).
[Crossref]

Ostaszewski, M.

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[Crossref]

Pierce, R.

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[Crossref]

Régnier, S.

F. Souvestre, M. Hafez, and S. Régnier, “DMD-based multi-target laser tracking for motion capturing,” Proc. SPIE 7596, 75960B (2010).
[Crossref]

Richard, B.

V. Lavigne and B. Richard, “Step-stare image gathering for high-resolution targeting,” RTO-MP-SET 092, 17 (2005).

Rosell, F. A.

Sanchez, M.

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[Crossref]

Sánchez, M.

M. Sánchez and D. Gutow, “Control laws for a 3-element Risley prism optical beam pointer,” Proc. SPIE 6304, 630403 (2006).
[Crossref]

Sergan, V.

P. J. Bos, H. Garcia, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasures and imaging applications: solving the singularity problem in the two-prism design,” Opt. Eng. 46(11), 113001 (2007).
[Crossref]

B. D. Duncan, P. J. Bos, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasure applications,” Opt. Eng. 42(4), 1038–1047 (2003).
[Crossref]

Souvestre, F.

F. Souvestre, M. Hafez, and S. Régnier, “DMD-based multi-target laser tracking for motion capturing,” Proc. SPIE 7596, 75960B (2010).
[Crossref]

Sun, J.

L. Wang, L. Liu, L. Zhu, J. Sun, Y. Zhou, and D. Liu, “The mechanical design of the large-optics double-shearing interferometer for the test of the diffraction-limited wavefront,” Proc. SPIE 7091, 70910S (2008).
[Crossref]

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

J. Sun, L. Liu, M. Yun, L. Wang, and M. Zhang, “The effect of the rotating double-prism wide-angle laser beam scanner on the beam shape,” Optik (Stuttg.) 116(12), 553–556 (2005).
[Crossref]

Sun, W.

Tao, X.

Wang, L.

L. Wang, L. Liu, L. Zhu, J. Sun, Y. Zhou, and D. Liu, “The mechanical design of the large-optics double-shearing interferometer for the test of the diffraction-limited wavefront,” Proc. SPIE 7091, 70910S (2008).
[Crossref]

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

J. Sun, L. Liu, M. Yun, L. Wang, and M. Zhang, “The effect of the rotating double-prism wide-angle laser beam scanner on the beam shape,” Optik (Stuttg.) 116(12), 553–556 (2005).
[Crossref]

Warger, W. C.

Xu, N.

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

Yan, A.

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

Yi, W.

Yun, M.

J. Sun, L. Liu, M. Yun, L. Wang, and M. Zhang, “The effect of the rotating double-prism wide-angle laser beam scanner on the beam shape,” Optik (Stuttg.) 116(12), 553–556 (2005).
[Crossref]

Zhang, M.

J. Sun, L. Liu, M. Yun, L. Wang, and M. Zhang, “The effect of the rotating double-prism wide-angle laser beam scanner on the beam shape,” Optik (Stuttg.) 116(12), 553–556 (2005).
[Crossref]

Zhong, X.

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

Zhou, Y.

Y. Zhou, Y. Lu, M. Hei, G. Liu, and D. Fan, “Motion control of the wedge prisms in Risley-prism-based beam steering system for precise target tracking,” Appl. Opt. 52(12), 2849–2857 (2013).
[Crossref] [PubMed]

L. Wang, L. Liu, L. Zhu, J. Sun, Y. Zhou, and D. Liu, “The mechanical design of the large-optics double-shearing interferometer for the test of the diffraction-limited wavefront,” Proc. SPIE 7091, 70910S (2008).
[Crossref]

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

Zhu, L.

L. Wang, L. Liu, L. Zhu, J. Sun, Y. Zhou, and D. Liu, “The mechanical design of the large-optics double-shearing interferometer for the test of the diffraction-limited wavefront,” Proc. SPIE 7091, 70910S (2008).
[Crossref]

Zuo, Q.

Appl. Opt. (6)

J. Opt. Soc. Am. (1)

Opt. Eng. (2)

B. D. Duncan, P. J. Bos, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasure applications,” Opt. Eng. 42(4), 1038–1047 (2003).
[Crossref]

P. J. Bos, H. Garcia, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasures and imaging applications: solving the singularity problem in the two-prism design,” Opt. Eng. 46(11), 113001 (2007).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Optik (Stuttg.) (1)

J. Sun, L. Liu, M. Yun, L. Wang, and M. Zhang, “The effect of the rotating double-prism wide-angle laser beam scanner on the beam shape,” Optik (Stuttg.) 116(12), 553–556 (2005).
[Crossref]

Proc. SPIE (8)

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

X. Tao and H. Cho, “Variable view imaging system and its application in vision based microassembly,” Proc. SPIE 6719, 67190L (2007).
[Crossref]

