Generalization of the first-order formula for analysis of scan patterns of Risley prisms

Yong-Geun Jeon

doi:10.1117/1.3655501

1 November 2011 Generalization of the first-order formula for analysis of scan patterns of Risley prisms

Yong-Geun Jeon

Author Affiliations +

Optical Engineering, Vol. 50, Issue 11, 113002 (November 2011). https://doi.org/10.1117/1.3655501

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.

1. Introduction

Risley prisms, composed of two prisms with a small apex angle, are widely used for beam scanning or steering in optical instruments^{1, 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 s_i and [TeX:] ${\bf s}_i^{\prime}$ $s_{i}^{'}$ be unit vectors in the direction of incident and transmitted rays at i’th surface, and N_i = (sin α_i cos ϕ_i, sin α_i sin ϕ_i, cos α_i) is the unit normal vector at this surface (see Fig. 1), then Snell's law can be written as¹⁴

Eq. 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}

n_{i} N_{i} \times s_{i} = n_{i}^{'} N_{i} \times s_{i}^{'},

where n_i and n_i^′ are refractive indices of the mediums before and after the i’th surface. Performing vector product with N_i on both sides of Eq. 1 and using the vector identity: A × (B × C) = (A·C)B − (A·B)C, we obtain

Eq. 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}

n_{i} [(N_{i} \cdot s_{i}) N_{i} - s_{i}] = n_{i}^{'} [(N_{i} \cdot s_{i}^{'}) N_{i} - s_{i}^{'}] .

Fig. 1

Notations related to the refraction equation.

This gives

Eq. 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}

s_{i}^{'} = γ_{i}^{'} N_{i} - \frac{n_{i}}{n_{i}^{'}} (γ_{i} N_{i} - s_{i}),

where the following definitions are used:

Eq. 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}

γ_{i} \equiv N_{i} \cdot s_{i} = \cos Θ_{i}, γ_{i}^{'} \equiv N_{i} \cdot s_{i}^{'} = \cos Θ_{i}^{'},

where Θ_i is the angle between N_i and s_i and Θ_i^′ is defined similarly. From Snell's law: [TeX:] ${n}_i^{\prime} \sin \Theta_i^\prime = n_i \sin \Theta_i$

n_{i}^{'} \sin Θ_{i}^{'} = n_{i} \sin Θ_{i}

and Eq. 4, we have

Eq. 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}

γ_{i}^{'} = \sqrt{1 - \sin^{2} Θ_{i}^{'}} = \frac{1}{n_{i}^{'}} \sqrt{n_{i}^{' 2} - n_{i}^{2} + n_{i}^{2} γ_{i}^{2}} .

Equation 3, with coefficients given by Eq. 5, is the refraction equation at i’th surface. For successive calculations, we put s_i+1 = [TeX:] ${\bf s}_i^{\prime}$

s_{i}^{'}

and n_i+1 = [TeX:] ${n}_i^{\prime}$

n_{i}^{'}

. 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 s_i = (s_xi, s_yi, s_zi) are direction cosines of the ray. The components of the refracted ray vector are represented as [TeX:] ${\bf s}_i^{\prime}$

s_{i}^{'}

= ( [TeX:] $s_{xi}^{\prime}$

s_{x i}^{'}

, [TeX:] $s_{yi}^{\prime}$

s_{y i}^{'}

, [TeX:] $s_{zi}^{\prime}$

s_{z i}^{'}

). Normal vectors and ray vectors for analysis of a typical Risley prism are shown in Fig. 2. We define the rotation angle φ_i 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

Eq. 6

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \phi _1 = \varphi _1,\quad\phi _4 = \pi + \varphi _2. \end{equation}\end{document}

φ_{1} = ϕ_{1}, φ_{4} = π + ϕ_{2} .

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.

