1 November 2011 Generalization of the first-order formula for analysis of scan patterns of Risley prisms
Author Affiliations +
Abstract
A first-order formula for calculations of the direction cosines of the rays refracted by Risley prisms was derived. The formula was obtained by representing the deviation of the ray passing through a prism by the product of rotation matrices, and using the series expansion of the product. It can be applied to the system of an arbitrary number of prisms or a combination of Risley prisms. Related errors were discussed and some numerical calculations were made and compared with the exact solutions using the refraction equation. The scan patterns of a single Risley prism or a combination of two Risley prisms calculated using the generalized first-order formula are in good agreement with the exact solutions.
Jeon: Generalization of the first-order formula for analysis of scan patterns of Risley prisms

1.

Introduction

Risley prisms, composed of two prisms with a small apex angle, are widely used for beam scanning or steering in optical instruments1, 2, 3 and other developing systems.4, 5, 6, 7 The beam steering using three prisms is also investigated for the same applications.7 The primary concern for analyses of Risley prisms is to calculate the deviations of the ray passing through them, thereby obtaining the steering or scan patterns. The basic properties of Risley prisms can be understood as the combination of deviations by each prism,8, 9 but the rigorous calculation is complicated because several refractions at planar surfaces are involved. In previous works, several methods have been used such as the three-dimensional model,2 analytic formula,10 and approximate formula.11, 12 Recently, approximate formulas up to third-order of the apex angle were obtained by expanding analytic solution.13 But all the analytic formulas and the approximate ones in those works were obtained for a single Risley prism composed of two prisms. In this paper, by representing the deviation of a ray passing through a prism by the product of rotation matrices, a generalized first-order formula was obtained. It can be applied to the system of an arbitrary number of prisms or combination of Risley prisms. Related errors were discussed and some numerical calculations were made and compared with the exact solutions using the refraction equation. The scan patterns of a single Risley prism or a combination of two Risley prisms were calculated using the generalized first-order formula, and the results were in good agreement with the exact solutions.

2.

Refraction Equation for Cascaded Planar Surfaces

Let si and [TeX:] ${\bf s}_i^{\prime}$ si be unit vectors in the direction of incident and transmitted rays at i’th surface, and Ni = (sin αi cos ϕi, sin αi sin ϕi, cos αi) is the unit normal vector at this surface (see Fig. 1), then Snell's law can be written as14

1

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} n_i {\bf N}_i \;\times\; {\bf s}_i = n'_i {\bf N}_i \;\times\; {\bf s}^{\prime}_i, \end{equation}\end{document} niNi×si=niNi×si,
where ni and ni are refractive indices of the mediums before and after the i’th surface. Performing vector product with Ni on both sides of Eq. 1 and using the vector identity: A × (B × C) = (A·C)B − (A·B)C, we obtain

2

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} n_i [({\bf N}_i \cdot {\bf s}_i){\bf N}_i - {\bf s}_i] = n'_i [({\bf N}_i \cdot {\bf s}^{\prime}_i){\bf N}_i - {\bf s}^{\prime}_i] .\end{equation}\end{document} ni[(Ni·si)Nisi]=ni[(Ni·si)Nisi].

Fig. 1

Notations related to the refraction equation.

113002_1_1.jpg

This gives

3

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} {\bf s}^{\prime}_i = \gamma '_i {\bf N}_i - \frac{{n_i }}{{n'_i }}(\gamma _i {\bf N}_i - {\bf s}_i), \end{equation}\end{document} si=γiNinini(γiNisi),
where the following definitions are used:

4

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \gamma _i \equiv {\bf N}_i \cdot {\bf s}_i = \cos \Theta _i,\quad\gamma '_i \equiv {\bf N}_i \cdot {\bf s}^{\prime}_i = \cos \Theta '_i, \end{equation}\end{document} γiNi·si=cosΘi,γiNi·si=cosΘi,
where Θi is the angle between Ni and si and Θi is defined similarly. From Snell's law: [TeX:] ${n}_i^{\prime} \sin \Theta_i^\prime = n_i \sin \Theta_i$ nisinΘi=nisinΘi and Eq. 4, we have

5

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \gamma ^{\prime}_i = \sqrt {1 - \sin ^2 \Theta'_i} = \frac{1}{{n^{\prime}_i }}\sqrt {n^{\prime2}_i - n_i ^2 + n_i ^2 \gamma _i ^2 }. \end{equation}\end{document} γi=1sin2Θi=1nini2ni2+ni2γi2.
Equation 3, with coefficients given by Eq. 5, is the refraction equation at i’th surface. For successive calculations, we put si+1 = [TeX:] ${\bf s}_i^{\prime}$ si and ni+1 = [TeX:] ${n}_i^{\prime}$ ni . By applying the refraction equation successively to each surface, we can obtain directions of a ray passing through an arbitrary number of surfaces. We notice that the components of the ray vector si = (sxi, syi, szi) are direction cosines of the ray. The components of the refracted ray vector are represented as [TeX:] ${\bf s}_i^{\prime}$ si = ( [TeX:] $s_{xi}^{\prime}$ sxi , [TeX:] $s_{yi}^{\prime}$ syi , [TeX:] $s_{zi}^{\prime}$ szi ). Normal vectors and ray vectors for analysis of a typical Risley prism are shown in Fig. 2. We define the rotation angle φi of each prism to be zero when the apex is directed downward along the x axis as shown in Fig. 2. When the second and third surfaces are orthogonal to the z axis, the azimuth angles of normal vectors N1 and N4 are related to the rotation angles by

