Design and analysis of a five-mirror derotator with minimal instrumental polarization in astronomical telescopes

Junfeng Hou; Ming Liang; Dongguang Wang; Yuanyong Deng

doi:10.1364/OE.26.019356

1. Introduction

Nowadays, the aperture of the astronomical telescopes becomes larger and larger so as to improve the spatial resolution, e.g., Thirty Meter Telescope (TMT) [1], Giant Magellan Telescope (GMT) [2], Daniel K. Inouye Solar Telescope (DKIST) [3], etc. An alt-azimuth mount have to be used for these large telescopes when their apertures are larger than 2 meter. However, one shortcoming of the alt-azimuth telescopes is the image rotation. An image derotation is necessary for compensating image rotation before the light entering the focal plane instruments. In general, two kinds of image rotations occur in optical telescopes. One is caused by the coordinate transformation between the horizontal coordinate system and equatorial coordinate system, while the other is caused by the mutual rotation among fold mirrors employed in the telescopes. Three usual methods, electrical derotation [4], physical derotation [5] and optical derotation [6], are used for derotation. Among them, the electrical derotation has time delay and cannot do integral in a long time. Similarly, the physical method, whose structure is complicated, mass and size are large due to the larger optical and electrical instruments equipped on the large optical telescopes, does not suit for these telescopes. By way of the above comparison, optical derotation is a better plan.

The principle of the optical derotation is to rotate the optical derotator installed in front of the imaging device so as to compensate for the image rotation of the target in the focal plane. Traditionally, there are two kinds of optical derotators, one is the derotation prism [7,8] and the other one is K-mirror including three mirrors [9]. By now, K-mirror is widely used in many large optical astronomical telescopes because of its advantage in wavelength coverage.

However, the classical derotator K-mirror has a high instrumental polarization, which is not suitable for the telescopes especially for polarimetry. Taking the aluminum-coated K-mirror at λ = 0.525um as an example, the crosstalk from unpolarized light into polarized light is about 13%, and the one between linear polarization and circular polarization is about 83%. The high instrumental polarization will give rise to a large polarization measurement error in polarimetry, particularly for solar telescopes, in which polarimetry is generally used for the measurement of magnetic fields and the corresponding polarization measurement accuracy should be very high. In that case, the instrumental polarization from the K-mirror is so large that can not be ignored. How to reduce its instrumental polarization has become a key problem in polarimetry for those telescopes.

In this paper, a five-mirror derotator with minimal instrumental polarization is theoretically designed at first. In order to eliminate the beam shift induced by the thickness of the reflector, a novel compensation method is studied then. Finally a theoretical polarization analysis is given to show the real polarization properties of the new designed derotator. The relationship of the instrumental polarization with rotation angle for beam shift compensation and the incident field of view is calculated to verify its validity and applicability in polarimetry.

2. Design of the five-mirror derotator

2.1. Design of the five-mirror derotator by minimizing instrumental polarization

Consider a partially polarized light, described by Stokes vector $S_{in} = {[\begin{matrix} I & Q & U & V \end{matrix}]}^{T}$ , traveling toward z direction, the output polarization after the reflection of one mirror can be described in Eq. (1) [10],

S_{out} = M_{i} S_{in} = R (- ϕ) M_{i}^{0} R (- ϕ) S_{in}

where S_in and S_out are the input and output Stokes vectors respectively. R(ϕ) is a rotation matrix of the mirror which makes a rotation angle ϕ with respect to the Stokes coordinate system and coincides with the mirror’s plane of incidence.

