Analytical forward and inverse solutions using a mathematics model for double wedge prisms in different configurations

Shian-Fu Lai; Cheng-Chung Lee

doi:10.1364/OSAC.2.002199

1. Introduction

Double wedge prism systems (DWPS) cascade two wedge prisms with different configurations and the two wedge prisms’ respective rotations. The Risley prisms [1–3] are two wedge prisms of equal power that are counter rotated around the optical axis at equal angular velocities, is equivalent to a single wedge prism of variable power. In the past several decades, DWPS was utilized in a large variety of applications including precision satellite positioning systems [4], raster scanners for confocal microscopy [5], interferometers [6], one-dimension scanners for imaging systems [7], laser radar [8], and shearing interferometers [9]. DWPS are particularly useful in precision tracking and pointing systems [10–15], in which DWPS can be used to move a laser beam or track a target over a wide angular range with high resolution and precision.

With regard to other scanning devices [16], such as the galvanometric scanner, rotating polygonal mirror scanner, and oscillating mirror scanner, DWPS offer tremendous advantages including compact size, stability, robustness, a large field of view, high precision and resolution, a high scan speed, and insensitivity to vibrations and wobble.

Using the exact patterns produced by DWPS when rotating the two wedge prisms, scan patterns are studied with regard to the characteristic parameters of the wedge prism material, apex angles, and rotational angular velocity. In the past several decades, many researchers explored the scan patterns of DWPS in precision tracking and pointing systems using different methods [10–13]. A specific prism configuration could use a monopolistic mathematical model to explain the scan patterns.

Conventional-method researchers focused on four special configurations: two elementary right triangle wedge prisms of Type 1 and Type 2, as shown in Fig. 1; the two elementary wedge prisms of type 1 and 2 through arrangement and assembly produced four permutations [11]. An incident ray passes through the wedge prisms. The vector form of Snell’s law is applied to derive the ray deviation formulas and obtain the pointing position of the emergent light ray. An inverse procedure is then performed based on the specified position in free space in order to derive the analytical inverse solutions using a two-step method. Thus, the orientations of both wedge prisms are determined. The analytical forward and inverse solutions are independently derived for each elementary configuration. The derived procedure is complicated.

Fig. 1. (a) Type 1, (b) Type2, and (c) Type1 and Type2 through arrangement and assembly produce four configurations [11].

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None of these methods is concerned with the inherent optical characteristics of wedge prisms. An incident ray passes through the wedge prism, the perpendicular component is deviated, and the parallel component is unaffected by the wedge prism. Recently, Lai and Lee [17] proposed a method to solve the problem. This method was not the same as the conventional method. Rather, Lai and Lee’s method was based on the scalar form of Snell’s law and 2D vector algebra. This method derived the analytical forward and reverse solutions for DWPS in four configurations. Four ray deviation formulas were obtained for the four configurations. The derived procedures same as conventional method are time-consuming and complicated.

DWPS possesses a diversity of configurations, applications, and development for different configurations. A specified mathematical model is needed to explain its optical characteristics. Following the conventional method for a specific configuration, a monopolistic mathematical model was proposed, but this approach derived the analytical forward and inverse formulas in a very complicated and time-consuming manner. Moreover, the mathematical model was confined to a specific configuration.

Because the geometrical shape of a right triangle wedge prism has only one inclined wedge plane surface (compared with a triangle wedge prism, which has two inclined wedge plane surfaces), the mathematical model of a right triangle wedge prism is simpler than that for the triangle wedge prism [18]. Almost all previous research focused on right triangle wedge prisms, and this tendency restricted the development of DWPS.

For further applications and developments of DWPS, creating an effective and efficient mathematical model is critical. This study proposes a mathematical model for DWPS application in different configurations. The analytical forward and inverse solutions are derived based on the mathematical model.

This study is organized as follows. In Section 2, for a triangle wedge prism, the ray deviation formulas are derived based on the scalar form of Snell’s law and 2D vector algebra. Section 3 applies the method described in Section 2 and derives the ray deviation formulas of DWPS. In addition, the analytical forward solutions are proposed. Section 4 applies the mathematical model to analyze the total internal reflection. Section 5 further extends the analytical forward solutions to obtain the analytical inverse solutions. Two examples are proposed, and the numerical results are compared with those of the conventional method. The final section draws our conclusions.

This study was conducted under the following assumptions: no prism tilt, no component errors, and no assembly errors were considered.

2. Ray deviation formulas for single wedge prism in far-field region

When a single wedge prism of Type 1 or Type 2 is composed of two right triangle wedge prisms, an incident ray travels in the direction specified by a unit ray vector and passes through the wedge prism. The direction of the incident rays in 2D vector form is (δ_1in, θ). θ₁ is the rotation angle of the wedge prism, δ_1in cos(θ₁ –θ) is a perpendicular component deviated by the wedge prism, and δ_1insin(θ₁ –θ) is a parallel component unaffected by the wedge prism. The refractive index for the prism is n₂; for air, it is n₁. This is based on the scalar form of Snell’s law and 2D vector algebra. The deviation formula of Type 1 is Eq. (1), and that of Type 2 is Eq. (2) [17].

(1)$$\begin{aligned}{\delta _{1out}} &= - {\alpha _{12}} + {\sin ^{ - 1}}(\sin ({\alpha _{12}})\sqrt {{{(\frac{{{n_{12}}}}{{{n_{11}}}})}^2} - {{\sin }^2}({\delta _{1in}}\cos ({\theta _1} - \theta ))} \\ &\quad+ \cos ({\alpha _{12}})\sin ({\delta _{1in}}\cos ({\theta _1} - \theta )) - {\delta _{1in}}\cos ({\theta _1} - \theta )\end{aligned}$$

(2)$$\begin{aligned}{\delta _{1out}} &= {\sin ^{ - 1}}(\sin ({\alpha _{11}})\sqrt {{{(\frac{{{n_{12}}}}{{{n_{11}}}})}^2} - {{\sin }^2}({\delta _{1in}}\cos ({\theta _1} - \theta ) - {\alpha _{11}})}\\ &\quad + \cos ({\alpha _{11}})\sin ({\delta _{1in}}\cos ({\theta _1} - \theta ) - {\alpha _{11}}) - {\delta _{1in}}\cos ({\theta _1} - \theta )\end{aligned}$$

2.1 Relation between the deviation formulas and geometrical shape for a triangle wedge prism

Referring to Fig. 2(a), (b), and (c), the apex angle of the triangle wedge prism is α = α₁₁ + α₁₂. For α₁₁ = 0°, we define a right triangle wedge prism of Type 1. For α12 = 0°, we define a right triangle wedge prism of Type 2.

Fig. 2. From geometry, a triangle is two combined right triangles. Wedge profiles: (a) Type1 at α11 = 0°, (b) Type2 at α12 = 0°, (c) Type3 where apex angle is sum of α₁₁ and α₁₂.