6

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \phi _1 = \varphi _1,\quad\phi _4 = \pi + \varphi _2. \end{equation}\end{document} φ1=ϕ1,φ4=π+ϕ2.
For the configuration of Fig. 2 with φ2 = π, we have ϕ4 = 2π, which means N4 = N1 when the apex angles are same, as expected. In Sec. 4, we will use Eqs. 3, 4, 5 for numerical calculations.

Fig. 2

Normal vectors and ray vectors for analysis of a Risley prism; the rotation angles are φ1 = 0 and φ2 = π.

113002_1_2.jpg

Substituting Eq. 5 into Eq. 3 gives [TeX:] ${\bf s}^{\prime}_i = n{\bf s}_i\break + \sqrt {1 - n^2 + n^2 \gamma _i ^2 } {\bf N}_i - n\gamma _i {\bf N}_i$ si=nsi+1n2+n2γi2NinγiNi where n is defined by n = [TeX:] $n_i/n_{i}^{\prime}$ ni/ni . The refraction equation in this form is used for analysis of the Risley prisms,13 and also for analyses of the plano convex or hyperboloidal focusing lenses.15, 16

3.

Derivation of the Approximate Formula

By applying Eqs. 3, 4, 5 to two successive surfaces of a prism with refractive index n, we obtain

7

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} {\bf s}^{\prime}_2 = (\gamma '_2 - n\gamma _2){\bf N}_2 + (n\gamma '_1 - \gamma _1){\bf N}_1 + {\bf s}_1, \end{equation}\end{document} s2=(γ2nγ2)N2+(nγ1γ1)N1+s1,
which means the vector [TeX:] ${\bf s}_{2}^{\prime}$ s2 s1 is in the plane generated by N1 and N2. So, if N1 and N2 are in the x-z plane, then the effect of the prism is the rotation of the ray vector s1 about the y-axis, and it is clear that the rotation is counter-clockwise when the rotation angle φ is zero. Let δ be the deviation angle of the prism, then this rotation can be represented in matrix form as follows17

8

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} R_y (\delta) = \left( {\begin{array}{c@{\quad}c@{\quad}c} {\cos \delta } & 0 & {\sin \delta } \\[3pt] 0 & 1 & 0 \\[3pt] { - \sin \delta } & 0 & {\cos \delta } \\ \end{array}} \right). \end{equation}\end{document} Ry(δ)=cosδ0sinδ010sinδ0cosδ.

To find the transmitted ray vector for a prism rotated by φ in azimuth, the components of the incident ray vector have to be transformed to the coordinate system with φ = 0, i.e., the coordinate system fixed to the prism, then rotated about the y axis by δ and transformed to the original coordinate system. This operation can be represented as follows:

9

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} M_\delta (\varphi) = R_z (\varphi)R_y (\delta)R_z ( - \varphi). \end{equation}\end{document} Mδ(ϕ)=Rz(ϕ)Ry(δ)Rz(ϕ).
Here the matrix Rz(φ) for the rotation about the z axis is

10

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} R_z (\varphi) = \left( {\begin{array}{c@{\quad}c@{\quad}c} {\cos \varphi } & { - \sin \varphi } & 0 \\ {\sin \varphi } & {\cos \varphi } & 0 \\ 0 & 0 & 1 \\ \end{array}} \right). \end{equation}\end{document} Rz(ϕ)=cosϕsinϕ0sinϕcosϕ0001.
The incident ray vector s1 = (s1x, s1y, s1z) and the transmitted ray vector s = (sx, sy, sz) which is equal to [TeX:] ${\bf s}_{2}^{\prime}$ s2 of Eq. 8 are related by

11

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} {\rm s}^T = M_\delta (\varphi){\rm s}_{\rm 1} ^T, \end{equation}\end{document} sT=Mδ(ϕ)s1T,
where the superscript T represents the transpose operation for the components of vector. Substituting Eq. 8 into Eq. 9, and using the series expansion about δ gives

12

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} M_\delta (\varphi) = M_\delta ^{(1)} (\varphi) - \frac{1}{2}\delta ^2 M_\delta ^{(2)} (\varphi) + O(\delta ^3)I, \end{equation}\end{document} Mδ(ϕ)=Mδ(1)(ϕ)12δ2Mδ(2)(ϕ)+O(δ3)I,
where the matrices are defined by