L. Wang, L. Liu, L. Zhu, J. Sun, Y. Zhou, and D. Liu, “The mechanical design of the large-optics double-shearing interferometer for the test of the diffraction-limited wavefront,” Proc. SPIE 7091, 70910S (2008).
[Crossref]

A. Li, X. Gao, and Y. Ding, “Comparison of refractive rotating dual-prism scanner used in near and far field,” Proc. SPIE 9192, 919216 (2014).
[Crossref]

G. F. Marshall, “Risley prism scan patterns,” Proc. SPIE 3787, 74–86 (1999).
[Crossref]

F. Souvestre, M. Hafez, and S. Régnier, “DMD-based multi-target laser tracking for motion capturing,” Proc. SPIE 7596, 75960B (2010).
[Crossref]

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[Crossref]

M. Sánchez and D. Gutow, “Control laws for a 3-element Risley prism optical beam pointer,” Proc. SPIE 6304, 630403 (2006).
[Crossref]

RTO-MP-SET (1)

V. Lavigne and B. Richard, “Step-stare image gathering for high-resolution targeting,” RTO-MP-SET 092, 17 (2005).

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Figures (9)

Fig. 1
Fig. 1 Theoretical model of the three-Risley-prism system.
Fig. 2
Fig. 2 Beam scan trajectories under different angular velocity ratios of three prisms, where ω r1 : ω r2 : ω r3 equals to (a) 1:1:1 , (b) 1:2:1.5 , (c) 1:2:1.5 , (d) 1:1.5:1.5 , (e) 1:1.5:1.5 and (f) 1:1.5:1 , respectively.
Fig. 3
Fig. 3 Schematic diagram illustrating the critical condition for a fully eliminated blind zone.
Fig. 4
Fig. 4 Comparison between beam scan regions (a) in the absence of prism 3 and (b) in the presence of prism 3.
Fig. 5
Fig. 5 The three-prism system separated into single-prism and double-prism subsystems.
Fig. 6
Fig. 6 Prism motion profiles for (a) a given heart-shaped trajectory, including (b) the first set and (c) the second set of inverse solutions.
Fig. 7
Fig. 7 Prism motion profiles for (a) a given pentacle trajectory, including (b) the first set and (c) the second set of inverse solutions.
Fig. 8
Fig. 8 Control singularities caused by (a) a linear scan trajectory passing through the center of scan region, which exist in both (b) the first set and (c) the second set of inverse solutions.
Fig. 9
Fig. 9 The continuous motion profiles of three prisms for (a) the given linear scan trajectory, including (b) the first set and (c) the second set of inverse solutions.

Tables (1)

Tables Icon

Table 1 Operation time for the inverse solutions to given beam scan trajectories

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

N 11 = ( cos θ r1 sinα,sin θ r1 sinα,cosα ) T , N 12 = ( 0,0,1 ) T , N 21 = ( 0,0,1 ) T , N 22 = ( cos θ r2 sinα,sin θ r2 sinα,cosα ) T , N 31 = ( cos θ r3 sinα,sin θ r3 sinα,cosα ) T , N 32 = ( 0,0,1 ) T .
A i =( n i1 n i ) A i-1 +{ 1 ( n i1 n i ) 2 [ 1 ( A i-1 N ) 2 ] ( n i1 n i ) A i-1 N }N= ( x i , y i , z i ) T .
r 1 =dtan δ 2 +( D 1 2d )tan δ 3 ,
r 2 = d r 1 tanα cot δ 5 +tanα ,
r 3 =[ d( r 1 + r 2 )tanα ]tan δ 10 + D 3 tan δ 11 .
R bz = r 1 + r 2 r 3 .
D c ={ ( r 1 + r 2 )[ d( r 1 + r 2 )tanα ]tan δ 2 }cot δ 3 .
λ 2 1( m 1 +1 ) λ 1 m 2 ,
λ 3 m 1 λ 1 + m 2 .
θ r1 =φ180°.
φ={ arccos( x 6 x 6 2 + y 6 2 ), y 6 0 2πarccos( x 6 x 6 2 + y 6 2 ), y 6 <0
[ x p y p ]=[ f x ( θ r2 , θ r3 ) f y ( θ r2 , θ r3 ) ]=F( θ ).
J=[ f x θ r2 f x θ r3 f y θ r2 f y θ r3 ]
L 4 ( θ r2 )= i=s2 s+2 [ j=s2,ji s+2 ( θ r2 θ j θ i θ j )g( θ i ) ].
g'( θ s )= 1 12h [ g( θ s2 )8g( θ s1 )+8g( θ s+1 )g( θ s+2 ) ] h 4 30 g ( 5 ) ( ξ )
θ r1 =( φ180° )±f( σ ).
θ r1 ( k )={ [ φ( k )180° ] f max 90° arccos[ σ( k ) σ max ], 1k<p [ φ( k )180° ]+ f max 90° arccos[ σ( k ) σ max ], p<kq θ r1 ( k1 )+ θ r1 ( k+1 ) 2 , k=p

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