Substituting Eq. 5 into Eq. 3 gives [TeX:] ${\bf s}^{\prime}_i = n{\bf s}_i\break + \sqrt {1 - n^2 + n^2 \gamma _i ^2 } {\bf N}_i - n\gamma _i {\bf N}_i$ $s_{i}^{'} = n s_{i} + \sqrt{1 - n^{2} + n^{2} γ_{i}^{2}} N_{i} - n γ_{i} N_{i}$ where n is defined by n = [TeX:] $n_i/n_{i}^{\prime}$ $n_{i} / n_{i}^{'}$ . 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

Eq. 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}

s_{2}^{'} = (γ_{2}^{'} - n γ_{2}) N_{2} + (n γ_{1}^{'} - γ_{1}) N_{1} + s_{1},

which means the vector [TeX:] ${\bf s}_{2}^{\prime}$

s_{2}^{'}

− 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

Eq. 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}

R_{y} (δ) = (\begin{matrix} \cos δ & 0 & \sin δ \\ 0 & 1 & 0 \\ - \sin δ & 0 & \cos δ \end{matrix}) .

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:

Eq. 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}

M_{δ} (ϕ) = R_{z} (ϕ) R_{y} (δ) R_{z} (- ϕ) .

Here the matrix R_z(φ) for the rotation about the z axis is

Eq. 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}

R_{z} (ϕ) = (\begin{matrix} \cos ϕ & - \sin ϕ & 0 \\ \sin ϕ & \cos ϕ & 0 \\ 0 & 0 & 1 \end{matrix}) .

The incident ray vector s₁ = (s_1x, s_1y, s_1z) and the transmitted ray vector s = (s_x, s_y, s_z) which is equal to [TeX:] ${\bf s}_{2}^{\prime}$

s_{2}^{'}

of Eq. 8 are related by

Eq. 11

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} {\rm s}^T = M_\delta (\varphi){\rm s}_{\rm 1} ^T, \end{equation}\end{document}

s^{T} = M_{δ} (ϕ) s_{1}^{T},

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

Eq. 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}

M_{δ} (ϕ) = M_{δ}^{(1)} (ϕ) - \frac{1}{2} δ^{2} M_{δ}^{(2)} (ϕ) + O (δ^{3}) I,

where the matrices are defined by

Eq. 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}

M_{δ}^{(1)} (ϕ) = (\begin{matrix} 1 & 0 & δ \cos ϕ \\ 0 & 1 & δ \sin ϕ \\ - δ \cos ϕ & - δ \sin ϕ & 1 \end{matrix}),

and

Eq. 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}

M_{δ}^{(2)} (ϕ) = (\begin{matrix} \cos^{2} ϕ & \cos ϕ \sin ϕ & 0 \\ \cos ϕ \sin ϕ & \sin^{2} ϕ & 0 \\ 0 & 0 & 1 \end{matrix}),

and I in Eq. 12 is the unit matrix.

When the incident ray is in the x-z plane, i.e., s_1y = 0, the deviation angle δ is given by¹⁸

Eq. 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}

\begin{matrix} δ = Θ - α + \sin^{- 1} [\sin α \cdot {(n^{2} - \sin^{2} Θ)}^{1 / 2} - \sin Θ \cos α] . \end{matrix}

When s_1y is not zero, the deviation angle must be defined to be the angle between the vectors projected on the x-z plane, i.e., s_1p ≡ (s_1x, 0, s_1z) and s_p ≡ (s_x, 0, s_z). If the angle between s_1p and N₁ is denoted by Θ_p, and the angle between s_1p and s₁ by Θ_v as shown in Fig. 3, then the corresponding angles [TeX:] $\Theta_p^{\prime}$

Θ_{p}^{'}

and [TeX:] $\Theta_v^{\prime}$

Θ_{v}^{'}

for the refracted ray [TeX:] ${\bf s}_1^{\prime}$

s_{1}^{'}

are given by Snell's law in the following form:¹⁹

Eq. 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}

n_{r} \sin Θ_{p} \cos Θ_{v} = n_{r}^{'} \sin Θ_{p}^{'} \cos Θ_{v}^{'}

Eq. 16b

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

n_{r} \sin Θ_{v} = n_{r}^{'} \sin Θ_{v}^{'},

where n_r and [TeX:] ${n}_r^{\prime}$

n_{r}^{'}

are refractive indices of the mediums before and after the surface. By using Eq. 16a with n_r = 1 and [TeX:] ${n}_r^{\prime}$

n_{r}^{'}

= n, and following the same method for derivation of Eq. 15, we obtain

Eq. 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}

\begin{matrix} δ & = & Θ_{p} - α + \sin^{- 1} [\sin α \cdot (n^{2} ψ^{2} \\ - {\sin^{2} Θ_{p})}^{1 / 2} - \sin Θ_{p} \cos α], \end{matrix}

where ψ = [TeX:] $\cos\Theta_v^{\prime}/\cos\Theta _v$

\cos Θ_{v}^{'} / \cos Θ_{v}

. Using Eq. 16b, we have [TeX:] $\psi \break\approx 1 + [(n^2 - 1)/(2n^2)]\Theta _v ^2$