13

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} M_\delta ^{(1)} (\varphi) = \left( {\begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & {\delta \cos \varphi } \\ 0 & 1 & {\delta \sin \varphi } \\ { - \delta \cos \varphi } & { - \delta \sin \varphi } & 1 \\ \end{array}} \right), \end{equation}\end{document} Mδ(1)(ϕ)=10δcosϕ01δsinϕδcosϕδsinϕ1,
and

14

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} M_\delta ^{(2)} (\varphi) = \left( {\begin{array}{c@{\quad}c@{\quad}c} {\cos ^2 \varphi } & {\cos \varphi \sin \varphi } & 0 \\ {\cos \varphi \sin \varphi } & {\sin ^2 \varphi } & 0 \\ 0 & 0 & 1 \\ \end{array}} \right), \end{equation}\end{document} Mδ(2)(ϕ)=cos2ϕcosϕsinϕ0cosϕsinϕsin2ϕ0001,
and I in Eq. 12 is the unit matrix.

When the incident ray is in the x-z plane, i.e., s1y = 0, the deviation angle δ is given by18

15

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{eqnarray} \delta = \Theta - \alpha + \sin ^{ - 1} [ {\sin \alpha \cdot (n^2 - \sin ^2 \Theta)^{1/2} - \sin \Theta \cos \alpha }].\nonumber\\ \end{eqnarray}\end{document} δ=Θα+sin1[sinα·(n2sin2Θ)1/2sinΘcosα].
When s1y is not zero, the deviation angle must be defined to be the angle between the vectors projected on the x-z plane, i.e., s1p ≡ (s1x, 0, s1z) and sp ≡ (sx, 0, sz). If the angle between s1p and N1 is denoted by Θp, and the angle between s1p and s1 by Θv as shown in Fig. 3, then the corresponding angles [TeX:] $\Theta_p^{\prime}$ Θp and [TeX:] $\Theta_v^{\prime}$ Θv for the refracted ray [TeX:] ${\bf s}_1^{\prime}$ s1 are given by Snell's law in the following form:19

16a

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} n_r \sin \Theta _p \cos \Theta _v = n'_r \sin \Theta '_p \cos \Theta '_v \end{equation}\end{document} nrsinΘpcosΘv=nrsinΘpcosΘv

16b

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \,n_r \sin \Theta _v = n'_r \sin \Theta '_v, \end{equation}\end{document} nrsinΘv=nrsinΘv,
where nr and [TeX:] ${n}_r^{\prime}$ nr are refractive indices of the mediums before and after the surface. By using Eq. 16a with nr = 1 and [TeX:] ${n}_r^{\prime}$ nr = n, and following the same method for derivation of Eq. 15, we obtain

17

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{eqnarray} \delta &=& \Theta _p - \alpha + \sin ^{ - 1} [ \sin \alpha \cdot (n^2 \psi ^2 \nonumber\\ &&- \sin ^2 \Theta _p)^{1/2} - \sin \Theta _p \cos \alpha ], \end{eqnarray}\end{document} δ=Θpα+sin1[sinα·(n2ψ2sin2Θp)1/2sinΘpcosα],
where ψ = [TeX:] $\cos\Theta_v^{\prime}/\cos\Theta _v$ cosΘv/cosΘv . Using Eq. 16b, we have [TeX:] $\psi \break\approx 1 + [(n^2 - 1)/(2n^2)]\Theta _v ^2$ ψ1+[(n21)/(2n2)]Θv2 . When Θv is zero, we have Θp = Θ and Eq. 17 reduces to Eq. 15. By expanding Eq. 17 up to the third-order terms, we obtain

18

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{eqnarray} \delta &=& (n - 1)\alpha + (\nu /n)\alpha \Theta _p ^2 - \nu \alpha ^2 \Theta _p\nonumber\\ && +\, (\nu /n)\alpha \Theta _v ^2 + (n/3)\nu \alpha ^3, \end{eqnarray}\end{document} δ=(n1)α+(ν/n)αΘp2να2Θp+(ν/n)αΘv2+(n/3)να3,
where the constant ν is defined by ν = (1/2)(n2 − 1). The first-order approximation of δ is δ(1) = (n − 1)α as known, and Eq. 18 can be written as δ = δ(1) + ε(3), where ε(3) is the third-order terms:

19

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \varepsilon ^{(3)} \!= (\nu /n)\alpha \Theta _p ^2 - \!\nu \alpha ^2 \Theta _p + (\nu /n)\alpha \Theta _v ^2 + (n/3)\nu \alpha ^3. \end{equation}\end{document} ɛ(3)=(ν/n)αΘp2να2Θp+(ν/n)αΘv2+(n/3)να3.

Fig. 3

Geometry for definitions of incident angles Θp and Θv for an oblique ray vector s1. Here the vectors s1p and N1 are on the x-z plane.