M_{i}^{0}

is the Mueller matrix of the mirror with ϕ = 0, representing its intrinsic polarization properties. Here we define M_i as the Mueller matrix of the mirror including its intrinsic instrumental polarization and rotation operation. If M_i is an identity matrix, the mirror has no instrumental polarization, otherwise reverse. The expressions of R(ϕ) and

M_{i}^{0}

are written in the following Eq. (2) and Eq. (3):

R (ϕ) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & cos 2 ϕ & - sin 2 ϕ & 0 \\ 0 & sin 2 ϕ & cos 2 ϕ & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

M_{i}^{0} = [\begin{matrix} 1 & \frac{X^{2} - 1}{X^{2} + 1} & 0 & 0 \\ \frac{X^{2} - 1}{X^{2} + 1} & 1 & 0 & 0 \\ 0 & 0 & \frac{2 X}{X^{2} + 1} cos δ & \frac{2 X}{X^{2} + 1} sin δ \\ 0 & 0 & - \frac{2 X}{X^{2} + 1} sin δ & \frac{2 X}{X^{2} + 1} cos δ \end{matrix}]

In the above Eq. (3), δ is the phase difference between s and p components of the electric field, while X is the ratio of the amplitude reflection coefficients between s and p components. Both of them can be calculated by Fresnel formula [11] referring to Eqs. (4) – (7), where δ and X depend not only on the incident angle θ_i of the incident beam entering in the mirror surface, but also on the complex refractive index ñ of the metallic layer coated on the mirror.

r_{s} = \frac{cos θ_{i} - \sqrt{{\tilde{n}}^{2} - {sin}^{2} θ_{i}}}{cos θ_{i} + \sqrt{{\tilde{n}}^{2} - {sin}^{2} θ_{i}}}

r_{p} = \frac{{\tilde{n}}^{2} cos θ_{i} - \sqrt{{\tilde{n}}^{2} - {sin}^{2} θ_{i}}}{{\tilde{n}}^{2} cos θ_{i} + \sqrt{{\tilde{n}}^{2} - {sin}^{2} θ_{i}}}

X = \frac{| r_{s} |}{| r_{p} |}

δ = arctan (\frac{Imag (r_{s})}{Real (r_{s})}) - arctan (\frac{Imag (r_{p})}{Real (r_{p})})

According to polarized theory [12], the Mueller matrix M of a system with m mirrors can be represented by the multiplication of the Mueller matrix of each mirror, shown in Eq. (8).

M = M_{m} \dots M_{2} M_{1} = R (- ϕ_{m}) M_{m}^{0} R (- ϕ_{m}) \dots R (- ϕ_{2}) M_{2}^{0} R (- ϕ_{2}) R (- ϕ_{1}) M_{1}^{0} R (- ϕ_{1})

For a two-mirror system, as we know, the minimum of the instrumental polarization is achieved when the incidence and reflection planes of the two flat mirrors are perpendicular to each other [13]. In this situation, there is no instrumental polarization because the instrumental polarization of one mirror is completely compensated by the other one. The effect is able to be proved by Eq. (9), where ϕ₁ can be reduced to zero and ϕ₂ is π/2. Both mirrors’ eigen Mueller matrix will be identical as $M_{1}^{0}$ if they have the same incident angles (θ₁ = θ₂). Consequently, the total Mueller matrix M of the two-mirror system becomes an identity matrix I multiplied by a constant.

M = M_{2} M_{1} = R (- π / 2) M_{1}^{0} R (- π / 2) M_{1}^{0} = \frac{4 X^{2}}{{(X^{2} + 1)}^{2}} I

The effect shown above can be applied to our design of a derotator with minimal instrumental polarization. The new designed derotator, called five-mirror derotator, contains five mirrors, and its simplified diagram can be found in Fig. 1. A detail description of the design principles of the five-mirror derotator is in the following:

Assuming the incident beam in a right-hand reference system, +z axis reprents its propagation direction.
A mirror M1 reflects the vertical incident beam to the other direction with incident angle θ₁. M1 is placed at the center of the reference system and its normal vector is in the y–z plane. Plane Π₁, composed of A⃗ and B⃗, is the incidence and reflection plane of M1, where A⃗ and B⃗ are the incidence and reflection vectors of light beam in M1 respectively.
The reflected beam after M1 is going to reflect from M2, M3 and M4 in turn, and finally arrived at M5. B⃗, C⃗, D⃗ and E⃗ are the corresponding incident or reflection vectors from M2 to M4. They constitute the incidence and reflection planes (Π₂, Π₃ and Π₄), which are the same but is perpendicular to Π₁. In other words, plane Π₂₃₄ is composed of B⃗, C⃗, D⃗ and E⃗, Π₂₃₄ = Π₂ = Π₃ = Π₃ and Π₂₃₄ ⊥ Π₁. According to ray tracing theory, B⃗ and E⃗ are in y–z plane and are coaxial. Furthermore, the angle of B⃗ and E⃗ relative to y axis is the same as the rotation angle of plane Π₂₃₄ relative to y axis.
M5 has an opposite normal vector but the same incidence and reflection plane Π₅ as M1 (Π₅ = Π₁). The final light beam after M5 goes back to the vertical direction again.
Because plane Π₁ and Π₅ are orthogonal to plane Π₂₃₄, similar to the two-mirror system described above, it is possible to minimize the instrumental polarization of five-mirror derotator by optimizing the incident angle of the light beam in each mirror.
More simplified operations can be used in order to optimize the design: M1 and M5 correspond to the double-sided coatings on a reflector, as shown in Fig. 1 to ensure both of them have the same incidence and reflection planes (thus θ₅ = θ₁). M2, M3, and M4 have the same incident angle (θ₂ = θ₃ = θ₄ = π/6) as well as side length, forming an equilateral triangle.
Finally, the instrumental polarization of the five-mirror derotator can be minimized by only optimizing the incident angle θ₁.

The Mueller matrix of the five-mirror derotator can be represented as Eq. (10),

\begin{array}{l} M & = M_{5} M_{4} M_{3} M_{2} M_{1} \\ = R (- ϕ_{5}) M_{5}^{0} R (- ϕ_{5} - ϕ_{4}) M_{4}^{0} R (- ϕ_{4} - ϕ_{3}) M_{3}^{0} R (- ϕ_{3} - ϕ_{2}) M_{2}^{0} R (- ϕ_{2} - ϕ_{1}) M_{1}^{0} R (- ϕ_{1}) \end{array}

where ϕ₁ = ϕ₅ = 0, and ϕ₂ = ϕ₃ = ϕ₄ = π/2. Also, Eq. (10) is simplified into Eq. (11).

M = M_{5}^{0} R (- π / 2) M_{4}^{0} M_{3}^{0} M_{2}^{0} R (- π / 2) M_{1}^{0}

Fig. 1 Simplified diagram of a five-mirror derotator.

Download Full Size | PDF

The instrumental polarization can be calculated and minimized once the coating material of mirrors is defined. Here we suppose an aluminum coating is applied. Figure 2 displays the relations of Mueller matrix of the five-mirror derotator with the incident angle θ₁, where the abscissa is θ₁’s values in degree and the ordinate is the corresponding value in each element of Mueller matrix M. The lines with different colors indicate the calculation in different wavelengths. The complex refractive index ñ used is 0.91–6.34i, 1.36–10.46i, and 16.68–70.90i, corresponding to the wavelength λ of 0.525um, 1.083um, and 12.00um, respectively. The result shows that the Mueller matrix almost works as a unit matrix when θ₁ = 36.3° and the minus signs in M₃₃ and M₄₄ are produced by the intrinsic characteristics of mirrors. The instrumental polarization, no matter from the unpolarized light into the polarized light (I→ QUV) or between linear polarization and circular polarization (QUV → QUV), is smaller than 0.01%. Besides, the optimized incident angle θ₁ is almost independent of wavelength, which means that the optimal design results of the five-mirror derotator is universal from visible to mid-infrared wavelengths. It has to be noted that the design values are not unique, any incident angle adjustment in M2, M3 and M4 can get other optimization results for the five-mirror derotator.

Fig. 2 Relationship of Mueller matrix of five-mirror derotator with the incident angle θ₁. x axis is the incident angle θ₁ in degree, y axis is each element of Mueller matrix. The blue, green and red lines represent the optimized results in wavelengths 0.525um, 1.083um and 12um respectively.

Download Full Size | PDF

2.2. Derotation feature of the five-mirror derotator

In the section, some calculation is given to see how the five-mirror derotator realizes its derotation function. The incident and output vectors of the light beam from the five-mirror derotator have the relationship described in Eq. (12), where F⃗ is the unit vector of the output light, A⃗ is that of the incident light, T is the eigen transformation matrix of the five-mirror derotator, and Rot_z(α) is its rotation matrix around +z axis.