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The deviation formula of a triangle wedge prism is deduced based on the geometrical shapes of Fig. 2 and Eqs. (1) – (2). From geometry, the triangle wedge prism has two inclined wedge plane surfaces. The inclined angles are α₁₁ and α₁₂. For α₁₁ = 0°, the triangle wedge prism changes to the right triangle wedge prism of Type 1. For α₁₂ = 0°, the triangle wedge prism changes to the right triangle wedge prism of Type 2. We refer to the geometrical shape when merging the deviation formulas of Type 1 and Type 2 into a formula to explain the power in the ray deviation for a triangle wedge prism. The deviation formula as follows:

(3)$$\begin{array}{l} {\delta _{1out}} = - ({\alpha _{12}}) - {\delta _{1in}} \cos ({\theta _1} - \theta ) + {\sin ^{ - 1}}\{ \sin ({\alpha _{11}} + {\alpha _{12}})\sqrt {{{(\frac{{{n_{12}}}}{{{n_{11}}}})}^2} - {{[\sin ({\delta _{1in}}\cos ({\theta _1} - \theta ) - {\alpha _{11}})]}^2}} \\ \quad \quad + \cos ({\alpha _{11}} + {\alpha _{12}})\sin ({\delta _{1in}}\cos ({\theta _1} - \theta ) - {\alpha _{11}})\} \end{array}$$

For Eq. (3) at α₁₁ = 0°, we obtain the deviation formula equal to Eq. (1). For Eq. (3) at α₁₂ = 0°, we obtain the deviation formula equal to Eq. (2).

In the following section, the formula of the power in the ray deviation for a single triangle wedge prism is derived based on geometry and the scalar form of Snell’s law [17]. This formula is compared with Eq. (3).

2.2 Formulas of power in ray deviation for single triangle wedge prism

A incident ray vector $\overrightarrow {o{a_1}} $ in a 3D Cartesian coordinate system, as shown in Fig. 3(a), α, β, and δ_1in are the angles the ray vector makes with the coordinate axes (x, y, z) and $|{\overrightarrow {o{a_1}} } |= 1$, it is transformed into the 2D vector form of (δ_1in, θ) [3,17,19], (K, L, M) expressed in direction cosines, where K = cos(α), L = cos(β) and M = cos(δ_1in) [11]; θ = tan⁻¹(L/K), δ_1in= cos⁻¹(M), by applying the 2D vector form, the pointing position of $\overrightarrow {o{a_1}} $ can be expressed as follows: δ_1in = deviation angle; θ = rotation angle.

Fig. 3. (a) Schematic diagrams illustrating the notation and coordinate system for an incident ray vector in a 3D coordinate system. (b) The incident ray vector from 3D vector form transformed into 2D vector form.

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Figure 3(a) is a schematic diagram that illustrates the notation and 3D coordinate system for the incident ray vector and refracted ray vector. The counterclockwise rotation angle of the wedge prism is defined as positive; the clockwise rotation angle is defined as negative.

δ_1in is the angle of incident ray $\overline {ao} $ relative to the z-axis. Incident ray $\overline {ao} $ and the z-axis determine a plane of [PQRS]. θ is the angle of this plane rotated around the z-axis as counted from the (x, 0, z) plane. δ_1in and θ are independent variables, and the direction of the incident ray vector in 2D vector form is as follows:

(4)$$\overrightarrow {{\delta _{1in}}} = {\delta _{1in}}\cos (\theta )\hat{x} + {\delta _{1in}}\sin (\theta )\hat{y}$$

In Fig. 4(a), based on 2D vector algebra [17], the incident ray vector is divided into two components: δ_1in cos(θ–θ₁) is a perpendicular component refracted by the wedge prism, and δ_1in sin(θ–θ₁) is a parallel component unaffected by the wedge prism.

Fig. 4. Schematic diagrams illustrating ray propagation paths for a triangle wedge prism. Refractive index: air = n₁₁ and prism = n₁₂, apex angle: α = α₁₁ +α₁₂, rotation angle = θ₁. (b), (c), and (d) illustrate incident ray from different directions entering and traveling through wedge prism. For (a)–(d), plane of (eb₁b₃) corresponds to profile of wedge prism perpendicular to thick-side.

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From Fig. 4(b), the perpendicular components of the incident rays point to the thick-side, and δ_1in cos(θ–θ₁) > α₁₁. The ray deviation formula is derived from the scalar form of Snell’s law and geometry as

(5)$$\begin{array}{l} {\delta _{1out}} = - ({\alpha _{12}}) + {\sin ^{ - 1}}\{ \sin ({\alpha _{11}} + {\alpha _{12}})\sqrt {{{(\frac{{{n_{12}}}}{{{n_{11}}}})}^2} - {{[\sin ({\delta _{1in}}\cos ({\theta _1} - \theta ) - {\alpha _{11}})]}^2}} \\ \quad \quad + \cos ({\alpha _{11}} + {\alpha _{12}})[\sin ({\delta _{1in}}\cos ({\theta _1} - \theta ) - {\alpha _{11}})]\} - {\delta _{1in}}\cos ({\theta _1} - \theta ) \end{array}$$

From Fig. 4(c), the perpendicular components of the incident rays point to the thick-side, and δ_1in cos(θ–θ₁) < α₁₁. The ray deviation formula is derived from the scalar form of Snell’s law and geometry as follows:

(6)$$\begin{array}{l} {\delta _{1out}} = - ({\alpha _{12}}) + {\sin ^{ - 1}}\{ \sin ({\alpha _{11}} + {\alpha _{12}})\sqrt {{{(\frac{{{n_{12}}}}{{{n_{11}}}})}^2} - {{[\sin ({\delta _{1in}}\cos ({\theta _1} - \theta ) - {\alpha _{11}})]}^2}} \\ \quad \quad - \cos ({\alpha _{11}} + {\alpha _{12}})[\sin ({\alpha _{11}} - {\delta _{1in}}\cos ({\theta _1} - \theta ))]\} - {\delta _{1in}}\cos ({\theta _1} - \theta ) \end{array}$$

From Fig. 4(d), the perpendicular components of the incident rays point to the thin-side, and the ray deviation formula is derived from the scalar form of Snell’s law and geometry as follows:

(7)$$\begin{array}{l} {\delta _{1out}} = - ({\alpha _{12}}) + {\sin ^{ - 1}}\{ \sin ({\alpha _{11}} + {\alpha _{12}})\sqrt {{{(\frac{{{n_{12}}}}{{{n_{11}}}})}^2} - {{[\sin ({\delta _{1in}}\cos ({\theta _1} - \theta ) + {\alpha _{11}})]}^2}} \\ \quad \quad - \cos ({\alpha _{11}} + {\alpha _{12}})[\sin ({\delta _{1in}}\cos ({\theta _1} - \theta ) + {\alpha _{11}})]\} + {\delta _{1in}}\cos ({\theta _1} - \theta ) \end{array}$$

Figure 5 shows the powers of ray deviation for the wedge prism at different refractive indices and apex angles. The curves are plotted according to Eqs. (4)–(7) by rotated the triangle wedge prism from 0° to 180°. The incident ray points to the thick-side. At δ_1in cos(θ₁ – θ) > α₁₁ or δ_1in cos(θ₁ – θ) < α₁₁, two different paths travel into the wedge prism. The ray deviation formula is derived from the two different paths described by Eqs. (5) and (6). From the two formulas that determine the same powers and are shown in Figs. 5(a)–(d), one can see that the tilt of the wedge plane surface in the triangle wedge prism does not influence the symmetrical distribution of the power curves around the z-axis. At θ = 0°, prism rotated from 0° to 90°, δ_1in cos(θ₁ – θ) > 0° ; prism rotated from 90° to 180°, δ_1in cos(θ₁ – θ) < 0°. The curves of Eqs. (5) and (6) are mirror images of Eq. (7). The two symmetric curves indicate that the coefficients of δ_1in cos(θ₁–θ) in these equations change their signs.