113002_1_3.jpg

Figure 4 shows the deviation angles calculated by Eqs. 17, 18. It is for the case of the apex angle α = 0.2 rad (≈11.5 deg) and the refractive index n = 1.5, so that δ(1) = 0.1 rad. It is seen that the errors of the first-order approximation are in the range of 0.6 to 2.5 mrad at Θv = 0.0 rad, and 1.5 to 3.5 mrad at Θv = 0.1 rad. The graph of the third-order approximation is symmetric about Θp = 0.15 rad because ε(3) is the quadratic equation of Θp and it has the minimum value at Θp = (1/2) = 0.15 rad which is the approximate value of the minimum deviation angle.

Fig. 4

Deviation angles of a single prism: α = 0.2 rad, n = 1.5.

113002_1_4.jpg

By substituting Eqs. 13, 14 into Eq. 12, the components of s can be written as

20

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{eqnarray} s_x &=& s_{1x} + \delta \cos \varphi \cdot s_{1z} - (1/2)\delta ^2 (s_{1x} \cos ^2 \beta + s_{1y} \cos \beta \sin \beta) + O(\delta ^3)s_{1x},\nonumber\\ s_y &=& s_{1y} + \delta \sin \varphi \cdot s_{1z} - (1/2)\delta ^2 (s_{1x} \cos \beta \sin \beta + s_{1y} \sin ^2 \beta) + O(\delta ^3)s_{1y},\\ s_z &=& s_{1z} - \delta \cos \varphi \cdot s_{1x} - \delta \sin \varphi \cdot s_{1y} \, - (1/2)\delta ^2 s_{1z} + O(\delta ^3)s_{1z}.\nonumber\\[-7pt]\nonumber \end{eqnarray}\end{document} sx=s1x+δcosϕ·s1z(1/2)δ2(s1xcos2β+s1ycosβsinβ)+O(δ3)s1x,sy=s1y+δsinϕ·s1z(1/2)δ2(s1xcosβsinβ+s1ysin2β)+O(δ3)s1y,sz=s1zδcosϕ·s1xδsinϕ·s1y(1/2)δ2s1z+O(δ3)s1z.
Using δ = δ(1) + ε(3), and s1xs1yΘ and s1z ≈ 1 in the higher order terms, Eq. 20 becomes

21

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{eqnarray} s_x &=& s_{1x} + \delta ^{(1)} \cos \varphi \cdot s_{1z} + \varepsilon ^{(3)} \cos \varphi \cdot s_{1z} + O((n - 1)^2 \alpha ^2 \Theta),\nonumber\\ s_y &=& s_{1y} + \delta ^{(1)} \sin \varphi \cdot s_{1z} + \varepsilon ^{(3)} \cos \varphi \cdot s_{1z} + O((n - 1)^2 \alpha ^2 \Theta),\\ s_z &=& s_{1z} - \delta ^{(1)} \cos \varphi \cdot s_{1x} - \delta ^{(1)} \sin \varphi \cdot s_{1y} \, + O((n - 1)^2 \alpha ^2).\nonumber \end{eqnarray}\end{document} sx=s1x+δ(1)cosϕ·s1z+ɛ(3)cosϕ·s1z+O((n1)2α2Θ),sy=s1y+δ(1)sinϕ·s1z+ɛ(3)cosϕ·s1z+O((n1)2α2Θ),sz=s1zδ(1)cosϕ·s1xδ(1)sinϕ·s1y+O((n1)2α2).

Equation 21 means that if we use the first-order approximation Mδ (φ), i.e.,

22

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} M_\delta (\varphi) = \left( {\begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & {\delta ^{(1)} \cos \varphi } \\ 0 & 1 & {\delta ^{(1)} \sin \varphi } \\ { - \delta ^{(1)} \cos \varphi } & { - \delta ^{(1)} \sin \varphi } & 1 \\ \end{array}} \right), \end{equation}\end{document} Mδ(ϕ)=10δ(1)cosϕ01δ(1)sinϕδ(1)cosϕδ(1)sinϕ1,

then the errors in calculating sx and sy are determined by the third-order terms of α and Θ, and the one for sz is determined by the second-order term of α. The last term in the equation of sx or sy in Eq. 21 depends on Θ, so that the errors do not vanish even in the case of ε(3) = 0. For example, when n = 1.5, α = 0.2 rad, and Θ = 0.1, it is (n − 1)2α2Θ = 1.0 mrad. The order of magnitude of the third terms including ɛ(3) are estimated by assuming φ = 0 rad and s1z = 1. Figure 4 shows that ɛ(3) ≈ 1.7 mrad when Θp = Θv = 0.1 rad, from which the total error in calculation of sx or sy using the first-order formula is estimated to be 2.7 mrad. When several prisms are involved, the total error depends on the relative orientations of the prisms, and the error analysis done here will give only the order of magnitudes.

Hereafter, we will use Eq. 22 for calculations of ray vectors, and also use δ = δ(1) by dropping the upper index. To investigate the scan patterns, only sx and sy are needed, so that the errors of our first-order formula are of third-order.