\vec{R} = {Rot}_{z} (α) T {Rot}_{z} (- α) \vec{A}

{Rot}_{z} (α) = [\begin{matrix} cos α & - sin α & 0 \\ sin α & cos α & 0 \\ 0 & 0 & 1 \end{matrix}]

T = T_{5} T_{4} T_{3} T_{2} T_{1}

Just like Mueller matrix M, the transformation matrix T of the five-mirror derotator can also be written in the multiplication of each mirror’s transformation matrix T_i, as shown from Eq. (15) to Eq. (20) [14]. Here N⃗_i is the normalized normal vector of each mirror and can be calculated based on the design values in above mentioned section.

T_{i} = [\begin{matrix} 1 - 2 N_{i x}^{2} & - 2 N_{i x} N_{i y} & - 2 N_{i x} N_{i z} \\ - 2 N_{i x} N_{i y} & 1 - 2 N_{i y}^{2} & - 2 N_{i y} N_{i z} \\ - 2 N_{i x} N_{i z} & - 2 N_{i y} N_{i z} & 1 - 2 N_{i z}^{2} \end{matrix}], i = 1 \dots 5

{\vec{N}}_{1} = (\begin{matrix} 0 \\ - sin θ_{1} \\ - cos θ_{1} \end{matrix}) = - {\vec{N}}_{5}

{\vec{N}}_{2} = {Rot}_{x} (90 ° - 2 θ_{1}) (\begin{matrix} sin 30 ° \\ cos 30 ° \\ 0 \end{matrix}) = (\begin{matrix} sin 30 ° \\ sin 2 θ_{1} cos 30 ° \\ cos 2 θ_{1} cos 30 ° \end{matrix})

{\vec{N}}_{3} = {Rot}_{x} (90 ° - 2 θ_{1}) (\begin{matrix} - 1 \\ 0 \\ 0 \end{matrix}) = (\begin{matrix} - 1 \\ 0 \\ 0 \end{matrix})

{\vec{N}}_{4} = {Rot}_{x} (90 ° - 2 θ_{1}) (\begin{matrix} sin 30 ° \\ - cos 30 ° \\ 0 \end{matrix}) = (\begin{matrix} sin 30 ° \\ - sin 2 θ_{1} cos 30 ° \\ - cos 2 θ_{1} cos 30 ° \end{matrix})

{Rot}_{x} (90 ° - 2 θ_{1}) = [\begin{matrix} 1 & 0 & 0 \\ 0 & cos (90 ° - 2 θ_{1}) & - sin (90 ° - 2 θ_{1}) \\ 0 & sin (90 ° - 2 θ_{1}) & cos (90 ° - 2 θ_{1}) \end{matrix}]

Inserting Eqs. (15)–(20) into Eq. (14), the transformation matrix T of the five-mirror derotator is:

T = [\begin{matrix} - 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]

If we suppose the incident beam of the five-mirror derotator has a rotation α₀ around z axis, A⃗ becomes Rot_z(α₀) A⃗₀. Substituting Eqs. (21) and (13) into Eq. (12), the output vector F⃗ can be obtained in Eq. (22), where A_x0 A_y0 and A_z0 are the original coordinates without rotation. It illustrates that if the incident beam rotates α₀ and the five-mirror derotator also rotates α₀/2 in the same direction, the output beam from derotator has no rotation at all relative to the original coordinates A⃗₀. The function of the five-mirror derotator, derotating the rotation of the incident beams, works well in the case.