Fig. 5. Triangle wedge prism is rotated from 0° to 180°, the incident angle is at rest. For BK-7, the incident angle δ_1in = 6°, θ = 0°, n₁₁ = 1, n₁₂ = 1.5, (a) α₁₁ = 5°, α₁₂ = 10°, δ_1in _> α₁₁; (b) α₁₁ = 15°, α₁₂ = 7°, δ_1in _< α₁₁. For Ge, the incident angle δ_1in = 4° , θ = 0°, n₁₁ = 1, n₁₂ = 4, (c) α₁₁ = 2°, α₁₂ = 5°, δ_1in _> α₁₁; and (d) α₁₁ = 6°, α₁₂ = 3°, δ_1in _< α₁₁.

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The powers of ray deviation in a triangle wedge prism can be derived. An equation is deduced from Eqs. (5)–(7) and from the equation describing the wedge prism rotated about the z-axis. The resulting equation is as follows [17]:

(8)$$\begin{array}{l} {\delta _{1out}} = - ({\alpha _{12}}) + {\sin ^{ - 1}}\{ \sin ({\alpha _{11}} + {\alpha _{12}})\sqrt {{{(\frac{{{n_{12}}}}{{{n_{11}}}})}^2} - {{[\sin ({\delta _{1in}}\cos ({\theta _1} - \theta ) - {\alpha _{11}})]}^2}} \\ \quad \quad + \cos ({\alpha _{11}} + {\alpha _{12}})[\sin ({\delta _{1in}}\cos ({\theta _1} - \theta ) - {\alpha _{11}})]\} - {\delta _{1in}}\cos ({\theta _1} - \theta ) \end{array}$$

Geometrical optics defines the limits for wavelengths close to zero in electromagnetic theory. When the wavelength is small enough, the sum-total optical phenomena may be deduced from geometrical considerations. In this approximation, the optical laws may be formulated in the language of geometry [19].

Referring to Fig. 2, the ray deviation formula for a triangle wedge prism is deduced from the formulas of Type 1 and Type 2. This is shown as Eq. (3). The ray deviation formulas are derived based on the geometry. The ray deviation formula of Eq. (8) is equal to Eq. (3).

For Fig. 2 and Fig. 5 at α₁₂ > α₁₁, in other cases, α₁₂ < α₁₁, α₁₂ = α₁₁, α₁₂ = 0° and α₁₁ = 0° share the same formula.

In 2D vector form, the total power of the ray deviation for a triangle wedge prism is as follows [17]:

(9)$$\; \overrightarrow {{\delta _{1out}}} = {\delta _{1out}}\cos ({\theta _1})\hat{x} + {\delta _{1out}}\sin ({\theta _1})\hat{y}$$

(10)$$\; \overrightarrow {{\delta _{1t}}} = \; \overrightarrow {{\delta _{1in}}} + \; \overrightarrow {{\delta _{1out}}} = {\delta _{1t}}\cos ({\beta _1})\hat{x} + {\delta _{1t}}\sin ({\beta _1})\hat{y}$$

(11)$$\left\{ \begin{array}{l} {\beta_1} = {\cos^{ - 1}}(\frac{{({\delta_{1in}}\cos \theta + {\delta_{1out}}\cos {\theta_1})}}{{{{[{{({\delta_{1in}})}^2} + {{({\delta_{1out}})}^2} + 2{\delta_{1in}}{\delta_{1out}}\cos ({\theta_1} - \theta )]}^{0.5}}}})\\ {\delta_{1t}} = {[{({\delta_{1in}})^2} + {({\delta_{1out}})^2} + 2{\delta_{1in}}{\delta_{1out}}\cos ({\theta_1} - \theta )]^{0.5}} \end{array} \right.$$

The ray deviation formula of Eqs. (8)–(11) defines the apex angle parameters of α₁₁ and α₁₂ and can be used for different types. For Type 1, α₁₁ = 0°; for Type 2, α₁₂ = 0°; and for a triangle wedge prism, α₁₁ and α₁₂ are not zero and may be defined as specified values.

3. Formulas of power in ray deviation for DWPS in the far-field region

Wedge prisms of different types are shown in Fig. 5. Type 1 and Type 2 are right triangle wedge prisms with one inclined wedge plane surface. Type a, Type b and Type c are triangle wedge prisms with two inclined wedge plane surfaces, Type a, Type b, and Type c belong to element type of Type 3.

The schematic configurations of DWPS are cascaded into two elementary wedge prisms. Through arrangement and assembly, a diverse group of configurations is produced.

For a triangle wedge prism, the four apex angle parameters of α₁₁, α₁₂, α₂₁, and α₂₂ define the types and configurations of DWPS. For double triangle wedge prisms, the orientations of the two wedge prisms are specified by their rotation angles of θ₁ and θ₂. θ₁ is of the S₁ prism, and θ₂ is of the S₂ prism.

In Fig. 7(a) and (b), the center axis of S₁ and S₂ is the z-axis. The two wedge prisms are rotated about the z-axis. The orientations of θ₁ and θ₂ are the angles around the z-axis counted from the (x, o, z) plane. In the (x, y, z) coordinate system, point o is the origin, and the optical axis is the z-axis in the center of two triangle wedge prisms. An incident ray travels in the direction specified by the ray vector $\overrightarrow {ao} $ and hits the center of the first surface of S₁ at point o. The ray emerges from the first wedge prism of point b. This shifts a distance from the origin, hits the first surface of the second wedge prism at point c, deviates by the second wedge prism, and emerges from point o₂.

For the two wedge prisms, points b, c, and o₂ in the 3D coordinate system shift by a certain distance from the origin. The method of Snell’s law in scalar form and 2D vector algebra does not include a position component, so parallel shifts do not affect the magnitude of each ray vector in the 2D coordinate system. In the far-field region, the total power of the ray deviation is obtained by using vector operations to add or subtract the 2D ray vectors, with the accuracy of the ray deviation formulas unaffected by these shifts.