ψ \approx 1 + [(n^{2} - 1) / (2 n^{2})] Θ_{v}^{2}

. When Θ_v is zero, we have Θ_p = Θ and Eq. 17 reduces to Eq. 15. By expanding Eq. 17 up to the third-order terms, we obtain

Eq. 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}

\begin{matrix} δ & = & (n - 1) α + (ν / n) α Θ_{p}^{2} - ν α^{2} Θ_{p} \\ + (ν / n) α Θ_{v}^{2} + (n / 3) ν α^{3}, \end{matrix}

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:

Eq. 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) α Θ_{p}^{2} - ν α^{2} Θ_{p} + (ν / n) α Θ_{v}^{2} + (n / 3) ν α^{3} .

Fig. 3

Geometry for definitions of incident angles Θ_p and Θ_v for an oblique ray vector s₁. Here the vectors s_1p and N₁ are on the x-z plane.

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 Θ_v = 0.0 rad, and 1.5 to 3.5 mrad at Θ_v = 0.1 rad. The graph of the third-order approximation is symmetric about Θ_p = 0.15 rad because ε⁽³⁾ is the quadratic equation of Θ_p and it has the minimum value at Θ_p = (1/2)nα = 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.

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

Eq. 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}

\begin{matrix} s_{x} & = & s_{1 x} + δ \cos ϕ \cdot s_{1 z} - (1 / 2) δ^{2} (s_{1 x} \cos^{2} β + s_{1 y} \cos β \sin β) + O (δ^{3}) s_{1 x}, \\ s_{y} & = & s_{1 y} + δ \sin ϕ \cdot s_{1 z} - (1 / 2) δ^{2} (s_{1 x} \cos β \sin β + s_{1 y} \sin^{2} β) + O (δ^{3}) s_{1 y}, \\ s_{z} & = & s_{1 z} - δ \cos ϕ \cdot s_{1 x} - δ \sin ϕ \cdot s_{1 y} - (1 / 2) δ^{2} s_{1 z} + O (δ^{3}) s_{1 z} . \end{matrix}

Using δ = δ⁽¹⁾ + ε⁽³⁾, and s_1x ≈ s_1y ≈ Θ and s_1z ≈ 1 in the higher order terms, Eq. 20 becomes

Eq. 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}

\begin{matrix} s_{x} & = & s_{1 x} + δ^{(1)} \cos ϕ \cdot s_{1 z} + ɛ^{(3)} \cos ϕ \cdot s_{1 z} + O ({(n - 1)}^{2} α^{2} Θ), \\ s_{y} & = & s_{1 y} + δ^{(1)} \sin ϕ \cdot s_{1 z} + ɛ^{(3)} \cos ϕ \cdot s_{1 z} + O ({(n - 1)}^{2} α^{2} Θ), \\ s_{z} & = & s_{1 z} - δ^{(1)} \cos ϕ \cdot s_{1 x} - δ^{(1)} \sin ϕ \cdot s_{1 y} + O ({(n - 1)}^{2} α^{2}) . \end{matrix}

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

Eq. 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}

M_{δ} (ϕ) = (\begin{matrix} 1 & 0 & δ^{(1)} \cos ϕ \\ 0 & 1 & δ^{(1)} \sin ϕ \\ - δ^{(1)} \cos ϕ & - δ^{(1)} \sin ϕ & 1 \end{matrix}),

then the errors in calculating s_x and s_y are determined by the third-order terms of α and Θ, and the one for s_z is determined by the second-order term of α. The last term in the equation of s_x or s_y 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 s_1z = 1. Figure 4 shows that ɛ⁽³⁾ ≈ 1.7 mrad when Θ_p = Θ_v = 0.1 rad, from which the total error in calculation of s_x or s_y 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 s_x and s_y 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 M_δ(φ) in Eq. 22 is independent of incident ray vectors, the transmitted ray vector s can be obtained from the following equation:

Eq. 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}

s^{T} = M_{δ_{2}} (ϕ_{2}) M_{δ_{1}} (ϕ_{1}) s_{1}^{T} \equiv \bar{M} s_{1}^{T} .