\vec{F} = {Rot}_{z} (α) T {Rot}_{z} (- α) {Rot}_{z} (α_{0}) {\vec{A}}_{0} = [\begin{matrix} - cos (2 α - α_{0}) & - sin (2 α - α_{0}) & 0 \\ - sin (2 α - α_{0}) & cos (2 α - α_{0}) & 0 \\ 0 & 0 & 1 \end{matrix}] (\begin{matrix} A_{x 0} \\ A_{y 0} \\ A_{z 0} \end{matrix})

3. Beam shift compensation

In the real design of a five-mirror derotator, it is inevitable that the reflector, double-side coated for M1 and M5, has limited thickness in order to get excellent image quality. Nevertheless, the thickness results in a displacement between the output and incident beam, as marked in red lines in Fig. 3, causing an additional shift of image even though it works well for image derotation. The phenomenon will become more and more serious especially in a converging optical path. Therefore, a novel method is approved here to compensate for the beam shift.

Fig. 3 Beam shift and compensation in the five-mirror derotator. The red line indicates a beam shift due to the thickness of the reflector composed of M1 and M5, and the blue dotted line shows the beam shift compensation after a rotation and a translation of M3 and M4.

Download Full Size | PDF

The main objective of the compensation is to make the output beam F⃗ and incident beam A⃗ coaxial. As A⃗ is not in the plane Π₂₃₄, an operator combined of both translation and rotation is essential to compensate for the beam shift.

Specifically, the plane Π₂₃₄ is split into two planes. One is Π₂ composed of B⃗ and C⃗ while the other is Π₄ composed of D⃗ and E⃗. Thus, the beam shift can be compensated by a translation of M4 along with the opposite direction of its normal vector N⃗₄ and a rotation of Π₄ around the axis parallel to B⃗ with the center of M3, which is drawn in blue lines in Fig. 3. The rotation operation compensates for the thickness of the reflector, and the translation operation makes sure that the beam E⃗′ is still in the y − z plane after the previous rotation. Let d and β be the translation distance and rotation angle respectively, both of them can be calculated by Eqs. (23) and (24), where h is thickness of the reflector and b is the side length of the equilateral triangle composed of centers of M2, M3 and M4 before rotation.

d = (b / 2 cos β - b / 2) tan 60 °

β = arcsin (\frac{h sin 2 θ_{1}}{b cos θ_{1} cos 30 °})

Now the beams D⃗, E⃗ and F⃗ are changed into D⃗′, E⃗′ and F⃗′ as shown in Fig. 3. A⃗ and F⃗′ becomes coaxial, which is different from the case of A⃗ and F⃗. What we want to mention is that Π₄ is still the incidence and reflection plane of M4 composed of D⃗′ and E⃗′, and the normal vector N′₄ of M4 can be modified to Eq. (25) from Eq. (19). However, the incidence and reflection plane Π₃ of M3 is not the same as Π₄ any more. The normal vector N⃗′₃ of M3 can be defined by the incident vector C⃗ and reflected vector D⃗′ of M3 based on the law of reflection (Eqs. (27) and (28)). Once the normal vectors of all mirrors are defined, the orientation of the five-mirror derotator will be completely determined.

{\vec{N}}_{4}^{'} = {Rot}_{x} (90 ° - 2 θ_{1}) {Rot}_{y} (β) (\begin{matrix} sin 30 ° \\ - cos 30 ° \\ 0 \end{matrix})

{Rot}_{y} (β) = [\begin{matrix} cos β & 0 & sin β \\ 0 & 1 & 0 \\ - sin β & 0 & cos β \end{matrix}]

\vec{C} = {Rot}_{x} (90 ° - 2 θ_{1}) (\begin{matrix} sin 60 ° \\ cos 60 ° \\ 0 \end{matrix}), {\vec{D}}^{'} = {Rot}_{x} (90 ° - 2 θ_{1}) {Rot}_{y} (β) (\begin{matrix} - sin 60 ° \\ cos 60 ° \\ 0 \end{matrix})

{\vec{N}}_{3}^{'} = \frac{{\vec{D}}^{'} - \vec{C}}{| {\vec{D}}^{'} - \vec{C} |}

A ZEMAX code is applied to validate the expected compensation effect. Figure 4 shows the simulation results based on ZEMAX, where b = 120mm and h = 8mm. A large displacement can be found from Fig. 4(a) between the output and incident beam without beam shift compensation, which disappears in Fig. 4(b) after compensation.