For the S₁ prism, the direction of the incident ray is specified by the ray $\overline {ao} $. δ_1in is the angle the ray makes with the z-axes. The orientation of the incident ray expressed in 2D vector form is (δ_1in, θ). The incident ray and z-axis determine a plane of [PQRS], and θ is the angle of this plane rotated around the z-axis as counted from the (x, 0, z) plane. $\overrightarrow {{\delta _{1in}}} $ is expressed in Eq. (4), and θ₁ is the rotated angle of S₁. δ_1out is the angle of $\overline {bc} $ with the z-axis. The orientation of $\overline {bc} $ expressed in 2D vector form is (δ_1out, θ₁). The power of the ray deviation from S₁ whose orientation of the emerged ray is expressed in 2D vector form is (δ_1in, θ) + (δ_1out, θ₁) = (δ_1t, β₁). The result is shown in Fig. 7.

For S₂, the orientation of the incident ray as expressed in 2D vector form is (δ_1t, β₁), and $\overrightarrow {{\delta _{1t}}} = \overrightarrow {{\delta _{2in}}} $. When S₁ and S₂ are rotated about the z-axis, the direction of the incident ray of S₂ varies continuously. From Eqs. (10) and (11), Δθ = θ₂ –β₁ or Δθ= β₁–θ₂.

The perpendicular components of the incident rays travel into the wedge prism from three different directions: (1) the incident ray points to the thick-side, with δ_2in cos(Δθ) > α₂₁; (2) the incident ray points to the thick- side, with δ_2in cos(Δθ) < α₂₁; and (3) the incident ray points to the thin-side.

The ray deviation formula for S₂ is derived for the three different directions similar to the case of the single triangle wedge prism.

When the incident ray points to the thick-side and δ_2in cos(Δθ) > α₂₁, the ray deviation formula is derived from the scalar form of Snell’s law and geometry as follows:

(12)$$\begin{aligned}&{A_2} = {\delta _{2in}}\cos (\varDelta \theta ) - {\alpha _{21}};\,{B_2} = {\delta _{2in}}\cos (\varDelta \theta ) + {\alpha _{22}}; {C_2} = \sin ({\alpha _{21}} + {\alpha _{22}});\, {D_2} = \cos ({\alpha _{21}} + {\alpha _{22}}) \\ &{\delta _{2out}} = - {B_2} + {\sin ^{ - 1}}\{ {C_2}{[{(\frac{{{n_{22}}}}{{{n_{11}}}})^2} - {(\sin {A_2})^2}]^{0.5}} + {D_2}\sin ({A_2})\}\end{aligned}$$

When the incident ray points to the thick-side and δ_2in cos(Δθ) < α₂₁, the ray deviation formula is as follows:

(13)$${\delta _{2out}} = - {B_2} + {\sin ^{ - 1}}\{ {C_2}{[{(\frac{{{n_{22}}}}{{{n_{11}}}})^2} - {(\sin {A_2})^2}]^{0.5}} - {D_2}\sin ( - {A_2})\}$$

When the incident ray points to the thin side, the ray deviation formula is as follows:

(14)$${\delta _{2out}} = {B_2} - 2{\alpha _{22}} + {\sin ^{ - 1}}\{ {C_2}{[{(\frac{{{n_{22}}}}{{{n_{11}}}})^2} - {(\sin {A_2} + 2{\alpha _{21}})^2}]^{0.5}} - {D_2}\sin ({A_2} + 2{\alpha _{21}})\}$$

The method is similar to that described in Section 2-B. An equation is deduced from Eqs. (12)–(14) and from the equation describing the second wedge prism rotated about the z-axis. The equation is as follows:

(15)$${\delta _{2out}} = - {B_2} + {\sin ^{ - 1}}\{ {C_2}{[{(\frac{{{n_{22}}}}{{{n_{11}}}})^2} - {(\sin {A_2})^2}]^{0.5}} + {D_2}\sin ({A_2})\}$$

Using the method with 2D vector algebra, in Fig. 7(b), ray $\overline {{o_2}{e_2}}$ is parallel-shifted to $\overline {{o_1}{e_1}}$, and δ_2out is the angle of $\overline {{o_1}{e_1}}$ on the z-axis. The ray directions are expressed in 2D vector form as (δ_2out, θ₂) and $\overrightarrow {{\delta _{2out}}} = \overrightarrow {{o_3}h} $,$\overrightarrow {{\delta _{1t}}} = \overrightarrow {{o_3}f} $, and $\overrightarrow {{\delta _{1t}}} + \overrightarrow {{\delta _{2out}}} = \overrightarrow {{o_3}g}, \, \overrightarrow {{o_3}g} = \overrightarrow {{\delta _{2t}}} $.

In the 2D coordinate system, plane (x₃, y₃, 0) is perpendicular to the z-axis because the 2D ray vector does not include the position component. In the far-field region, we can parallel-shift the plane of (x₃, y₃, L) along the z-axis to any position. L is a variable, as shown in Fig. 7(b).

From Fig. 8, for S₁, the incident ray travels in the direction specified by the ray vector, and the ray vector in the 2D vector form is (δ_1in, θ). The incident ray vector is separated into two components: δ_1in cos(θ–θ₁) is a perpendicular component refracted by S₁, and δ_1in sin(θ–θ₁) is a parallel component unaffected by S₁. The ray deviation angles and orientations expressed in 2D vector form are shown in Eqs. (8)–(11).

For S₂, the incident ray vector is $\overrightarrow {{\delta _{2in}}} = \overrightarrow {{\delta _{1t}}} $ and is separated into two components:

δ_1t cos(β₁–θ₂) is a perpendicular component refracted by S₂, andδ_1t sin(β₁–θ₂) is a parallel component unaffected by S₂. The ray deviation formula is derived as follows [17]:

(16)$$\overrightarrow {{\delta _{2out}}} = {\delta _{2out}}\cos ({\theta _2})\hat{x} + {\delta _{2out}}\sin ({\theta _2})\hat{y}$$

(17)$$\overrightarrow {{\delta _{2t}}} = \overrightarrow {{\delta _{1t}}} + \overrightarrow {{\delta _{2out}}} $$

(18)$$\overrightarrow {{\delta _{2t}}} = \; \overrightarrow {{\delta _{1t}}} + \; \overrightarrow {{\delta _{2out}}} = {\delta _{2t}}\cos ({\beta _2})\hat{x} + {\delta _{2t}}\sin ({\beta _2})\hat{y}$$

(19)$$\left\{ \begin{array}{l} {\beta_2} = {\cos^{ - 1}}(\frac{{({\delta_{1t}}\cos {\beta_1} + {\delta_{2out}}\cos {\theta_2})}}{{{{[{{({\delta_{1t}})}^2} + {{({\delta_{2out}})}^2} + 2{\delta_{1t}}{\delta_{2out}}\cos ({\theta_2} - {\beta_1})]}^{0.5}}}})\\ {\delta_{2t}} = {[{({\delta_{1t}})^2} + {({\delta_{2out}})^2} + 2{\delta_{1t}}{\delta_{2out}}\cos ({\theta_2} - {\beta_1})]^{0.5}} \end{array} \right.$$

δ_2t: altitude; β₂: azimuth.