For analysis of a Risley prism, let the deviation angle of the first prism be δ1, and the one of the second prism be δ2, and the rotation angle of each prism be φ1 and φ2. Since Mδ(φ) in Eq. 22 is independent of incident ray vectors, the transmitted ray vector s can be obtained from the following equation:

23

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} {\bf s}^T = M_{\delta _2 } (\varphi _2)M_{\delta _1 } (\varphi _1){\bf s}_1 ^T \equiv \bar M{\bf s}_1 ^T. \end{equation}\end{document} sT=Mδ2(ϕ2)Mδ1(ϕ1)s1TM¯s1T.

Substituting Eq. 22 into 23 and keeping the terms of the first-order with respect to δ1 or δ2, we have

24

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \bar M = \left( {\begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & {\delta _1 \cos \varphi _1 + \delta _2 \cos \varphi _2 } \\[3pt] 0 & 1 & {\delta _1 \sin \varphi _1 + \delta _2 \sin \varphi _2 } \\[3pt] { - \delta _1 \cos \varphi _1 - \delta _2 \cos \varphi _2 } & { - \delta _1 \sin \varphi _1 - \delta _2 \sin \varphi _2 } & 1 \\ \end{array}} \right). \end{equation}\end{document} M¯=10δ1cosϕ1+δ2cosϕ201δ1sinϕ1+δ2sinϕ2δ1cosϕ1δ2cosϕ2δ1sinϕ1δ2sinϕ21.

Using Eq. 23 gives

25

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{eqnarray} s_x &=& s_{x1} + (\delta _1 \cos \varphi _1 + \delta _2 \cos \varphi _2)s_{z1},\nonumber\\ s_y &=& s_{y1} + (\delta _1 \sin \varphi _1 + \delta _2 \sin \varphi _2)s_{z1}, \nonumber\\[-8pt]\\[-8pt] s_z &=& - (\delta _1 \cos \varphi _1 + \delta _2 \cos \varphi _2)s_{x1} \nonumber\\ && - (\delta _1 \sin \varphi _1 + \delta _2 \sin \varphi _2)s_{y1} + s_{z1}.\nonumber \end{eqnarray}\end{document} sx=sx1+(δ1cosϕ1+δ2cosϕ2)sz1,sy=sy1+(δ1sinϕ1+δ2sinϕ2)sz1,sz=(δ1cosϕ1+δ2cosϕ2)sx1(δ1sinϕ1+δ2sinϕ2)sy1+sz1.
Equation 25 is the first-order formula for the beam deviation of an arbitrary incident ray s1. There is no restriction on the incident ray s1 in Eq. 25, so that it can be applied to the axial ray or oblique rays. When the incident ray is axial; s1 = (0, 0, 1), Eq. 25 gives

26

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{eqnarray} s_x &=& \delta _1 \cos \varphi _1 + \delta _2 \cos \varphi _2,\nonumber\\ s_y &=& \delta _1 \sin \varphi _1 + \delta _2 \sin \varphi _2,\\ s_z &=& 1\,.\nonumber \end{eqnarray}\end{document} sx=δ1cosϕ1+δ2cosϕ2,sy=δ1sinϕ1+δ2sinϕ2,sz=1.
Equation 26 has been used for analyses of scan patterns of Risley prisms.8, 9, 12

When δ2 = δ1, φ1 = 0, and φ2 = φ, Eq. 26 gives

27

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} s_x = \delta _1 (1 + \cos \varphi '),\,\,s_y = \delta _1 \sin \varphi ',\,\,s_z = 1. \end{equation}\end{document} sx=δ1(1+cosϕ),sy=δ1sinϕ,sz=1.
A formula equivalent to Eq. 27 can be obtained by using the polar angle θ and the azimuth angle χ of the ray vector s = (sx, sy, sz). Since sin2θ = sx2 + sy2 and sin2θθ 2, Eq. 27 gives

28

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \theta \approx \sqrt {2\delta _1 ^2 (1 + \cos \varphi ')} = 2\delta _1 \cos \frac{{\varphi '}}{2}, \end{equation}\end{document} θ2δ12(1+cosϕ)=2δ1cosϕ2,
and since tanχ = sy / sx, it gives

29

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \tan \chi = \frac{{\sin \varphi '}}{{1 + \cos \varphi '}} = \tan \frac{{\varphi '}}{2}. \end{equation}\end{document} tanχ=sinϕ1+cosϕ=tanϕ2.
Equations 28, 29 were obtained directly by the vector-summation of the deviations.8

It is straightforward to generalize Eq. 25 to the formula for a system composed of arbitrary number of prisms, and the result is

30

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{eqnarray} s_x &=& s_{x1} + \left(\sum_{i = 1}^N {\delta _i \cos \varphi _i } \right)s_{z1}, \nonumber\\ s_y &=& s_{y1} + \left(\sum_{i = 1}^N {\delta _i \sin \varphi _i } \right)s_{z1}, \nonumber\\[-8pt]\\[-8pt] s_z &=& - \left(\sum_{i = 1}^N {\delta _i \cos \varphi _i } \right)s_{x1} \nonumber\\ &&- \left(\sum_{i = 1}^N {\delta _i \sin \varphi _i }\right)s_{y1} + s_{z1} \,,\nonumber \end{eqnarray}\end{document} sx=sx1+i=1Nδicosϕisz1,sy=sy1+i=1Nδisinϕisz1,sz=i=1Nδicosϕisx1i=1Nδisinϕisy1+sz1,
where N is the number of prisms, and δi is the deviation angle, and φi is the rotation angle for the i’th prism.