Fig. 4 ZEMAX simulation for compensation effect. (a) and (b) are the simulation results without and with beam shift compensation respectively.

Download Full Size | PDF

4. Polarization analysis

The instrumental polarization of the five-mirror derotator after beam shift compensation is sure to be different with that in the ideal design described in Sec. 2. Furthermore, some other unavoidable effects will also lead to some instrumental polarization. Hence a polarization analysis is necessary to be done to evaluate the validity and applicability of the five-mirror derotator in polarimetry.

In order to calculate the real instrumental polarization of the five-mirror derotator, the original Eq. (10) must be adopted, which includes ten unknown parameters, θ_i and ϕ_i, where i = 1...5. All the parameters can be calculated by the reflection law and vector geometry as shown in Eqs. (29) – (33), where ${\vec{V}}_{i}^{i}$ and ${\vec{V}}_{i}^{o}$ represent the incident and output vectors of the i_th mirror respectively. N⃗_i is the normal vector of the i_th mirror, N⃗_{Π_i} is the normal vector of the plane Π_i, and also the incident vector of the (i + 1)_th mirror is equal to the output vector of the i_th mirror.

θ_{i} = π - arccos (\frac{{\vec{V}}_{i}^{i} \cdot {\vec{N}}_{i}}{| {\vec{V}}_{i}^{i} | | {\vec{N}}_{i} |})

ϕ_{i} = arccos (\frac{\vec{N_{Π_{i}}} \cdot \vec{N_{Π 1}}}{| \vec{N_{Π i}} | | \vec{N_{Π_{1}}} |})

\vec{N_{Π_{i}}} = {\vec{V}}_{i}^{i} \times {\vec{N}}_{i}

{\vec{V}}_{i}^{o} = {\vec{V}}_{i}^{i} - 2 \frac{{\vec{V}}_{i}^{i} \cdot {\vec{N}}_{i}}{| {\vec{V}}_{i}^{i} | {\vec{N}}_{i} |}

{\vec{V_{i + 1}}}^{i} = {\vec{V}}_{i}^{o}

The Mueller matrix M in Eq. (10) can be calculated based on the above formulae once the vector ${\vec{V}}_{1}^{i}$ of the incident beam and the refractive index ñ of mirrors are determined. We define here ΔM = |M − M0| as an error matrix describing the instrumental polarization of the five-mirror derotator. M0 = diag(1, 1, −1, −1) is its theoretical Mueller matrix. Different kinds of telescopes have their own individual requirements for instrumental polarization. Taking the solar telescope with a large aperture used for the measurement of magnetic fields as an example, the crosstalk from unpolarized light into polarized light is required to be less than 0.001 and the one between linear polarization and circular polarization is required to be less than 0.01 [15]. Then, the error matrix have the following form, which gives the basis of polarization analysis for the five-mirror derotator.

Δ M < [\begin{matrix} 0.001 & 0.01 & 0.01 & 0.01 \\ 0.001 & 0.01 & 0.01 & 0.01 \\ 0.001 & 0.01 & 0.01 & 0.01 \\ 0.001 & 0.01 & 0.01 & 0.01 \end{matrix}]

The real instrumental polarization of the five-mirror derotator is mainly caused by the rotation effect for beam shift compensation and the field of view (FOV) of incident beam, where FOV is defined as angles relative with +z axis in Fig. 1. In addition, the instrumental polarization also has a large variation from visible to mid-infrared wavelengths. Figures 5–7 show error matrix ΔM of the five-mirror derotator in different wavelengths at 0.525um, 1.083um and 12um respectively. Each figure displays the relationships between error matrix ΔM, the field of view (FOV) and the rotation angle β.

Fig. 5 The contour map of error matrix ΔM at λ = 0.525um. the horizontal coordinate is the field of view (FOV) in degrees, the vertical coordinate is the rotation angle β in degrees, and the colorized subgraph displays variations of each element of ΔM with FOV and β (see colorbar).

Download Full Size | PDF

Fig. 6 The contour map of error matrix ΔM at λ = 1.083um, the same as Fig. 5.

Download Full Size | PDF

Fig. 7 The contour map of error matrix ΔM at λ = 12um, the same as Fig. 5.