The orientations of the two wedge prisms are specified by their rotation angles θ₁ and θ₂. θ₁ is of S₁, and θ₂ is of S₂. The ray deviation formulas are derived and shown in Eqs. (4), (8)–(11), and (15)–(19).

An incident ray travels in the direction specified by a ray vector, and the ray is parallel to the z-axis. Thus, Eq. (4) is equal to zero. For Eqs. (10)–(11), δ_1in = 0°, θ = 0°, and β₁ = θ₁. For a DWPS under the condition of two identical wedge prisms aligned at θ₁ = θ₂ = 0°, the maximum ray deviation angles δ_2t are obtained from Eq. (19). Using the conventional method, when the apex angles of Type 1 and Type 2 are equal, the two elementary right triangle wedge prisms through arrangement and assembly produced four configurations. These are schematically described in Fig. 1(c) [11,17]. From geometry and by referring to Fig. 6 and Fig. 7, the deviation formulas for the four configurations can be obtained by defining the four variables of α₁₁, α₁₂, α₂₁, and α₂₂. For the (1, 1) configuration: α₁₁ = 0°, α₂₁ = 0° and α₁₂ = α₂₂. For the (1, 2) configuration: α₁₁ = 0°, α₂₂ = 0° and α₁₂ = α₂₁. For the (2, 1) configuration: α₁₂ = 0°, α₂₁ = 0° and α₁₁ = α₂₂. For the (2, 2) configuration: α₁₂ = 0°, α₂₂ = 0° and α₁₁ = α₂₁.

Fig. 6. Schematic diagrams illustrating five element types, in Type 1, α₁₁ = 0°; Type 2, α₁₂ = 0°; Type a, α₁₁<α₁₂; Type b, α₁₁> α₁₂; and Type c, α₁₁ = α₁₂.

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Fig. 7. (a) Schematic diagram illustrating notation and coordinate system for a configuration of double triangle wedge prisms. Refractive index of S₁ = n₁₂, S₂ = n₂₂ and air = n₁₁. (b) Two planes (k₁, m₁, u₁) and (k₂, m₂, u₂) correspond to profile perpendicular to base side. In far-field region, plane of (x³, y³, 0) is perpendicular to z-axis. For S₂, ray vector emerged from point o₂ and parallel shift to point o₁.

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Fig. 8. Schematic diagram illustrating notation and coordinate systems for DWPS in form of 2D vector algebra.

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In Fig. 7, for a specific prism configuration of (1, 1), the parameters of α₁₁ = α₂₁ = 0° and α₁₂ = α₂₂ = argument are used. These four parameters are substituted into Eqs. (4), (8)–(11), and (15)–(19). A curve is plotted as a function of the apex angles, and is shown in Fig. 9 as a line curve. The curve is compared with the curve plotted from the previous formulas of the (1, 1) configuration [17], marked as “+” and is shown in Fig. 9. The two curves are a complete match. The formulas of Eqs. (4), (8)–(11), and (15)–(19) define the parameters of α₁₁ = α₂₁ = 0° and α₁₂ = α₂₂ = argument; the results are equal to the formulas of the (1, 1) configuration.

Fig. 9. Curves of maximum ray deviation angles are plotted as a function of α₁₁ + α₁₂ (= α₂₁ + α₂₂). (a) and (b) are of BK-7 and n₁₁ = 1, n₁₂ = n₂₂=1.5. (b) Enlargement of partial region of (a); (c) and (d) are of Ge and n₁₁ = 1, n₁₂ = n₂₂ = 4. (d) Enlargement of partial region of (c).

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For the optical characteristics of (1, 2), (2, 1), and (2, 2) the same method can be derived from Eqs. (4), (8)–(11), and (15)–(19) by defining the four apex angle parameters of α₁₁, α₁₂, α₂₁, and α₂₂. For (1, 2), α₁₁ = α₂₂ = 0° and α₁₂ = α₂₁ = argument; for (2, 1), α₁₂ = α₂₁ = 0° and α₁₁ = α₂₂ = argument; and for (2, 2), α₁₂ = α₂₂ = 0° and α₁₁ = α₂₁ = argument. Curves are plotted as a function of the apex angles, and are shown in Fig. 9 as line curves. The three curves are compared with the curves plotted from the previous formulas of the (1, 2), (2, 1) and (2, 2) configuration [17], marked as “+” and are shown in Fig. 9.

For isosceles triangle wedge prisms, the maximum ray deviation angles of δ_2t are plotted from Eqs. (4), (8)–(11) and (15)–(19) by defining the four apex angle parameters of α₁₁ = α₁₂, α₂₁ = α₂₂. For BK-7 and n₁₂ = n₂₂ = 1.5, two identical triangle wedge prisms, the maximum ray deviation angles of δ_2t are plotted from Eqs. (4), (8)–(11) and (15)–(19) by defining the four apex angle parameters of α₁₂ = 2.3α₁₁, α₂₂ = 2.3α₂₁ and α₁₂ = 0.3α₁₁; α₂₂ = 0.3α₂₁. For Ge and n₁₂ = n₂₂= 4, two identical triangle wedge prisms, the maximum ray deviation angles of δ_2t are plotted from Eqs. (4), (8)–(11) and (15)–(19) by defining the four apex angle parameters of α₁₁ = 0.35 α₁₂, α₂₁ = 0.35 α₂₂ and α₁₁ = 2.5 α₁₂, α₂₁ = 2.5 α₂₂. These are shown in Fig. 9 as a line curve.

In the first-order paraxial method [17], the deflective angles depend on the apex angles and refractive indices. The power in the ray deviation is a straight line and is shown in Fig. 9.

4. Total internal reflection

The maximum ray deviation angles and apex angles of DWPS are limited by the total internal reflection, and the optical characteristic limits DWPS applications in wind angle beam steering systems.

When the incident light ray is parallel to the z-axis and the apex angles of two wedge prisms are aligned at θ₁ = θ₂ = 0°, the ray maximum power is obtained, as shown in Fig. 9.

For a specified prism material, when the apex angles increase to a certain limit value, the power in the ray deviation reaches an extreme value. When using the conventional method to obtain the extreme value of the power and the limit value of the apex angles, the derived procedure becomes very complicated [20].

In this research, analytical forward solutions for DWPS are proposed. When the apex angle is higher than the limit value, from the ray deviation formulas we obtain the power in the ray deviations as an imaginary number, and are shown in Fig. 10. When the apex angles are lower than the limit value, the powers in the ray deviation are real numbers. For these real numbers, the physical meaning is that the incident light ray can exit the prisms. When the powers in the ray deviation are imaginary numbers, the incident light ray cannot exit from the prisms.