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

Eq. 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}

\bar{M} = (\begin{matrix} 1 & 0 & δ_{1} \cos ϕ_{1} + δ_{2} \cos ϕ_{2} \\ 0 & 1 & δ_{1} \sin ϕ_{1} + δ_{2} \sin ϕ_{2} \\ - δ_{1} \cos ϕ_{1} - δ_{2} \cos ϕ_{2} & - δ_{1} \sin ϕ_{1} - δ_{2} \sin ϕ_{2} & 1 \end{matrix}) .

Using Eq. 23 gives

Eq. 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}

\begin{matrix} s_{x} & = & s_{x 1} + (δ_{1} \cos ϕ_{1} + δ_{2} \cos ϕ_{2}) s_{z 1}, \\ s_{y} & = & s_{y 1} + (δ_{1} \sin ϕ_{1} + δ_{2} \sin ϕ_{2}) s_{z 1}, \\ s_{z} & = & - (δ_{1} \cos ϕ_{1} + δ_{2} \cos ϕ_{2}) s_{x 1} \\ - (δ_{1} \sin ϕ_{1} + δ_{2} \sin ϕ_{2}) s_{y 1} + s_{z 1} . \end{matrix}

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

Eq. 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}

\begin{matrix} s_{x} & = & δ_{1} \cos ϕ_{1} + δ_{2} \cos ϕ_{2}, \\ s_{y} & = & δ_{1} \sin ϕ_{1} + δ_{2} \sin ϕ_{2}, \\ s_{z} & = & 1 . \end{matrix}

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

Eq. 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}

s_{x} = δ_{1} (1 + \cos ϕ^{'}), s_{y} = δ_{1} \sin ϕ^{'}, s_{z} = 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 = (s_x, s_y, s_z). Since sin²θ = s_x² + s_y² and sin²θ ≈ θ ², Eq. 27 gives

Eq. 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}

θ \approx \sqrt{2 δ_{1}^{2} (1 + \cos ϕ^{'})} = 2 δ_{1} \cos \frac{ϕ^{'}}{2},

and since tanχ = s_y / s_x, it gives

Eq. 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}

\tan χ = \frac{\sin ϕ^{'}}{1 + \cos ϕ^{'}} = \tan \frac{ϕ^{'}}{2} .

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

Eq. 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}

\begin{matrix} s_{x} & = & s_{x 1} + (\sum_{i = 1}^{N} δ_{i} \cos ϕ_{i}) s_{z 1}, \\ s_{y} & = & s_{y 1} + (\sum_{i = 1}^{N} δ_{i} \sin ϕ_{i}) s_{z 1}, \\ s_{z} & = & - (\sum_{i = 1}^{N} δ_{i} \cos ϕ_{i}) s_{x 1} \\ - (\sum_{i = 1}^{N} δ_{i} \sin ϕ_{i}) s_{y 1} + s_{z 1}, \end{matrix}

where N is the number of prisms, and δ_i is the deviation angle, and φ_i 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

Eq. 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}

ϕ_{1} (t) = ϕ_{1}^{(i)} + 2 π f_{1} t, ϕ_{2} (t) = ϕ_{2}^{(i)} + 2 π f_{2} t,

with time t and initial angles [TeX:] $\varphi_1^{(i)}$

ϕ_{1}^{(i)}

and [TeX:] $\varphi_2^{(i)}$

ϕ_{2}^{(i)}

. The scan patterns depend on the initial angles of rotations and the following ratios:¹¹

Eq. 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}

k \equiv δ_{2} / δ_{1}, M \equiv f_{2} / f_{1} .