Download Full Size | PDF

It is found that there is a strictest limit in ΔM₃₁ among all the elements of ΔM by comparing Eq. (34) and Figs. 5–7. Three linear inequalities are used to describe the relations of FOV and β when ΔM is required to satisfy Eq. (34). The equalities are showed in Table 1. It is noted that FOV is calculated from β which is determined by the ratio of h and b according to Eq. (24). β is usually between 0° and 4°, thus FOVs will be in [0.12°0.8°], [0.66°1.3°] and [4.8° 6.4°] at wavelengths of 0.525um, 1.083um and 12.00um respectively. By contrast, K-mirror induces a more serious instrumental polarization no matter what wavelengths (Eqs. (35)–(37)), which is far greater than the requirement on Eq. (34). It indicates that the five-mirror derotator has a significant advantage for small FOV telescopes in polarimetry. Furthermore, the observed wavelength becomes longer, the acceptable incident FOV is larger.

Δ M_{K - mirror}^{0.525 u m} = [\begin{matrix} 0.0053 & 0.1357 & 0 & 0 \\ 0.1357 & 0.0053 & 0 & 0 \\ 0 & 0 & 0.4565 & 0.5348 \\ 0 & 0 & 0.8348 & 0.4565 \end{matrix}]

Δ M_{K - mirror}^{1.083 u m} = [\begin{matrix} 0.0018 & 0.0782 & 0 & 0 \\ 0.0782 & 0.0018 & 0 & 0 \\ 0 & 0 & 0.1830 & 0.5745 \\ 0 & 0 & 0.5745 & 0.1830 \end{matrix}]

Δ M_{K - mirror}^{12.00 u m} = [\begin{matrix} 0.0001 & 0.0207 & 0 & 0 \\ 0.0207 & 0.0001 & 0 & 0 \\ 0 & 0 & 0.0040 & 0.0878 \\ 0 & 0 & 0.0878 & 0.0040 \end{matrix}]

Table 1. Linear relationships between FOV and β in different wavelengths.

View Table

5. Conclusion

Derotator is commonly used to prevent image rotation in astronomical telescopes. How to reduce the instrumental polarization of the derotator becomes an extremely important problem in polarimetry for large telescopes, especially for those requires very high polarization measurement accuracy. In this paper, we proposed a new design of derotator with minimal instrumental polarization based on the Mueller matrix transformation theory of mirrors in polarized light. The designed derotator contains five mirrors and is called five-mirror derotator. Its detailed description are summarized in the following: M1 and M5 have the same incident angle θ₁ and plane of incidence Π₁. M2, M3 and M4 have the same incident angle 30° and the same incident plane Π₂₃₄. Moreover, Π₁ is perpendicular to Π₂₃₄, which makes it possible to minimize the instrumental polarization of five-mirror derotator by optimizing θ₁ only. The optimization results of θ₁ show that the instrumental polarization of the five-mirror derotator is less than 0.01% when θ₁ = 36.3°, and the optimization results are independent of wavelengths. Besides, we also analyze the derotation features of five-mirror deratator and show it works well.

The thickness of the reflector composed of M1 and M5 induces a shift between the incident and output beams, affecting the real derotation result. We applied a novel compensation method which combines both translation and rotation to eliminate the beam shift, and deduced all the parameters of the five-mirror derotator. A ZEMAX example is done to verify the correctness and effectiveness of the compensation method.

In fact, the instrumental polarization of the five-mirror derotator is sure to be different with the original design after considering the beam shift compensation. We analyze relationships between the instrumental polarization, the rotation angle β and the incident field of view FOV. The results show that the instrumental polarization of the five-mirror derotator is far less than K-mirror and has a greater advantage for small FOV telescopes in polarimetry. Furthermore, the observed wavelength becomes longer, the acceptable incident FOV is larger. It should be noted that this paper mainly focuses on the design and theoretical analysis of five-mirror derotator. How to adjust it correctly is the future work we are going to carry out.