Fig. 10. Curve for two identical prisms in two different prism materials, zoomed-in circular region of Fig. 9(a) and (c). For BK-7, n₁₁ = 1, n₁₂ = n₂₂=1.5. (A_1.5) α₁₁ = α₂₁ = 0 and α₁₂ = α₂₂; (B_1.5) α₁₁ = α₂₂ = 0 and α₁₂ = α₂₁; (C_1.5) α₁₂ = α₂₁ = 0 and α₁₁ = α₂₂; (D_1.5) α₁₂ = α₂₂ = 0 and α₁₁ = α₂₁; (E_1.5) α₁₁ = α₂₁ = α₁₂ = α₂₂ (F_1.5) α₁₂=2.3α₁₁, α₂₂=2.3α₂₁; (G_1.5) α₁₂=0.3α₁₁; α₂₂=0.3α₂₁. For Ge, n₁₁ = 1, n₁₂ = n₂₂ = 4. (A₄) α₁₁ = α₂₁ = 0 and α₁₂ = α₂₂; (B₄) α₁₁ = α₂₂ = 0 and α₁₂ = α₂₁; (C₄) α₁₂ = α₂₁ = 0 and α₁₁ = α₂₂; (D₄) α₁₂ = α₂₂ = 0 and α₁₁ = α₂₁; (E₄) α₁₁ = α₂₁ = α₁₂ = α₂₂; (F₄) α₁₁=2.5α₁₂, α₂₁=2.5α₂₂; (G₄) α₁₁=0.35α₁₂; α₂₁=0.35α₂₂. For each curve, imaginary parts of complex X and/or Y arguments ignored.

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The numerical limit values and extreme values are derived from Eqs. (4), (8)–(11), and (15)–(19) for different configurations by defining different apex angle parameters. Referring to Fig. 10, these values are listed in Table 1 and 2 for the two different prism materials. The numerical limit values and extreme values can help demonstrate the accuracy provided by these formulas as derived from the triangle wedge prisms.

Table 1. Referring to Fig. 10 for n₁₁ =1, n₁₂ = n₂₂ = 1.5.

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Table 2. Referring to Fig. 10 for n₁₁ =1, n₁₂ = n₂₂ = 4.

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These formulas define different parameters of α₁₁, α₁₂, α₂₁, and α₂₂ to represent different configurations: For n₁₁ = 1, n₁₂ = n₂₂ =1.5 and n₁₂ = n₂₂ = 4, A_1.5 and A₄ are of (1, 1) configuration, α₁₁ = α₂₁ = 0 and α₁₂ = α₂₂; B_1.5 and B₄ are of (1, 2) configuration, α₁₁ = α₂₂ = 0 and α₁₂ = α₂₁; C_1.5 and C₄ are of (2, 1) configuration, α₁₂ = α₂₁ = 0 and α₁₁ = α₂₂; D_1.5 and D₄ are of (2, 2) configuration, α₁₂ = α₂₂ = 0 and α₁₁ = α₂₁; E_1.5 and E₄ are of isosceles triangle configuration, α₁₁ = α₂₁ = α₁₂ = α₂₂. For n₁₁ = 1, n₁₂ = n₂₂ = 1.5, F_1.5: α₁₂=2.3α₁₁, α₂₂=2.3α₂₁. G_1.5: α₁₂=0.3α₁₁; α₂₂=0.3α₂₁. For n₁₁ = 1, n₁₂ = n₂₂ = 4, F₄: α₁₁=2.5α₁₂; α₂₁=2.5α₂₂; G₄: α₁₁=0.35α₁₂; α₂₁=0.35α₂₂.

When the apex angles of the two wedge prisms are aligned at θ₁ = θ₂ = 0°, the power in the ray deviation is at its maximum, and the extreme value is not affected by the optical characteristics: for an incident ray, the perpendicular component is refracted and the parallel component is unaffected by the two wedge prisms. In Table 1 and 2, these numerical limit values and extreme values of (A_1.5, B_1.5, C_1.5, D_1.5) and (A₄, B₄, C₄, D₄) are consistent with the results obtained from the conventional method [17,20].

5. Analytical inverse solution for DWPS in the far-field region

The ray deviation formulas for a single triangle wedge prism are derived based on the scalar form of Snell’s law and 2D vector algebra. For the double triangle wedge prisms shown in Fig. 7, the ray deviation formulas are deduced from the corresponding formulas for a single triangle wedge prism. The two wedge prism parameters are specified by α₁₁, α₁₂, α₂₁, α₂₂, n₁₁, n₁₂, and n₂₂. The rotation angles of the two wedge prisms are θ₁ and θ₂.

In the analytical forward solutions, a laser beam along the z-axis enters the first wedge prism. The laser beam is steered to any given position within the angular range of the system. Equations (4), (8)–(11), and (15)–(19) are used to obtain the pointing position with altitude δ_2t and azimuth β₂ for different configurations by defining the four parameters of α₁₁, α₁₂, α₂₁, and α₂₂.

In an inverse procedure, given a pointing position of altitude δ_h and azimuth Ω, with two orientations for the same pointing position, we need to steer the wedge prisms for the different configurations by defining the four parameters of α₁₁, α₁₂, α₂₁, and α₂₂. Since δ_h = δ_2t, from Eqs. (4), (8), (11), (15), and (19), we obtain Δθ. Δθ= θ₁–θ₂ or Δθ= θ₂–θ₁ at β₁ = θ₁ [3,17].

(20)$$\varDelta \theta = {\cos ^{ - 1}}(\frac{{{{({\delta _h})}^2} - {{({\delta _{1t}})}^2} - {{({\delta _{2out}})}^2}}}{{2{\delta _{1t}}{\delta _{2out}}}})$$

For Δθ = θ₁ – θ₂ > 0, F₂₁ is the angle of $\overrightarrow {{\delta _{2t}}} $ relative to $\overrightarrow {{\delta _{2out}}} $. From conventional geometry, we obtain

(21)$${F_{21}} = {\cos ^{ - 1}}(\frac{{{\delta _{2out}} + {\delta _{1t}}\cos \varDelta \theta )}}{{{{[{{({\delta _{1t}})}^2} + {{({\delta _{2out}})}^2} + 2{\delta _{1t}}{\delta _{2out}}\cos (\varDelta \theta )]}^{0.5}}}})$$

From Eqs. (20) and (21), we obtain (θ₁, θ₂) as follows:

(22)$${\theta _2} = \Omega - {F_{21}}, \, {\theta _1} = \varDelta \theta + {\theta _2}$$

For Δθ = θ₂ – θ₁ > 0, F₂₂ is the angle of $\overrightarrow {{\delta _{2t}}} $ relative to $\overrightarrow {{\delta _{1t}}} $. From conventional geometry, we obtain

(23)$${F_{22}} = {\cos ^{ - 1}}(\frac{{{\delta _{1t}} + {\delta _{2out}}\cos \varDelta \theta )}}{{{{[{{({\delta _{1t}})}^2} + {{({\delta _{2out}})}^2} + 2{\delta _{1t}}{\delta _{2out}}\cos (\varDelta \theta )]}^{0.5}}}})$$

From Eqs. (20) and (23), we obtain(θ₁, θ₂) as follows:

(24)$${\theta _1} = \Omega - {F_{22}}, \, {\theta _2} = \varDelta \theta + {\theta _1}$$

Inverse solution for the case of a visible laser, the laser beam is pointed to the direction specified by altitude δ_h = 4.523° and azimuth Ω = 120°, with two orientations for the same pointing position, and need to steer the wedge prisms for the four configurations.