4.

Numerical Calculations for Sample Cases

To obtain the scan patterns of a Risley prism, we put the rotational frequencies of the prisms to be f1 and f2 each so that the angles of rotations are

31

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \varphi _1 (t) = \varphi _1 ^{(i)} + 2\pi f_1 t,\,\,\varphi _2 (t) = \varphi _2 ^{(i)} + 2\pi f_2 t, \end{equation}\end{document} ϕ1(t)=ϕ1(i)+2πf1t,ϕ2(t)=ϕ2(i)+2πf2t,
with time t and initial angles [TeX:] $\varphi_1^{(i)}$ ϕ1(i) and [TeX:] $\varphi_2^{(i)}$ ϕ2(i) . The scan patterns depend on the initial angles of rotations and the following ratios:11

32

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} k \equiv \delta _2 /\delta _1,\,\,\,M \equiv f_2 /f_1. \end{equation}\end{document} kδ2/δ1,Mf2/f1.
For the first case, we consider the configuration where the second prism rotates in the opposite direction with the same frequency, i.e., f2 = −f1, and the initial angles are [TeX:] $\varphi_1^{(i)}$ ϕ1(i) = 0 and [TeX:] $\varphi_2^{(i)}$ ϕ2(i) = π so that we have φ2(t) = πφ1(t). Let the deviation angles be the same; δ2 = δ1. This case corresponds to k = 1 and M = −1. From Eq. 26, we obtain the approximate solutions for the components of the transmitted ray vector as follows:

33

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} s_x = 0,\quad s_y = 2\delta _1 \sin (2\pi f_1 t),\quad s_z = 1, \end{equation}\end{document} sx=0,sy=2δ1sin(2πf1t),sz=1,
which means the linear scan along the y axis. In this paper, all the refractive indices are assumed to be n = 1.5, and the apex angle of prism 1 to be α1 = 0.2 rad, so that δ1 = (n − 1)α1 = 0.1 rad. The exact numerical calculation using Eqs. 3, 4, 5 can be performed straightforwardly, where the azimuth angles of normal vectors are determined by Eq. 6. Figure 5 shows the scan patterns generated during one period of rotation for prism 1. The maximum error in the x direction is about 1.6 mrad at sy ≈ 0.17, which is comparable with the total error (=2.7 mrad) obtained for a single prism in Sec. 3. The bow tie pattern of the exact solution is one of the typical properties of the Risley prisms with the same apex angle.20

Fig. 5

Scan patterns of Risley prism : φ1(i) = 0, φ2(i) = π, M = −1, k = 1.

113002_1_5.jpg

Figure 6 is an example of scan patterns obtained by the approximate calculations using Eq. 25 and the exact numerical calculations using Eqs. 3, 4, 5, in which we consider the axial and the oblique incident rays which are specified by s1 = (sinθ, 0, cosθ) with the polar angles θ = 0.0 and 0.1 rad each. The case of θ = 0.0 rad is comparable to the one given in Ref. 11. It can be seen that the error of the approximate solution for θ = 0.0 rad at sx = 0.0 and sy = 0.17 is approximately 2.8 mrad in the y direction, and the one for θ = 0.1 rad is approximately 4.2 mrad in the x direction. The errors in the x direction are approximately zero in both cases. The approximate solutions are in reasonably good agreement with the exact solutions even though the errors increase with the polar angle θ.

Fig. 6

Scan patterns of Risley prism for axial (θ = 0) and oblique (θ = 0.1 rad) incident rays: [TeX:] $\varphi_1^{(i)}$ ϕ1(i) = 0, [TeX:] $\varphi_2^{(i)}$ ϕ2(i) = π, M = −1, k = 0.7.