Funding

National Natural Science Foundation of China (NSFC) (11427901, 11773040, 11403047 and 11427803); The Strategic Pioneer Program on Space Science, Chinese Academy of Sciences (XDA15010800 and XDA15320102).

Acknowledgments

The authors would like to thank Prof. Wenda Cao and Dr. Nicolas Gorceix in Big Bear Solar Observatory for helpful discussions with derotator.

References and links

1. G. H. Sanders, “The Thirty Meter Telescope (TMT): An International Observatory,” J. Astrophys. Astr. 34(2), 81–86 (2013). [CrossRef]

2. M. Johnsa, R. Angelb, S. Shectmana, R. Bernsteind, D. Fabricantc, P. McCarthya, and M. Phillips, “Status of the Giant Magellan Telescope (GMT) Project,” Proc. SPIE 5489, 441–453 (2004). [CrossRef]

3. M. Warner, J. Mcmullin, T. Rimmele, and T. Berger, “The Advanced Technology Solar Telescope (ATST) project: a construction update,” Proc. SPIE 8862, 88620D (2013). [CrossRef]

4. S. X. Tao, W. Ma, and L. Z. Qian, “Rotation-canceling real time system of color video image,” Proc. SPIE 6279, 627959 (2007). [CrossRef]

5. Z. Liu, J. Xu, B. Z. Gu, S. Wang, J. Q. You, L. X. Shen, R. W. Lu, Z. Y. Jin, L. F. Chen, K. Lou, Z. Li, G. Q. Liu, Z. Xu, C. H. Rao, Q. Qian, H. R. Feng, L. H. Wen, F. F. Wang, M. X. Bao, M. C. Wu, and B. R. Zhang, “New vacuum solar telescope and observations with high resolution,” RAA 14(6), 705–718 (2014).

6. D. S. L. Durie, “A compact derotator design,” Opt. Eng. 13(1), 19–22 (1973).

7. H. Z. Sar-EI, “Revised Dove prism formulas,” Appl Opt. 30(4), 375–376 (1991). [CrossRef]

8. M. Padgett and J. Paullesso, “Dove prisms and polarized light,” Mod. Opt. 46(2), 175–179 (1999). [CrossRef]

9. Z. C. Wang, Y. Z. Zhao, and C. Zhou, “Design of K Mirror for Alt-az Telescope,” Acta Photonica Sinica 41(7), 762–765 (2012). [CrossRef]

10. S. C. West, “Polarimetry with multiple mirror telescopes,” Proc. SPIE 628, 50–59 (1986). [CrossRef]

11. M. Born and E. Wolf, Principles of Optics (Pergamon, 1965).

12. D. Goldstein and D. H. Goldstein, Polarized Light, Revised and Expanded (Lasers Optics & Photonics, 2003). [CrossRef]

13. J. S. Almeida, V. M. Pillet, and A. D. Wittmann, “The instrumental polarization of a Gregory-Coude telescope,” Solar Phys. 134(1), 1–13 (1991). [CrossRef]

14. J. Polasek, “Matrix Analysis of Gimbaled Mirror and Prism System,” J. Opt. Soc. Am. 57(10), 1193–1201 (1967). [CrossRef]

15. J. Schou, J. M. Borrero, A. A. Norton, S. Tomczyk, D. Elmore, and G. L. Card, “Polarization Calibration of the Helioseismic and Magnetic Imager (HMI) onborad the Solar Dynamics Observatory (SDO),” Solar Physics 275(1–2), 327–355 (2012). [CrossRef]

λ	FOV [°]	I → QUV	QUV → QUV
0.525um	<−0.17β+0.8	<0.001	<0.01
1.083um	<−0.16β+1.3	<0.001	<0.01
12.00um	<−0.40β+6.4	<0.001	<0.01

Design and analysis of a five-mirror derotator with minimal instrumental polarization in astronomical telescopes

Abstract

1. Introduction

2. Design of the five-mirror derotator

2.1. Design of the five-mirror derotator by minimizing instrumental polarization

2.2. Derotation feature of the five-mirror derotator

3. Beam shift compensation

4. Polarization analysis

5. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (7)

Tables (1)

Equations (37)

Optics Express