For the different configurations of double right-triangle wedge prisms, the refractive indices are n₁₁ = 1 and n₁₂ = n₂₂ = 1.5. The apex angles are A for the (1, 1) configuration, α₁₁ = α₂₁ = 0 and α₁₂ = α₂₂ = 5°; A₊ configuration, α₁₁ = α₂₁ = 0.1° and α₁₂ = α₂₂ = 4.9°. B for the (1, 2) configuration,α₁₁ = α₂₂ = 0 and α₁₂ = α₂₁ = 5°; B₊ configuration,α₁₁ = α₂₂ = 0.1° and α₁₂ = α₂₁ = 4.9°. C for the (2, 1) configuration, α₁₂ = α₂₁ = 0 and α₁₁ = α₂₂ = 5°; C₊ configuration, α₁₂ = α₂₁ =0.1° and α₁₁ = α₂₂ = 4.9°. D for the (2, 2) configuration, α₁₂ = α₂₂ = 0 and α₁₁ =α₂₁ = 5°. D₊ configuration, α₁₂ = α₂₂ = 0.1° and α₁₁ =α₂₁ = 4.9°. E for the double isosceles-triangle wedge prism configuration, α₁₁ = 2.5°, α₁₂ = 2.5°, α₂₁ = 2.5°, α₂₂ = 2.5°; E_{+ +} configuration, α₁₁ = 2.6°, α₁₂ = 2.4°, α₂₁ = 2.6°, α₂₂ = 2.4°; E_{− +} configuration, α₁₁ = 2.4°, α₁₂ = 2.6°, α₂₁ = 2.6°, α₂₂ = 2.4°; E_{+ −} configuration, α₁₁ = 2.6°, α₁₂ = 2.4°, α₂₁ = 2.4°, α₂₂ = 2.6°; E_{− −} configuration, α₁₁ = 2.4°, α₁₂ = 2.6°, α₂₁ = 2.4°, α₂₂ = 2.6°. For the different configurations, the relative rotation angle of Δθ is obtained from Eqs. (4), (8), (11), (15), (19), and (20).

When Δθ= θ₁–θ₂ > 0, from Eqs. (20), (21), and (22) obtain (θ₁, θ₂) as shown in Table 3. When Δθ = θ₂–θ₁ > 0, from Eqs. (20), (23), and (24) obtain (θ₁, θ₂) as shown in Table 4.

Table 3. n₁₁ =1; n₁₂ = n₂₂ = 1.5; Δθ = θ₁ – θ₂ > 0.

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Table 4. n₁₁ =1; n₁₂ = n₂₂ = 1.5; Δθ = θ₂ – θ₁ > 0.

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For the case of an invisible laser, the laser beam is pointed in the direction specified by altitude δ_h = 4.5° and azimuth Ω = 120°, with two orientations for the same pointing position, and need to steer the wedge prisms for the four configurations. The refractive indices are n₁₁ = 1 and n₁₂ = n₂₂ = 4. In addition, in A for the (1, 1) configuration, α₁₁ = α₂₁ = 0 and α₁₂ = α₂₂ = 5°; A₊ configuration, α₁₁ = α₂₁ = 0.1° and α₁₂ = α₂₂ = 4.9°. B for the (1, 2) configuration,α₁₁ = α₂₂ = 0 and α₁₂ = α₂₁ = 5°; B₊ configuration, α₁₁ = α₂₂ = 0.1° and α₁₂ = α₂₁ = 4.9°. C for the (2, 1) configuration, α₁₂ = α₂₁ = 0 and α₁₁ = α₂₂ = 5°; C₊ configuration, α₁₂ = α₂₁ =0.1° and α₁₁ = α₂₂ = 4.9°. D for the (2, 2) configuration, α₁₂ = α₂₂ = 0 and α₁₁ =α₂₁ = 5°. D₊ configuration, α₁₂ = α₂₂ = 0.1° and α₁₁ =α₂₁ = 4.9°. E for the double isosceles-triangle wedge prism configuration, α₁₁ = 2.5°, α₁₂ = 2.5°, α₂₁ = 2.5°, α₂₂ = 2.5°; E_{+ +} configuration, α₁₁ = 2.6°, α₁₂ = 2.4°, α₂₁ = 2.6°, α₂₂ = 2.4°; E_{− +} configuration, α₁₁ = 2.4°, α₁₂ = 2.6°, α₂₁ = 2.6°, α₂₂ = 2.4°; E_{+ −} configuration, α₁₁ = 2.6°, α₁₂ = 2.4°, α₂₁ = 2.4°, α₂₂ = 2.6°; E_{− −} configuration, α₁₁ = 2.4°, α₁₂ = 2.6°, α₂₁ = 2.4°, α₂₂ = 2.6°.

When Δθ= θ₁–θ₂> 0, from Eqs. (20), (21), and (22) obtain (θ₁, θ₂) as shown in Table 5. When Δθ = θ₂–θ₁> 0, from Eqs. (20), (23), and (24) obtain (θ₁, θ₂) as shown in Table 6.

Table 5. n₁₁ =1; n₁₂ = n₂₂ = 4; Δθ = θ₁ – θ₂ > 0.

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Table 6. n₁₁ =1; n₁₂ = n₂₂ = 4; Δθ = θ₂ – θ₁ > 0.

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This Section proposed two examples. The conditions of the two examples are identical as that in [11,17]. The configurations of A, B, C, and D from the proposed mathematical model are used to obtain the orientations of Δθ, θ₁, and θ₂. In Table 3 to Table 6, the numerical results are equal to those of [17]. The results prove the mathematical model for double triangle wedge prisms by showing that different parameters of the apex angle can be applied to different configurations. The configurations: A₊ to A; B₊ to B; C₊ to C; D₊ to D; E_{+ +}, E_{+ −}, E_{− +} and E_{− −} to E, these configurations have equal apex angles (α₁₁ +α₁₂ =α₂₁+ α₂₂), the capital letters with footnote labels expressed the shape of right triangle wedge prisms have imperceptible deformation relative to not footnote labels.

6. Conclusion

Using the conventional method to investigate the optical characteristics of DWPS for the forward and inverse solutions based on the vector form of Snell’s law and a two-step method, a specific configuration was developed with a monopolistic mathematical model to explain the optical characteristics.

An incident ray travels through the wedge prism, the perpendicular component is deviated, and the parallel component is unaffected by the wedge prism. The conventional method is not concerned with the inherent optical characteristics and results in a problem in precise pointing and tracking systems.

The problem was solved by Lai and Lee [17] based on the theory of Snell’s law in scalar form and 2D vector algebra. For a specific configuration, a monopolistic mathematical model was developed to explain the optical characteristics. For the forward and inverse solutions, the developed procedures were the same as those of the conventional method and were tedious, time-consuming, and complicated.