113002_1_6.jpg

When two Risley prisms are combined as in Fig. 7, it can generate more general two-dimensional scan patterns. As an example, we use the configuration in which the second Risley prism is rotated by 90 deg, and the apex angles of prisms in each Risley prism are the same. Here the directions of rotation in each Risley prism are opposite. Let the deviation angles of the prisms in each Risley prism be δA and δB, and the rotation angle be φA and φB. Therefore we put δ1 = δ2 = δA, φ1 = φA, φ2 = πφA for Risley prism A. Using Eq. 24, we obtain

34

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \bar M_A (\varphi _A) = \left( {\begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & 0 \\[3pt] 0 & 1 & {2\delta _A \sin \varphi _A } \\[3pt] 0 & { - 2\delta _A \sin \varphi _A } & 1 \\ \end{array}} \right). \end{equation}\end{document} M¯A(ϕA)=100012δAsinϕA02δAsinϕA1.
For Risley prism B, we put δ1 = δ2 = δB, φ1 = (π/2) + φB, φ2 = −(π/2)φB, so that we obtain

35

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \bar M_B (\varphi _B) = \left( {\begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & { - 2\delta _B \cos \varphi _B } \\ [2pt] 0 & 1 & 0 \\ [2pt] {2\delta _B \cos \varphi _B } & 0 & 1 \\ \end{array}} \right). \end{equation}\end{document} M¯B(ϕB)=102δBcosϕB0102δBcosϕB01.
The transmitted ray vector s = (sx, sy, sz), when the incident ray vector is s1, is given by the following equation:

36

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} {\bf s}^T = M_{\delta _B } (\varphi _B)M_{\delta _A } (\varphi _A){\bf s}_1 ^T. \end{equation}\end{document} sT=MδB(ϕB)MδA(ϕA)s1T.
When the incident ray is axial, substituting Eqs. 34, 35 into Eq. 36 gives

37

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} s_x = - 2\delta _B \sin \varphi _B,\quad s_y = 2\delta _A \sin \varphi _A,\quad s_z = 1. \end{equation}\end{document} sx=2δBsinϕB,sy=2δAsinϕA,sz=1.
To obtain a scan pattern, we put the rotational frequencies to be fA and fB for each Risley prism so that

38

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \varphi _A (t) = 2\pi f_A t,\quad \varphi _B (t) = 2\pi f_B t. \end{equation}\end{document} ϕA(t)=2πfAt,ϕB(t)=2πfBt.
The ratios similar to the ones in Eq. 32 can be defined by

39

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} k \equiv \delta _B /\delta _A,\quad M \equiv f_B /f_A. \end{equation}\end{document} kδB/δA,MfB/fA.
For exact calculation using the refraction equation, we need to specify the rotation angles of each prism. Let φA1 and φA2 be rotation angles of prisms in Risley prism A (referring to Fig. 7), and φB1 and φB2 be the ones of prisms in Risley prism B, then we have

40

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{eqnarray} \varphi _{A1} (t) &=& 2\pi f_A t,\quad \varphi _{A2} (t) = \pi - 2\pi f_A t,\,\,\nonumber\\ \varphi _{B1} (t) &=& \frac{\pi }{2} + 2\pi f_B t,\,\,\\ \varphi _{B2} (t) &=& - \frac{\pi }{2} - 2\pi f_B t\,.\nonumber \end{eqnarray}\end{document} ϕA1(t)=2πfAt,ϕA2(t)=π2πfAt,ϕB1(t)=π2+2πfBt,ϕB2(t)=π22πfBt.
Using Eqs. 6, 40, we can determine all the components of the normal vectors, and perform exact calculations using Eqs. 3, 4, 5.

Fig. 7

Geometry of a combination of two Risley prisms.

113002_1_7.jpg

Figure 8 shows the scan patterns generated during one period of rotation of the prism A1 for the two cases of M = 6, k = 1, and M = 7, k = 1.

Fig. 8

Scan patterns of a combination of Risley prisms: [TeX:] $\varphi_{A1}^{(i)}$ ϕA1(i) = 0, [TeX:] $\varphi_{A2}^{(i)}$ ϕA2(i) = π, [TeX:] $\varphi_{B1}^{(i)}$ ϕB1(i) = π/2, [TeX:] $\varphi_{B2}^{(i)}$ ϕB2(i) = −π/2. (a) M = 6, k = 1, (b) M = 7, k = 1.

113002_1_8.jpg

It can be seen that the errors of the approximate solutions for M = 6 are about 10 mrad in the x direction and 7 mrad in the y direction at the point of sx = 0.2 and sy = 0.2. The errors of the same level of magnitudes are obtained for the case of M = 7. We can notice that the approximate solutions have reasonable accuracies for describing the scan patterns. It is also observed that the scan patterns obtained from this configuration are composed of closed curves when M is even.

5.

Conclusion

A first-order formula for calculations of the direction cosines of the rays refracted by Risley prisms was derived. The formula was obtained by representing the deviation of the ray passing through a prism by the product of rotation matrices, and using the series expansion of the product. It can be applied to the system of arbitrary number of prisms or combination of Risley prisms. It permits the calculations of the direction cosines of the transmitted ray vectors for arbitrary incident rays such as oblique rays. The errors associated with the first-order formula were analyzed by using the series expansion of the expression for the deviation angle. It showed that the errors are of third-order of the prism's apex angle and the incidence angle. The numerical estimation using examples showed that the total error for a single prism is reasonably small, approximately 2.7 mrad for incident angles of 0.2 rad. The generalized first-order formula was applied to the numerical calculations of the scan patterns of a single Risley prism and a combination of two Risley prisms, and the results were compared with the exact solutions using the formulation based on the refraction equations. The maximum error in the scan patterns of the single Risley prism in the example was 2.8 mrad for axial incident rays and 4.2 mrad for oblique rays with the polar angle of 0.1 rad. The maximum error in the scan patterns of the combination of two Risley prisms in the example were about 10 mrad in the x direction at the point of sx = 0.2 and sy = 0.2. Even though the errors tend to increase with the number of prisms, the first-order approximate formula will be useful for analyzing the scan patterns of Risley prisms.