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:] $\varphi_1^{(i)}$

ϕ_{1}^{(i)}

= 0 and [TeX:] $\varphi_2^{(i)}$

ϕ_{2}^{(i)}

= π so that we have φ₂(t) = π − φ₁(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:

Eq. 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}

s_{x} = 0, s_{y} = 2 δ_{1} \sin (2 π f_{1} t), s_{z} = 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 s_y ≈ 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.

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 s_x = 0.0 and s_y = 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:] $\varphi_1^{(i)}$ $ϕ_{1}^{(i)}$ = 0, [TeX:] $\varphi_2^{(i)}$ $ϕ_{2}^{(i)}$ = π, M = −1, k = 0.7.

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 δ_A and δ_B, and the rotation angle be φ_A and φ_B. Therefore we put δ₁ = δ₂ = δ_A, φ₁ = φ_A, φ₂ = π − φ_A for Risley prism A. Using Eq. 24, we obtain

Eq. 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}

{\bar{M}}_{A} (ϕ_{A}) = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 2 δ_{A} \sin ϕ_{A} \\ 0 & - 2 δ_{A} \sin ϕ_{A} & 1 \end{matrix}) .

For Risley prism B, we put δ₁ = δ₂ = δ_B, φ₁ = (π/2) + φ_B, φ₂ = −(π/2) − φ_B, so that we obtain

Eq. 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}

{\bar{M}}_{B} (ϕ_{B}) = (\begin{matrix} 1 & 0 & - 2 δ_{B} \cos ϕ_{B} \\ 0 & 1 & 0 \\ 2 δ_{B} \cos ϕ_{B} & 0 & 1 \end{matrix}) .

The transmitted ray vector s = (s_x, s_y, s_z), when the incident ray vector is s₁, is given by the following equation:

Eq. 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}

s^{T} = M_{δ_{B}} (ϕ_{B}) M_{δ_{A}} (ϕ_{A}) s_{1}^{T} .

When the incident ray is axial, substituting Eqs. 34, 35 into Eq. 36 gives

Eq. 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}

s_{x} = - 2 δ_{B} \sin ϕ_{B}, s_{y} = 2 δ_{A} \sin ϕ_{A}, s_{z} = 1 .

To obtain a scan pattern, we put the rotational frequencies to be f_A and f_B for each Risley prism so that

Eq. 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}

ϕ_{A} (t) = 2 π f_{A} t, ϕ_{B} (t) = 2 π f_{B} t .

The ratios similar to the ones in Eq. 32 can be defined by

Eq. 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}

k \equiv δ_{B} / δ_{A}, M \equiv f_{B} / f_{A} .

For exact calculation using the refraction equation, we need to specify the rotation angles of each prism. Let φ_A₁ and φ_A₂ be rotation angles of prisms in Risley prism A (referring to Fig. 7), and φ_B1 and φ_B2 be the ones of prisms in Risley prism B, then we have

Eq. 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}

\begin{matrix} ϕ_{A 1} (t) & = & 2 π f_{A} t, ϕ_{A 2} (t) = π - 2 π f_{A} t, \\ ϕ_{B 1} (t) & = & \frac{π}{2} + 2 π f_{B} t, \\ ϕ_{B 2} (t) & = & - \frac{π}{2} - 2 π f_{B} t . \end{matrix}

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.

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:] $\varphi_{A1}^{(i)}$ $ϕ_{A 1}^{(i)}$ = 0, [TeX:] $\varphi_{A2}^{(i)}$ $ϕ_{A 2}^{(i)}$ = π, [TeX:] $\varphi_{B1}^{(i)}$ $ϕ_{B 1}^{(i)}$ = π/2, [TeX:] $\varphi_{B2}^{(i)}$ $ϕ_{B 2}^{(i)}$ = −π/2. (a) M = 6, k = 1, (b) M = 7, k = 1.

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 s_x = 0.2 and s_y = 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 s_x = 0.2 and s_y = 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.

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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.

Citation Download Citation

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

Published: 1 November 2011

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KEYWORDS

Prisms

Refraction

Error analysis

Optical engineering

Matrices

Refractive index

3D modeling

1.

Introduction

2.

Refraction Equation for Cascaded Planar Surfaces