In this study, an innovative mathematics model was proposed for DWPS. The configuration of double triangle wedge prisms possesses four apex angle parameters of α₁₁, α₁₂, α₂₁, and α₂₂. Particular configurations are created by assigning a set of suitable apex angle parameters. For example, in the (1, 1) configuration, the analytical forward and inverse solutions can apply the mathematical model by assigning the apex angle parameters of α₁₁ = α₂₁ = 0° and α₁₂ =α₂₂. In DWPS, different configurations created by defining the four apex angle parameters share the same mathematical model, and the novel mathematics model is fitted to different configurations.

The total internal reflection is an important optical characteristic in DWPS. The limit value of the apex angle and extreme value of the power can offer useful suggestions for the design of wide-angle ray pointing and tracking systems. For different configurations, by defining different apex angle parameters, the numerical limit values and extreme values are directly obtained from the Eq. (19) and are shown in Table 1, Table 2, Fig. 9 and Fig. 10. In Table 1 and 2, the numerical results of (A_1.5, B_1.5, C_1.5, D_1.5) and (A₄, B₄, C₄, D₄) are equal to the results that obtained using the conventional method [17,20].

For DWPS, the numerical results in Table 3 to Table 6 help demonstrate the mathematical model by defining the apex angle parameters that can be applied to different configurations. For DWPS applied in precise pointing and tracking system, the shape of right triangle wedge prisms may be produced imperceptible deformation, the mathematical model is an useful tool for researched and analyzed the optical characteristics, the results are shown in Table 3 to Table 6. For DWPS in different configurations, from the proposed mathematics model to analyze the deformation of geometrical shapes, the method is direct, concise, and precise.

References

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8. V. Vuthea and H. Toshiyoshi, “A Design of Risley Scanner for LiDAR Applications,” International Conference on Optical MEMS and Nanophotonics (OMN, 135–136 (2018).

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configuration	A_1.5	B_1.5	C_1.5	D_1.5	E_1.5	F_1.5	G_1.5
Limit value	29.716°	41.478°	30.923°	55.655°	40.072°	35.31°	48.295°
Extreme value	60.284°	89.99°	59.077°	89.99°	69.964°	65.39°	78.855°

configuration	A₄	B₄	C₄	D₄	E₄	F₄	G₄
Limit value	8.122°	9.232°	8.270°	9.503°	8.76°	9.065°	8.44°
Extreme value	81.878°	89.99°	81.730°	89.99°	85.62°	87.41°	83.75°

configuration	Δθ	F₂₁	θ₂	θ₁
A	52.0320°	25.9659°	94.0341°	146.0661°
A₊	51.9790°	25.9406°	94.0594°	146.0384°
B	51.1850°	25.6414°	94.3586°	145.5436°
B₊	51.1650°	25.6285°	94.3715°	145.5365°
C	51.6590°	25.7353°	94.2647°	145.9237°
C₊	51.6200°	25.7187°	94.2813°	145.9013°
D	50.8011°	25.4055°	94.5945°	145.3955°
D₊	50.7960°	25.4019°	94.5981°	145.3941°
E	51.0430°	25.4991°	94.5009°	145.5439°
E₊₊	51.0190°	25.4882°	94.5118°	145.5308°
E_{+ −}	51.0530°	25.5013°	94.4987°	145.5517°
E_{− +}	51.0340°	25.4974°	94.5026°	145.5366°
E_{− −}	51.0690°	25.5110°	94.4890°	145.5580°

configuration	Δθ	F₂₂	θ₂	θ₁
A	52.0320°	26.0661°	145.9659°	93.9339°
A₊	51.9790°	26.0384°	145.9406°	93.9616°
B	51.1850°	25.5436°	145.6414°	94.4564°
B₊	51.1650°	25.5365°	145.6285°	94.4635°
C	51.6590°	25.9237°	145.7353°	94.0763°
C₊	51.6200°	25.9013°	145.7187°	94.0987°
D	50.8011°	25.3955°	145.4055°	94.6045°
D₊	50.7960°	25.3941°	145.4019°	94.6059°
E	51.0430°	25.5439°	145.4991°	94.4561°
E₊₊	51.0190°	25.5308°	145.4882°	94.4692°
E_{+ −}	51.0530°	25.5517°	145.5013°	94.4483°
E_{− +}	51.0340°	25.5366°	145.4974°	94.4634°
E_{− −}	51.0690°	25.5580°	145.5110°	94.4420°

configuration	Δθ	F₂₁	θ₂	θ₁
A	163.0910°	84.6125°	35.3875°	198.4785°
A₊	163.0870°	84.4995°	35.5005°	198.5875°
B	163.1810°	82.0657°	37.9343°	201.1153°
B₊	163.1740°	82.0573°	37.9427°	201.1167°
C	162.9330°	81.6972°	38.3028°	201.2358°
C₊	162.9360°	81.7036°	38.2964°	201.2324°
D	163.0580°	79.2509°	40.7491°	203.8071°
D₊	163.0560°	79.3542°	40.6458°	203.7018°
E	163.0350°	81.8703°	38.1297°	201.1647°
E₊₊	163.0340°	81.7634°	38.2366°	201.2706°
E_{+ −}	163.0300°	81.8629°	38.1371°	201.1671°
E_{− +}	163.0400°	81.8777°	38.1223°	201.1623°
E_{− −}	163.0360°	81.9774°	38.0226°	201.0586°

Analytical forward and inverse solutions using a mathematics model for double wedge prisms in different configurations

Abstract

1. Introduction

2. Ray deviation formulas for single wedge prism in far-field region

2.1 Relation between the deviation formulas and geometrical shape for a triangle wedge prism

2.2 Formulas of power in ray deviation for single triangle wedge prism

3. Formulas of power in ray deviation for DWPS in the far-field region

4. Total internal reflection

5. Analytical inverse solution for DWPS in the far-field region

6. Conclusion

References

Cited By

Figures (10)

Tables (6)

Equations (24)

OSA Continuum

configuration	Δθ	F₂₂	θ₂	θ₁
A	163.0910°	78.4785°	204.6125°	41.5215°
A₊	163.0870°	78.5875°	204.4995°	41.4125°
B	163.1810°	81.1153°	202.0657°	38.8847°
B₊	163.1740°	81.1167°	202.0573°	38.8833°
C	162.9330°	81.2358°	201.6972°	38.7642°
C₊	162.9330°	81.2324°	201.7036°	38.7676°
D	163.0580°	83.8071°	199.2509°	36.1929°
D₊	163.0560°	83.7018°	199.3542°	36.2982°
E	163.0350°	81.1647°	201.8703°	38.8353°
E₊₊	163.0340°	81.2706°	201.7634°	38.7294°
E_{+ −}	163.0300°	81.1671°	201.8629°	38.8329°
E_{− +}	163.0400°	81.1623°	201.8777°	38.8377°
E_{− −}	163.0360°	81.0586°	201.9774°	38.9414°