References

1.  C. Schwarze, “A new look at Risley prisms,” Photonics Spectra 40(6), 67–70 (2006). Google Scholar

2.  V. Lavigne and B. Ricard, “Fast Risley prisms camera steering system: calibration and image distortions correction through the use of a three-dimensional refraction model,” Opt. Eng. 46, 043201 (2007). 10.1117/1.2722754 Google Scholar

3.  G. Garcia-Torales, M. Strojnik, and G. Paez, “Risley prisms to control wave-front tilt and displacement in a vectorial shearing interferometer,” Appl. Opt. 41, 1380–1384 (2002). 10.1364/AO.41.001380 Google Scholar

4.  C. T. Amirault and C. A. DiMarzio, “Precision pointing using a dual-wedge scanner,” Appl. Opt. 24, 1302–1308 (1985). 10.1364/AO.24.001302 Google Scholar

5.  W. C. Stone, M. Juberts, N. Dagalakis, J. Stone, and J. Gorman, “Performance analysis of next-generation LADAR for manufacturing, construction, and mobility,” NISTIR 7117, National Institute of Standards and Technology (2004). Google Scholar

6.  B. D. Duncan, P. J. Bos, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasures applications,” Opt. Eng. 42, 1038–1047 (2003). 10.1117/1.1556393 Google Scholar

7.  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, 113001 (2007). 10.1117/1.2802147 Google Scholar

8.  F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th Ed., Sec. 2.7, McGraw-Hill Book Co., New York (1981). Google Scholar

9.  F. A. Rosell, “Prism scanner,” J. Opt. Soc. Am. 50, 521–526 (1960). 10.1364/JOSA.50.000521 Google Scholar

10.  M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006). 10.1117/12.679097 Google Scholar

11.  G. F. Marshall, “Risley prism scan patterns,” part of the SPIE Conference on Optical Scanning: Design and Application, Denver, Colorado, Vol. 3787, pp. 74–86 (1999). Google Scholar

12.  W. L. Wolfe and G. J. Zissis, Eds., The Infrared Handbook, revised ed., Chapter 10, The Infrared Information Analysis (IRIA) Center, Environmental Research Institute of Michigan, Prepared for the Office of Naval Research, Department of the Navy, Washington, DC (1989). Google Scholar

13.  Y. Li, “Third-order theory of the Risley-prism-based beam steering system,” Appl. Opt. 50, 679–686 (2011). 10.1364/AO.50.000679 Google Scholar

14.  M. Born and E. Wolf, Principles of Optics, 7th Ed., Sec. 3.2.2, Cambridge University Press, Cambridge, UK (1999). Google Scholar

15.  A. Ghaffar and Q. A. Naqvi, “Focusing of electromagnetic plane wave into uniaxial crystal by a three dimensional plano convex lens,” PIER 83, 25–42 (2008). 10.2528/PIER08041404 Google Scholar

16.  A. Ghaffar, A. A. Rizvi, and Q. A. Naqvi, “Fields in the focal space of symmetrical hyperboloidal focusing lens,” PIER 89, 255–273 (2009). 10.2528/PIER08122207 Google Scholar

17.  G. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th Ed., Sec. 3.3, Academic Press, San Diego, CA (1995). Google Scholar

18.  E. Hecht, Optics, 4th Ed., Sec. 5.5.1, Addison Wesley, San Francisco, CA (2002). Google Scholar

19.  H. S. Uhler, “Oblique deviation and refraction produced by prisms,” Am. Math. Monthly 28(1), 1–10 (1921). 10.2307/2974201 Google Scholar

20.  R. E. Juhala and G. Dube, “Refractive beam steering,” Proc. SPIE 5528, 282–292 (2004). 10.1117/12.559865 Google Scholar

Biography

Yong-Geun Jeon received his PhD degree in physics from Korea Advanced Institute of Science and Technology in 1994. The topic of his PhD research was the stimulated Raman and Brillouin scattering in high pressure gases. He is now a principal researcher of the Agency for Defense Development. He has been working on the developments of electro-optical systems. His main research interests are solid-state lasers, nonlinear optics, and the optical design for laser systems. He is a member of the OSA.

Yong-Geun Jeon, "Generalization of the first-order formula for analysis of scan patterns of Risley prisms," Optical Engineering 50(11), 113002 (1 November 2011). https://doi.org/10.1117/1.3655501
JOURNAL ARTICLE
8 PAGES


SHARE
Back to Top